Wikiversity enwikiversity https://en.wikiversity.org/wiki/Wikiversity:Main_Page MediaWiki 1.39.0-wmf.22 first-letter Media Special Talk User User talk Wikiversity Wikiversity talk File File talk MediaWiki MediaWiki talk Template Template talk Help Help talk Category Category talk School School talk Portal Portal talk Topic Topic talk Collection Collection talk Draft Draft talk TimedText TimedText talk Module Module talk Gadget Gadget talk Gadget definition Gadget definition talk 0 2563 2410329 2380551 2022-07-29T23:53:27Z 24.52.159.126 why doe sthedkdnf wikitext text/x-wiki {{Prereq||Introduction to Limits}} == Resources == [[b:Calculus/Differentiation|Wikibooks entry for Differentiation]] ==Prelude== Arithmetic is about what you can do with numbers. Algebra is about what you can do with variables. Calculus is about what you can do with functions. Just as in arithmetic there are things you can do to a number to give another number, such as square it or add it to another number, in calculus there are two basic operations that given a function yield new and intimately related functions. The first of these operations is called differentiation, and the new function is called the ''derivative'' of the original function. This set of notes deals with the fundamentals of differentiation. For information about the second functional operator of calculus, visit [[Integration by Substitution]] after completing this unit. Before we dive in, we will warm up with an excursion into the mathematical workings of interest in banking. ===Compound Interest=== Let us suppose that we deposit an amount <math>A_0</math> in the bank on New Year's Day, and furthermore that every year on the year the amount is augmented by a rate <math>r</math> times the present amount. Then the amount <math>A</math> in the bank on any given New Year's Day, <math>t</math> years after the first is given by the expression :<math>A=A_0(1+r)^t\,\!</math>. Unfortunately, if we withdraw the money three days before the New Year, we don't get any of the interest payment for that year. A fairer system would involve calculating interest <math>n</math> times a year at the rate <math>r/n</math>. In fact this gives us a shitty different value even if we take our money out on a New Year's Day, because every time we calculate interest, we receive interest on our previous interest. The amount <math>A</math> we receive with this improved system is given by the expression :<math>A=A_0\left(1+\frac{r}{n}\right)^{nt}.</math> With this flexible system, we could set <math>n</math> to <math>12</math> to compound every month, or to <math>365</math> to compound every day or to about <math>31536000</math> to compound every second. But why stop there? Why not compound the interest every ''moment''? What is really meant by that is this: as we increase <math>n</math> does the value for <math>A</math> get ever greater with <math>n</math> or does it approach some reasonable quantity? If the latter is the case, then it is meaningful to ask, "What does <math>A</math> approach?" As we can see from the following table with sample values, this is in fact the case. :{| | <math>n</math> | <math>A</math> |- | 1 | 1.02500 |- | 12 | 1.02529 |- | 365 | 1.02531 |- | 31536000 | 1.02532 |- | 100000000 | 1.02532 |- ! colspan="2" | <math>A_0=1</math>, <math>r=.025</math>, <math>t=1</math> |} As we can see, as <math>n</math> goes off toward infinity, <math>A</math> approaches a finite value. Taking this to heart, we may come to our final system in which we define <math>A</math> as follows: :<math>A=\lim_{n\rightarrow \infty}A_0\left(1+\frac{r}{n}\right)^{nt}.</math> Thus we set <math>A</math> now not to <math>A_0\left(1+\frac{r}{n}\right)^{nt}</math> evaluated for some large <math>n</math>, but rather to the ''limit'' of that value as <math>n</math> approaches infinity. This is the formula for ''continually'' compounded interest. To clean up this formula, note that neither <math>A_0</math> nor <math>t</math> "interfere" in any way with the evaluation of the limit, and may consequently be moved outside of the limit without affecting the value of the expression: :<math>A=A_0B^t\,\!</math>, where :<math>B=\lim_{n\rightarrow \infty}\left(1+\frac{r}{n}\right)^n.</math> We can see from the form of the expression that <math>A</math> increases exponentially with <math>t</math> much as it did in our very first equation. The difference is that the original base <math>(1+r)</math> has been replaced with the base <math>B</math> which we have yet to simplify. Take a moment to step back and do the following exercises: #Without looking back, see if you can write down the expressions that represent #*yearly interest #*semiannual interest #*monthly interest #*interest <math>n</math> times a year #*continually compounded interest #Think about how much money you have. Figure out how long you would have to leave your money in a bank that compounds interest monthly before you became a millionaire, with a yearly interest rate of #*.02 (common for a savings account) #*.07 (average gain in the US stock market over a reasonably long period). ===Finding the Base=== In order to shed some light on the expression whose value we call <math>B</math>, we shall make use of the following expansion, known as the Binomial Theorem: :<math>(a + b)^n = a^n + na^{n-1}b + \frac{n(n-1)}{2}a^{n-2}b^2 + \dots + \frac{n(n-1)(n-2)\dots(n-(k-1))}{k!}a^{n-k}b^k + \dots + nab^{n-1} + b^n.</math> By applying it to our limit, we get :<math>B=\lim_{n\rightarrow \infty}\left(1+\frac{r}{n}\right)^n = \lim_{n\rightarrow \infty}\left(1+n\left(\frac{r}{n}\right)+\frac{n(n-1)}{2}\left(\frac{r}{n}\right)^2 +\frac{n(n-1)(n-2)}{3!}\left(\frac{r}{n}\right)^3 + \dots\right) =1+r+\frac{r^2}{2}+\frac{r^3}{3!}+\frac{r^4}{4!}+\dots</math>. This last step may seem mystifying at first. What happened to the limit? And where did all of the <math>n</math>'s go? In fact it was the evaluation of the limit that allowed us to remove the <math>n</math>'s. More exactly, as <math>n\rightarrow\infty</math>, so too <math>n(n-1)\rightarrow n^2</math>, <math>n(n-1)(n-2)\rightarrow n^3</math>, etc., so that the top left and bottom right of each term cancel to produce the last expression. Take a moment to look over the following exercises. Take the time to follow the trains of thought that are newest to you. {| | #The Pascal triangle, one of the world's most famous number patterns, popularized by the Seventeenth-Century mathematician Blaise Pascal, is shown on the right. What does this have to do with the Binomial Theorem? #The factorial (!) may seem like a silly operation to have its own name, but as it turns out it is one of the most common operations in both statistics and pure math. #*What is <math>100!/98!</math>? #*Which is bigger <math>1000!</math> or <math>2^{1000}</math>? #The Binomial Theorem is sometimes stated <math>(a+b)^n=\sum_{k=0}^n\frac{n!}{k!(n-k)!}a^{n-k}b^k</math>. Is this the same as the formula we used? #In the proof, we made use of the fact that <math>\lim_{n\rightarrow\infty}\frac{n(n-1)\dots(n-(k-1))}{k!}\frac{r^k}{n^k}=\frac{r^k}{k!}</math>. Does this make sense based on what you know about limits? #*What is <math>\lim_{x\rightarrow\infty}\frac{x^5+x^4+x^3+x^2+x+1}{x^5}</math>? #Without looking back, can you remember how it is that we used binomial expansion to show that <math>\lim_{n\rightarrow \infty}\left(1+\frac{r}{n}\right)^n = 1+r+\frac{r^2}{2}+\frac{r^3}{3!}+\frac{r^4}{4!}+\dots</math>? | {{center top}} {|border=0 cellpadding=2 align=center style="font-size: 8pt" !colspan="6" | !colspan="2" | 1 !colspan="6" | |- !colspan="5" | !colspan="2" | 1 !colspan="2" | 1 !colspan="5" | |- !colspan="4" | !colspan="2" | 1 !colspan="2" | 2 !colspan="2" | 1 !colspan="4" | |- !colspan="3" | !colspan="2" | 1 !colspan="2" | 3 !colspan="2" | 3 !colspan="2" | 1 !colspan="3" | |- !colspan="2" | !colspan="2" | 1 !colspan="2" | 4 !colspan="2" | 6 !colspan="2" | 4 !colspan="2" | 1 !colspan="2" | |- !colspan="1" | !colspan="2" | 1 !colspan="2" | 5 !colspan="2" | 10 !colspan="2" | 10 !colspan="2" | 5 !colspan="2" | 1 !colspan="1" | |- !colspan="2" | 1 !colspan="2" | 6 !colspan="2" | 15 !colspan="2" | 20 !colspan="2" | 15 !colspan="2" | 6 !colspan="2" | 1 |} {{center bottom}} |} ===The Birth of ''e''=== Now comes a real surprise. As it turns out, the infinite polynomial above is in fact ''exponential'' in <math>r</math>. That is, <math>B = 1+r+\frac{r^2}{2}+\frac{r^3}{3!}+\frac{r^4}{4!}+\dots = b^r</math>, for some <math>b</math>. In order to show this far-from-obvious fact, I offer the following. :<math>1+r+\frac{r^2}{2}+\frac{r^3}{3!}+\frac{r^4}{4!}+\dots = \lim_{n\rightarrow \infty}\bigg(1+nr\bigg(\frac{1}{n}\bigg)+\frac{nr(nr-1)}{2}\bigg(\frac{1}{n}\bigg)^2 +\frac{nr(nr-1)(nr-2)}{3!}\bigg(\frac{1}{n}\bigg)^3 + \dots \bigg)</math> :<math> = \lim_{n\rightarrow\infty}\bigg(1+\frac{1}{n}\bigg)^{nr} = \Bigg(\lim_{n\rightarrow\infty}\bigg(1+\frac{1}{n}\bigg)^n\Bigg)^r = \bigg(1+1+\frac{1}{2}+\frac{1}{3!}+\frac{1}{4!}+\dots \bigg)^r</math> To this last infinite series of numbers, define the quantity to be <math>e</math>: :<math>e=1+1+\frac{1}{2}+\frac{1}{3!}+\frac{1}{4!}+\dots</math>. <math>e</math>, an irrational (and in fact transcendental) number, has the approximate value 2.71828, which you may easily verify on a standard pocket or graphing calculator. There are a few things to think about. #The first line in the preceding derivation was motivated by my knowledge of the outcome. ##Convince yourself that the two expressions are in fact equal to one another. ##*Evaluate the term <math>\frac{nr(nr-1)(nr-2)}{3!}\bigg(\frac{1}{n}\bigg)^3</math> for <math>r=.02</math> and <math>n=10000</math>. How does that compare to <math>\frac{r^3}{3!}</math>? How about with <math>n=10000000</math>? ##Now that you have convinced yourself that I ''may'' do it, ask yourself why I would do it. ##*Using the reverse Binomial Theorem, do you understand how it leads to the next expression? #Is the equation <math>1+r+\frac{r^2}{2}+\frac{r^3}{3!}+\frac{r^4}{4!}+\dots = \bigg(1+1+\frac{1}{2}+\frac{1}{3!}+\frac{1}{4!}+\dots\bigg)^r</math> something that one would predict merely from the rules of exponents or distribution? #What makes certain seemingly uninteresting numbers so profoundly central to mathematics, such as <math>0</math>, <math>1</math>, <math>\pi</math>, and <math>e</math>? ===Back to the Start=== From here, everything cascades back to our original goal, namely to find a usable formula for continually compounded interest. <math>B = e^r \rightarrow A=A_0(e^r)^t \rightarrow A=A_0e^{rt}</math>. And there she is. Take a moment to do the following exercises. #Think about how much money you have. How long will it take to become millionaire if you leave the money in a bank with yearly interest of .025 #*that compounds interest yearly? #*that compounds interest continually? #Seeing as the values with and without continually compounded interest are very close to one another, what does that tell you about the two equations used? #*Both formulas are of the form <math>A=A_0(</math>_____<math>)^t</math>. Compare the various values that we have put in this blank, especially the in the equations for yearly and continually compounded interest. #*How close in value is <math>(1+r)</math> to <math>e^r</math>? Does that surprise you? #*Now look at the infinite series version of the function <math>e^r</math>. Does it still surprise you that <math>(1+r)</math> and <math>e^r</math> are so close in value? ===Commentary=== The formula itself, however, is quite forgettable. In fact, as you may have guessed, the importance of compounded interest pales in comparison to the importance of the ideas we stumbled upon on the way, namely limits and <math>e</math>. It is these two things that beg for us to go further into the heart of the life and being of functions. That wish is called calculus. And it all starts rather innocently with the derivative… ==Notion of secant & slope== Imagine a straight line plotted on square graph paper. For the sake of our discussion, suppose this line goes off your sheet of paper on both sides, and keeps going forever. What can you say about this line? Take your page and look at it. A line might be flat, parallel to the bottom of the page. It might be vertical, parallel to the sides of the page. Or it might lie somewhere between these two extremes, not as flat as the first, and not as steep as the second. The first thing to understand about 'slope' is that it is a measure of steepness. We often measure slope as a ''ratio''. If you drive 30 kilometers in the timespan of one hour, we say your ''speed'' is the ratio of distance over time: 30 kilometers ''per'' hour. Similarly, the slope is the change in vertical distance over the change in horizontal distance. How steep is a horizontal line? Draw a horizontal line, and then place one finger on one end of the line. Take another finger and slowly move it along the line. As you change the ''horizontal distance'' (as you move your finger from side to side), you will notice that the ''vertical distance'' does not change at all- you don't have to move your finger up or down. The change in vertical distance is always 0, regardless of the change in horizontal distance. So the slope is 0/x where x is the change in horizontal distance (you can choose whatever number you want for this), meaning the slope is 0. Our flat, horizontal line has a slope of zero - nothing happens to the y's whatever you do to the x's, think of cycling in parts of the Netherlands for example. A line at 45 degrees to the horizontal (half way between vertical (90 degrees) and horizontal (0 degrees)) has a slope of 1 (this would be a brutal hill for cycling, and very tough on foot). As you increase the horizontal distance by one unit, you also increase the vertical distance by one unit. This makes the slope 1/1=1. Our vertical line is incredibly steep (and much harder to cycle on). The slope is undefined, and as our line gets closer and closer to vertical, the slope gets bigger and bigger without limit. The second part of slope captures the idea of direction. Look at your line again. As it goes from left to right, does it go up the page, or down the page? If it was a road going up a hill, would it be hard to follow on a bicycle (going up), or very easy (going down)? This is expressed by saying that a line has a positive slope if, as it goes across, it also goes up, (or as the y's increase, the x's also increase). A line has a negative slope if it goes down as it goes across (or as the y's increase, the x's decrease). As a cyclist, you want a negative slope, unless you're in training. ==The Derivative== === Definition === Given a function <math>y = f(x)</math>, we define the derivative <math>f'(x)</math> to be :<math>f'(x)=\lim_{h\rightarrow 0}{\frac{f(x+h)-f(x)}{h}}</math>. This definition is motivated by the proportion <math>\frac{\Delta y}{\Delta x}=\frac{f(x+h)-f(x)}{h}</math>, which for any ''h'' defines the slope of a line, when ''f'' is linear. Because of the nature of the calculation, the derivative can be figuratively thought of as the ratio between an infinitesimal ''dy'' and an infinitesimal ''dx'' and is often written <math>\frac{dy}{dx}</math>. Both functional notation <math>\left ( f'(x) \right )</math> and infinitesimal or Leibniz notation <math>\left ( \frac{dy}{dx} \right )</math> have their virtues. In operator theory, the derivative of a function <math>f</math> is sometimes written as <math>D_x f \,\!</math>. #Using the definition above, what is <math>\frac{d(x^2)}{dx}</math>? #*Note that this is a short way of asking, if <math>f(x)=x^2</math>, what is <math>f'(x)</math>? One may also ask, what is <math>[x^2]'</math>? #If you have trouble remembering the definition of the derivative, it's much more important to know what it means, that is, ''why'' it's defined how it is. Remember it like this: #*<math>f'(x)=\frac{dy}{dx}=\lim_{\Delta x\rightarrow 0}\frac{\Delta y}{\Delta x}=\lim_{\Delta x\rightarrow 0}\frac{f(x+\Delta x)-f(x)}{(x+\Delta x) - x}</math>. #*From this we get the definition as stated above, <math>f'(x)=\lim_{h\rightarrow 0}{\frac{f(x+h)-f(x)}{h}}</math>. #What kinds of functions have derivatives? What would a function need to have, for it ''not'' to have a derivative at some point? === Properties === The derivative satisfies a number of fundamental properties ==== Linearity ==== An operator <math>L</math> is called ''linear'' if <math>L(f(x)+g(x))=L(f(x)) +L(g(x))</math> and <math>L(kf(x))=kL(f(x))</math> for any constant <math>k</math>. To show that differentiation is a linear operator, we must show that <math>[f(x)+g(x)]'=f'(x)+g'(x)</math> and <math>[kf(x)]'=kf'(x)</math> for any constant <math>k</math>. :<math>[f(x)+g(x)]'=\lim_{h\rightarrow 0}\frac{(f(x+h)+g(x+h))-(f(x)+g(x))}{h}</math> :<math>=\lim_{h\rightarrow 0}(\frac{f(x+h)-f(x)}{h}+\frac{g(x+h)-g(x)}{h})=\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h} + \lim_{h\rightarrow 0}\frac{g(x+h)-g(x)}{h}=f'(x)+g'(x)</math>. In other words, the differential operator (e.g., <math>\frac{d}{dx}</math>) ''distributes'' over addition. :<math>[kf(x)]'=\lim_{h\rightarrow 0}\frac{kf(x+h)-kf(x)}{h}=k\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}=kf'(x)</math>. In other words, addition before and after doing differentiation are equivalent. ==Fundamental Rules of Differentiation== Along with linearity, which is so simple that one hardly thinks of it as a rule, the following are essential to finding the derivative of arbitrary functions. ===The Product Rule=== It may be shown that for functions ''f'' and ''g'', <math>[f(x)g(x)]'=f(x)g'(x) + f'(x)g(x)</math>. Like the other two rules, this one is not a new axiom: it is directly provable from the definition of the derivative. <math>[f(x)g(x)]' = \lim_{h\rightarrow 0} \frac{f(x+h)g(x+h) - f(x)g(x)}{h} = \lim_{h\rightarrow 0} \frac{f(x+h)g(x+h) - f(x+h)g(x) + f(x+h)g(x) - f(x)g(x)}{h}</math> <math>= \lim_{h\rightarrow 0} \left( f(x+h)\frac{g(x+h)-g(x)}{h} + g(x)\frac{f(x+h)-f(x)}{h} \right)</math> <math> = f(x)\left( \lim_{h\rightarrow 0}\frac{g(x+h)-g(x)}{h} \right) + g(x) \left( \lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{h} \right) = f(x)g'(x) + g(x)f'(x)</math>. ===Chain Rule=== If a function ''f''(''x'') can be written as a compound function ''f''(''g''(''x'')), one can obtain its derivative using the chain rule. The chain rule states that the derivative of ''f''(''x'') will equal the derivative of ''f''(''g'') with respect to ''g'', multiplied by the derivative of ''g(x)'' with respect to ''x''. In mathematical terms: <math>[ \, f(g(x)) \, ]'=f'(g(x))g'(x).</math> This is commonly written as <math>\textstyle \frac{df}{dx} = \frac{df}{dg}\frac{dg}{dx}</math>, or more explicitly <math>\tfrac{d}{dx}f(x) = \tfrac{d}{dg}f(g(x)) \, \tfrac{d}{dx} g(x).</math> The proof makes use of an alternate but patently equivalent definition of the derivative: <math>f'(x) = \lim_{p\rightarrow x}\tfrac{f(p)-f(x)}{p-x}</math>. The first step is to write the derivative of the compound function in this form; one then manipulates it and obtains the chain rule. <math> \begin{align} \left[f(g(x))\right]' & = \lim_{p\rightarrow x}\frac{f(g(p))-f(g(x))}{p-x} \\ & = \lim_{p\rightarrow x} \frac{f(g(p))-f(g(x))}{g(p)-g(x)} \; \frac{g(p)-g(x)}{p-x} \\ & = \lim_{g(p)\rightarrow g(x)} \frac{f(g(p))-f(g(x))}{g(p)-g(x)} \; \lim_{p\rightarrow x} \frac{g(p)-g(x)}{p-x} \\ & = f'(g(x)) g'(x). \end{align} </math> In the third step, the first limit changes from ''p''&rarr;''x'' to ''g''(''p'')&rarr;''g''(''x''). This is valid because if ''g'' is continuous at ''x'', which it must be to have a derivative at ''x'', then of course as ''p'' approaches ''x'' the value of ''g(p)'' approaches that of ''g(x)''. Differentiating a nested function occurs very frequently, which makes this rule very useful. ===The Power Rule=== We may now readily show the relation <math>\frac{d(x^n)}{dx} = n x^{n-1}</math> as follows: <math>\frac{d(x^n)}{dx} = \lim_{h\rightarrow 0}\frac{(x+h)^{n}-x^n}{h} = \lim_{h\rightarrow 0}\frac{(x^n+nx^{n-1}h+\frac{n(n-1)}{2}x^{n-2}h^2+\cdots)-x^n}{h}</math> <math>= \lim_{h\rightarrow 0}(nx^{n-1}+\frac{n(n-1)}{2}x^{n-2}h+\cdots)=nx^{n-1}+0=nx^{n-1}.</math> While this derivation assumes that <math>n</math> is an positive integer, it turns out that the same rule holds for all real <math>n</math>. For example, <math>\frac{d}{dx}\left[\frac{1}{x}\right]=\frac{d(x^{-1})}{dx}=(-1)x^{-2}=-\frac{1}{x^2}</math>. Take a moment to do the following exercises. #Using the <math>x^n</math> rule and linearity, find the derivatives of the following: ##<math>x^{14} + 8</math> ##<math>3x^2+x^5</math> ##<math>\frac{1}{\sqrt{x}}</math> #What functions have the following derivatives? ##<math>3x^2</math> ##<math>15x^{14} + 2x</math> ##<math>x^2</math> == Exponentials and logarithms == Exponentials and logarithms involve a special number denoted ''e''. ===Differentiating ''e''^x=== Now, recall that <math>e^x = 1 + x + \frac{x^2}{2} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots.</math> Using the three basic rules established above we can differentiate any polynomial, even one of infinite degree: <math>\frac{d}{dx}e^x = 0 + 1 + \frac{2x}{2} + \frac{3x^2}{3!} + \frac{4x^3}{4!} + \cdots = 1 + x + \frac{x^2}{2} + \frac{x^3}{3!} + \cdots = e^x.</math> <math>e^x</math> is the remarkable function that is its own derivative. In other words, <math>e^x</math> is an ''eigenfunction'' of the differential operator. Which means that the application of the differential operator on <math>e^x</math> has the same effect as multiplication by a real number. For example, these concepts are useful in quantum mechanics. ===Differentiating ln(''x'')=== The natural logarithm is the function such that if <math>e^y=x</math> then <math>y=\ln(x)</math>; in other words, it is the inverse function of <math>e^x</math>. We will make use of the chain rule (marked by the brace) in order to find its derivative: <math>y=\ln(x) \rightarrow x=e^y \rightarrow \frac{dx}{dx}=\frac{d(e^y)}{dx} \rightarrow 1=\underbrace{\frac{d(e^y)}{dy}\frac{dy}{dx}}=e^y\frac{dy}{dx}=e^{\ln(x)}\frac{dy}{dx}=x\frac{dy}{dx} \rightarrow \frac{1}{x}=\frac{dy}{dx}=\frac{d(\ln(x))}{dx}.</math> This conclusion, that the derivative of <math>\ln(x)</math> is <math>\frac{1}{x}</math>, is remarkable: it ties together two seemingly unrelated functions. Be careful, this derivative has definite values only when x > 0! (Examine the <math>ln(x)</math> to understand why.) === Differentiating functions which are not immediately related to base ''e'' === ==== Exponentials ==== Supppose we have the function :<math>y = 3^x \,\!</math> To differentiate this, we rewrite this as :<math>y = 3^x = \left ( e^{\ln 3} \right )^x = e^{x \ln 3}</math> Since <math>\ln 3</math> is a constant, :<math>\frac{dy}{dx} = e^{x \ln 3} \cdot \ln 3 = 3^x \ln 3.</math> In other words, for a constant ''a'', we have :<math>\frac{dy}{dx} = a^x \ln a </math> whenever <math>y = a^x \,\!</math> This re-enforces the special place that <math>e</math> has in calculus - it is the unique number for which the constant <math>\ln a</math> is precisely equal to one. ==== Logarithms ==== Let us differentiate the function :<math>y = \log_a x\!</math> We already know how to differentiate <math>\ln x\!</math>, so let's change it into another form with the base ''e''. :<math>y = \log_a x = \frac{1}{\ln a} \cdot \ln x.</math> Because <math>\frac {1}{\ln a}</math> is a constant, :<math>\frac{dy}{dx} = \frac{1}{\ln a} \cdot \ln' x = \frac{1}{x\ln a}.</math> In conclusion, for any constant ''a'', the derivative of <math>\log_a x\!</math> is <math>\frac{1}{x\ln a}</math> === Implicit Differentiation === Let's suppose that :<math>y(x) = \frac{x^3 + x^2 + 1}{2x^5 -x^4 +2} </math> One could find <math>\frac{dy}{dx}</math> with the quotient rule, but for more complicated functions, it may be better to use what is called "implicit differentiation". In this case, we take the logarithm of both sides, to obtain :<math>\ln y = \ln \left ( \frac{x^3 + x^2 + 1}{2x^5 -x^4 +2} \right ) = \ln \left( x^3 + x^2 + 1 \right ) - \ln \left( 2x^5 -x^4 +2 \right )</math> or, in other words, just simply :<math>\ln y = \ln \left( x^3 + x^2 + 1 \right ) - \ln \left( 2x^5 -x^4 +2 \right )</math> Differentiating the left and right hand side, we get :<math>\frac{1}{y} \frac{dy}{dx} = \frac{3x^2+2x}{x^3 + x^2 + 1} - \frac{10x^4 - 4x^3}{2x^5 -x^4 +2 }</math> Now, multiply both sides by y, which we know is just <math>y(x) = \frac{x^3 + x^2 + 1}{2x^5 -x^4 +2} </math> to obtain the answer: :<math>{dy \over dx} = \frac{x^3 + x^2 + 1}{2x^5 -x^4 +2} \left ( \frac{3x^2+2x}{x^3 + x^2 + 1} - \frac{10x^4 - 4x^3}{2x^5 -x^4 +2 } \right )</math> which of course can be simplified further. You should verify that this result agrees with the quotient rule. Differentials of logarithms of functions occur frequently in places like [[statistical mechanics]]. === General exponentials and logarithms === Consider the function :<math>y(x) = u(x)^{v(x)} = e^{v(x) \ln u(x)} \,\!</math> It can be immediately seen that :<math>y'(x) = u^v \left ( \frac{v}{u} \frac{du}{dx} + \ln u \frac{dv}{dx} \right ) = v u^{v-1} \frac{du}{dx} + u^v \ln u \frac{dv}{dx} </math> Compare this result to the chain rule and power rule results. The first term results in treating v constant. The second term results in treating u constant. == Trigonometric functions == Consider the function <math>f(x) = \sin x</math>. To find the derivative of <math>f(x)</math>, we use the definition of the derivative, as well as some trigonometric identities and the linearity of the limit operator. :<math>\lim_{h \to 0} \dfrac{f(x+h) - f(x)}{h} = \lim_{h \to 0} \dfrac{\sin(x+h) - \sin x}{h} = \lim_{h \to 0}\dfrac{\cos x\sin h + \sin x \cos h - \sin x}{h}</math> :<math>=\lim_{h \to 0} \cos x \dfrac{\sin h}{h} - \sin x \dfrac{1 - \cos h}{h} = \cos x\left[\lim_{h \to 0} \dfrac{\sin h}{h}\right] - \sin x \left[\lim_{h \to 0}\dfrac{1 - \cos h}{h}\right],</math> and since <math>\lim_{h \to 0}\dfrac{1 - \cos h}{h} = 0</math> and <math>\lim_{h \to 0} \dfrac{\sin h}{h}=1</math>, the above expression simplifies to <math>\cos x\!</math>. Thus, the derivative of <math>\sin x\!</math> is <math>\cos x\!</math>. We perform the same process to find the derivatives of the other trigonometric functions (try to derive them on your own as an exercise). Since these derivatives come up quite often, it would behoove (advantageous to) you to memorize them. <math>\dfrac{d}{dx}\sin x = \cos x</math> <math>\dfrac{d}{dx}\cos x = -\sin x</math> <math>\dfrac{d}{dx}\tan x = \sec^2 x</math> <math>\dfrac{d}{dx}\sec x = \sec x \tan x</math> <math>\dfrac{d}{dx}\csc x = -\csc x \cot x</math> <math>\dfrac{d}{dx}\cot x = -\csc^2 x</math> == Hyperbolic functions == The rules for differentiation involving hyperbolic functions behave very much like their trigonometric counterparts, with the notable difference being in the sign of the derivative. Here, <math>\textrm{sinh}(x) = \frac{e^{x} - e^{-x}}{2} </math> <math>\cosh(x) = \frac{e^{x} + e^{-x}}{2} </math> so it can be seen that <math>\frac{d}{dx} \sinh(x) = \cosh(x) </math> and <math>\frac{d}{dx} \cosh(x) = \sinh(x) </math> [[Category:Introductions]] [[Category:Differentiation]] as7fv1nqruv6uisx0w93kza951a7myu 2410330 2410329 2022-07-29T23:53:45Z PhantomTech 1293967 Reverted edits by [[Special:Contributions/24.52.159.126|24.52.159.126]] ([[User talk:24.52.159.126|talk]]) ([[Project:Huggle|HG]]) (3.4.10) wikitext text/x-wiki {{Prereq||Introduction to Limits}} == Resources == [[b:Calculus/Differentiation|Wikibooks entry for Differentiation]] ==Prelude== Arithmetic is about what you can do with numbers. Algebra is about what you can do with variables. Calculus is about what you can do with functions. Just as in arithmetic there are things you can do to a number to give another number, such as square it or add it to another number, in calculus there are two basic operations that given a function yield new and intimately related functions. The first of these operations is called differentiation, and the new function is called the ''derivative'' of the original function. This set of notes deals with the fundamentals of differentiation. For information about the second functional operator of calculus, visit [[Integration by Substitution]] after completing this unit. Before we dive in, we will warm up with an excursion into the mathematical workings of interest in banking. ===Compound Interest=== Let us suppose that we deposit an amount <math>A_0</math> in the bank on New Year's Day, and furthermore that every year on the year the amount is augmented by a rate <math>r</math> times the present amount. Then the amount <math>A</math> in the bank on any given New Year's Day, <math>t</math> years after the first is given by the expression :<math>A=A_0(1+r)^t\,\!</math>. Unfortunately, if we withdraw the money three days before the New Year, we don't get any of the interest payment for that year. A fairer system would involve calculating interest <math>n</math> times a year at the rate <math>r/n</math>. In fact this gives us a slightly different value even if we take our money out on a New Year's Day, because every time we calculate interest, we receive interest on our previous interest. The amount <math>A</math> we receive with this improved system is given by the expression :<math>A=A_0\left(1+\frac{r}{n}\right)^{nt}.</math> With this flexible system, we could set <math>n</math> to <math>12</math> to compound every month, or to <math>365</math> to compound every day or to about <math>31536000</math> to compound every second. But why stop there? Why not compound the interest every ''moment''? What is really meant by that is this: as we increase <math>n</math> does the value for <math>A</math> get ever greater with <math>n</math> or does it approach some reasonable quantity? If the latter is the case, then it is meaningful to ask, "What does <math>A</math> approach?" As we can see from the following table with sample values, this is in fact the case. :{| | <math>n</math> | <math>A</math> |- | 1 | 1.02500 |- | 12 | 1.02529 |- | 365 | 1.02531 |- | 31536000 | 1.02532 |- | 100000000 | 1.02532 |- ! colspan="2" | <math>A_0=1</math>, <math>r=.025</math>, <math>t=1</math> |} As we can see, as <math>n</math> goes off toward infinity, <math>A</math> approaches a finite value. Taking this to heart, we may come to our final system in which we define <math>A</math> as follows: :<math>A=\lim_{n\rightarrow \infty}A_0\left(1+\frac{r}{n}\right)^{nt}.</math> Thus we set <math>A</math> now not to <math>A_0\left(1+\frac{r}{n}\right)^{nt}</math> evaluated for some large <math>n</math>, but rather to the ''limit'' of that value as <math>n</math> approaches infinity. This is the formula for ''continually'' compounded interest. To clean up this formula, note that neither <math>A_0</math> nor <math>t</math> "interfere" in any way with the evaluation of the limit, and may consequently be moved outside of the limit without affecting the value of the expression: :<math>A=A_0B^t\,\!</math>, where :<math>B=\lim_{n\rightarrow \infty}\left(1+\frac{r}{n}\right)^n.</math> We can see from the form of the expression that <math>A</math> increases exponentially with <math>t</math> much as it did in our very first equation. The difference is that the original base <math>(1+r)</math> has been replaced with the base <math>B</math> which we have yet to simplify. Take a moment to step back and do the following exercises: #Without looking back, see if you can write down the expressions that represent #*yearly interest #*semiannual interest #*monthly interest #*interest <math>n</math> times a year #*continually compounded interest #Think about how much money you have. Figure out how long you would have to leave your money in a bank that compounds interest monthly before you became a millionaire, with a yearly interest rate of #*.02 (common for a savings account) #*.07 (average gain in the US stock market over a reasonably long period). ===Finding the Base=== In order to shed some light on the expression whose value we call <math>B</math>, we shall make use of the following expansion, known as the Binomial Theorem: :<math>(a + b)^n = a^n + na^{n-1}b + \frac{n(n-1)}{2}a^{n-2}b^2 + \dots + \frac{n(n-1)(n-2)\dots(n-(k-1))}{k!}a^{n-k}b^k + \dots + nab^{n-1} + b^n.</math> By applying it to our limit, we get :<math>B=\lim_{n\rightarrow \infty}\left(1+\frac{r}{n}\right)^n = \lim_{n\rightarrow \infty}\left(1+n\left(\frac{r}{n}\right)+\frac{n(n-1)}{2}\left(\frac{r}{n}\right)^2 +\frac{n(n-1)(n-2)}{3!}\left(\frac{r}{n}\right)^3 + \dots\right) =1+r+\frac{r^2}{2}+\frac{r^3}{3!}+\frac{r^4}{4!}+\dots</math>. This last step may seem mystifying at first. What happened to the limit? And where did all of the <math>n</math>'s go? In fact it was the evaluation of the limit that allowed us to remove the <math>n</math>'s. More exactly, as <math>n\rightarrow\infty</math>, so too <math>n(n-1)\rightarrow n^2</math>, <math>n(n-1)(n-2)\rightarrow n^3</math>, etc., so that the top left and bottom right of each term cancel to produce the last expression. Take a moment to look over the following exercises. Take the time to follow the trains of thought that are newest to you. {| | #The Pascal triangle, one of the world's most famous number patterns, popularized by the Seventeenth-Century mathematician Blaise Pascal, is shown on the right. What does this have to do with the Binomial Theorem? #The factorial (!) may seem like a silly operation to have its own name, but as it turns out it is one of the most common operations in both statistics and pure math. #*What is <math>100!/98!</math>? #*Which is bigger <math>1000!</math> or <math>2^{1000}</math>? #The Binomial Theorem is sometimes stated <math>(a+b)^n=\sum_{k=0}^n\frac{n!}{k!(n-k)!}a^{n-k}b^k</math>. Is this the same as the formula we used? #In the proof, we made use of the fact that <math>\lim_{n\rightarrow\infty}\frac{n(n-1)\dots(n-(k-1))}{k!}\frac{r^k}{n^k}=\frac{r^k}{k!}</math>. Does this make sense based on what you know about limits? #*What is <math>\lim_{x\rightarrow\infty}\frac{x^5+x^4+x^3+x^2+x+1}{x^5}</math>? #Without looking back, can you remember how it is that we used binomial expansion to show that <math>\lim_{n\rightarrow \infty}\left(1+\frac{r}{n}\right)^n = 1+r+\frac{r^2}{2}+\frac{r^3}{3!}+\frac{r^4}{4!}+\dots</math>? | {{center top}} {|border=0 cellpadding=2 align=center style="font-size: 8pt" !colspan="6" | !colspan="2" | 1 !colspan="6" | |- !colspan="5" | !colspan="2" | 1 !colspan="2" | 1 !colspan="5" | |- !colspan="4" | !colspan="2" | 1 !colspan="2" | 2 !colspan="2" | 1 !colspan="4" | |- !colspan="3" | !colspan="2" | 1 !colspan="2" | 3 !colspan="2" | 3 !colspan="2" | 1 !colspan="3" | |- !colspan="2" | !colspan="2" | 1 !colspan="2" | 4 !colspan="2" | 6 !colspan="2" | 4 !colspan="2" | 1 !colspan="2" | |- !colspan="1" | !colspan="2" | 1 !colspan="2" | 5 !colspan="2" | 10 !colspan="2" | 10 !colspan="2" | 5 !colspan="2" | 1 !colspan="1" | |- !colspan="2" | 1 !colspan="2" | 6 !colspan="2" | 15 !colspan="2" | 20 !colspan="2" | 15 !colspan="2" | 6 !colspan="2" | 1 |} {{center bottom}} |} ===The Birth of ''e''=== Now comes a real surprise. As it turns out, the infinite polynomial above is in fact ''exponential'' in <math>r</math>. That is, <math>B = 1+r+\frac{r^2}{2}+\frac{r^3}{3!}+\frac{r^4}{4!}+\dots = b^r</math>, for some <math>b</math>. In order to show this far-from-obvious fact, I offer the following. :<math>1+r+\frac{r^2}{2}+\frac{r^3}{3!}+\frac{r^4}{4!}+\dots = \lim_{n\rightarrow \infty}\bigg(1+nr\bigg(\frac{1}{n}\bigg)+\frac{nr(nr-1)}{2}\bigg(\frac{1}{n}\bigg)^2 +\frac{nr(nr-1)(nr-2)}{3!}\bigg(\frac{1}{n}\bigg)^3 + \dots \bigg)</math> :<math> = \lim_{n\rightarrow\infty}\bigg(1+\frac{1}{n}\bigg)^{nr} = \Bigg(\lim_{n\rightarrow\infty}\bigg(1+\frac{1}{n}\bigg)^n\Bigg)^r = \bigg(1+1+\frac{1}{2}+\frac{1}{3!}+\frac{1}{4!}+\dots \bigg)^r</math> To this last infinite series of numbers, define the quantity to be <math>e</math>: :<math>e=1+1+\frac{1}{2}+\frac{1}{3!}+\frac{1}{4!}+\dots</math>. <math>e</math>, an irrational (and in fact transcendental) number, has the approximate value 2.71828, which you may easily verify on a standard pocket or graphing calculator. There are a few things to think about. #The first line in the preceding derivation was motivated by my knowledge of the outcome. ##Convince yourself that the two expressions are in fact equal to one another. ##*Evaluate the term <math>\frac{nr(nr-1)(nr-2)}{3!}\bigg(\frac{1}{n}\bigg)^3</math> for <math>r=.02</math> and <math>n=10000</math>. How does that compare to <math>\frac{r^3}{3!}</math>? How about with <math>n=10000000</math>? ##Now that you have convinced yourself that I ''may'' do it, ask yourself why I would do it. ##*Using the reverse Binomial Theorem, do you understand how it leads to the next expression? #Is the equation <math>1+r+\frac{r^2}{2}+\frac{r^3}{3!}+\frac{r^4}{4!}+\dots = \bigg(1+1+\frac{1}{2}+\frac{1}{3!}+\frac{1}{4!}+\dots\bigg)^r</math> something that one would predict merely from the rules of exponents or distribution? #What makes certain seemingly uninteresting numbers so profoundly central to mathematics, such as <math>0</math>, <math>1</math>, <math>\pi</math>, and <math>e</math>? ===Back to the Start=== From here, everything cascades back to our original goal, namely to find a usable formula for continually compounded interest. <math>B = e^r \rightarrow A=A_0(e^r)^t \rightarrow A=A_0e^{rt}</math>. And there she is. Take a moment to do the following exercises. #Think about how much money you have. How long will it take to become millionaire if you leave the money in a bank with yearly interest of .025 #*that compounds interest yearly? #*that compounds interest continually? #Seeing as the values with and without continually compounded interest are very close to one another, what does that tell you about the two equations used? #*Both formulas are of the form <math>A=A_0(</math>_____<math>)^t</math>. Compare the various values that we have put in this blank, especially the in the equations for yearly and continually compounded interest. #*How close in value is <math>(1+r)</math> to <math>e^r</math>? Does that surprise you? #*Now look at the infinite series version of the function <math>e^r</math>. Does it still surprise you that <math>(1+r)</math> and <math>e^r</math> are so close in value? ===Commentary=== The formula itself, however, is quite forgettable. In fact, as you may have guessed, the importance of compounded interest pales in comparison to the importance of the ideas we stumbled upon on the way, namely limits and <math>e</math>. It is these two things that beg for us to go further into the heart of the life and being of functions. That wish is called calculus. And it all starts rather innocently with the derivative… ==Notion of secant & slope== Imagine a straight line plotted on square graph paper. For the sake of our discussion, suppose this line goes off your sheet of paper on both sides, and keeps going forever. What can you say about this line? Take your page and look at it. A line might be flat, parallel to the bottom of the page. It might be vertical, parallel to the sides of the page. Or it might lie somewhere between these two extremes, not as flat as the first, and not as steep as the second. The first thing to understand about 'slope' is that it is a measure of steepness. We often measure slope as a ''ratio''. If you drive 30 kilometers in the timespan of one hour, we say your ''speed'' is the ratio of distance over time: 30 kilometers ''per'' hour. Similarly, the slope is the change in vertical distance over the change in horizontal distance. How steep is a horizontal line? Draw a horizontal line, and then place one finger on one end of the line. Take another finger and slowly move it along the line. As you change the ''horizontal distance'' (as you move your finger from side to side), you will notice that the ''vertical distance'' does not change at all- you don't have to move your finger up or down. The change in vertical distance is always 0, regardless of the change in horizontal distance. So the slope is 0/x where x is the change in horizontal distance (you can choose whatever number you want for this), meaning the slope is 0. Our flat, horizontal line has a slope of zero - nothing happens to the y's whatever you do to the x's, think of cycling in parts of the Netherlands for example. A line at 45 degrees to the horizontal (half way between vertical (90 degrees) and horizontal (0 degrees)) has a slope of 1 (this would be a brutal hill for cycling, and very tough on foot). As you increase the horizontal distance by one unit, you also increase the vertical distance by one unit. This makes the slope 1/1=1. Our vertical line is incredibly steep (and much harder to cycle on). The slope is undefined, and as our line gets closer and closer to vertical, the slope gets bigger and bigger without limit. The second part of slope captures the idea of direction. Look at your line again. As it goes from left to right, does it go up the page, or down the page? If it was a road going up a hill, would it be hard to follow on a bicycle (going up), or very easy (going down)? This is expressed by saying that a line has a positive slope if, as it goes across, it also goes up, (or as the y's increase, the x's also increase). A line has a negative slope if it goes down as it goes across (or as the y's increase, the x's decrease). As a cyclist, you want a negative slope, unless you're in training. ==The Derivative== === Definition === Given a function <math>y = f(x)</math>, we define the derivative <math>f'(x)</math> to be :<math>f'(x)=\lim_{h\rightarrow 0}{\frac{f(x+h)-f(x)}{h}}</math>. This definition is motivated by the proportion <math>\frac{\Delta y}{\Delta x}=\frac{f(x+h)-f(x)}{h}</math>, which for any ''h'' defines the slope of a line, when ''f'' is linear. Because of the nature of the calculation, the derivative can be figuratively thought of as the ratio between an infinitesimal ''dy'' and an infinitesimal ''dx'' and is often written <math>\frac{dy}{dx}</math>. Both functional notation <math>\left ( f'(x) \right )</math> and infinitesimal or Leibniz notation <math>\left ( \frac{dy}{dx} \right )</math> have their virtues. In operator theory, the derivative of a function <math>f</math> is sometimes written as <math>D_x f \,\!</math>. #Using the definition above, what is <math>\frac{d(x^2)}{dx}</math>? #*Note that this is a short way of asking, if <math>f(x)=x^2</math>, what is <math>f'(x)</math>? One may also ask, what is <math>[x^2]'</math>? #If you have trouble remembering the definition of the derivative, it's much more important to know what it means, that is, ''why'' it's defined how it is. Remember it like this: #*<math>f'(x)=\frac{dy}{dx}=\lim_{\Delta x\rightarrow 0}\frac{\Delta y}{\Delta x}=\lim_{\Delta x\rightarrow 0}\frac{f(x+\Delta x)-f(x)}{(x+\Delta x) - x}</math>. #*From this we get the definition as stated above, <math>f'(x)=\lim_{h\rightarrow 0}{\frac{f(x+h)-f(x)}{h}}</math>. #What kinds of functions have derivatives? What would a function need to have, for it ''not'' to have a derivative at some point? === Properties === The derivative satisfies a number of fundamental properties ==== Linearity ==== An operator <math>L</math> is called ''linear'' if <math>L(f(x)+g(x))=L(f(x)) +L(g(x))</math> and <math>L(kf(x))=kL(f(x))</math> for any constant <math>k</math>. To show that differentiation is a linear operator, we must show that <math>[f(x)+g(x)]'=f'(x)+g'(x)</math> and <math>[kf(x)]'=kf'(x)</math> for any constant <math>k</math>. :<math>[f(x)+g(x)]'=\lim_{h\rightarrow 0}\frac{(f(x+h)+g(x+h))-(f(x)+g(x))}{h}</math> :<math>=\lim_{h\rightarrow 0}(\frac{f(x+h)-f(x)}{h}+\frac{g(x+h)-g(x)}{h})=\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h} + \lim_{h\rightarrow 0}\frac{g(x+h)-g(x)}{h}=f'(x)+g'(x)</math>. In other words, the differential operator (e.g., <math>\frac{d}{dx}</math>) ''distributes'' over addition. :<math>[kf(x)]'=\lim_{h\rightarrow 0}\frac{kf(x+h)-kf(x)}{h}=k\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}=kf'(x)</math>. In other words, addition before and after doing differentiation are equivalent. ==Fundamental Rules of Differentiation== Along with linearity, which is so simple that one hardly thinks of it as a rule, the following are essential to finding the derivative of arbitrary functions. ===The Product Rule=== It may be shown that for functions ''f'' and ''g'', <math>[f(x)g(x)]'=f(x)g'(x) + f'(x)g(x)</math>. Like the other two rules, this one is not a new axiom: it is directly provable from the definition of the derivative. <math>[f(x)g(x)]' = \lim_{h\rightarrow 0} \frac{f(x+h)g(x+h) - f(x)g(x)}{h} = \lim_{h\rightarrow 0} \frac{f(x+h)g(x+h) - f(x+h)g(x) + f(x+h)g(x) - f(x)g(x)}{h}</math> <math>= \lim_{h\rightarrow 0} \left( f(x+h)\frac{g(x+h)-g(x)}{h} + g(x)\frac{f(x+h)-f(x)}{h} \right)</math> <math> = f(x)\left( \lim_{h\rightarrow 0}\frac{g(x+h)-g(x)}{h} \right) + g(x) \left( \lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{h} \right) = f(x)g'(x) + g(x)f'(x)</math>. ===Chain Rule=== If a function ''f''(''x'') can be written as a compound function ''f''(''g''(''x'')), one can obtain its derivative using the chain rule. The chain rule states that the derivative of ''f''(''x'') will equal the derivative of ''f''(''g'') with respect to ''g'', multiplied by the derivative of ''g(x)'' with respect to ''x''. In mathematical terms: <math>[ \, f(g(x)) \, ]'=f'(g(x))g'(x).</math> This is commonly written as <math>\textstyle \frac{df}{dx} = \frac{df}{dg}\frac{dg}{dx}</math>, or more explicitly <math>\tfrac{d}{dx}f(x) = \tfrac{d}{dg}f(g(x)) \, \tfrac{d}{dx} g(x).</math> The proof makes use of an alternate but patently equivalent definition of the derivative: <math>f'(x) = \lim_{p\rightarrow x}\tfrac{f(p)-f(x)}{p-x}</math>. The first step is to write the derivative of the compound function in this form; one then manipulates it and obtains the chain rule. <math> \begin{align} \left[f(g(x))\right]' & = \lim_{p\rightarrow x}\frac{f(g(p))-f(g(x))}{p-x} \\ & = \lim_{p\rightarrow x} \frac{f(g(p))-f(g(x))}{g(p)-g(x)} \; \frac{g(p)-g(x)}{p-x} \\ & = \lim_{g(p)\rightarrow g(x)} \frac{f(g(p))-f(g(x))}{g(p)-g(x)} \; \lim_{p\rightarrow x} \frac{g(p)-g(x)}{p-x} \\ & = f'(g(x)) g'(x). \end{align} </math> In the third step, the first limit changes from ''p''&rarr;''x'' to ''g''(''p'')&rarr;''g''(''x''). This is valid because if ''g'' is continuous at ''x'', which it must be to have a derivative at ''x'', then of course as ''p'' approaches ''x'' the value of ''g(p)'' approaches that of ''g(x)''. Differentiating a nested function occurs very frequently, which makes this rule very useful. ===The Power Rule=== We may now readily show the relation <math>\frac{d(x^n)}{dx} = n x^{n-1}</math> as follows: <math>\frac{d(x^n)}{dx} = \lim_{h\rightarrow 0}\frac{(x+h)^{n}-x^n}{h} = \lim_{h\rightarrow 0}\frac{(x^n+nx^{n-1}h+\frac{n(n-1)}{2}x^{n-2}h^2+\cdots)-x^n}{h}</math> <math>= \lim_{h\rightarrow 0}(nx^{n-1}+\frac{n(n-1)}{2}x^{n-2}h+\cdots)=nx^{n-1}+0=nx^{n-1}.</math> While this derivation assumes that <math>n</math> is an positive integer, it turns out that the same rule holds for all real <math>n</math>. For example, <math>\frac{d}{dx}\left[\frac{1}{x}\right]=\frac{d(x^{-1})}{dx}=(-1)x^{-2}=-\frac{1}{x^2}</math>. Take a moment to do the following exercises. #Using the <math>x^n</math> rule and linearity, find the derivatives of the following: ##<math>x^{14} + 8</math> ##<math>3x^2+x^5</math> ##<math>\frac{1}{\sqrt{x}}</math> #What functions have the following derivatives? ##<math>3x^2</math> ##<math>15x^{14} + 2x</math> ##<math>x^2</math> == Exponentials and logarithms == Exponentials and logarithms involve a special number denoted ''e''. ===Differentiating ''e''^x=== Now, recall that <math>e^x = 1 + x + \frac{x^2}{2} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots.</math> Using the three basic rules established above we can differentiate any polynomial, even one of infinite degree: <math>\frac{d}{dx}e^x = 0 + 1 + \frac{2x}{2} + \frac{3x^2}{3!} + \frac{4x^3}{4!} + \cdots = 1 + x + \frac{x^2}{2} + \frac{x^3}{3!} + \cdots = e^x.</math> <math>e^x</math> is the remarkable function that is its own derivative. In other words, <math>e^x</math> is an ''eigenfunction'' of the differential operator. Which means that the application of the differential operator on <math>e^x</math> has the same effect as multiplication by a real number. For example, these concepts are useful in quantum mechanics. ===Differentiating ln(''x'')=== The natural logarithm is the function such that if <math>e^y=x</math> then <math>y=\ln(x)</math>; in other words, it is the inverse function of <math>e^x</math>. We will make use of the chain rule (marked by the brace) in order to find its derivative: <math>y=\ln(x) \rightarrow x=e^y \rightarrow \frac{dx}{dx}=\frac{d(e^y)}{dx} \rightarrow 1=\underbrace{\frac{d(e^y)}{dy}\frac{dy}{dx}}=e^y\frac{dy}{dx}=e^{\ln(x)}\frac{dy}{dx}=x\frac{dy}{dx} \rightarrow \frac{1}{x}=\frac{dy}{dx}=\frac{d(\ln(x))}{dx}.</math> This conclusion, that the derivative of <math>\ln(x)</math> is <math>\frac{1}{x}</math>, is remarkable: it ties together two seemingly unrelated functions. Be careful, this derivative has definite values only when x > 0! (Examine the <math>ln(x)</math> to understand why.) === Differentiating functions which are not immediately related to base ''e'' === ==== Exponentials ==== Supppose we have the function :<math>y = 3^x \,\!</math> To differentiate this, we rewrite this as :<math>y = 3^x = \left ( e^{\ln 3} \right )^x = e^{x \ln 3}</math> Since <math>\ln 3</math> is a constant, :<math>\frac{dy}{dx} = e^{x \ln 3} \cdot \ln 3 = 3^x \ln 3.</math> In other words, for a constant ''a'', we have :<math>\frac{dy}{dx} = a^x \ln a </math> whenever <math>y = a^x \,\!</math> This re-enforces the special place that <math>e</math> has in calculus - it is the unique number for which the constant <math>\ln a</math> is precisely equal to one. ==== Logarithms ==== Let us differentiate the function :<math>y = \log_a x\!</math> We already know how to differentiate <math>\ln x\!</math>, so let's change it into another form with the base ''e''. :<math>y = \log_a x = \frac{1}{\ln a} \cdot \ln x.</math> Because <math>\frac {1}{\ln a}</math> is a constant, :<math>\frac{dy}{dx} = \frac{1}{\ln a} \cdot \ln' x = \frac{1}{x\ln a}.</math> In conclusion, for any constant ''a'', the derivative of <math>\log_a x\!</math> is <math>\frac{1}{x\ln a}</math> === Implicit Differentiation === Let's suppose that :<math>y(x) = \frac{x^3 + x^2 + 1}{2x^5 -x^4 +2} </math> One could find <math>\frac{dy}{dx}</math> with the quotient rule, but for more complicated functions, it may be better to use what is called "implicit differentiation". In this case, we take the logarithm of both sides, to obtain :<math>\ln y = \ln \left ( \frac{x^3 + x^2 + 1}{2x^5 -x^4 +2} \right ) = \ln \left( x^3 + x^2 + 1 \right ) - \ln \left( 2x^5 -x^4 +2 \right )</math> or, in other words, just simply :<math>\ln y = \ln \left( x^3 + x^2 + 1 \right ) - \ln \left( 2x^5 -x^4 +2 \right )</math> Differentiating the left and right hand side, we get :<math>\frac{1}{y} \frac{dy}{dx} = \frac{3x^2+2x}{x^3 + x^2 + 1} - \frac{10x^4 - 4x^3}{2x^5 -x^4 +2 }</math> Now, multiply both sides by y, which we know is just <math>y(x) = \frac{x^3 + x^2 + 1}{2x^5 -x^4 +2} </math> to obtain the answer: :<math>{dy \over dx} = \frac{x^3 + x^2 + 1}{2x^5 -x^4 +2} \left ( \frac{3x^2+2x}{x^3 + x^2 + 1} - \frac{10x^4 - 4x^3}{2x^5 -x^4 +2 } \right )</math> which of course can be simplified further. You should verify that this result agrees with the quotient rule. Differentials of logarithms of functions occur frequently in places like [[statistical mechanics]]. === General exponentials and logarithms === Consider the function :<math>y(x) = u(x)^{v(x)} = e^{v(x) \ln u(x)} \,\!</math> It can be immediately seen that :<math>y'(x) = u^v \left ( \frac{v}{u} \frac{du}{dx} + \ln u \frac{dv}{dx} \right ) = v u^{v-1} \frac{du}{dx} + u^v \ln u \frac{dv}{dx} </math> Compare this result to the chain rule and power rule results. The first term results in treating v constant. The second term results in treating u constant. == Trigonometric functions == Consider the function <math>f(x) = \sin x</math>. To find the derivative of <math>f(x)</math>, we use the definition of the derivative, as well as some trigonometric identities and the linearity of the limit operator. :<math>\lim_{h \to 0} \dfrac{f(x+h) - f(x)}{h} = \lim_{h \to 0} \dfrac{\sin(x+h) - \sin x}{h} = \lim_{h \to 0}\dfrac{\cos x\sin h + \sin x \cos h - \sin x}{h}</math> :<math>=\lim_{h \to 0} \cos x \dfrac{\sin h}{h} - \sin x \dfrac{1 - \cos h}{h} = \cos x\left[\lim_{h \to 0} \dfrac{\sin h}{h}\right] - \sin x \left[\lim_{h \to 0}\dfrac{1 - \cos h}{h}\right],</math> and since <math>\lim_{h \to 0}\dfrac{1 - \cos h}{h} = 0</math> and <math>\lim_{h \to 0} \dfrac{\sin h}{h}=1</math>, the above expression simplifies to <math>\cos x\!</math>. Thus, the derivative of <math>\sin x\!</math> is <math>\cos x\!</math>. We perform the same process to find the derivatives of the other trigonometric functions (try to derive them on your own as an exercise). Since these derivatives come up quite often, it would behoove (advantageous to) you to memorize them. <math>\dfrac{d}{dx}\sin x = \cos x</math> <math>\dfrac{d}{dx}\cos x = -\sin x</math> <math>\dfrac{d}{dx}\tan x = \sec^2 x</math> <math>\dfrac{d}{dx}\sec x = \sec x \tan x</math> <math>\dfrac{d}{dx}\csc x = -\csc x \cot x</math> <math>\dfrac{d}{dx}\cot x = -\csc^2 x</math> == Hyperbolic functions == The rules for differentiation involving hyperbolic functions behave very much like their trigonometric counterparts, with the notable difference being in the sign of the derivative. Here, <math>\textrm{sinh}(x) = \frac{e^{x} - e^{-x}}{2} </math> <math>\cosh(x) = \frac{e^{x} + e^{-x}}{2} </math> so it can be seen that <math>\frac{d}{dx} \sinh(x) = \cosh(x) </math> and <math>\frac{d}{dx} \cosh(x) = \sinh(x) </math> [[Category:Introductions]] [[Category:Differentiation]] mjnb3scm1x1bfd7nnem8k2a3x1ruw4k 0 3761 2410335 2410220 2022-07-30T00:08:23Z Dave Braunschweig 426084 Reverted edits by [[Special:Contributions/45.118.167.74|45.118.167.74]] ([[User_talk:45.118.167.74|talk]]) to last version by [[User:Dave Braunschweig|Dave Braunschweig]] using [[Wikiversity:Rollback|rollback]] wikitext text/x-wiki {{Uncited|article|date=September 2010}} [[File:Gustave Le Gray - Brig upon the Water - Google Art Project.jpg|alt=Challenge me|border|right|frameless|400x400px|Shreenith]] {{TOCright}} '''Web Design''' is an incredibly '''fun''' skill to learn—combining the latest toys of technology with the creativity of design! On top of that, learning web design is unique in that we can learn directly from '''current professionals''' who publish their techniques for all to read on their own Web-logs! The idea of the Web started with {{w|Tim Berners-Lee}} during 1989, while he was working at {{w|CERN}}. His vision was to create a global level hypertext based project, and his implementation with a browser was with [[w:WorldWideWeb|WorldWideWeb]] in 1991. During the early days only text based pages could be used on single lined web browsers.There was no integration of multimedia elements like images, sound and others. However the arrival of the {{w|Mosaic (web browser)|Mosaic web browser}} allowed integration of multimedia elements. In October 1994, {{w|W3C}} was founded to develop standards for the Web, which process still continues. Since about 2000, many browsers like Microsoft Edge, Mozilla Firefox, and Google Chrome have been released into the market, and new browsers are still coming. You'll find below a growing number of topics that we think provide a '''good foundation''' for any web designer. We're also working on the requirements for formal [[Web design qualifications|qualifications]], so you can start collecting evidence of your skills towards a '''formal qualification''' in your country. Of course, if you have anything to add or improve then please join us and contribute! (Read [[Web Design/About_the_web_design_learning_project|More about the web design learning project]]) == What you can learn here== The following topics have been ordered to help provide a pathway for you to learn the main skills of web design. Of course, you might deviate from that path—or create a new path! * [[Web Design/Build a basic web page|Build a basic web page]] [[Image:75%.svg]]—Get started creating your own web pages and learning how to style them! You will also find valuable information on what makes a good web design. All of this information is fundamental to getting started. * [[Web Design/Build a small website|Build a small website]] [[Image:25%.svg]]—Building on your skills to create structured HTML content that can be styled and laid out with your stylesheets (Includes [[CSS challenges|10 CSS Challenges]]!) * [[Web Design/Developing a Client Project|Developing a Client Project]] [[Image:00%.svg]]—Applying your HTML/CSS skills to your first real client project where you'll learn some project management, information architecture and usability along the way! Now featuring the [[Information Architecture Challenges]] * [[Web Design/An Introduction to Programming with JavaScript|An Introduction to Programming with JavaScript]] [[Image:50%.svg]]—Learn some of the fundamentals of computer programming (sequence, selection, repetition and variables) with your own web pages! Now with [[Web Design/Introductory algorithm challenges|Introductory algorithm challenges]] and [[Web Design/JavaScript Challenges|JavaScript Challenges]]! * [[Web Design/CSS3 Animations|CSS3 Animations]] [[Image:100%.svg]] − Animate your website with the CSS3 animations * [[Web Design/Dynamic_websites_with_PHP|An Introduction to Dynamic Websites with PHP]][[Image:25%.svg]]—Learn the basics of server-side scripting, including page templates and form handling. Now including [[Web Design/PHP_challenges|PHP Challenges]]! * What to consider when creating [[Web Design/IDs|resource identifiers]]. Each topic includes an outline, suggested activities and learning resources to help you along your way. == Topics under development == * [[Web Design/Using the Internet as a Learning Tool|Using the Internet as a Learning Tool]] [[Image:25%.svg]] – Start learning how you can keep up-to-date with the world of web-design. * [[Web Design/Design Principles for Web Design|Design Principles for Web Design]] [[Image:00%.svg]] – Before getting too technical, get started with some all important graphic design principles as they're applied to the web! * [[Web Design/Emerging Technologies|Emerging Technologies]] – There are some pretty nifty tools and technologies constantly working their way into Web design... start to find out about them! Now with [[Web Design/XML challenges|XML Challenges]]. * [[Web Design/Accessibility|Accessibility]] – Start using the right practices from the start! * [[Web_Design/Design_Suggestions|Design Suggestions]] – Suggestions for Design resources. * [[Web Design/Getting Your Site On the Web|Getting Your Site On the Web]] [[Image:25%.svg]] – So you've got your site, now you need to get it on the web for everyone to see! * [[Web Design/Useful Applications|Useful Applications]] – Make sure you have the right tools for the job! Here you'll find a listing of HTML/CSS editors both WYSIWYG and text-based. * [[Web Design/CSS|CSS]] – Cascading Style Sheets (CSS) * [[Web Design/Dynamic websites|Dynamic websites]] * [[Google Web Designer]] == Qualifications == Some countries offer official Web design certificates and [[Web Design/Qualifications|qualifications]]. This course is intended to help those who would like to obtain such a qualification. ==See also== * [[Web development]] ===Wikipedia=== * [[w:{{PAGENAME}}|{{PAGENAME}}]] (in Wikipedia) * [[w:OpenStack|OpenStack]] == External Links == * [http://www.w3schools.com W3 Schools Online Web Tutorials] – A resource for learning HTML 4.0, XHTML 1.0, CSS and scripting (JavaScript, PHP, etc.) languages. * [http://www.w3resource.com/index.php w3resource Online Web Tutorials] – A resource for learning HTML(4.01), CSS(2.1), JavaScript(1.5), PHP 5, SQL 2003 standard and MySQL 5 tutorials. * [http://www.techved.com/uae/blog/web-design Web designing explained] - Web design Explained in detail * [http://htmldog.com/ HTMLDog] - A resource for best practice guides to XHTML and CSS. * [https://www.digitar.co.uk/web-design eCommerce / Shop] - Store Management and selling online * [https://thinktoshare.com/website-creating-tips Website Development Tutorial] - Guide to Create a Website [[Category:Web design]] e4dbkrbubgkxc2grju0bq4x6t42gggx 13 9521 2410245 2377050 2022-07-29T15:49:53Z Praxidicae 2805549 Reverted edits by [[Special:Contributions/41.114.211.28|41.114.211.28]] ([[User_talk:41.114.211.28|talk]]) to last version by [[User:Hasley|Hasley]] using [[Wikiversity:Rollback|rollback]] wikitext text/x-wiki == Help == This page is used to discuss [[Help:Talk]] and how to use Talk pages. For help using Wikiversity, see either [[Wikiversity:Colloquium]] or [[Wikiversity:Request custodian action]]. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 13:39, 8 August 2020 (UTC) osxlgvjod1pljdtuamfgd3jzaa4xlmr 0 40903 2410327 2060852 2022-07-29T23:20:18Z Abigail Richie 2942343 wikitext text/x-wiki COMMUNICATION SKILLS This has to do with the method, way or strategy through which communication is done successfully. One thing about communication is that we have the speakers and listeners, there is no way you would communicate without speaking, and you can't speak when there is no listener; there are various communication skills, we have some of them below : *listening skills You can't communicate to someone who doesn't listen it's just like screaming in a dessert. A good listener must pay good attention to the speaker, not just pay attention to hear but to understand what ever that is being said. A good listener must be able to provide answers when ever a question is being asked about the topic discussed. Once again he or she must have a good listening ears. Expression: A good speaker must be able to express him or herself very well in a suitable language which the listener will understand. A good speaker is important during communication, a good speaker must be audible and bold, a good speaker must speak fluently and express themselves in a good way. Interpretation: if the listeners can't interprete the information passed to them then the communication here is void and useless. ==Models== ===Simple, one-way=== [[Image:Communication sender-message-reciever.png|center|450px]]<br> <!-- [[Image:Communication sender-message-reciever en.png|200px|center]] --> ===Feedback-loop (Shannon-Weaver)=== [[Image:Communication shannon-weaver2.svg|center|600px]] ==See also== * [[Communication]] * [[Feedback]] {{commons|Communication}} {{wikipedia}} * [[w:Image:Communication emisor.jpg|Communication major dimensions scheme]] * [[w:Image:Encoding communication.jpg|Communication code scheme]] ==External links== # [http://michelle-socialpsychology.blogspot.com/2007/10/effective-interpersonal-communication.html Effective interpersonal communication] (Michelle, blogspot, 2007) [[Category:Communication]] 0zzj71mul4dlani0tb1xrqjnn14cl5e 1 84062 2410235 473288 2022-07-29T14:03:30Z 41.116.87.149 /* Shona */ new section wikitext text/x-wiki I first tackled translation back when Wikiversity and beta first opened. Here we go again for [http://strategy.wikimedia.org Strategic Planning] purposes. [[User:CQ|CQ]] 04:53, 30 July 2009 (UTC) == Czech == Note to self: See [[w:T-V distinction|T-V distinction]], [[w:Czech declension|Czech declension]] and '''[[Transwiki]]''' per convo with apergos and juandev (#wikiversity hub) --[[User:CQ|CQ]] 00:10, 6 September 2009 (UTC) == Shona == Translate words in shona to english so foreigners can also understand english and be able to do better communication [[Special:Contributions/41.116.87.149|41.116.87.149]] ([[User talk:41.116.87.149|discuss]]) 14:03, 29 July 2022 (UTC) 32hdxbklfohtm761k8whl75g93sopx9 3 117349 2410236 2384326 2022-07-29T14:05:26Z 41.114.82.169 /* Physical sciences,life science and maths. */ new section wikitext text/x-wiki {{Robelbox|theme=9|title=Welcome!|width=100%}} <div style="{{Robelbox/pad}}"> '''Hello Marshallsumter, and [[Wikiversity:Welcome|welcome]] to [[Wikiversity:What is Wikiversity?|Wikiversity]]!''' If you need [[Help:Contents|help]], feel free to visit my talk page, or [[Wikiversity:Contact|contact us]] and [[Wikiversity:Questions|ask questions]]. After you leave a comment on a [[Wikiversity:Talk page|talk page]], remember to [[Wikiversity:Signature|sign and date]]; it helps everyone follow the threads of the discussion. The signature icon [[File:Signature icon.png]] in the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy policy]], [[Wikiversity:Civility|Civility policy]], and the [[Foundation:Terms of Use|Terms of Use]] while at Wikiversity. To [[Wikiversity:Introduction|get started]], you may <!-- The Left column --> <div style="width:50.0%; float:left"> * [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]]. * Visit a (kind of) [[Wikiversity:Random|random project]]. * [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]]. * Find out about [[Wikiversity:Research|research]] activities on Wikiversity. * [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left. </div> <!-- The Right column --> <div style="width:50.0%; float:left"> * Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity. * Give [[Wikiversity:Feedback|feedback]] about your initial observations * Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]]. * [[Wikiversity:Chat|Chat]] with other Wikiversitans on [irc://irc.freenode.net/wikiversity-en <kbd>#wikiversity-en</kbd>]. * Follow Wikiversity on [[twitter]] (http://twitter.com/Wikiversity) and [[identi.ca]] (http://identi.ca/group/wikiversity). </div> <br clear="both"/> You don't need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Abd|Abd]] 04:25, 24 August 2011 (UTC)</div> {{Robelbox/close}} == Most Active Wikiversity User for January 2013 == {| style="border: 1px solid gray; background-color: #ffffff;" |rowspan="2" valign="middle" | [[Image:Learningcycle.png|100px]] |rowspan="2" | |style="font-size: x-large; padding: 0; vertical-align: middle; height: 1.1em;" | '''The Learning Cycle Barnstar''' |- |style="vertical-align: middle; border-top: 1px solid gray;" | Most Active Wikiversity User for January 2013 |} Marshallsumter, I was reviewing the list of active users for this past month and noticed you had by far the most edits in January. Keep up the good work! -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 00:04, 1 February 2013 (UTC) == Barnstar for you! == {| style="border: 1px solid gray; background-color: #ffffff;" |rowspan="2" valign="middle" | [[Image:Star constellation.png|100px]] |rowspan="2" | |style="font-size: x-large; padding: 0; vertical-align: middle; height: 1.1em;" | '''The astronomy barnstar''' |- |style="vertical-align: middle; border-top: 1px solid gray;" | Thank you for the massive edits on astronomy! [[User:Goldenburg111|Goldenburg111]] ([[User talk:Goldenburg111|talk]]|[[Special:Contributions/Goldenburg111|contribs]]) 18:49, 25 December 2013 (UTC) |} {| style="border: 1px solid gray; background-color: #ffffff;" |rowspan="2" valign="middle" | [[Image:Original_Barnstar.png|100px]] |rowspan="2" | |style="font-size: x-large; padding: 0; vertical-align: middle; height: 1.1em;" | '''The Original Barnstar''' |- |style="vertical-align: middle; border-top: 1px solid gray;" | Thank you for your help with [[Research in programming Wikidata]]! -- [[User:AKA MBG|Andrew Krizhanovsky]] ([[User talk:AKA MBG|discuss]] • [[Special:Contributions/AKA MBG|contribs]]) 05:45, 30 May 2017 (UTC) |} ==See also== {{Archive box non-auto}} {{clear}} == Physical sciences,life science and maths. == I wish to become a madical doctor and have my own patients whom will say I am a good doctor who is caring and loving [[Special:Contributions/41.114.82.169|41.114.82.169]] ([[User talk:41.114.82.169|discuss]]) 14:05, 29 July 2022 (UTC) kgae11nsvn1gfuanmymgnr9ls2r7o1w 2410237 2410236 2022-07-29T14:10:51Z Praxidicae 2805549 Reverted edits by [[Special:Contributions/41.114.82.169|41.114.82.169]] ([[User_talk:41.114.82.169|talk]]) to last version by [[User:Marshallsumter|Marshallsumter]] using [[Wikiversity:Rollback|rollback]] wikitext text/x-wiki {{Robelbox|theme=9|title=Welcome!|width=100%}} <div style="{{Robelbox/pad}}"> '''Hello Marshallsumter, and [[Wikiversity:Welcome|welcome]] to [[Wikiversity:What is Wikiversity?|Wikiversity]]!''' If you need [[Help:Contents|help]], feel free to visit my talk page, or [[Wikiversity:Contact|contact us]] and [[Wikiversity:Questions|ask questions]]. After you leave a comment on a [[Wikiversity:Talk page|talk page]], remember to [[Wikiversity:Signature|sign and date]]; it helps everyone follow the threads of the discussion. The signature icon [[File:Signature icon.png]] in the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy policy]], [[Wikiversity:Civility|Civility policy]], and the [[Foundation:Terms of Use|Terms of Use]] while at Wikiversity. To [[Wikiversity:Introduction|get started]], you may <!-- The Left column --> <div style="width:50.0%; float:left"> * [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]]. * Visit a (kind of) [[Wikiversity:Random|random project]]. * [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]]. * Find out about [[Wikiversity:Research|research]] activities on Wikiversity. * [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left. </div> <!-- The Right column --> <div style="width:50.0%; float:left"> * Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity. * Give [[Wikiversity:Feedback|feedback]] about your initial observations * Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]]. * [[Wikiversity:Chat|Chat]] with other Wikiversitans on [irc://irc.freenode.net/wikiversity-en <kbd>#wikiversity-en</kbd>]. * Follow Wikiversity on [[twitter]] (http://twitter.com/Wikiversity) and [[identi.ca]] (http://identi.ca/group/wikiversity). </div> <br clear="both"/> You don't need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Abd|Abd]] 04:25, 24 August 2011 (UTC)</div> {{Robelbox/close}} == Most Active Wikiversity User for January 2013 == {| style="border: 1px solid gray; background-color: #ffffff;" |rowspan="2" valign="middle" | [[Image:Learningcycle.png|100px]] |rowspan="2" | |style="font-size: x-large; padding: 0; vertical-align: middle; height: 1.1em;" | '''The Learning Cycle Barnstar''' |- |style="vertical-align: middle; border-top: 1px solid gray;" | Most Active Wikiversity User for January 2013 |} Marshallsumter, I was reviewing the list of active users for this past month and noticed you had by far the most edits in January. Keep up the good work! -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 00:04, 1 February 2013 (UTC) == Barnstar for you! == {| style="border: 1px solid gray; background-color: #ffffff;" |rowspan="2" valign="middle" | [[Image:Star constellation.png|100px]] |rowspan="2" | |style="font-size: x-large; padding: 0; vertical-align: middle; height: 1.1em;" | '''The astronomy barnstar''' |- |style="vertical-align: middle; border-top: 1px solid gray;" | Thank you for the massive edits on astronomy! [[User:Goldenburg111|Goldenburg111]] ([[User talk:Goldenburg111|talk]]|[[Special:Contributions/Goldenburg111|contribs]]) 18:49, 25 December 2013 (UTC) |} {| style="border: 1px solid gray; background-color: #ffffff;" |rowspan="2" valign="middle" | [[Image:Original_Barnstar.png|100px]] |rowspan="2" | |style="font-size: x-large; padding: 0; vertical-align: middle; height: 1.1em;" | '''The Original Barnstar''' |- |style="vertical-align: middle; border-top: 1px solid gray;" | Thank you for your help with [[Research in programming Wikidata]]! -- [[User:AKA MBG|Andrew Krizhanovsky]] ([[User talk:AKA MBG|discuss]] • [[Special:Contributions/AKA MBG|contribs]]) 05:45, 30 May 2017 (UTC) |} ==See also== {{Archive box non-auto}} {{clear}} nmv7d0fhw2iqiblob6owz5x1ovb0xch 2410239 2410237 2022-07-29T14:21:52Z 41.114.82.169 /* Physical science,maths and life science */ new section wikitext text/x-wiki {{Robelbox|theme=9|title=Welcome!|width=100%}} <div style="{{Robelbox/pad}}"> '''Hello Marshallsumter, and [[Wikiversity:Welcome|welcome]] to [[Wikiversity:What is Wikiversity?|Wikiversity]]!''' If you need [[Help:Contents|help]], feel free to visit my talk page, or [[Wikiversity:Contact|contact us]] and [[Wikiversity:Questions|ask questions]]. After you leave a comment on a [[Wikiversity:Talk page|talk page]], remember to [[Wikiversity:Signature|sign and date]]; it helps everyone follow the threads of the discussion. The signature icon [[File:Signature icon.png]] in the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy policy]], [[Wikiversity:Civility|Civility policy]], and the [[Foundation:Terms of Use|Terms of Use]] while at Wikiversity. To [[Wikiversity:Introduction|get started]], you may <!-- The Left column --> <div style="width:50.0%; float:left"> * [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]]. * Visit a (kind of) [[Wikiversity:Random|random project]]. * [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]]. * Find out about [[Wikiversity:Research|research]] activities on Wikiversity. * [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left. </div> <!-- The Right column --> <div style="width:50.0%; float:left"> * Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity. * Give [[Wikiversity:Feedback|feedback]] about your initial observations * Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]]. * [[Wikiversity:Chat|Chat]] with other Wikiversitans on [irc://irc.freenode.net/wikiversity-en <kbd>#wikiversity-en</kbd>]. * Follow Wikiversity on [[twitter]] (http://twitter.com/Wikiversity) and [[identi.ca]] (http://identi.ca/group/wikiversity). </div> <br clear="both"/> You don't need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Abd|Abd]] 04:25, 24 August 2011 (UTC)</div> {{Robelbox/close}} == Most Active Wikiversity User for January 2013 == {| style="border: 1px solid gray; background-color: #ffffff;" |rowspan="2" valign="middle" | [[Image:Learningcycle.png|100px]] |rowspan="2" | |style="font-size: x-large; padding: 0; vertical-align: middle; height: 1.1em;" | '''The Learning Cycle Barnstar''' |- |style="vertical-align: middle; border-top: 1px solid gray;" | Most Active Wikiversity User for January 2013 |} Marshallsumter, I was reviewing the list of active users for this past month and noticed you had by far the most edits in January. Keep up the good work! -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 00:04, 1 February 2013 (UTC) == Barnstar for you! == {| style="border: 1px solid gray; background-color: #ffffff;" |rowspan="2" valign="middle" | [[Image:Star constellation.png|100px]] |rowspan="2" | |style="font-size: x-large; padding: 0; vertical-align: middle; height: 1.1em;" | '''The astronomy barnstar''' |- |style="vertical-align: middle; border-top: 1px solid gray;" | Thank you for the massive edits on astronomy! [[User:Goldenburg111|Goldenburg111]] ([[User talk:Goldenburg111|talk]]|[[Special:Contributions/Goldenburg111|contribs]]) 18:49, 25 December 2013 (UTC) |} {| style="border: 1px solid gray; background-color: #ffffff;" |rowspan="2" valign="middle" | [[Image:Original_Barnstar.png|100px]] |rowspan="2" | |style="font-size: x-large; padding: 0; vertical-align: middle; height: 1.1em;" | '''The Original Barnstar''' |- |style="vertical-align: middle; border-top: 1px solid gray;" | Thank you for your help with [[Research in programming Wikidata]]! -- [[User:AKA MBG|Andrew Krizhanovsky]] ([[User talk:AKA MBG|discuss]] • [[Special:Contributions/AKA MBG|contribs]]) 05:45, 30 May 2017 (UTC) |} ==See also== {{Archive box non-auto}} {{clear}} == Physical science,maths and life science == By Innocent :What is on my mind is I would also like to be on this page because it encourage me very respectful and good and another thing is I want to be a medical doctor who is caring , loving and respected and also responsible for people's lives [[Special:Contributions/41.114.82.169|41.114.82.169]] ([[User talk:41.114.82.169|discuss]]) 14:21, 29 July 2022 (UTC) 4usfyvdrk8pwcyw5u8r76o12q69iqz1 2410240 2410239 2022-07-29T14:22:12Z Praxidicae 2805549 Reverted edits by [[Special:Contributions/41.114.82.169|41.114.82.169]] ([[User_talk:41.114.82.169|talk]]) to last version by [[User:Praxidicae|Praxidicae]] using [[Wikiversity:Rollback|rollback]] wikitext text/x-wiki {{Robelbox|theme=9|title=Welcome!|width=100%}} <div style="{{Robelbox/pad}}"> '''Hello Marshallsumter, and [[Wikiversity:Welcome|welcome]] to [[Wikiversity:What is Wikiversity?|Wikiversity]]!''' If you need [[Help:Contents|help]], feel free to visit my talk page, or [[Wikiversity:Contact|contact us]] and [[Wikiversity:Questions|ask questions]]. After you leave a comment on a [[Wikiversity:Talk page|talk page]], remember to [[Wikiversity:Signature|sign and date]]; it helps everyone follow the threads of the discussion. The signature icon [[File:Signature icon.png]] in the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy policy]], [[Wikiversity:Civility|Civility policy]], and the [[Foundation:Terms of Use|Terms of Use]] while at Wikiversity. To [[Wikiversity:Introduction|get started]], you may <!-- The Left column --> <div style="width:50.0%; float:left"> * [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]]. * Visit a (kind of) [[Wikiversity:Random|random project]]. * [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]]. * Find out about [[Wikiversity:Research|research]] activities on Wikiversity. * [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left. </div> <!-- The Right column --> <div style="width:50.0%; float:left"> * Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity. * Give [[Wikiversity:Feedback|feedback]] about your initial observations * Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]]. * [[Wikiversity:Chat|Chat]] with other Wikiversitans on [irc://irc.freenode.net/wikiversity-en <kbd>#wikiversity-en</kbd>]. * Follow Wikiversity on [[twitter]] (http://twitter.com/Wikiversity) and [[identi.ca]] (http://identi.ca/group/wikiversity). </div> <br clear="both"/> You don't need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Abd|Abd]] 04:25, 24 August 2011 (UTC)</div> {{Robelbox/close}} == Most Active Wikiversity User for January 2013 == {| style="border: 1px solid gray; background-color: #ffffff;" |rowspan="2" valign="middle" | [[Image:Learningcycle.png|100px]] |rowspan="2" | |style="font-size: x-large; padding: 0; vertical-align: middle; height: 1.1em;" | '''The Learning Cycle Barnstar''' |- |style="vertical-align: middle; border-top: 1px solid gray;" | Most Active Wikiversity User for January 2013 |} Marshallsumter, I was reviewing the list of active users for this past month and noticed you had by far the most edits in January. Keep up the good work! -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 00:04, 1 February 2013 (UTC) == Barnstar for you! == {| style="border: 1px solid gray; background-color: #ffffff;" |rowspan="2" valign="middle" | [[Image:Star constellation.png|100px]] |rowspan="2" | |style="font-size: x-large; padding: 0; vertical-align: middle; height: 1.1em;" | '''The astronomy barnstar''' |- |style="vertical-align: middle; border-top: 1px solid gray;" | Thank you for the massive edits on astronomy! [[User:Goldenburg111|Goldenburg111]] ([[User talk:Goldenburg111|talk]]|[[Special:Contributions/Goldenburg111|contribs]]) 18:49, 25 December 2013 (UTC) |} {| style="border: 1px solid gray; background-color: #ffffff;" |rowspan="2" valign="middle" | [[Image:Original_Barnstar.png|100px]] |rowspan="2" | |style="font-size: x-large; padding: 0; vertical-align: middle; height: 1.1em;" | '''The Original Barnstar''' |- |style="vertical-align: middle; border-top: 1px solid gray;" | Thank you for your help with [[Research in programming Wikidata]]! -- [[User:AKA MBG|Andrew Krizhanovsky]] ([[User talk:AKA MBG|discuss]] • [[Special:Contributions/AKA MBG|contribs]]) 05:45, 30 May 2017 (UTC) |} ==See also== {{Archive box non-auto}} {{clear}} nmv7d0fhw2iqiblob6owz5x1ovb0xch 2410241 2410240 2022-07-29T14:33:27Z 41.114.82.169 /* Maths,life science */ new section wikitext text/x-wiki {{Robelbox|theme=9|title=Welcome!|width=100%}} <div style="{{Robelbox/pad}}"> '''Hello Marshallsumter, and [[Wikiversity:Welcome|welcome]] to [[Wikiversity:What is Wikiversity?|Wikiversity]]!''' If you need [[Help:Contents|help]], feel free to visit my talk page, or [[Wikiversity:Contact|contact us]] and [[Wikiversity:Questions|ask questions]]. After you leave a comment on a [[Wikiversity:Talk page|talk page]], remember to [[Wikiversity:Signature|sign and date]]; it helps everyone follow the threads of the discussion. The signature icon [[File:Signature icon.png]] in the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy policy]], [[Wikiversity:Civility|Civility policy]], and the [[Foundation:Terms of Use|Terms of Use]] while at Wikiversity. To [[Wikiversity:Introduction|get started]], you may <!-- The Left column --> <div style="width:50.0%; float:left"> * [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]]. * Visit a (kind of) [[Wikiversity:Random|random project]]. * [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]]. * Find out about [[Wikiversity:Research|research]] activities on Wikiversity. * [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left. </div> <!-- The Right column --> <div style="width:50.0%; float:left"> * Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity. * Give [[Wikiversity:Feedback|feedback]] about your initial observations * Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]]. * [[Wikiversity:Chat|Chat]] with other Wikiversitans on [irc://irc.freenode.net/wikiversity-en <kbd>#wikiversity-en</kbd>]. * Follow Wikiversity on [[twitter]] (http://twitter.com/Wikiversity) and [[identi.ca]] (http://identi.ca/group/wikiversity). </div> <br clear="both"/> You don't need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Abd|Abd]] 04:25, 24 August 2011 (UTC)</div> {{Robelbox/close}} == Most Active Wikiversity User for January 2013 == {| style="border: 1px solid gray; background-color: #ffffff;" |rowspan="2" valign="middle" | [[Image:Learningcycle.png|100px]] |rowspan="2" | |style="font-size: x-large; padding: 0; vertical-align: middle; height: 1.1em;" | '''The Learning Cycle Barnstar''' |- |style="vertical-align: middle; border-top: 1px solid gray;" | Most Active Wikiversity User for January 2013 |} Marshallsumter, I was reviewing the list of active users for this past month and noticed you had by far the most edits in January. Keep up the good work! -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 00:04, 1 February 2013 (UTC) == Barnstar for you! == {| style="border: 1px solid gray; background-color: #ffffff;" |rowspan="2" valign="middle" | [[Image:Star constellation.png|100px]] |rowspan="2" | |style="font-size: x-large; padding: 0; vertical-align: middle; height: 1.1em;" | '''The astronomy barnstar''' |- |style="vertical-align: middle; border-top: 1px solid gray;" | Thank you for the massive edits on astronomy! [[User:Goldenburg111|Goldenburg111]] ([[User talk:Goldenburg111|talk]]|[[Special:Contributions/Goldenburg111|contribs]]) 18:49, 25 December 2013 (UTC) |} {| style="border: 1px solid gray; background-color: #ffffff;" |rowspan="2" valign="middle" | [[Image:Original_Barnstar.png|100px]] |rowspan="2" | |style="font-size: x-large; padding: 0; vertical-align: middle; height: 1.1em;" | '''The Original Barnstar''' |- |style="vertical-align: middle; border-top: 1px solid gray;" | Thank you for your help with [[Research in programming Wikidata]]! -- [[User:AKA MBG|Andrew Krizhanovsky]] ([[User talk:AKA MBG|discuss]] • [[Special:Contributions/AKA MBG|contribs]]) 05:45, 30 May 2017 (UTC) |} ==See also== {{Archive box non-auto}} {{clear}} == Maths,life science == To be honest,ahem Marshall is telling the truth which is making sence and she has the right to be listened and let us hear what she wants to say. [[Special:Contributions/41.114.82.169|41.114.82.169]] ([[User talk:41.114.82.169|discuss]]) 14:33, 29 July 2022 (UTC) glp7gwuq01018nlfgaetrgmw1oprpop 2410244 2410241 2022-07-29T15:49:05Z Praxidicae 2805549 Reverted edits by [[Special:Contributions/41.114.82.169|41.114.82.169]] ([[User_talk:41.114.82.169|talk]]) to last version by [[User:Praxidicae|Praxidicae]] using [[Wikiversity:Rollback|rollback]] wikitext text/x-wiki {{Robelbox|theme=9|title=Welcome!|width=100%}} <div style="{{Robelbox/pad}}"> '''Hello Marshallsumter, and [[Wikiversity:Welcome|welcome]] to [[Wikiversity:What is Wikiversity?|Wikiversity]]!''' If you need [[Help:Contents|help]], feel free to visit my talk page, or [[Wikiversity:Contact|contact us]] and [[Wikiversity:Questions|ask questions]]. After you leave a comment on a [[Wikiversity:Talk page|talk page]], remember to [[Wikiversity:Signature|sign and date]]; it helps everyone follow the threads of the discussion. The signature icon [[File:Signature icon.png]] in the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy policy]], [[Wikiversity:Civility|Civility policy]], and the [[Foundation:Terms of Use|Terms of Use]] while at Wikiversity. To [[Wikiversity:Introduction|get started]], you may <!-- The Left column --> <div style="width:50.0%; float:left"> * [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]]. * Visit a (kind of) [[Wikiversity:Random|random project]]. * [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]]. * Find out about [[Wikiversity:Research|research]] activities on Wikiversity. * [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left. </div> <!-- The Right column --> <div style="width:50.0%; float:left"> * Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity. * Give [[Wikiversity:Feedback|feedback]] about your initial observations * Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]]. * [[Wikiversity:Chat|Chat]] with other Wikiversitans on [irc://irc.freenode.net/wikiversity-en <kbd>#wikiversity-en</kbd>]. * Follow Wikiversity on [[twitter]] (http://twitter.com/Wikiversity) and [[identi.ca]] (http://identi.ca/group/wikiversity). </div> <br clear="both"/> You don't need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Abd|Abd]] 04:25, 24 August 2011 (UTC)</div> {{Robelbox/close}} == Most Active Wikiversity User for January 2013 == {| style="border: 1px solid gray; background-color: #ffffff;" |rowspan="2" valign="middle" | [[Image:Learningcycle.png|100px]] |rowspan="2" | |style="font-size: x-large; padding: 0; vertical-align: middle; height: 1.1em;" | '''The Learning Cycle Barnstar''' |- |style="vertical-align: middle; border-top: 1px solid gray;" | Most Active Wikiversity User for January 2013 |} Marshallsumter, I was reviewing the list of active users for this past month and noticed you had by far the most edits in January. Keep up the good work! -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 00:04, 1 February 2013 (UTC) == Barnstar for you! == {| style="border: 1px solid gray; background-color: #ffffff;" |rowspan="2" valign="middle" | [[Image:Star constellation.png|100px]] |rowspan="2" | |style="font-size: x-large; padding: 0; vertical-align: middle; height: 1.1em;" | '''The astronomy barnstar''' |- |style="vertical-align: middle; border-top: 1px solid gray;" | Thank you for the massive edits on astronomy! [[User:Goldenburg111|Goldenburg111]] ([[User talk:Goldenburg111|talk]]|[[Special:Contributions/Goldenburg111|contribs]]) 18:49, 25 December 2013 (UTC) |} {| style="border: 1px solid gray; background-color: #ffffff;" |rowspan="2" valign="middle" | [[Image:Original_Barnstar.png|100px]] |rowspan="2" | |style="font-size: x-large; padding: 0; vertical-align: middle; height: 1.1em;" | '''The Original Barnstar''' |- |style="vertical-align: middle; border-top: 1px solid gray;" | Thank you for your help with [[Research in programming Wikidata]]! -- [[User:AKA MBG|Andrew Krizhanovsky]] ([[User talk:AKA MBG|discuss]] • [[Special:Contributions/AKA MBG|contribs]]) 05:45, 30 May 2017 (UTC) |} ==See also== {{Archive box non-auto}} {{clear}} nmv7d0fhw2iqiblob6owz5x1ovb0xch 0 137559 2410313 2102248 2022-07-29T21:58:28Z 196.191.221.90 Yes wikitext text/x-wiki {{TOCright}} ipconfig /renew renews the lease on dynamic (DHCP-assigned) IP addresses. ipconfig /release releases dynamic addresses. These activities will show you how to use ipconfig /renew and ipconfig /release. == Preparation == To prepare for this activity: # Start Windows. # Log in if necessary. == Activity 1 - Display DHCP Configuration Information == To display DHCP configuration information: # [[Command_Prompt/Open | Open a command prompt]].<ref>{{Cite web|url=https://www.google.com/|title=Google|website=www.google.com|access-date=2022-07-29}}fast internet speed</ref> # Use '''[[Ipconfig/All | ipconfig /all]]''' to display all IP configuration information. # Observe whether you have any network adapters that are '''DHCP Enabled'''. If so, identify your '''DHCP Server''', when it shows '''Lease Obtained''', and when it shows '''Lease Expires'''.<br>Note: If none of your network adapters are DHCP enabled, you will not be able to complete the following activities.8.8.8.8 == Activity 2 - Renew DHCP-Assigned Address Lease == To renew the lease on DHCP-assigned IP address(es): # Type '''ipconfig /renew''' and press '''Enter'''. # Use '''ipconfig /all''' to display all IP configuration information. # Observe the updated DHCP lease information. Dynamic IP addresses attempt to renew themselves automatically beginning at one-half of lease time. It is generally not necessary to renew DHCP-assigned addresses manually, but this approach may be used to test a DHCP server configuration from one or more clients. == Activity 3 - Release DHCP-Assigned Address == '''Note: You will not be able to access the Internet or other devices on your network from this point forward until you complete Activity 4.''' <br>To release the DCHP-assigned IP address: # Type '''ipconfig /release''' and press '''Enter'''.<br>Note that it now shows no IP address for the adapter(s). Dynamic IP addresses are released automatically when the system shuts down. It is generally not necessary to release DHCP-assigned addresses manually. # Wait ten seconds and then use '''ipconfig /all''' to display all IP configuration information. # Observe the updated IP address information. It may show no IP address (0.0.0.0), or it may show an Automatic Private IP Address (APIPA) address, one that begins with 169.254.n.n. Systems will typically assign an APIPA address if they cannot obtain an address from a DHCP server after a few seconds. APIPA addresses can work on the local network to connect with other devices that have an APIPA address, but do not have a default gateway and are unable to access the Internet. == Activity 4 - Obtain a DHCP-Assigned Address == To obtain a DHCP-assigned address: # Type '''ipconfig /renew''' and press '''Enter''' to obtain a DHCP-assigned IP address. # Use '''ipconfig /all''' to display all IP configuration information. # Observe the updated DHCP lease information. The system should now have a valid IP address on the network, and the address will very likely be the same address as the one displayed in Activity 1. # Close the command prompt to complete this activity. == Readings == * [[w:Ipconfig | Wikipedia: ipconfig]] * [[w:Internet_Protocol | Wikipedia: Internet Protocol]] * [[w:Dynamic_Host_Configuration_Protocol | Wikipedia Dynamic Host Configuration Protocol]] * [[w:Link-local_address | Wikipedia: Link-local address]] == References == * [http://technet.microsoft.com/en-us/library/bb490921.aspx Microsoft TechNet: Ipconfig] [[Category:Ipconfig]] [[Category:Activities]] [[Category:Dynamic Host Configuration Protocol]] ffso65q8v3aiaoz3k3k0k0h6kbnujgi 2410332 2410313 2022-07-30T00:01:42Z Dave Braunschweig 426084 Reverted edits by [[Special:Contributions/196.191.221.90|196.191.221.90]] ([[User_talk:196.191.221.90|talk]]) to last version by [[User:Dave Braunschweig|Dave Braunschweig]] using [[Wikiversity:Rollback|rollback]] wikitext text/x-wiki {{TOCright}} ipconfig /renew renews the lease on dynamic (DHCP-assigned) IP addresses. ipconfig /release releases dynamic addresses. These activities will show you how to use ipconfig /renew and ipconfig /release. == Preparation == To prepare for this activity: # Start Windows. # Log in if necessary. == Activity 1 - Display DHCP Configuration Information == To display DHCP configuration information: # [[Command_Prompt/Open | Open a command prompt]]. # Use '''[[Ipconfig/All | ipconfig /all]]''' to display all IP configuration information. # Observe whether you have any network adapters that are '''DHCP Enabled'''. If so, identify your '''DHCP Server''', when it shows '''Lease Obtained''', and when it shows '''Lease Expires'''.<br>Note: If none of your network adapters are DHCP enabled, you will not be able to complete the following activities. == Activity 2 - Renew DHCP-Assigned Address Lease == To renew the lease on DHCP-assigned IP address(es): # Type '''ipconfig /renew''' and press '''Enter'''. # Use '''ipconfig /all''' to display all IP configuration information. # Observe the updated DHCP lease information. Dynamic IP addresses attempt to renew themselves automatically beginning at one-half of lease time. It is generally not necessary to renew DHCP-assigned addresses manually, but this approach may be used to test a DHCP server configuration from one or more clients. == Activity 3 - Release DHCP-Assigned Address == '''Note: You will not be able to access the Internet or other devices on your network from this point forward until you complete Activity 4.''' <br>To release the DCHP-assigned IP address: # Type '''ipconfig /release''' and press '''Enter'''.<br>Note that it now shows no IP address for the adapter(s). Dynamic IP addresses are released automatically when the system shuts down. It is generally not necessary to release DHCP-assigned addresses manually. # Wait ten seconds and then use '''ipconfig /all''' to display all IP configuration information. # Observe the updated IP address information. It may show no IP address (0.0.0.0), or it may show an Automatic Private IP Address (APIPA) address, one that begins with 169.254.n.n. Systems will typically assign an APIPA address if they cannot obtain an address from a DHCP server after a few seconds. APIPA addresses can work on the local network to connect with other devices that have an APIPA address, but do not have a default gateway and are unable to access the Internet. == Activity 4 - Obtain a DHCP-Assigned Address == To obtain a DHCP-assigned address: # Type '''ipconfig /renew''' and press '''Enter''' to obtain a DHCP-assigned IP address. # Use '''ipconfig /all''' to display all IP configuration information. # Observe the updated DHCP lease information. The system should now have a valid IP address on the network, and the address will very likely be the same address as the one displayed in Activity 1. # Close the command prompt to complete this activity. == Readings == * [[w:Ipconfig | Wikipedia: ipconfig]] * [[w:Internet_Protocol | Wikipedia: Internet Protocol]] * [[w:Dynamic_Host_Configuration_Protocol | Wikipedia Dynamic Host Configuration Protocol]] * [[w:Link-local_address | Wikipedia: Link-local address]] == References == * [http://technet.microsoft.com/en-us/library/bb490921.aspx Microsoft TechNet: Ipconfig] [[Category:Ipconfig]] [[Category:Activities]] [[Category:Dynamic Host Configuration Protocol]] q3882u68a82a3l477xgy7bttiam0zat 0 139384 2410228 2409993 2022-07-29T13:26:53Z Young1lim 21186 /* Adder */ wikitext text/x-wiki {{nocat}} == Adder == * Binary Adder Architecture Exploration ( [[Media:adder.20131113.pdf |pdf]] ) {| class="wikitable" |- ! Adder type !! Overview !! Analysis !! VHDL Level Design !! CMOS Level Design |- | '''1. Ripple Carry Adder''' || [[Media:VLSI.Arith.1A.RCA.20211108.pdf |pdf]] || || [[Media:adder.rca.20140313.pdf |pdf]] || [[Media:VLSI.Arith.1D.RCA.CMOS.20211108.pdf |pdf]] |- | '''2. Carry Lookahead Adder''' || [[Media:VLSI.Arith.1.A.CLA.20211106.pdf |pdf]] || || [[Media:adder.cla.20140313.pdf |pdf]] || |- | '''3. Carry Save Adder''' || [[Media:VLSI.Arith.1.A.CSave.20151209.pdf |pdf]] || || || |- || '''4. Carry Select Adder''' || [[Media:VLSI.Arith.1.A.CSelA.20191002.pdf |pdf]] || || || |- || '''5. Carry Skip Adder''' || [[Media:VLSI.Arith.5A.CSkip.20211111.pdf |pdf]] || || || [[Media:VLSI.Arith.5D.CSkip.CMOS.20211108.pdf |pdf]] |- || '''6. Carry Chain Adder''' || [[Media:VLSI.Arith.6A.CCA.20211109.pdf |pdf]] || || [[Media:VLSI.Arith.6C.CCA.VHDL.20211109.pdf |pdf]], [[Media:adder.cca.20140313.pdf |pdf]] || [[Media:VLSI.Arith.6D.CCA.CMOS.20211109.pdf |pdf]] |- || '''7. Kogge-Stone Adder''' || [[Media:VLSI.Arith.1.A.KSA.20140315.pdf |pdf]] || || [[Media:adder.ksa.20140409.pdf |pdf]] || |- || '''8. Prefix Adder''' || [[Media:VLSI.Arith.1.A.PFA.20140314.pdf |pdf]] || || || |- || '''9. Variable Block Adder''' || [[Media:VLSI.Arith.1.A.VBA.20220728.pdf |pdf]] || || || |} </br> === Adder Architectures Suitable for FPGA === * FPGA Carry-Chain Adder ([[Media:VLSI.Arith.1.A.FPGA-CCA.20210421.pdf |pdf]]) * FPGA Carry Select Adder ([[Media:VLSI.Arith.1.B.FPGA-CarrySelect.20210522.pdf |pdf]]) * FPGA Variable Block Adder ([[Media:VLSI.Arith.1.C.FPGA-VariableBlock.20220125.pdf |pdf]]) * FPGA Carry Lookahead Adder ([[Media:VLSI.Arith.1.D.FPGA-CLookahead.20210304.pdf |pdf]]) * Carry-Skip Adder </br> == Barrel Shifter == * Barrel Shifter Architecture Exploration ([[Media:bshift.20131105.pdf |bshfit.vhdl]], [[Media:bshift.makefile.20131109.pdf |bshfit.makefile]]) </br> '''Mux Based Barrel Shifter''' * Analysis ([[Media:Arith.BShfiter.20151207.pdf |pdf]]) * Implementation </br> == Multiplier == === Array Multipliers === * Analysis ([[Media:VLSI.Arith.1.A.Mult.20151209.pdf |pdf]]) </br> === Tree Mulltipliers === * Lattice Multiplication ([[Media:VLSI.Arith.LatticeMult.20170204.pdf |pdf]]) * Wallace Tree ([[Media:VLSI.Arith.WallaceTree.20170204.pdf |pdf]]) * Dadda Tree ([[Media:VLSI.Arith.DaddaTree.20170701.pdf |pdf]]) </br> === Booth Multipliers === * [[Media:RNS4.BoothEncode.20161005.pdf |Booth Encoding Note]] * Booth Multiplier Note ([[Media:BoothMult.20160929.pdf |H1.pdf]]) </br> == Divider == * Binary Divider ([[Media:VLSI.Arith.1.A.Divider.20131217.pdf |pdf]])</br> </br> </br> go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] [[Category:Computer architecture]] kg2hxljv0vntz7jluu26sj0cmllt15s 0 157379 2410250 2395023 2022-07-29T17:29:48Z 70.54.122.39 wikitext text/x-wiki ==Les Nombres (numbers) 1-20== {| class="prettytable" !number !in French !pronunciation !listen !number !in French !pronunciation !listen |- |1 |un |(uhn) |[[Image:Fr-Un-fr FR-Paris.ogg]] |11 |onze |(ohnze) |[[Image:Fr-onze.ogg]] |- |2 |deux |(duh) |[[Image:Fr-deux-fr.ogg]] |12 |douze |(dooze) |[[Image:Fr-douze.ogg]] |- |3 |trois |(trwah) |[[Image:Fr-Trois-fr-Paris.ogg]] |13 |treize |(trez) |[[Image:Fr-treize.ogg]] |- |4 |quatre |(katr) |[[Image:Fr-quatre.ogg]] |14 |quatorze |(katorz) |[[Image:Fr-quatorze.ogg]] |- |5 |cinq |(sank) |[[Image:Fr-cinq-fr.ogg]] |15 |quinze |(kanz) |[[Image:Fr-quinze.ogg]] |- |6 |six |(sees) |[[Image:Fr-Six-fr-Paris.ogg]] |16 |seize |(sez) |[[Image:Fr-seize.ogg]] |- |7 |sept |(set) |[[Image:Fr-sept.ogg]] |17 |dix-sept |(dees-set) |[[Image:Fr-dix-sept.ogg]] |- |8 |huit |(weet) |[[Image:Fr-huit.ogg]] |18 |dix-huit |(dees-weet) |[[Image:Fr-dix-huit.ogg]] |- |9 |neuf |(nuff) |[[Image:Fr-Neuf-fr-Paris.ogg]] |19 |dix-neuf |(dees-nuff) |[[Image:Fr-dix-neuf.ogg]] |- |10 |dix |(dees) |[[Image:Fr-dix.ogg]] |20 |vingt |(van) |[[Image:Fr-vingt.ogg]] |} ==Les Nombres (numbers) 60-100 == {| class="prettytable" !number !in French !pronunciation !listen !number !in French !pronunciation !listen |- | 60 || soixante || || | 80 || '''quatre-vingts'''|| || |- | 61 || soixante-et-un || || | 81 || quatre-vingt-un || || |- | 62 || soixante-deux || || | 82 || quatre-vingt-deux || || |- | 70 || '''soixante-dix'''|| || | 90 || '''quatre-vingt-dix'''|| || |- | 71 || soixante-et-onze || || | 91 || quatre-vingt-onze || || |- | 72 || soixante-douze || || | 92 || quatre-vingt-douze || || |} ==Les Nombres (numbers) 100+ == {| class="prettytable" !number !in French !pronunciation !listen !number !in French !pronunciation !listen |- | 100 || '''cent'''|| || | {{formatnum:1000}} || '''mille'''|| || |- | 101 || cent-un || || | {{formatnum:10000}} || dix mille || || |- | 125 || cent-vingt-cinq || || | {{formatnum:100000}} || cent mille || || |- | 200 || deux cents || || | {{formatnum:1000000}} || un million || || |- | 201 || deux cent-un || || | {{formatnum:2000000}} || deux millions || || |- | 243 || deux cent-quarante-trois || || | {{formatnum:1000000000}} || '''un milliard'''|| || |} NB: in plural ''mille'' doesn't take -s. [[Category:French]] 5ut9apujfz6qg41fekmrdfyqjto01c0 2410334 2410250 2022-07-30T00:08:01Z Dave Braunschweig 426084 Reverted edits by [[Special:Contributions/70.54.122.39|70.54.122.39]] ([[User_talk:70.54.122.39|talk]]) to last version by [[User:Dave Braunschweig|Dave Braunschweig]] using [[Wikiversity:Rollback|rollback]] wikitext text/x-wiki ==Les Nombres (numbers) 1-20== {| class="prettytable" !number !in French !pronunciation !listen !number !in French !pronunciation !listen |- |1 |un |(uhn) |[[Image:Fr-Un-fr FR-Paris.ogg]] |11 |onze |(ohnze) |[[Image:Fr-onze.ogg]] |- |2 |deux |(duh) |[[Image:Fr-deux-fr.ogg]] |12 |douze |(dooze) |[[Image:Fr-douze.ogg]] |- |3 |trois |(trwah) |[[Image:Fr-Trois-fr-Paris.ogg]] |13 |treize |(trez) |[[Image:Fr-treize.ogg]] |- |4 |quatre |(katr) |[[Image:Fr-quatre.ogg]] |14 |quatorze |(katorz) |[[Image:Fr-quatorze.ogg]] |- |5 |cinq |(sank) |[[Image:Fr-cinq-fr.ogg]] |15 |quinze |(kanz) |[[Image:Fr-quinze.ogg]] |- |6 |six |(sees) |[[Image:Fr-Six-fr-Paris.ogg]] |16 |seize |(sez) |[[Image:Fr-seize.ogg]] |- |7 |sept |(set) |[[Image:Fr-sept.ogg]] |17 |dix-sept |(dees-set) |[[Image:Fr-dix-sept.ogg]] |- |8 |huit |(weet) |[[Image:Fr-huit.ogg]] |18 |dix-huit |(dees-weet) |[[Image:Fr-dix-huit.ogg]] |- |9 |neuf |(nuff) |[[Image:Fr-Neuf-fr-Paris.ogg]] |19 |dix-neuf |(dees-nuff) |[[Image:Fr-dix-neuf.ogg]] |- |10 |dix |(dees) |[[Image:Fr-dix.ogg]] |20 |vingt |(van) |[[Image:Fr-vingt.ogg]] |} ==Les Nombres (numbers) 20-60 == {| class="prettytable" !number !in French !pronunciation !listen !number !in French !listen !pronunciation |- | 20 || vingt || van|| || | 30 || '''trente'''|| || trente |- | 21 || vingt-et-un || van-eh-un|| || | 31 || trente-et-un || || trente-et-un |- | 22 || vingt-deux || van duh|| || | 32 || trente-deux || || trente-deux |- | 23 || vingt-trois || van trwah|| || | 40 || '''quarante'''|| || quarante |- | 24 || vingt-quatre || van katr|| || | 41 || quarante-et-un || || quarante-et-un |- | 25 || vingt-cinq || van sank|| || | 42 || quarante-deux || || quarante-deux |- | 26 || vingt-six || van sees|| || | 50 || '''cinquante'''|| || cinquanre |- | 27 || vingt-sept || van set|| || | 51 || cinquante-et-un || || cinquante-et-un |- | 28 || vingt-huit || van weet|| || | 60 || '''soixante'''|| || soixante |- | 29 || vingt-neuf || van nuff|| || | 61 || soixante-et-un || || soixante-et-un |} ==Les Nombres (numbers) 60-100 == {| class="prettytable" !number !in French !pronunciation !listen !number !in French !pronunciation !listen |- | 60 || soixante || || | 80 || '''quatre-vingts'''|| || |- | 61 || soixante-et-un || || | 81 || quatre-vingt-un || || |- | 62 || soixante-deux || || | 82 || quatre-vingt-deux || || |- | 70 || '''soixante-dix'''|| || | 90 || '''quatre-vingt-dix'''|| || |- | 71 || soixante-et-onze || || | 91 || quatre-vingt-onze || || |- | 72 || soixante-douze || || | 92 || quatre-vingt-douze || || |} ==Les Nombres (numbers) 100+ == {| class="prettytable" !number !in French !pronunciation !listen !number !in French !pronunciation !listen |- | 100 || '''cent'''|| || | {{formatnum:1000}} || '''mille'''|| || |- | 101 || cent-un || || | {{formatnum:10000}} || dix mille || || |- | 125 || cent-vingt-cinq || || | {{formatnum:100000}} || cent mille || || |- | 200 || deux cents || || | {{formatnum:1000000}} || un million || || |- | 201 || deux cent-un || || | {{formatnum:2000000}} || deux millions || || |- | 243 || deux cent-quarante-trois || || | {{formatnum:1000000000}} || '''un milliard'''|| || |} NB: in plural ''mille'' doesn't take -s. [[Category:French]] qrs4u1v1pxs5yaih8lm5lq920ajbq3r 0 171005 2410226 2409994 2022-07-29T13:25:37Z Young1lim 21186 /* Geometric Series Examples */ wikitext text/x-wiki Many of the functions that arise naturally in mathematics and real world applications can be extended to and regarded as complex functions, meaning the input, as well as the output, can be complex numbers <math>x+iy</math>, where <math>i=\sqrt{-1}</math>, in such a way that it is a more natural object to study. '''Complex analysis''', which used to be known as '''function theory''' or '''theory of functions of a single complex variable''', is a sub-field of analysis that studies such functions (more specifically, '''holomorphic''' functions) on the complex plane, or part (domain) or extension (Riemann surface) thereof. It notably has great importance in number theory, e.g. the [[Riemann zeta function]] (for the distribution of primes) and other <math>L</math>-functions, modular forms, elliptic functions, etc. <blockquote>The shortest path between two truths in the real domain passes through the complex domain. — [[wikipedia:Jacques_Hadamard|Jacques Hadamard]]</blockquote>In a certain sense, the essence of complex functions is captured by the principle of [[analytic continuation]].{{mathematics}} ==''' Complex Functions '''== * Complex Functions ([[Media:CAnal.1.A.CFunction.20140222.Basic.pdf|1.A.pdf]], [[Media:CAnal.1.B.CFunction.20140111.Octave.pdf|1.B.pdf]], [[Media:CAnal.1.C.CFunction.20140111.Extend.pdf|1.C.pdf]]) * Complex Exponential and Logarithm ([[Media:CAnal.5.A.CLog.20131017.pdf|5.A.pdf]], [[Media:CAnal.5.A.Octave.pdf|5.B.pdf]]) * Complex Trigonometric and Hyperbolic ([[Media:CAnal.7.A.CTrigHyper..pdf|7.A.pdf]], [[Media:CAnal.7.A.Octave..pdf|7.B.pdf]]) '''Complex Function Note''' : 1. Exp and Log Function Note ([[Media:ComplexExp.29160721.pdf|H1.pdf]]) : 2. Trig and TrigH Function Note ([[Media:CAnal.Trig-H.29160901.pdf|H1.pdf]]) : 3. Inverse Trig and TrigH Functions Note ([[Media:CAnal.Hyper.29160829.pdf|H1.pdf]]) ==''' Complex Integrals '''== * Complex Integrals ([[Media:CAnal.2.A.CIntegral.20140224.Basic.pdf|2.A.pdf]], [[Media:CAnal.2.B.CIntegral.20140117.Octave.pdf|2.B.pdf]], [[Media:CAnal.2.C.CIntegral.20140117.Extend.pdf|2.C.pdf]]) ==''' Complex Series '''== * Complex Series ([[Media:CPX.Series.20150226.2.Basic.pdf|3.A.pdf]], [[Media:CAnal.3.B.CSeries.20140121.Octave.pdf|3.B.pdf]], [[Media:CAnal.3.C.CSeries.20140303.Extend.pdf|3.C.pdf]]) ==''' Residue Integrals '''== * Residue Integrals ([[Media:CAnal.4.A.Residue.20140227.Basic.pdf|4.A.pdf]], [[Media:CAnal.4.B.pdf|4.B.pdf]], [[Media:CAnal.4.C.Residue.20140423.Extend.pdf|4.C.pdf]]) ==='''Residue Integrals Note'''=== * Laurent Series with the Residue Theorem Note ([[Media:Laurent.1.Residue.20170713.pdf|H1.pdf]]) * Laurent Series with Applications Note ([[Media:Laurent.2.Applications.20170327.pdf|H1.pdf]]) * Laurent Series and the z-Transform Note ([[Media:Laurent.3.z-Trans.20170831.pdf|H1.pdf]]) * Laurent Series as a Geometric Series Note ([[Media:Laurent.4.GSeries.20170802.pdf|H1.pdf]]) === Laurent Series and the z-Transform Example Note === * Overview ([[Media:Laurent.4.z-Example.20170926.pdf|H1.pdf]]) ====Geometric Series Examples==== * Causality ([[Media:Laurent.5.Causality.1.A.20191026n.pdf|A.pdf]], [[Media:Laurent.5.Causality.1.B.20191026.pdf|B.pdf]]) * Time Shift ([[Media:Laurent.5.TimeShift.2.A.20191028.pdf|A.pdf]], [[Media:Laurent.5.TimeShift.2.B.20191029.pdf|B.pdf]]) * Reciprocity ([[Media:Laurent.5.Reciprocity.3A.20191030.pdf|A.pdf]], [[Media:Laurent.5.Reciprocity.3B.20191031.pdf|B.pdf]]) * Combinations ([[Media:Laurent.5.Combination.4A.20200702.pdf|A.pdf]], [[Media:Laurent.5.Combination.4B.20201002.pdf|B.pdf]]) * Properties ([[Media:Laurent.5.Property.5A.20220105.pdf|A.pdf]], [[Media:Laurent.5.Property.5B.20220126.pdf|B.pdf]]) * Applications ([[Media:Laurent.6.Application.6A.20220728.pdf|A.pdf]], [[Media:Laurent.5.Application.6B.20220723.pdf|B.pdf]]) * Double Pole Case :- Examples ([[Media:Laurent.5.DPoleEx.7A.20220722.pdf|A.pdf]], [[Media:Laurent.5.DPoleEx.7B.20220720.pdf|B.pdf]]) :- Properties ([[Media:Laurent.5.DPoleProp.5A.20190226.pdf|A.pdf]], [[Media:Laurent.5.DPoleProp.5B.20190228.pdf|B.pdf]]) ====The Case Examples==== * Example Overview : ([[Media:Laurent.4.Example.0.A.20171208.pdf|0A.pdf]], [[Media:Laurent.6.CaseExample.0.B.20180205.pdf|0B.pdf]]) * Example Case 1 : ([[Media:Laurent.4.Example.1.A.20171107.pdf|1A.pdf]], [[Media:Laurent.4.Example.1.B.20171227.pdf|1B.pdf]]) * Example Case 2 : ([[Media:Laurent.4.Example.2.A.20171107.pdf|2A.pdf]], [[Media:Laurent.4.Example.2.B.20171227.pdf|2B.pdf]]) * Example Case 3 : ([[Media:Laurent.4.Example.3.A.20171017.pdf|3A.pdf]], [[Media:Laurent.4.Example.3.B.20171226.pdf|3B.pdf]]) * Example Case 4 : ([[Media:Laurent.4.Example.4.A.20171017.pdf|4A.pdf]], [[Media:Laurent.4.Example.4.B.20171228.pdf|4B.pdf]]) * Example Summary : ([[Media:Laurent.4.Example.5.A.20171212.pdf|5A.pdf]], [[Media:Laurent.4.Example.5.B.20171230.pdf|5B.pdf]]) ==''' Conformal Mapping '''== * Conformal Mapping ([[Media:CAnal.6.A.Conformal.20131224.pdf|6.A.pdf]], [[Media:CAnal.6.A.Octave..pdf|6.B.pdf]]) go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] [[Category:Complex analysis]] 4e733g0ai2ejp45mey9tnn5onl8shdu 0 171008 2410294 2409828 2022-07-29T20:05:27Z Young1lim 21186 /* The Temporal Characteristics of Random Processes */ wikitext text/x-wiki ==''' Random Variables '''== === Single Random Variables === * Random Variables ([[Media:RV1.RVariable.1.A.20200427.pdf |A.pdf]]) * Distribution Function ([[Media:RV1.Distribution.2.A.201200428.pdf |A.pdf]]) * Density Function ([[Media:RV1.Density.3.A.20200429.pdf |A.pdf]]) * Functions of Random Variables ([[Media:RV1.RVFunction.4.A.20220317.pdf |A.pdf]]) * Gaussian Random Variables ([[Media:RV1.4.Gaussian.20200430.pdf |A.pdf]], [[Media:RV1.4B.Gaussian.20180314.pdf |B.pdf]]) * Other Distribution and Density Functions ([[Media:RV1.5.Other.20200501.pdf |A.pdf]]) * Conditional Distribution and Density Functions ([[Media:RV1.6.Conditional.20200506.pdf |A.pdf]]) </br> === The Characteristics of a Single Random Variable === * Expected Value ([[Media:RV2.Expectation.1.A.20200506.pdf |A.pdf]]) * Moments ([[Media:RV2.Moment.2.B.20200507.pdf |A.pdf]], [[Media:RV2.Moment.2.B.20180320.pdf |B.pdf]]) * Moment Generating Functions ([[Media:RV2.MFunctions.3.A.20200508.pdf |A.pdf]]) * Transformations of Random Variables ([[Media:RV2.Transform.4.A.20200514.pdf |A.pdf]]) </br> === Multiple Random Variables === * Vector Random Variables ([[Media:3MRV.1A.VectorRV.20200515.pdf |A.pdf]]) * Joint Distribution ([[Media:3MRV.2A.JointDist.20200518.pdf |A.pdf]]) * Joint Density ([[Media:3MRV.3A.JointDensity.20200521.pdf |A.pdf]]) * Conditional Joint Distribution and Density ([[Media:3MRV.4A.CondDistrib.20200527.pdf |A.pdf]]) * Statistical Independence ([[Media:3MRV.5A.StatIndep.20200423.pdf |A.pdf]]) * Sums of Random Variables ([[Media:3MRV.6A.RVSum.20200528.pdf |A.pdf]]) * Central Limit Theorem ([[Media:3MRV.7A.CLimit.20200612.pdf |A.pdf]]) </br> === The Characteristics of Multiple Random Variables === * Expected Values ([[Media:4MRV.1A.Expect.20200617.pdf |A.pdf]]) * Joint Characteristic Functions ([[Media:4MRV.2A.JChar.20200618.pdf |A.pdf]]) * Jointly Gaussian Random Variables ([[Media:4MRV.3A.JGauss.20200619.pdf |A.pdf]]) * Transformations of Multiple Random Variables ([[Media:4MRV.4A.Transform.20200620.pdf |A.pdf]]) * Linear Transformation of Gaussian Random Variables ([[Media:4MRV.5A.LinearTrans.20200623.pdf |A.pdf]]) * Simulating Multiple Random Variables ([[Media:4MRV.6A.Simulation.20200624.pdf |A.pdf]]) * Sampling and Some Limit Theorem ([[Media:4MRV.7A.LimitTheorem.20200625.pdf |A.pdf]]) * Complex Random Variables ([[Media:4MRV.8A.ComplexRV.20200626.pdf |A.pdf]]) </br> ==''' Random Processes '''== === The Temporal Characteristics of Random Processes === * Random Processes ([[Media:5MRV.1A.RandomProcess.20210216.pdf |A.pdf]]) * Joint Distribution, Independence, Processes ([[Media:5MRV.2A.JointDistribution.20210220.pdf |A.pdf]]) * Stationary Random Processes ([[Media:5MRV.3A.Stationary.20220402.pdf |A.pdf]], [[Media:5MRV.3B.Stationary.20220727.pdf |B.pdf]]) * Covariance & Correlation of Random Variables ([[Media:5MRV.4A.CovCorrRV.20210910.pdf |A.pdf]]) * Covariance & Correlation of Random Processses ([[Media:5MRV.5A.CovCorrRP.20210911.pdf |A.pdf]]) * Example Random Processes ([[Media:5MRV.4A.Example.20210227.pdf |A.pdf]]) * Ergodic Random Processes ([[Media:5MRV.5A.Ergodicity.20211022.pdf |A.pdf]], [[Media:5MRV.7B.Ergodicity.20211215.pdf |B.pdf]]) * Measurement of Correlation Functions ([[Media:5MRV.6A.Measure.20201013.pdf |A.pdf]]) * Complex Random Processes ([[Media:5MRV.7A.Complex.20201022.pdf |A.pdf]]) </br> === The Spectral Characteristics of Random Processes === * Power Density Spectrum - Continuous Time ([[Media:6MRV.1A.PSpecCT.20210204.pdf |A.pdf]]) * Auto Correlation Function ([[Media:6MRV.2A.AutoCor.20201218.pdf |A.pdf]]) * Power Density Spectrum - Discrete Time ([[Media:6MRV.3A.PSpecDT.20201203.pdf |A.pdf]]) * Cross Power Density Spectrum ([[Media:6MRV.4A.CPSpec.20191108.pdf |A.pdf]]) * Cross Correlation Function ([[Media:6MRV.5A.CCorrel.20191114.pdf |A.pdf]]) * Noise Definitions ([[Media:6MRV.6A.Noise.20191121.pdf |A.pdf]]) * Power Spectrum of Complex Random Processes ([[Media:6MRV.7A.ComplexProc.20191125.pdf |A.pdf]]) </br> === Linear System with Random Inputs === * Continuous Time LTI System ([[Media:7LTI.1A.CTime.20191203.pdf |A.pdf]]) * Discrete Time LTI System ([[Media:7LTI.2A.DTime.20191211.pdf |A.pdf]]) * System Response ([[Media:7LTI.3A.Response.20191224.pdf |A.pdf]]) * Spectral Characteristics ([[Media:7LTI.4A.Spectral.20200104.pdf |A.pdf]]) * Noise Modeling ([[Media:7LTI.5A.Noise.20200122.pdf |A.pdf]]) <br> === Optimum Linear System === * Maximum SNR ([[Media:8OPT.1A.MaxSNR.20200128.pdf |A.pdf]]) * Minimum Squared Error ([[Media:8OPT.2A.MinSE.20200207.pdf |A.pdf]]) <br> === Noise in Some Application Systems === * AM Communication Systems ([[Media:9APP.1A.AM.20200212.pdf |A.pdf]]) * FM Communication Systems ([[Media:9APP.2A.FM.20200221.pdf |A.pdf]]) * Control Systems ([[Media:9APP.3A.Control.20200225.pdf |A.pdf]]) * PLL Systems ([[Media:9APP.4A.PLL.20200305.pdf |A.pdf]]) * Random Waveforms ([[Media:9APP.5A.RandWave.20200311.pdf |A.pdf]]) * Radar Systems ([[Media:9APP.6A.Radar.20200313.pdf |A.pdf]]) <br> ==''' Correlation and Power Spectra '''== # Correlation Functions of Random Signals ([[Media:RAND.1.A.Correlation.20121106.pdf |pdf]]) # Spectra of Random Signals ([[Media:RAND.2.A.Spectra.20121108.pdf |pdf]]) </br> ==''' Ergodicity, Statistics, Estimation '''== </br> ==''' Random Processes and Linear Systems '''== </br> # Time Domain Techniques for Noisy Signals ([[Media:RAND.3.A.Time.20130205.pdf |A.pdf]], [[Media:Dirichlet.pdf |B.pdf]]) # Frequency Domain Techniques for Noisy Signals # Correlation v.s. Convolution for Noisy Signals </br> # System Identification ([[Media:RP.SysId.1.A.pdf |pdf]]) # Matched Filter <ref>[[Understanding Digital Communications]], See Baseband Mod/Demod Section</ref> </br> </br> go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] 3202r2lj6bga4vhv3qpt77z0vdkl98o 2410297 2410294 2022-07-29T20:07:06Z Young1lim 21186 /* The Temporal Characteristics of Random Processes */ wikitext text/x-wiki ==''' Random Variables '''== === Single Random Variables === * Random Variables ([[Media:RV1.RVariable.1.A.20200427.pdf |A.pdf]]) * Distribution Function ([[Media:RV1.Distribution.2.A.201200428.pdf |A.pdf]]) * Density Function ([[Media:RV1.Density.3.A.20200429.pdf |A.pdf]]) * Functions of Random Variables ([[Media:RV1.RVFunction.4.A.20220317.pdf |A.pdf]]) * Gaussian Random Variables ([[Media:RV1.4.Gaussian.20200430.pdf |A.pdf]], [[Media:RV1.4B.Gaussian.20180314.pdf |B.pdf]]) * Other Distribution and Density Functions ([[Media:RV1.5.Other.20200501.pdf |A.pdf]]) * Conditional Distribution and Density Functions ([[Media:RV1.6.Conditional.20200506.pdf |A.pdf]]) </br> === The Characteristics of a Single Random Variable === * Expected Value ([[Media:RV2.Expectation.1.A.20200506.pdf |A.pdf]]) * Moments ([[Media:RV2.Moment.2.B.20200507.pdf |A.pdf]], [[Media:RV2.Moment.2.B.20180320.pdf |B.pdf]]) * Moment Generating Functions ([[Media:RV2.MFunctions.3.A.20200508.pdf |A.pdf]]) * Transformations of Random Variables ([[Media:RV2.Transform.4.A.20200514.pdf |A.pdf]]) </br> === Multiple Random Variables === * Vector Random Variables ([[Media:3MRV.1A.VectorRV.20200515.pdf |A.pdf]]) * Joint Distribution ([[Media:3MRV.2A.JointDist.20200518.pdf |A.pdf]]) * Joint Density ([[Media:3MRV.3A.JointDensity.20200521.pdf |A.pdf]]) * Conditional Joint Distribution and Density ([[Media:3MRV.4A.CondDistrib.20200527.pdf |A.pdf]]) * Statistical Independence ([[Media:3MRV.5A.StatIndep.20200423.pdf |A.pdf]]) * Sums of Random Variables ([[Media:3MRV.6A.RVSum.20200528.pdf |A.pdf]]) * Central Limit Theorem ([[Media:3MRV.7A.CLimit.20200612.pdf |A.pdf]]) </br> === The Characteristics of Multiple Random Variables === * Expected Values ([[Media:4MRV.1A.Expect.20200617.pdf |A.pdf]]) * Joint Characteristic Functions ([[Media:4MRV.2A.JChar.20200618.pdf |A.pdf]]) * Jointly Gaussian Random Variables ([[Media:4MRV.3A.JGauss.20200619.pdf |A.pdf]]) * Transformations of Multiple Random Variables ([[Media:4MRV.4A.Transform.20200620.pdf |A.pdf]]) * Linear Transformation of Gaussian Random Variables ([[Media:4MRV.5A.LinearTrans.20200623.pdf |A.pdf]]) * Simulating Multiple Random Variables ([[Media:4MRV.6A.Simulation.20200624.pdf |A.pdf]]) * Sampling and Some Limit Theorem ([[Media:4MRV.7A.LimitTheorem.20200625.pdf |A.pdf]]) * Complex Random Variables ([[Media:4MRV.8A.ComplexRV.20200626.pdf |A.pdf]]) </br> ==''' Random Processes '''== === The Temporal Characteristics of Random Processes === * Random Processes ([[Media:5MRV.1A.RandomProcess.20210216.pdf |A.pdf]]) * Joint Distribution, Independence, Processes ([[Media:5MRV.2A.JointDistribution.20210220.pdf |A.pdf]]) * Stationary Random Processes ([[Media:5MRV.3A.Stationary.20220402.pdf |A.pdf]], [[Media:5MRV.3B.Stationary.20220728.pdf |B.pdf]]) * Covariance & Correlation of Random Variables ([[Media:5MRV.4A.CovCorrRV.20210910.pdf |A.pdf]]) * Covariance & Correlation of Random Processses ([[Media:5MRV.5A.CovCorrRP.20210911.pdf |A.pdf]]) * Example Random Processes ([[Media:5MRV.4A.Example.20210227.pdf |A.pdf]]) * Ergodic Random Processes ([[Media:5MRV.5A.Ergodicity.20211022.pdf |A.pdf]], [[Media:5MRV.7B.Ergodicity.20211215.pdf |B.pdf]]) * Measurement of Correlation Functions ([[Media:5MRV.6A.Measure.20201013.pdf |A.pdf]]) * Complex Random Processes ([[Media:5MRV.7A.Complex.20201022.pdf |A.pdf]]) </br> === The Spectral Characteristics of Random Processes === * Power Density Spectrum - Continuous Time ([[Media:6MRV.1A.PSpecCT.20210204.pdf |A.pdf]]) * Auto Correlation Function ([[Media:6MRV.2A.AutoCor.20201218.pdf |A.pdf]]) * Power Density Spectrum - Discrete Time ([[Media:6MRV.3A.PSpecDT.20201203.pdf |A.pdf]]) * Cross Power Density Spectrum ([[Media:6MRV.4A.CPSpec.20191108.pdf |A.pdf]]) * Cross Correlation Function ([[Media:6MRV.5A.CCorrel.20191114.pdf |A.pdf]]) * Noise Definitions ([[Media:6MRV.6A.Noise.20191121.pdf |A.pdf]]) * Power Spectrum of Complex Random Processes ([[Media:6MRV.7A.ComplexProc.20191125.pdf |A.pdf]]) </br> === Linear System with Random Inputs === * Continuous Time LTI System ([[Media:7LTI.1A.CTime.20191203.pdf |A.pdf]]) * Discrete Time LTI System ([[Media:7LTI.2A.DTime.20191211.pdf |A.pdf]]) * System Response ([[Media:7LTI.3A.Response.20191224.pdf |A.pdf]]) * Spectral Characteristics ([[Media:7LTI.4A.Spectral.20200104.pdf |A.pdf]]) * Noise Modeling ([[Media:7LTI.5A.Noise.20200122.pdf |A.pdf]]) <br> === Optimum Linear System === * Maximum SNR ([[Media:8OPT.1A.MaxSNR.20200128.pdf |A.pdf]]) * Minimum Squared Error ([[Media:8OPT.2A.MinSE.20200207.pdf |A.pdf]]) <br> === Noise in Some Application Systems === * AM Communication Systems ([[Media:9APP.1A.AM.20200212.pdf |A.pdf]]) * FM Communication Systems ([[Media:9APP.2A.FM.20200221.pdf |A.pdf]]) * Control Systems ([[Media:9APP.3A.Control.20200225.pdf |A.pdf]]) * PLL Systems ([[Media:9APP.4A.PLL.20200305.pdf |A.pdf]]) * Random Waveforms ([[Media:9APP.5A.RandWave.20200311.pdf |A.pdf]]) * Radar Systems ([[Media:9APP.6A.Radar.20200313.pdf |A.pdf]]) <br> ==''' Correlation and Power Spectra '''== # Correlation Functions of Random Signals ([[Media:RAND.1.A.Correlation.20121106.pdf |pdf]]) # Spectra of Random Signals ([[Media:RAND.2.A.Spectra.20121108.pdf |pdf]]) </br> ==''' Ergodicity, Statistics, Estimation '''== </br> ==''' Random Processes and Linear Systems '''== </br> # Time Domain Techniques for Noisy Signals ([[Media:RAND.3.A.Time.20130205.pdf |A.pdf]], [[Media:Dirichlet.pdf |B.pdf]]) # Frequency Domain Techniques for Noisy Signals # Correlation v.s. Convolution for Noisy Signals </br> # System Identification ([[Media:RP.SysId.1.A.pdf |pdf]]) # Matched Filter <ref>[[Understanding Digital Communications]], See Baseband Mod/Demod Section</ref> </br> </br> go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] 4ry47okbi91tatz7qco08f1kwqrirzo 0 203942 2410308 2409902 2022-07-29T20:22:53Z Young1lim 21186 /* Monads III : Mutable State Monads */ wikitext text/x-wiki ==Introduction== * Overview I ([[Media:HSKL.Overview.1.A.20160806.pdf |pdf]]) * Overview II ([[Media:HSKL.Overview.2.A.20160926.pdf |pdf]]) * Overview III ([[Media:HSKL.Overview.3.A.20161011.pdf |pdf]]) * Overview IV ([[Media:HSKL.Overview.4.A.20161104.pdf |pdf]]) * Overview V ([[Media:HSKL.Overview.5.A.20161108.pdf |pdf]]) </br> ==Applications== * Sudoku Background ([[Media:Sudoku.Background.0.A.20161108.pdf |pdf]]) * Bird's Implementation :- Specification ([[Media:Sudoku.1Bird.1.A.Spec.20170425.pdf |pdf]]) :- Rules ([[Media:Sudoku.1Bird.2.A.Rule.20170201.pdf |pdf]]) :- Pruning ([[Media:Sudoku.1Bird.3.A.Pruning.20170211.pdf |pdf]]) :- Expanding ([[Media:Sudoku.1Bird.4.A.Expand.20170506.pdf |pdf]]) </br> ==Using GHCi== * Getting started ([[Media:GHCi.Start.1.A.20170605.pdf |pdf]]) </br> ==Using Libraries== * Library ([[Media:Library.1.A.20170605.pdf |pdf]]) </br> </br> ==Function Oriented Typeclasses== === Background === * Constructors ([[Media:Background.1.A.Constructor.20180904.pdf |pdf]]) * TypeClasses ([[Media:Background.1.B.TypeClass.20180904.pdf |pdf]]) * Functions ([[Media:Background.1.C.Function.20180712.pdf |pdf]]) * Expressions ([[Media:Background.1.D.Expression.20180707.pdf |pdf]]) * Operators ([[Media:Background.1.E.Operator.20180707.pdf |pdf]]) === Functors === * Functor Overview ([[Media:Functor.1.A.Overview.20180802.pdf |pdf]]) * Function Functor ([[Media:Functor.2.A.Function.20180804.pdf |pdf]]) * Functor Lifting ([[Media:Functor.2.B.Lifting.20180721.pdf |pdf]]) === Applicatives === * Applicatives Overview ([[Media:Applicative.3.A.Overview.20180606.pdf |pdf]]) * Applicatives Methods ([[Media:Applicative.3.B.Method.20180519.pdf |pdf]]) * Function Applicative ([[Media:Applicative.3.A.Function.20180804.pdf |pdf]]) * Applicatives Sequencing ([[Media:Applicative.3.C.Sequencing.20180606.pdf |pdf]]) === Monads I : Background === * Side Effects ([[Media:Monad.P1.1A.SideEffect.20190316.pdf |pdf]]) * Monad Overview ([[Media:Monad.P1.2A.Overview.20190308.pdf |pdf]]) * Monadic Operations ([[Media:Monad.P1.3A.Operations.20190308.pdf |pdf]]) * Maybe Monad ([[Media:Monad.P1.4A.Maybe.201900606.pdf |pdf]]) * IO Actions ([[Media:Monad.P1.5A.IOAction.20190606.pdf |pdf]]) * Several Monad Types ([[Media:Monad.P1.6A.Types.20191016.pdf |pdf]]) === Monads II : State Transformer Monads === * State Transformer : - State Transformer Basics ([[Media:MP2.1A.STrans.Basic.20191002.pdf |pdf]]) : - State Transformer Generic Monad ([[Media:MP2.1B.STrans.Generic.20191002.pdf |pdf]]) : - State Transformer Monads ([[Media:MP2.1C.STrans.Monad.20191022.pdf |pdf]]) * State Monad : - State Monad Basics ([[Media:MP2.2A.State.Basic.20190706.pdf |pdf]]) : - State Monad Methods ([[Media:MP2.2B.State.Method.20190706.pdf |pdf]]) : - State Monad Examples ([[Media:MP2.2C.State.Example.20190706.pdf |pdf]]) === Monads III : Mutable State Monads === * Mutability Background : - Types ([[Media:MP3.1A.Mut.Type.20200721.pdf |pdf]]) : - Primitive Types ([[Media:MP3.1B.Mut.PrimType.20200611.pdf |pdf]]) : - Polymorphic Types ([[Media:MP3.1C.Mut.Polymorphic.20201212.pdf |pdf]]) : - Continuation Passing Style ([[Media:MP3.1D.Mut.Continuation.20220110.pdf |pdf]]) : - Expressions ([[Media:MP3.1E.Mut.Expression.20220628.pdf |pdf]]) : - Lambda Calculus ([[Media:MP3.1F.Mut.LambdaCal.20220728.pdf |pdf]]) : - Non-terminating Expressions ([[Media:MP3.1F.Mut.Non-terminating.20220616.pdf |pdf]]) : - Inhabitedness ([[Media:MP3.1F.Mut.Inhabited.20220319.pdf |pdf]]) : - Existential Types ([[Media:MP3.1E.Mut.Existential.20220128.pdf |pdf]]) : - forall Keyword ([[Media:MP3.1E.Mut.forall.20210316.pdf |pdf]]) : - Mutability and Strictness ([[Media:MP3.1C.Mut.Strictness.20200613.pdf |pdf]]) : - Strict and Lazy Packages ([[Media:MP3.1D.Mut.Package.20200620.pdf |pdf]]) * Mutable Objects : - Mutable Variables ([[Media:MP3.1B.Mut.Variable.20200224.pdf |pdf]]) : - Mutable Data Structures ([[Media:MP3.1D.Mut.DataStruct.20191226.pdf |pdf]]) * IO Monad : - IO Monad Basics ([[Media:MP3.2A.IO.Basic.20191019.pdf |pdf]]) : - IO Monad Methods ([[Media:MP3.2B.IO.Method.20191022.pdf |pdf]]) : - IORef Mutable Variable ([[Media:MP3.2C.IO.IORef.20191019.pdf |pdf]]) * ST Monad : - ST Monad Basics ([[Media:MP3.3A.ST.Basic.20191031.pdf |pdf]]) : - ST Monad Methods ([[Media:MP3.3B.ST.Method.20191023.pdf |pdf]]) : - STRef Mutable Variable ([[Media:MP3.3C.ST.STRef.20191023.pdf |pdf]]) === Monads IV : Reader and Writer Monads === * Function Monad ([[Media:Monad.10.A.Function.20180806.pdf |pdf]]) * Monad Transformer ([[Media:Monad.3.I.Transformer.20180727.pdf |pdf]]) * MonadState Class :: - State & StateT Monads ([[Media:Monad.9.A.MonadState.Monad.20180920.pdf |pdf]]) :: - MonadReader Class ([[Media:Monad.9.B.MonadState.Class.20180920.pdf |pdf]]) * MonadReader Class :: - Reader & ReaderT Monads ([[Media:Monad.11.A.Reader.20180821.pdf |pdf]]) :: - MonadReader Class ([[Media:Monad.12.A.MonadReader.20180821.pdf |pdf]]) * Control Monad ([[Media:Monad.9.A.Control.20180908.pdf |pdf]]) === Monoid === * Monoids ([[Media:Monoid.4.A.20180508.pdf |pdf]]) === Arrow === * Arrows ([[Media:Arrow.1.A.20190504.pdf |pdf]]) </br> ==Polymorphism== * Polymorphism Overview ([[Media:Poly.1.A.20180220.pdf |pdf]]) </br> ==Concurrent Haskell == </br> go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] ==External links== * [http://learnyouahaskell.com/introduction Learn you Haskell] * [http://book.realworldhaskell.org/read/ Real World Haskell] * [http://www.scs.stanford.edu/14sp-cs240h/slides/ Standford Class Material] [[Category:Computer programming]] ggvvgc6zcc4mw4gyfvaj2vxu16avxv1 0 207105 2410307 2408569 2022-07-29T20:21:39Z Aherman012 2943941 /* Psychometric properties of screening measures for ODD */ Updated reference wikitext text/x-wiki <noinclude>{{Helping Give Away Psychological Science Banner}}</noinclude> {{medical disclaimer}} {{:{{BASEPAGENAME}}/Sidebar}} ==[[Evidence based assessment/Portfolio template/What is a "portfolio"|'''What is a "portfolio"?''']]== * For background information on what assessment portfolios are, click the link in the heading above. ==[[Evidence based assessment/Preparation phase|'''Preparation phase''']]== === Diagnostic criteria for oppositional defiant disorder=== {{blockquotetop}} <big>ICD-11 Diagnostic Criteria</big><br> <br> '''General Description:''' Oppositional defiant disorder is a persistent pattern (e.g., 6 months or more) of markedly defiant, disobedient, provocative or spiteful behaviour that occurs more frequently than is typically observed in individuals of comparable age and developmental level and that is not restricted to interaction with siblings. Oppositional defiant disorder may be manifest in prevailing, persistent angry or irritable mood, often accompanied by severe temper outbursts or in headstrong, argumentative and defiant behaviour. The behavior pattern is of sufficient severity to result in significant impairment in personal, family, social, educational, occupational or other important areas of functioning <br> <br> '''Oppositional Defiant Disorder With Chronic Irritability-Anger:''' All definitional requirements for oppositional defiant disorder are met. This form of oppositional defiant disorder is characterized by prevailing, persistent angry or irritable mood that may be present independent of any apparent provocation. The negative mood is often accompanied by regularly occurring severe temper outbursts that are grossly out of proportion in intensity or duration to the provocation. Chronic irritability and anger are characteristic of the individual’s functioning nearly every day, are observable across multiple settings or domains of functioning (e.g., home, school, social relationships), and are not restricted to the individual’s relationship with his/her parents or guardians. The pattern of chronic irritability and anger is not limited to occasional episodes (e.g., developmentally typical irritability) or discrete periods (e.g., irritable mood in the context of manic or depressive episodes). *Note: The ICD-11 lists 3 additional subcategories of oppositional defiant disorder with chronic irritability-anger (i.e., with limited prosocial emotions, with typical prosocial emotions, and unspecified). They can be found [https://icd.who.int/browse11/l-m/en#/http%3a%2f%2fid.who.int%2ficd%2fentity%2f818792113 here]. <br> '''Oppositional Defiant Disorder Without Chronic Irritability-Anger:''' Meets all definitional requirements for oppositional defiant disorder. This form of oppositional defiant disorder is not characterized by prevailing, persistent, angry or irritable mood, but does feature headstrong, argumentative, and defiant behavior. *Note: The ICD-11 lists 3 additional subcategories of oppositional defiant disorder without chronic irritability-anger (i.e., with limited prosocial emotions, with typical prosocial emotions, and unspecified). They can be found [https://icd.who.int/browse11/l-m/en#/http%3a%2f%2fid.who.int%2ficd%2fentity%2f736540987 here]. <br> <big>Changes in DSM-5</big> <br> The diagnostic criteria for ADHD changed slightly from DSM-IV to DSM-5. See the changes [https://www.ncbi.nlm.nih.gov/books/NBK519712/table/ch3.t14/ here]. {{blockquotebottom}}<ref>{{Cite web|url=https://icd.who.int/browse11/l-m/en#/http://id.who.int/icd/entity/1545599508|title=ICD-11 for Mortality and Morbidity Statistics|website=icd.who.int|access-date=2022-07-11}}</ref> ===Base rates of ODD in different clinical settings=== This section describes the demographic setting of the population(s) sampled, base rates of diagnosis, country/region sampled and the diagnostic method that was used. Using this information, clinicians will be able to anchor the rate of ODD that they are likely to see in their clinical practice. * '''''To see prevalence rates across multiple disorders, [[Evidence-based assessment/Preparation phase|click here.]]''''' {|class="wikitable sortable" border="1" |- ! Demography ! Setting !! Base Rate !! Diagnostic Method |- || All of the U.S. | Nationally representative large-scale study (N = 3,119) <ref>Nock, M. K., Kazdin, A. E., Hiripi, E., & Kessler, R. C. (2007). Lifetime prevalence, correlates, and persistence of oppositional defiant disorder: results from the National Comorbidity Survey Replication. ''Journal of Child Psychology and Psychiatry, 48''(7), 703-713.</ref> || 10.2% (overall) 11.2% (males) 9.2% (females) || World Health Organization (WHO) Composite International Diagnostic Interview (CIDI)^r |- || Semi-rural North Carolina | Preschool-aged children from pediatric practices (N = 306; age 2 - 5 years old)<ref>Egger, H.L., & Angold, A. (2006). Common emotional and behavioral disorders in preschool children: Presentation, nosology, and epidemiology. ''Journal of Child Psychiatry and Psychology, 47'', 313–337.</ref> || 6.6% || Preschool Age Psychiatric Assessment (PAPA)^p |- || Western North Carolina | The Great Smoky Mountains Study - longitudinal, population-based study of community sample<ref>Costello, E. J., Mustillo, S., Erkanli, A., Keeler, G., & Angold, A. (2003) Prevalence and development of psychiatric disorders in adolescence. ''Arch Gen Psychiatry, 60'', 837-844.</ref> || 2.33% (overall) 3.16% (males) 2.75% (females) || Child and Adolescent Psychiatric Assessment (CAPA)^{p, y} |- || Chicago | Preschool-aged children from inner city schools and pediatric practices (N = 796; age 2 - 5 years old)<ref>Lavigne, J. V., LeBailly, S. A., Hopkins, J., Gouze, K. R., & Binns, H. J. (2009). The prevalence of ADHD, ODD, depression, and anxiety in a community sample of 4-year-olds. ''Journal Of Clinical Child And Adolescent Psychology, 38''(3), 315-328. doi:10.1080/15374410902851382</ref> || 8.3% || Diagnostic Interview Schedule for Children–Parent Scale–Young Child Version (DISC-YC) ^p |- || Various locations across USA | Meta-analysis of 38 studies<ref>Canino G, Polanczyk G, Bauermeister JJ, et al. (2010) Does the prevalence of CD and ODD vary across cultures? Social Psychiatry Epidemiology, 45(7):695–704.</ref> || 3.3% || Varied |} ^p Parent interviewed as part of diagnostic assessment; ^y youth interviewed as part of diagnostic assessment, ^r adult interviewed for retrospective report as part of diagnostic assessment '''Note:''' Mash and Barkley note that prevalence rates of ODD must be qualified, because the definition of ODD has changed at a fast rate, the rates adolescents meeting criteria in any cross-sectional evaluation may be misleading because of the developmental progressions with and between ODD and Conduct Disorder, and categorical definitions of aggressive patterns may reflect arbitrary numbers of constituent estimates. These factors may lead to misleading prevalence rates. In addition, few studies have investigated the prevalence of ODD in preschool-aged children, and early onset of these behaviors is associated with more severe and stable impairment.<ref>Mash, E., & Barkley, R. (Eds.). (2003). ''Child Psychopathology''. 2nd Edition. New York: Guilford Press.</ref> ==[[Evidence based assessment/Prediction phase|'''Prediction phase''']]== The following section contains a list of screening and diagnostic instruments for ODD. The section includes administration information, psychometric data, and PDFs or links to the screenings. * Screenings are used as part of the [[Evidence based assessment/Prediction phase|prediction phase]] of assessment; for more information on interpretation of this data, or how screenings fit in to the assessment process, click [[Evidence based assessment/Prediction phase|here]]. * '''''For a list of more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&wteswitched=1#Psychometric_properties_of_common_screening_instruments click here.]''''' {| class="wikitable sortable" border="1" ! colspan="10" |Screening measures for ODD |- ! Measure ! Format (Reporter) ! Age Range ! Administration/ Completion Time ! Interrater Reliability ! Test-Retest Reliability ! Construct Validity ! Content Validity ! Highly Recommended !Where to access |- |Achenbach System of Empirically Based Assessments (ASEBA): Child Behavior Checklist (CBCL) |Parent report |6-18 <ref name=":9">{{Cite book|url=https://www.worldcat.org/oclc/1130319849|title=Assessment of disorders in childhood and adolescence|date=2020|others=Eric Arden Youngstrom, Mitchell J. Prinstein, Eric J. Mash, Russell A. Barkley|isbn=978-1-4625-4363-2|edition=Fifth edition|location=New York, NY|oclc=1130319849}}</ref> |10 - 15 minutes<ref name=":9" /> |A<ref name=":0">{{Cite book|url=https://www.worldcat.org/oclc/314222270|title=A guide to assessments that work|author=Hunsley, John|author2=Mash, Eric J.|date=2008|publisher=Oxford University Press|isbn=9780195310641|location=New York|oclc=314222270}}</ref> |E<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://store.aseba.org/School-Age-6-18-Materials/departments/11/ Purchase] |- |ASEBA: Teacher Report Form (TRF) |Teacher report |6-18 <ref name=":9" /> |10 - 15 minutes<ref name=":9" /> |A<ref name=":0" /> |E<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://store.aseba.org/School-Age-6-18-Materials/departments/11/ Purchase] |- |ASEBA: Youth Self-Report (YSR) |Youth self-report |11-18 <ref name=":9" /> |10 - 15 minutes<ref name=":9" /> |A<ref name=":0" /> |E<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://store.aseba.org/School-Age-6-18-Materials/departments/11/ Purchase] |- |Behavior Assessment for Children, Third Edition (BASC-3): Parent Rating Scale |Parent report |2-21 |10 - 20 minutes |A<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |E<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://www.pearsonassessments.com/store/usassessments/en/Store/Professional-Assessments/Behavior/Comprehensive/Behavior-Assessment-System-for-Children-%7C-Third-Edition-/p/100001402.html#:~:text=The%20BASC%E2%84%A2%20holds%20an,or%20adolescent's%20behavior%20and%20emotions. Purchase] |- |BASC-3: Teacher Rating Scale |Teacher report |2-21 |10 - 20 minutes |A<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |E<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://www.pearsonassessments.com/store/usassessments/en/Store/Professional-Assessments/Behavior/Comprehensive/Behavior-Assessment-System-for-Children-%7C-Third-Edition-/p/100001402.html#:~:text=The%20BASC%E2%84%A2%20holds%20an,or%20adolescent's%20behavior%20and%20emotions. Purchase] |- |BASC-3: Self-Report of Personality |Youth self-report |6 - college age |30 minutes |A<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |E<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://www.pearsonassessments.com/store/usassessments/en/Store/Professional-Assessments/Behavior/Comprehensive/Behavior-Assessment-System-for-Children-%7C-Third-Edition-/p/100001402.html#:~:text=The%20BASC%E2%84%A2%20holds%20an,or%20adolescent's%20behavior%20and%20emotions. Purchase] |- |Eyberg Child Behavior Inventory (ECBI) |Parent report |2-16 |5 minutes |A<ref name=":0" /> |G<ref name=":0" /> |E<ref name=":0" /> |E<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://www.parinc.com/products/pkey/97 Purchase] |- |Sutter-Eyberg Student Behavior Inventory - Revised (SESBI-R) |Teacher report |2-16 |5 minutes |A<ref name=":0" /> |G<ref name=":0" /> |E<ref name=":0" /> |E<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://www.parinc.com/products/pkey/97 Purchase] |- |Child and Adolescent Behavior Inventory (CABI) |Parent Report |3 - 18 |5 - 10 minutes | |A<ref>{{Cite journal|last=Burns|first=G. Leonard|last2=Preszler|first2=Jonathan|last3=Becker|first3=Stephen P.|date=2022-07-04|title=Psychometric and Normative Information on the Child and Adolescent Behavior Inventory in a Nationally Representative Sample of United States Children|url=https://doi.org/10.1080/15374416.2020.1852943|journal=Journal of Clinical Child & Adolescent Psychology|volume=51|issue=4|pages=443–452|doi=10.1080/15374416.2020.1852943|issn=1537-4416|pmc=PMC8272731|pmid=33428463}}</ref> | | | |PDF |- |Strengths and Difficulties Questionnaire (SDQ) |Parent report |3 - 16 |3 - 5 minutes |E<ref name=":10">{{Cite journal|last=Goodman|first=Robert|date=2001-11-01|title=Psychometric Properties of the Strengths and Difficulties Questionnaire|url=https://www.jaacap.org/article/S0890-8567(09)60543-8/abstract|journal=Journal of the American Academy of Child & Adolescent Psychiatry|language=English|volume=40|issue=11|pages=1337–1345|doi=10.1097/00004583-200111000-00015|issn=0890-8567|pmid=11699809}}</ref> |G<ref name=":10" /> | | | |[https://www.sdqinfo.org/a0.html SDQ Homepage][https://www.sdqinfo.org/py/sdqinfo/b0.py PDFs] |- |Disruptive Behavior Disorder Rating Scale |Parent report, teacher report |5 - 17 |5 - 10 minutes | | | | | |[https://web.archive.org/web/20151123022653/http://ccf.buffalo.edu/pdf/DBD_rating_scale.pdf PDF] |} '''Note:''' L = Less than adequate; A = Adequate; G = Good; E = Excellent; U = Unavailable; NA = Not applicable === Likelihood ratios and AUCs of screening instruments for ODD === * '''''For a list of the likelihood ratios for more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&wteswitched=1#Likelihood_ratios_and_AUCs_of_common_screening_instruments click here.]''''' {| class="wikitable sortable" border="1" ! Screening Measure (Primary Reference) ! Area Under Curve (AUC) ! LR+ (Score) ! LR- (Score) ! Clinical Generalizability !Where to Access |- |CBCL Aggression ''T''-score<ref name=":7" /> |.803 (''N''=370)<ref name=":2" /> |4.18 (55+)<ref name=":2" /> |.35 (<55)<ref name=":2" /> | | |- | |.71 (''N''=475)<ref name=":3" /> | ------- | ------- | | |- |CBCL DSM-Oriented Scales<ref name=":7" /> |.71 (N=475)<ref name=":3" /> | ------- | ------- | | |- | | ------- |2.46 (60+ to 70+)*<ref name=":4" /> |.54 (<60 to <70)*<ref name=":4" /> | | |- |SDQ- Conduct Problems Scale<ref name=":6" /> | ------- |8.33 (Not specified)*<ref name=":4" /> |.27 (Not specified)*<ref name=":4" /> | | |- |ECBI- Intensity Scale<ref name=":8" /> | ------- |6.92 (131+)<ref name=":5" /> |.11 (<131)<ref name=":5" /> | | |} '''Note:''' “LR+” refers to the change in likelihood ratio associated with a positive test score, and “LR-” is the likelihood ratio for a low score. Likelihood ratios of 1 indicate that the test result did not change impressions at all. LRs larger than 10 or smaller than .10 are frequently clinically decisive; 5 or .20 are helpful, and between 2.0 and .5 are small enough that they rarely result in clinically meaningful changes of formulation<ref>Sackett, D. L., Straus, S. E., Richardson, W. S., Rosenberg, W., & Haynes, R. B. (2000). Evidence-based medicine: How to practice and teach EBM. Edinburgh: Churchill Livingstone.</ref>. '''Search terms:''' [Oppositional Defiant Disorder] AND [sensitivity OR specificity] in GoogleScholar and PsychINFO === Interpreting ODD screening measure scores === * For information on interpreting screening measure scores, click [[Evidence based assessment/Prediction phase#Interpreting screening measure scores|here.]] == [[Evidence based assessment/Prescription phase|'''Prescription phase''']] == === Gold standard diagnostic interviews === * For a list of broad reaching diagnostic interviews sortable by disorder with PDFs (if applicable), [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prescription_phase&wteswitched=1#Common_Diagnostic_Interviews click here.] === Recommended diagnostic instruments for ODD === {| class="wikitable sortable" ! colspan="10" |Diagnostic instruments for ODD |- !Measure !Format (Reporter) !Age Range !Administration/ Completion Time !Interrater Reliability !Test-Retest Reliability !Construct Validity !Content Validity !Highly Recommended !Where to Access |- |Kiddie Schedule for Affective Disorders and Schizophrenia Present and Lifetime Version (KSADS-PL) |Structured interview |6-28 |45-75 minutes |E<ref name=":11">{{Cite journal|last=KAUFMAN|first=JOAN|last2=BIRMAHER|first2=BORIS|last3=BRENT|first3=DAVID|last4=RAO|first4=UMA|last5=FLYNN|first5=CYNTHIA|last6=MORECI|first6=PAULA|last7=WILLIAMSON|first7=DOUGLAS|last8=RYAN|first8=NEAL|date=1997-07|title=Schedule for Affective Disorders and Schizophrenia for School-Age Children-Present and Lifetime Version (K-SADS-PL): Initial Reliability and Validity Data|url=https://doi.org/10.1097/00004583-199707000-00021|journal=Journal of the American Academy of Child &amp; Adolescent Psychiatry|volume=36|issue=7|pages=980–988|doi=10.1097/00004583-199707000-00021|issn=0890-8567}}</ref> |G<ref name=":11" /> | | | |[https://www.pediatricbipolar.pitt.edu/resources/instruments Website to access] |- |Diagnostic Interview Schedule for Children (DISC-5) |Structured Interview (Self report and parent) |6-17 | | | | | |<ref name=":0" />[[File:Light green check.svg|center|frameless|36x36px]] |[https://telesage.com/netdisc-5/# Coming soon] |} '''Note:''' L = Less than adequate; A = Adequate; G = Good; E = Excellent; U = Unavailable; NA = Not applicable ===Recommended diagnostic interviews=== [http://narr.bmap.ucla.edu/docs/MINI_v5_002006.pdf "The Mini International Neuropsychiatric Interview for Children&Prime;] (MINI-Kids)-available and &Prime;the Kiddie-SADS-Present and Lifetime Version" [https://mfr.osf.io/render?url=https://osf.io/r47d9/?action=download%26mode=render KSADS-PL DSM-5 November 2016: Supplemental #4: Neurodevelopmental, Disruptive, and Conduct Disorders Supplement]. ===Screening instruments=== The following table provides diagnostic efficiency information for the Child Behavior Checklist (CBCL; Achenbach, 1991; hardcopy and scoring system available at the Finley Clinic); the Strengths and Difficulties Questionnaire<ref name=":6">{{Cite journal|last=Shabani|first=Amir|last2=Masoumian|first2=Samira|last3=Zamirinejad|first3=Somayeh|last4=Hejri|first4=Maryam|last5=Pirmorad|first5=Tahereh|last6=Yaghmaeezadeh|first6=Hooman|date=2021-05|title=Psychometric properties of Structured Clinical Interview for DSM‐5 Disorders‐Clinician Version (SCID‐5‐CV)|url=https://onlinelibrary.wiley.com/doi/10.1002/brb3.1894|journal=Brain and Behavior|language=en|volume=11|issue=5|doi=10.1002/brb3.1894|issn=2162-3279|pmc=PMC8119811|pmid=33729681}}</ref>; see http://www.sdqinfo.com/ to access the questionnaire and scoring information; and the Eyberg Child Behavior Index (ECBI)<ref name=":8">Eyberg, S. M., & Robinson, E. A. (1983). Conduct problem behavior: Standardization of a behavioral rating scale with adolescents. Journal of Clinical Child Psychology, 12 (3), 347-354.</ref>. '''Appendix 2''' includes a copy of the Eyberg Child Behavior Checklist<ref name=":8" />. ===Psychometric properties of screening measures for ODD=== {| class="wikitable" |- ! Screening Measure (Primary Preference) !! AUC !! LR+ (Score) !! LR- (Score) !! Citation |- | MINI-Kids || .81<ref name=":1">{{Cite journal|date=2001-08-01|title=Test-Retest Reliability of Anxiety Symptoms and Diagnoses With the Anxiety Disorders Interview Schedule for DSM-IV: Child and Parent Versions|url=https://www.sciencedirect.com/science/article/pii/S0890856709603427|journal=Journal of the American Academy of Child & Adolescent Psychiatry|language=en|volume=40|issue=8|pages=937–944|doi=10.1097/00004583-200108000-00016|issn=0890-8567}}</ref>|| 3.00<ref name=":1" />|| .65<ref name=":1" />|| |- | colspan="8" span style="font-size:110%; text-align:center;" | <b> Children and Adolescents (6 to 18 years)</b> |- | CBCL Aggression ''T''-score<ref name=":7">Achenbach, T. M. (1991a). ''Manual for the Child Behavior Checklist/4–18 and 1991 Profile''. Burlington , VT : University of Vermont Department of Psychiatry.</ref>|| .803 (''N''=370)<ref name=":2">{{Cite journal|last=Fresco|first=David M.|last2=Mennin|first2=Douglas S.|last3=Heimberg|first3=Richard G.|last4=Turk|first4=Cynthia L.|title=Using the Penn State Worry Questionnaire to identify individuals with generalized anxiety disorder: a receiver operating characteristic analysis|url=http://linkinghub.elsevier.com/retrieve/pii/S0005791603000569|journal=Journal of Behavior Therapy and Experimental Psychiatry|volume=34|issue=3-4|pages=283–291|doi=10.1016/j.jbtep.2003.09.001}}</ref>|| 4.18 (55+)<ref name=":2" />|| .35 (<55)<ref name=":2" />|| |- | || .71 (''N''=475)<ref name=":3">Ebesutani, C., Bernstein, A., Nakamura, B. J., Chorpita, B. F., Higa-McMillan, C. K., & Weisz, J. R. (2010). Concurrent validity of the child behavior checklist DSM-oriented scales: Correspondence with DSM diagnoses and comparison to syndrome scales. ''Journal of Psychopathology and Behavioral Assessment, 32''(3), 373-384.</ref>|| ------- || ------- || |- | CBCL DSM-Oriented Scales<ref name=":7" />|| .71 (N=475)<ref name=":3" />|| ------- || ------- || |- | || ------- || 2.46 (60+ to 70+)*<ref name=":4">Warnick, E. M., Bracken, M. B., & Kasl, S. (2008). Screening efficiency of the Child Behavior Checklist and Strengths and Difficulties Questionnaire: a systematic review. ''Child and Adolescent Mental Health, 13''(3), 140-147.</ref>|| .54 (<60 to <70)*<ref name=":4" />|| |- | colspan="8" span style="font-size:110%; text-align:center;" | <b> Children and Adolescents (4 to 12 years)</b> |- | SDQ- Conduct Problems Scale<ref name=":6" />|| ------- || 8.33 (Not specified)*<ref name=":4" />|| .27 (Not specified)*<ref name=":4" />|| |- | colspan="8" span style="font-size:110%; text-align:center;" | <b> Children and Adolescents (2 to 16 years)</b> |- | ECBI- Intensity Scale<ref name=":8" />|| ------- || 6.92 (131+)<ref name=":5">{{Cite journal|last=LYNEHAM|first=HEIDI J.|last2=ABBOTT|first2=MAREE J.|last3=RAPEE|first3=RONALD M.|date=2007-06|title=Interrater Reliability of the Anxiety Disorders Interview Schedule for DSM-IV: Child and Parent Version|url=https://doi.org/10.1097/chi.0b013e3180465a09|journal=Journal of the American Academy of Child &amp; Adolescent Psychiatry|volume=46|issue=6|pages=731–736|doi=10.1097/chi.0b013e3180465a09|issn=0890-8567}}</ref>|| .11 (<131)<ref name=":5" />|| |} '''Note:''' “LR+” refers to the change in likelihood ratio associated with a positive test score, and “LR-” is the likelihood ratio for a low score. Likelihood ratios of 1 indicate that the test result did not change impressions at all. LRs larger than 10 or smaller than .10 are frequently clinically decisive; 5 or .20 are helpful, and between 2.0 and .5 are small enough that they rarely result in clinically meaningful changes of formulation<ref>Sackett D, Strauss S, Richardson W, et al. Evidence-Based Medicine: How to Practice and Teach EBM.2nd ed. Churchill Livingstone; Edinburgh: 2000.</ref>. ''Searches (specified below) did not yield any data about sensitivity, specificity, AUC, or ROC for the Externalizing scale of the CBCL. Searches also did not yield data about TRF or YSR scales for Aggression or Externalizing: Achenbach and Rescorla (2001) provide data about clinically referred vs. non-referred samples but not about samples with oppositional disorder specifically; thus, only AUC and LRs for the Aggression scale are reported. In addition, searches did not yield any information on the AUC for Oppositional Defiant Disorder, however, there are studies looking at the AUC for the SDQ at differentiating clinical from non-clinical samples. Also, there was no information on the Problem Scale of the ECBI, and no information on the AUC for the ECBI.'' ==Treatment== ===Behavioral parent training=== Behavioral Parent Training is considered the most effective treatment for childhood disruptive behavior disorders (e.g., Oppositional Defiant Disorder), especially for younger children (i.e., 3-8 year-olds). See http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/rule-breaking-defiance-and-acting-out/ a website sponsored by The Society for Child and Adolescent Psychology (APA, Division 53) and the Association for Behavioral and Cognitive Therapies (ABCT), for current summary of evidence-based treatments. ===Overview of recommendations for assessment and treatment=== See the [https://pathways.nice.org.uk/pathways/antisocial-behaviour-and-conduct-disorders-in-children-and-young-people#content=view-info-category%3Aview-about-menu National Institute for Health and Care Excellence (NICE) Practice Guidelines for Childhood Conduct Disorders], for an overview of recommendations for both assessment and treatment of Oppositional Defiant Disorder. ==Process and outcome measures== ===Severity and outcome=== ====Clinically significant change benchmarks with common instruments==== {| class="wikitable sortable" border="1" |- | rowspan=1" style="text-align:center;font-size:130%;" | <b> Measure</b> | style="text-align:center;font-size:130%;" | <b> Subscale</b> | colspan="3" style="text-align:center;font-size:130%" width="300" | <b> Cut-off scores</b> | colspan="3" style="text-align:center;font-size:120%" | <b> Critical Change <br> (unstandardized scores)</b> |- | colspan="8" span style="font-size:110%; text-align:center;" | <b> Benchmarks Based on Published Norms</b> |- | colspan="2" | | style="text-align:center;font-size:110%" | <b> A</b> | style="text-align:center;font-size:110%" | <b> B</b> | style="text-align:center;font-size:110%" | <b> C</b> | style="text-align:center;font-size:110%" | <b> 95%</b> | style="text-align:center;font-size:110%" | <b> 90%</b> | style="text-align:center;font-size:110%" | <b> SE_difference</b> |- |rowspan="1" style="text-align:center;" | <b> CBCL T-scores <br> (2001 Norms)</b> | style="text-align:right;" | <i> Externalizing</i> | style="text-align:center;"| 49 | style="text-align:center;"| 70 | style="text-align:center;"| 58 | style="text-align:center;"| 7 | style="text-align:center;"| 6 | style="text-align:center;"| 3.4 |- | colspan="8" span style="font-size:110%; text-align:center;" | <b> CBCL Benchmarks Based on Oppositional Defiant Disorder Samples Were Not Found in Searches*</b> |- | rowspan="1" style="text-align:center;" | <b> ECBI Scaled Scores <br> (1983 Norms)</b> | style="text-align:right;" | <i> Intensity</i> | style="text-align:center;"| 80.1 | style="text-align:center;"| 169.5 | style="text-align:center;"| 112.9 | style="text-align:center;"| 9.5 | style="text-align:center;"| 8 | style="text-align:center;"| 4.8 |- | rowspan="1" style="text-align:center;" | <b> ECBI Scaled Scores <br> (1983 Norms)</b> | style="text-align:right;" | <i> Problem</i> | style="text-align:center;"| 3.9 | style="text-align:center;"| 17.7 | style="text-align:center;"| 11.5 | style="text-align:center;"| 2.1 | style="text-align:center;"| 1.8 | style="text-align:center;"| 1.1 |} '''Note:''' “A” = Away from the clinical range, “B” = Back into the nonclinical range, “C” = Closer to the nonclinical than clinical mean. '''Search terms:''' (1) “Strengths and Difficulties Questionnaire,” (2) Strengths and Difficulties Questionnaire AND benchmarks, searches previously mentioned. === Process measures=== See Section 1.1 for overview of evidence-based measures to use depending on etiology and symptomatology of Oppositional Defiant Disorder. == External Links == *[https://sccap53.org Society of Clinical Child and Adolescent Psychology] *http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/rule-breaking-defiance-and-acting-out/ == References == {{collapse top|Click here for references}} {{Reflist|30em}} [[Category:Psychological disorder portfolios|{{SUBPAGENAME}}]] {{collapse bottom}} [[Category:Psychological disorder portfolios|{{SUBPAGENAME}}]] qcr34723fre266drrsyajlso4qspv0g 2410312 2410307 2022-07-29T20:37:18Z Aherman012 2943941 /* Recommended diagnostic interviews */ Updated format wikitext text/x-wiki <noinclude>{{Helping Give Away Psychological Science Banner}}</noinclude> {{medical disclaimer}} {{:{{BASEPAGENAME}}/Sidebar}} ==[[Evidence based assessment/Portfolio template/What is a "portfolio"|'''What is a "portfolio"?''']]== * For background information on what assessment portfolios are, click the link in the heading above. ==[[Evidence based assessment/Preparation phase|'''Preparation phase''']]== === Diagnostic criteria for oppositional defiant disorder=== {{blockquotetop}} <big>ICD-11 Diagnostic Criteria</big><br> <br> '''General Description:''' Oppositional defiant disorder is a persistent pattern (e.g., 6 months or more) of markedly defiant, disobedient, provocative or spiteful behaviour that occurs more frequently than is typically observed in individuals of comparable age and developmental level and that is not restricted to interaction with siblings. Oppositional defiant disorder may be manifest in prevailing, persistent angry or irritable mood, often accompanied by severe temper outbursts or in headstrong, argumentative and defiant behaviour. The behavior pattern is of sufficient severity to result in significant impairment in personal, family, social, educational, occupational or other important areas of functioning <br> <br> '''Oppositional Defiant Disorder With Chronic Irritability-Anger:''' All definitional requirements for oppositional defiant disorder are met. This form of oppositional defiant disorder is characterized by prevailing, persistent angry or irritable mood that may be present independent of any apparent provocation. The negative mood is often accompanied by regularly occurring severe temper outbursts that are grossly out of proportion in intensity or duration to the provocation. Chronic irritability and anger are characteristic of the individual’s functioning nearly every day, are observable across multiple settings or domains of functioning (e.g., home, school, social relationships), and are not restricted to the individual’s relationship with his/her parents or guardians. The pattern of chronic irritability and anger is not limited to occasional episodes (e.g., developmentally typical irritability) or discrete periods (e.g., irritable mood in the context of manic or depressive episodes). *Note: The ICD-11 lists 3 additional subcategories of oppositional defiant disorder with chronic irritability-anger (i.e., with limited prosocial emotions, with typical prosocial emotions, and unspecified). They can be found [https://icd.who.int/browse11/l-m/en#/http%3a%2f%2fid.who.int%2ficd%2fentity%2f818792113 here]. <br> '''Oppositional Defiant Disorder Without Chronic Irritability-Anger:''' Meets all definitional requirements for oppositional defiant disorder. This form of oppositional defiant disorder is not characterized by prevailing, persistent, angry or irritable mood, but does feature headstrong, argumentative, and defiant behavior. *Note: The ICD-11 lists 3 additional subcategories of oppositional defiant disorder without chronic irritability-anger (i.e., with limited prosocial emotions, with typical prosocial emotions, and unspecified). They can be found [https://icd.who.int/browse11/l-m/en#/http%3a%2f%2fid.who.int%2ficd%2fentity%2f736540987 here]. <br> <big>Changes in DSM-5</big> <br> The diagnostic criteria for ADHD changed slightly from DSM-IV to DSM-5. See the changes [https://www.ncbi.nlm.nih.gov/books/NBK519712/table/ch3.t14/ here]. {{blockquotebottom}}<ref>{{Cite web|url=https://icd.who.int/browse11/l-m/en#/http://id.who.int/icd/entity/1545599508|title=ICD-11 for Mortality and Morbidity Statistics|website=icd.who.int|access-date=2022-07-11}}</ref> ===Base rates of ODD in different clinical settings=== This section describes the demographic setting of the population(s) sampled, base rates of diagnosis, country/region sampled and the diagnostic method that was used. Using this information, clinicians will be able to anchor the rate of ODD that they are likely to see in their clinical practice. * '''''To see prevalence rates across multiple disorders, [[Evidence-based assessment/Preparation phase|click here.]]''''' {|class="wikitable sortable" border="1" |- ! Demography ! Setting !! Base Rate !! Diagnostic Method |- || All of the U.S. | Nationally representative large-scale study (N = 3,119) <ref>Nock, M. K., Kazdin, A. E., Hiripi, E., & Kessler, R. C. (2007). Lifetime prevalence, correlates, and persistence of oppositional defiant disorder: results from the National Comorbidity Survey Replication. ''Journal of Child Psychology and Psychiatry, 48''(7), 703-713.</ref> || 10.2% (overall) 11.2% (males) 9.2% (females) || World Health Organization (WHO) Composite International Diagnostic Interview (CIDI)^r |- || Semi-rural North Carolina | Preschool-aged children from pediatric practices (N = 306; age 2 - 5 years old)<ref>Egger, H.L., & Angold, A. (2006). Common emotional and behavioral disorders in preschool children: Presentation, nosology, and epidemiology. ''Journal of Child Psychiatry and Psychology, 47'', 313–337.</ref> || 6.6% || Preschool Age Psychiatric Assessment (PAPA)^p |- || Western North Carolina | The Great Smoky Mountains Study - longitudinal, population-based study of community sample<ref>Costello, E. J., Mustillo, S., Erkanli, A., Keeler, G., & Angold, A. (2003) Prevalence and development of psychiatric disorders in adolescence. ''Arch Gen Psychiatry, 60'', 837-844.</ref> || 2.33% (overall) 3.16% (males) 2.75% (females) || Child and Adolescent Psychiatric Assessment (CAPA)^{p, y} |- || Chicago | Preschool-aged children from inner city schools and pediatric practices (N = 796; age 2 - 5 years old)<ref>Lavigne, J. V., LeBailly, S. A., Hopkins, J., Gouze, K. R., & Binns, H. J. (2009). The prevalence of ADHD, ODD, depression, and anxiety in a community sample of 4-year-olds. ''Journal Of Clinical Child And Adolescent Psychology, 38''(3), 315-328. doi:10.1080/15374410902851382</ref> || 8.3% || Diagnostic Interview Schedule for Children–Parent Scale–Young Child Version (DISC-YC) ^p |- || Various locations across USA | Meta-analysis of 38 studies<ref>Canino G, Polanczyk G, Bauermeister JJ, et al. (2010) Does the prevalence of CD and ODD vary across cultures? Social Psychiatry Epidemiology, 45(7):695–704.</ref> || 3.3% || Varied |} ^p Parent interviewed as part of diagnostic assessment; ^y youth interviewed as part of diagnostic assessment, ^r adult interviewed for retrospective report as part of diagnostic assessment '''Note:''' Mash and Barkley note that prevalence rates of ODD must be qualified, because the definition of ODD has changed at a fast rate, the rates adolescents meeting criteria in any cross-sectional evaluation may be misleading because of the developmental progressions with and between ODD and Conduct Disorder, and categorical definitions of aggressive patterns may reflect arbitrary numbers of constituent estimates. These factors may lead to misleading prevalence rates. In addition, few studies have investigated the prevalence of ODD in preschool-aged children, and early onset of these behaviors is associated with more severe and stable impairment.<ref>Mash, E., & Barkley, R. (Eds.). (2003). ''Child Psychopathology''. 2nd Edition. New York: Guilford Press.</ref> ==[[Evidence based assessment/Prediction phase|'''Prediction phase''']]== The following section contains a list of screening and diagnostic instruments for ODD. The section includes administration information, psychometric data, and PDFs or links to the screenings. * Screenings are used as part of the [[Evidence based assessment/Prediction phase|prediction phase]] of assessment; for more information on interpretation of this data, or how screenings fit in to the assessment process, click [[Evidence based assessment/Prediction phase|here]]. * '''''For a list of more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&wteswitched=1#Psychometric_properties_of_common_screening_instruments click here.]''''' {| class="wikitable sortable" border="1" ! colspan="10" |Screening measures for ODD |- ! Measure ! Format (Reporter) ! Age Range ! Administration/ Completion Time ! Interrater Reliability ! Test-Retest Reliability ! Construct Validity ! Content Validity ! Highly Recommended !Where to access |- |Achenbach System of Empirically Based Assessments (ASEBA): Child Behavior Checklist (CBCL) |Parent report |6-18 <ref name=":9">{{Cite book|url=https://www.worldcat.org/oclc/1130319849|title=Assessment of disorders in childhood and adolescence|date=2020|others=Eric Arden Youngstrom, Mitchell J. Prinstein, Eric J. Mash, Russell A. Barkley|isbn=978-1-4625-4363-2|edition=Fifth edition|location=New York, NY|oclc=1130319849}}</ref> |10 - 15 minutes<ref name=":9" /> |A<ref name=":0">{{Cite book|url=https://www.worldcat.org/oclc/314222270|title=A guide to assessments that work|author=Hunsley, John|author2=Mash, Eric J.|date=2008|publisher=Oxford University Press|isbn=9780195310641|location=New York|oclc=314222270}}</ref> |E<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://store.aseba.org/School-Age-6-18-Materials/departments/11/ Purchase] |- |ASEBA: Teacher Report Form (TRF) |Teacher report |6-18 <ref name=":9" /> |10 - 15 minutes<ref name=":9" /> |A<ref name=":0" /> |E<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://store.aseba.org/School-Age-6-18-Materials/departments/11/ Purchase] |- |ASEBA: Youth Self-Report (YSR) |Youth self-report |11-18 <ref name=":9" /> |10 - 15 minutes<ref name=":9" /> |A<ref name=":0" /> |E<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://store.aseba.org/School-Age-6-18-Materials/departments/11/ Purchase] |- |Behavior Assessment for Children, Third Edition (BASC-3): Parent Rating Scale |Parent report |2-21 |10 - 20 minutes |A<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |E<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://www.pearsonassessments.com/store/usassessments/en/Store/Professional-Assessments/Behavior/Comprehensive/Behavior-Assessment-System-for-Children-%7C-Third-Edition-/p/100001402.html#:~:text=The%20BASC%E2%84%A2%20holds%20an,or%20adolescent's%20behavior%20and%20emotions. Purchase] |- |BASC-3: Teacher Rating Scale |Teacher report |2-21 |10 - 20 minutes |A<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |E<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://www.pearsonassessments.com/store/usassessments/en/Store/Professional-Assessments/Behavior/Comprehensive/Behavior-Assessment-System-for-Children-%7C-Third-Edition-/p/100001402.html#:~:text=The%20BASC%E2%84%A2%20holds%20an,or%20adolescent's%20behavior%20and%20emotions. Purchase] |- |BASC-3: Self-Report of Personality |Youth self-report |6 - college age |30 minutes |A<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |E<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://www.pearsonassessments.com/store/usassessments/en/Store/Professional-Assessments/Behavior/Comprehensive/Behavior-Assessment-System-for-Children-%7C-Third-Edition-/p/100001402.html#:~:text=The%20BASC%E2%84%A2%20holds%20an,or%20adolescent's%20behavior%20and%20emotions. Purchase] |- |Eyberg Child Behavior Inventory (ECBI) |Parent report |2-16 |5 minutes |A<ref name=":0" /> |G<ref name=":0" /> |E<ref name=":0" /> |E<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://www.parinc.com/products/pkey/97 Purchase] |- |Sutter-Eyberg Student Behavior Inventory - Revised (SESBI-R) |Teacher report |2-16 |5 minutes |A<ref name=":0" /> |G<ref name=":0" /> |E<ref name=":0" /> |E<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://www.parinc.com/products/pkey/97 Purchase] |- |Child and Adolescent Behavior Inventory (CABI) |Parent Report |3 - 18 |5 - 10 minutes | |A<ref>{{Cite journal|last=Burns|first=G. Leonard|last2=Preszler|first2=Jonathan|last3=Becker|first3=Stephen P.|date=2022-07-04|title=Psychometric and Normative Information on the Child and Adolescent Behavior Inventory in a Nationally Representative Sample of United States Children|url=https://doi.org/10.1080/15374416.2020.1852943|journal=Journal of Clinical Child & Adolescent Psychology|volume=51|issue=4|pages=443–452|doi=10.1080/15374416.2020.1852943|issn=1537-4416|pmc=PMC8272731|pmid=33428463}}</ref> | | | |PDF |- |Strengths and Difficulties Questionnaire (SDQ) |Parent report |3 - 16 |3 - 5 minutes |E<ref name=":10">{{Cite journal|last=Goodman|first=Robert|date=2001-11-01|title=Psychometric Properties of the Strengths and Difficulties Questionnaire|url=https://www.jaacap.org/article/S0890-8567(09)60543-8/abstract|journal=Journal of the American Academy of Child & Adolescent Psychiatry|language=English|volume=40|issue=11|pages=1337–1345|doi=10.1097/00004583-200111000-00015|issn=0890-8567|pmid=11699809}}</ref> |G<ref name=":10" /> | | | |[https://www.sdqinfo.org/a0.html SDQ Homepage][https://www.sdqinfo.org/py/sdqinfo/b0.py PDFs] |- |Disruptive Behavior Disorder Rating Scale |Parent report, teacher report |5 - 17 |5 - 10 minutes | | | | | |[https://web.archive.org/web/20151123022653/http://ccf.buffalo.edu/pdf/DBD_rating_scale.pdf PDF] |} '''Note:''' L = Less than adequate; A = Adequate; G = Good; E = Excellent; U = Unavailable; NA = Not applicable === Likelihood ratios and AUCs of screening instruments for ODD === * '''''For a list of the likelihood ratios for more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&wteswitched=1#Likelihood_ratios_and_AUCs_of_common_screening_instruments click here.]''''' {| class="wikitable sortable" border="1" ! Screening Measure (Primary Reference) ! Area Under Curve (AUC) ! LR+ (Score) ! LR- (Score) ! Clinical Generalizability !Where to Access |- |CBCL Aggression ''T''-score<ref name=":7">Achenbach, T. M. (1991a). ''Manual for the Child Behavior Checklist/4–18 and 1991 Profile''. Burlington , VT : University of Vermont Department of Psychiatry.</ref> |.803 (''N''=370)<ref name=":2">{{Cite journal|last=Fresco|first=David M.|last2=Mennin|first2=Douglas S.|last3=Heimberg|first3=Richard G.|last4=Turk|first4=Cynthia L.|title=Using the Penn State Worry Questionnaire to identify individuals with generalized anxiety disorder: a receiver operating characteristic analysis|url=http://linkinghub.elsevier.com/retrieve/pii/S0005791603000569|journal=Journal of Behavior Therapy and Experimental Psychiatry|volume=34|issue=3-4|pages=283–291|doi=10.1016/j.jbtep.2003.09.001}}</ref> |4.18 (55+)<ref name=":2" /> |.35 (<55)<ref name=":2" /> | | |- | |.71 (''N''=475)<ref name=":3">Ebesutani, C., Bernstein, A., Nakamura, B. J., Chorpita, B. F., Higa-McMillan, C. K., & Weisz, J. R. (2010). Concurrent validity of the child behavior checklist DSM-oriented scales: Correspondence with DSM diagnoses and comparison to syndrome scales. ''Journal of Psychopathology and Behavioral Assessment, 32''(3), 373-384.</ref> | ------- | ------- | | |- |CBCL DSM-Oriented Scales<ref name=":7" /> |.71 (N=475)<ref name=":3" /> | ------- | ------- | | |- | | ------- |2.46 (60+ to 70+)*<ref name=":4">Warnick, E. M., Bracken, M. B., & Kasl, S. (2008). Screening efficiency of the Child Behavior Checklist and Strengths and Difficulties Questionnaire: a systematic review. ''Child and Adolescent Mental Health, 13''(3), 140-147.</ref> |.54 (<60 to <70)*<ref name=":4" /> | | |- |SDQ- Conduct Problems Scale<ref name=":6">{{Cite journal|last=Shabani|first=Amir|last2=Masoumian|first2=Samira|last3=Zamirinejad|first3=Somayeh|last4=Hejri|first4=Maryam|last5=Pirmorad|first5=Tahereh|last6=Yaghmaeezadeh|first6=Hooman|date=2021-05|title=Psychometric properties of Structured Clinical Interview for DSM‐5 Disorders‐Clinician Version (SCID‐5‐CV)|url=https://onlinelibrary.wiley.com/doi/10.1002/brb3.1894|journal=Brain and Behavior|language=en|volume=11|issue=5|doi=10.1002/brb3.1894|issn=2162-3279|pmc=PMC8119811|pmid=33729681}}</ref> | ------- |8.33 (Not specified)*<ref name=":4" /> |.27 (Not specified)*<ref name=":4" /> | | |- |ECBI- Intensity Scale<ref name=":8">Eyberg, S. M., & Robinson, E. A. (1983). Conduct problem behavior: Standardization of a behavioral rating scale with adolescents. Journal of Clinical Child Psychology, 12 (3), 347-354.</ref> | ------- |6.92 (131+)<ref name=":5">{{Cite journal|last=LYNEHAM|first=HEIDI J.|last2=ABBOTT|first2=MAREE J.|last3=RAPEE|first3=RONALD M.|date=2007-06|title=Interrater Reliability of the Anxiety Disorders Interview Schedule for DSM-IV: Child and Parent Version|url=https://doi.org/10.1097/chi.0b013e3180465a09|journal=Journal of the American Academy of Child &amp; Adolescent Psychiatry|volume=46|issue=6|pages=731–736|doi=10.1097/chi.0b013e3180465a09|issn=0890-8567}}</ref> |.11 (<131)<ref name=":5" /> | | |} '''Note:''' “LR+” refers to the change in likelihood ratio associated with a positive test score, and “LR-” is the likelihood ratio for a low score. Likelihood ratios of 1 indicate that the test result did not change impressions at all. LRs larger than 10 or smaller than .10 are frequently clinically decisive; 5 or .20 are helpful, and between 2.0 and .5 are small enough that they rarely result in clinically meaningful changes of formulation<ref>Sackett, D. L., Straus, S. E., Richardson, W. S., Rosenberg, W., & Haynes, R. B. (2000). Evidence-based medicine: How to practice and teach EBM. Edinburgh: Churchill Livingstone.</ref>. '''Search terms:''' [Oppositional Defiant Disorder] AND [sensitivity OR specificity] in GoogleScholar and PsychINFO; ''Searches (specified below) did not yield any data about sensitivity, specificity, AUC, or ROC for the Externalizing scale of the CBCL. Searches also did not yield data about TRF or YSR scales for Aggression or Externalizing: Achenbach and Rescorla (2001) provide data about clinically referred vs. non-referred samples but not about samples with oppositional disorder specifically; thus, only AUC and LRs for the Aggression scale are reported. In addition, searches did not yield any information on the AUC for Oppositional Defiant Disorder, however, there are studies looking at the AUC for the SDQ at differentiating clinical from non-clinical samples. Also, there was no information on the Problem Scale of the ECBI, and no information on the AUC for the ECBI.'' === Interpreting ODD screening measure scores === * For information on interpreting screening measure scores, click [[Evidence based assessment/Prediction phase#Interpreting screening measure scores|here.]] == [[Evidence based assessment/Prescription phase|'''Prescription phase''']] == === Gold standard diagnostic interviews === * For a list of broad reaching diagnostic interviews sortable by disorder with PDFs (if applicable), [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prescription_phase&wteswitched=1#Common_Diagnostic_Interviews click here.] === Recommended diagnostic instruments for ODD === {| class="wikitable sortable" ! colspan="10" |Diagnostic instruments for ODD |- !Measure !Format (Reporter) !Age Range !Administration/ Completion Time !Interrater Reliability !Test-Retest Reliability !Construct Validity !Content Validity !Highly Recommended !Where to Access |- |Kiddie Schedule for Affective Disorders and Schizophrenia Present and Lifetime Version (KSADS-PL) |Structured interview |6-28 |45-75 minutes |E<ref name=":11">{{Cite journal|last=KAUFMAN|first=JOAN|last2=BIRMAHER|first2=BORIS|last3=BRENT|first3=DAVID|last4=RAO|first4=UMA|last5=FLYNN|first5=CYNTHIA|last6=MORECI|first6=PAULA|last7=WILLIAMSON|first7=DOUGLAS|last8=RYAN|first8=NEAL|date=1997-07|title=Schedule for Affective Disorders and Schizophrenia for School-Age Children-Present and Lifetime Version (K-SADS-PL): Initial Reliability and Validity Data|url=https://doi.org/10.1097/00004583-199707000-00021|journal=Journal of the American Academy of Child &amp; Adolescent Psychiatry|volume=36|issue=7|pages=980–988|doi=10.1097/00004583-199707000-00021|issn=0890-8567}}</ref> |G<ref name=":11" /> | | | |[https://www.pediatricbipolar.pitt.edu/resources/instruments Website to access] |- |Diagnostic Interview Schedule for Children (DISC-5) |Structured Interview (Self report and parent) |6-17 | | | | | |<ref name=":0" />[[File:Light green check.svg|center|frameless|36x36px]] |[https://telesage.com/netdisc-5/# Coming soon] |} '''Note:''' L = Less than adequate; A = Adequate; G = Good; E = Excellent; U = Unavailable; NA = Not applicable == [[Evidence based assessment/Process phase|'''Process phase''']] == The following section contains a list of process and outcome measures for oppositional defiant disorder. The section includes benchmarks based on published norms for several outcome and severity measures, as well as information about commonly used process measures. Process and outcome measures are used as part of the [[Evidence based assessment/Process phase|process phase]] of assessment. For more information of differences between process and outcome measures, see the page on the [[Evidence based assessment/Process phase|process phase of assessment]]. === Outcome and severity measures === * This table includes clinically significant benchmarks for generalized anxiety disorder specific outcome measures * Information on how to interpret this table can be [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase found here]. * Additionally, these [[Evidence based assessment/Vignettes|vignettes]] might be helpful resources for understanding appropriate adaptation of outcome measures in practice. * ''<u>For clinically significant change benchmarks for the CBCL, YSR, and TRF total, externalizing, internalizing, and attention benchmarks,</u>'' [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase&wteswitched=1#Clinically_significant_change_benchmarks_for_widely-used_outcome_measures see here.] {| class="wikitable sortable" border="1" |- | rowspan="1&quot;" style="text-align:center;font-size:130%;" | <b> Measure</b> | style="text-align:center;font-size:130%;" | <b> Subscale</b> | colspan="3" style="text-align:center;font-size:130%" width="300" | <b> Cut-off scores</b> | colspan="3" style="text-align:center;font-size:120%" | <b> Critical Change <br> (unstandardized scores)</b> |- | colspan="8" style="font-size:110%; text-align:center;" span | <b> Benchmarks Based on Published Norms</b> |- | colspan="2" | | style="text-align:center;font-size:110%" | <b> A</b> | style="text-align:center;font-size:110%" | <b> B</b> | style="text-align:center;font-size:110%" | <b> C</b> | style="text-align:center;font-size:110%" | <b> 95%</b> | style="text-align:center;font-size:110%" | <b> 90%</b> | style="text-align:center;font-size:110%" | <b> SE_difference</b> |- | rowspan="1" style="text-align:center;" | <b> CBCL T-scores <br> (2001 Norms)</b> | style="text-align:right;" | <i> Externalizing</i> | style="text-align:center;" | 49 | style="text-align:center;" | 70 | style="text-align:center;" | 58 | style="text-align:center;" | 7 | style="text-align:center;" | 6 | style="text-align:center;" | 3.4 |- | rowspan="1" style="text-align:center;" | <b> ECBI Scaled Scores <br> (1983 Norms)</b> | style="text-align:right;" | <i> Intensity</i> | style="text-align:center;" | 80.1 | style="text-align:center;" | 169.5 | style="text-align:center;" | 112.9 | style="text-align:center;" | 9.5 | style="text-align:center;" | 8 | style="text-align:center;" | 4.8 |- | rowspan="1" style="text-align:center;" | <b> ECBI Scaled Scores <br> (1983 Norms)</b> | style="text-align:right;" | <i> Problem</i> | style="text-align:center;" | 3.9 | style="text-align:center;" | 17.7 | style="text-align:center;" | 11.5 | style="text-align:center;" | 2.1 | style="text-align:center;" | 1.8 | style="text-align:center;" | 1.1 |} '''Note:''' “A” = Away from the clinical range, “B” = Back into the nonclinical range, “C” = Closer to the nonclinical than clinical mean. '''Search terms:''' (1) “Strengths and Difficulties Questionnaire,” (2) Strengths and Difficulties Questionnaire AND benchmarks, searches previously mentioned. === Process measures=== See Section 1.1 for overview of evidence-based measures to use depending on etiology and symptomatology of Oppositional Defiant Disorder. ==Treatment== ===Behavioral parent training=== Behavioral Parent Training is considered the most effective treatment for childhood disruptive behavior disorders (e.g., Oppositional Defiant Disorder), especially for younger children (i.e., 3-8 year-olds). See http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/rule-breaking-defiance-and-acting-out/ a website sponsored by The Society for Child and Adolescent Psychology (APA, Division 53) and the Association for Behavioral and Cognitive Therapies (ABCT), for current summary of evidence-based treatments. ===Overview of recommendations for assessment and treatment=== See the [https://pathways.nice.org.uk/pathways/antisocial-behaviour-and-conduct-disorders-in-children-and-young-people#content=view-info-category%3Aview-about-menu National Institute for Health and Care Excellence (NICE) Practice Guidelines for Childhood Conduct Disorders], for an overview of recommendations for both assessment and treatment of Oppositional Defiant Disorder. == External Links == *[https://sccap53.org Society of Clinical Child and Adolescent Psychology] *http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/rule-breaking-defiance-and-acting-out/ == References == {{collapse top|Click here for references}} {{Reflist|30em}} [[Category:Psychological disorder portfolios|{{SUBPAGENAME}}]] {{collapse bottom}} [[Category:Psychological disorder portfolios|{{SUBPAGENAME}}]] 42hl1y69cdugsmmwi48vzdfsncqpdhb 2410314 2410312 2022-07-29T22:17:34Z Aherman012 2943941 /* Likelihood ratios and AUCs of screening instruments for ODD */ Added in likelihood data AND updated the base rates wikitext text/x-wiki <noinclude>{{Helping Give Away Psychological Science Banner}}</noinclude> {{medical disclaimer}} {{:{{BASEPAGENAME}}/Sidebar}} ==[[Evidence based assessment/Portfolio template/What is a "portfolio"|'''What is a "portfolio"?''']]== * For background information on what assessment portfolios are, click the link in the heading above. ==[[Evidence based assessment/Preparation phase|'''Preparation phase''']]== === Diagnostic criteria for oppositional defiant disorder=== {{blockquotetop}} <big>ICD-11 Diagnostic Criteria</big><br> <br> '''General Description:''' Oppositional defiant disorder is a persistent pattern (e.g., 6 months or more) of markedly defiant, disobedient, provocative or spiteful behaviour that occurs more frequently than is typically observed in individuals of comparable age and developmental level and that is not restricted to interaction with siblings. Oppositional defiant disorder may be manifest in prevailing, persistent angry or irritable mood, often accompanied by severe temper outbursts or in headstrong, argumentative and defiant behaviour. The behavior pattern is of sufficient severity to result in significant impairment in personal, family, social, educational, occupational or other important areas of functioning <br> <br> '''Oppositional Defiant Disorder With Chronic Irritability-Anger:''' All definitional requirements for oppositional defiant disorder are met. This form of oppositional defiant disorder is characterized by prevailing, persistent angry or irritable mood that may be present independent of any apparent provocation. The negative mood is often accompanied by regularly occurring severe temper outbursts that are grossly out of proportion in intensity or duration to the provocation. Chronic irritability and anger are characteristic of the individual’s functioning nearly every day, are observable across multiple settings or domains of functioning (e.g., home, school, social relationships), and are not restricted to the individual’s relationship with his/her parents or guardians. The pattern of chronic irritability and anger is not limited to occasional episodes (e.g., developmentally typical irritability) or discrete periods (e.g., irritable mood in the context of manic or depressive episodes). *Note: The ICD-11 lists 3 additional subcategories of oppositional defiant disorder with chronic irritability-anger (i.e., with limited prosocial emotions, with typical prosocial emotions, and unspecified). They can be found [https://icd.who.int/browse11/l-m/en#/http%3a%2f%2fid.who.int%2ficd%2fentity%2f818792113 here]. <br> '''Oppositional Defiant Disorder Without Chronic Irritability-Anger:''' Meets all definitional requirements for oppositional defiant disorder. This form of oppositional defiant disorder is not characterized by prevailing, persistent, angry or irritable mood, but does feature headstrong, argumentative, and defiant behavior. *Note: The ICD-11 lists 3 additional subcategories of oppositional defiant disorder without chronic irritability-anger (i.e., with limited prosocial emotions, with typical prosocial emotions, and unspecified). They can be found [https://icd.who.int/browse11/l-m/en#/http%3a%2f%2fid.who.int%2ficd%2fentity%2f736540987 here]. <br> <big>Changes in DSM-5</big> <br> The diagnostic criteria for ADHD changed slightly from DSM-IV to DSM-5. See the changes [https://www.ncbi.nlm.nih.gov/books/NBK519712/table/ch3.t14/ here]. {{blockquotebottom}}<ref>{{Cite web|url=https://icd.who.int/browse11/l-m/en#/http://id.who.int/icd/entity/1545599508|title=ICD-11 for Mortality and Morbidity Statistics|website=icd.who.int|access-date=2022-07-11}}</ref> ===Base rates of ODD in different clinical settings=== This section describes the demographic setting of the population(s) sampled, base rates of diagnosis, country/region sampled and the diagnostic method that was used. Using this information, clinicians will be able to anchor the rate of ODD that they are likely to see in their clinical practice. * '''''To see prevalence rates across multiple disorders, [[Evidence-based assessment/Preparation phase|click here.]]''''' {|class="wikitable sortable" border="1" |- ! Demography ! Setting !! Base Rate !! Diagnostic Method |- || Various locations across USA | Meta-analysis of 38 studies<ref>Canino G, Polanczyk G, Bauermeister JJ, et al. (2010) Does the prevalence of CD and ODD vary across cultures? Social Psychiatry Epidemiology, 45(7):695–704.</ref> || 3.3% || Varied |- || All of the U.S. | Nationally representative large-scale study (N = 3,119) <ref>Nock, M. K., Kazdin, A. E., Hiripi, E., & Kessler, R. C. (2007). Lifetime prevalence, correlates, and persistence of oppositional defiant disorder: results from the National Comorbidity Survey Replication. ''Journal of Child Psychology and Psychiatry, 48''(7), 703-713.</ref> || 10.2% (overall) 11.2% (males) 9.2% (females) || World Health Organization (WHO) Composite International Diagnostic Interview (CIDI)^r |- |Suburban and urban Colorado |Project to Learn about Youth-Mental health, school-based study for children from kindergarten to high-school (n = 236)<ref name=":1">{{Cite journal|last=Danielson|first=Melissa L.|last2=Bitsko|first2=Rebecca H.|last3=Holbrook|first3=Joseph R.|last4=Charania|first4=Sana N.|last5=Claussen|first5=Angelika H.|last6=McKeown|first6=Robert E.|last7=Cuffe|first7=Steven P.|last8=Owens|first8=Julie Sarno|last9=Evans|first9=Steven W.|date=2021-06-01|title=Community-Based Prevalence of Externalizing and Internalizing Disorders among School-Aged Children and Adolescents in Four Geographically Dispersed School Districts in the United States|url=https://doi.org/10.1007/s10578-020-01027-z|journal=Child Psychiatry & Human Development|language=en|volume=52|issue=3|pages=500–514|doi=10.1007/s10578-020-01027-z|issn=1573-3327|pmc=PMC8016018|pmid=32734339}}</ref> |6.8% |Diagnostic Interview Schedule for Children (DISC) |- |Urban and suburban Florida |Project to Learn about Youth-Mental health, school-based study for children from kindergarten to high-school (n = 289)<ref name=":1" /> |6.9% |Diagnostic Interview Schedule for Children (DISC) |- |Rural and suburban Ohio |Project to Learn about Youth-Mental health, school-based study for children from elementary to high-school (n = 152)<ref name=":1" /> |17.3% |Diagnostic Interview Schedule for Children (DISC) |- |Suburban and rural South Carolina |Project to Learn about Youth-Mental health, school-based study for children from elementary to high-school (n = 270)<ref name=":1" /> |5.7% |Diagnostic Interview Schedule for Children (DISC) |- || Semi-rural North Carolina | Preschool-aged children from pediatric practices (N = 306; age 2 - 5 years old)<ref>Egger, H.L., & Angold, A. (2006). Common emotional and behavioral disorders in preschool children: Presentation, nosology, and epidemiology. ''Journal of Child Psychiatry and Psychology, 47'', 313–337.</ref> || 6.6% || Preschool Age Psychiatric Assessment (PAPA)^p |- || Western North Carolina | The Great Smoky Mountains Study - longitudinal, population-based study of community sample<ref>Costello, E. J., Mustillo, S., Erkanli, A., Keeler, G., & Angold, A. (2003) Prevalence and development of psychiatric disorders in adolescence. ''Arch Gen Psychiatry, 60'', 837-844.</ref> || 2.33% (overall) 3.16% (males) 2.75% (females) || Child and Adolescent Psychiatric Assessment (CAPA)^{p, y} |- || Chicago | Preschool-aged children from inner city schools and pediatric practices (N = 796; age 2 - 5 years old)<ref>Lavigne, J. V., LeBailly, S. A., Hopkins, J., Gouze, K. R., & Binns, H. J. (2009). The prevalence of ADHD, ODD, depression, and anxiety in a community sample of 4-year-olds. ''Journal Of Clinical Child And Adolescent Psychology, 38''(3), 315-328. doi:10.1080/15374410902851382</ref> || 8.3% || Diagnostic Interview Schedule for Children–Parent Scale–Young Child Version (DISC-YC) ^p |- |Germany (Saarbrücken County) |All school-aged children were examined during a routine school-entry medical examination (N = 1676, mean age = 5.7)<ref>{{Cite journal|last=Niemczyk|first=Justine|last2=Equit|first2=Monika|last3=Braun-Bither|first3=Katrin|last4=Klein|first4=Anna-Maria|last5=von Gontard|first5=Alexander|date=2015-07-01|title=Prevalence of incontinence, attention deficit/hyperactivity disorder and oppositional defiant disorder in preschool children|url=https://doi.org/10.1007/s00787-014-0628-6|journal=European Child & Adolescent Psychiatry|language=en|volume=24|issue=7|pages=837–843|doi=10.1007/s00787-014-0628-6|issn=1435-165X}}</ref> |7.3% (males) 5.1%(females) |DISYPS-II |- |South Korea (Seoul) |Children were randomly surveyed across 6 school districts in Seoul (N = 1645, age 6 - 12 years old) |5.8% (males) 4.1%(females) |Diagnostic Interview Schedule for Children–Parent Scale IV (DISC-IV) |} ^p Parent interviewed as part of diagnostic assessment; ^y youth interviewed as part of diagnostic assessment, ^r adult interviewed for retrospective report as part of diagnostic assessment '''Note:''' Mash and Barkley note that prevalence rates of ODD must be qualified, because the definition of ODD has changed at a fast rate, the rates adolescents meeting criteria in any cross-sectional evaluation may be misleading because of the developmental progressions with and between ODD and Conduct Disorder, and categorical definitions of aggressive patterns may reflect arbitrary numbers of constituent estimates. These factors may lead to misleading prevalence rates. In addition, few studies have investigated the prevalence of ODD in preschool-aged children, and early onset of these behaviors is associated with more severe and stable impairment.<ref>Mash, E., & Barkley, R. (Eds.). (2003). ''Child Psychopathology''. 2nd Edition. New York: Guilford Press.</ref> ==[[Evidence based assessment/Prediction phase|'''Prediction phase''']]== The following section contains a list of screening and diagnostic instruments for ODD. The section includes administration information, psychometric data, and PDFs or links to the screenings. * Screenings are used as part of the [[Evidence based assessment/Prediction phase|prediction phase]] of assessment; for more information on interpretation of this data, or how screenings fit in to the assessment process, click [[Evidence based assessment/Prediction phase|here]]. * '''''For a list of more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&wteswitched=1#Psychometric_properties_of_common_screening_instruments click here.]''''' {| class="wikitable sortable" border="1" ! colspan="10" |Screening measures for ODD |- ! Measure ! Format (Reporter) ! Age Range ! Administration/ Completion Time ! Interrater Reliability ! Test-Retest Reliability ! Construct Validity ! Content Validity ! Highly Recommended !Where to access |- |Achenbach System of Empirically Based Assessments (ASEBA): Child Behavior Checklist (CBCL) |Parent report |6-18 <ref name=":9">{{Cite book|url=https://www.worldcat.org/oclc/1130319849|title=Assessment of disorders in childhood and adolescence|date=2020|others=Eric Arden Youngstrom, Mitchell J. Prinstein, Eric J. Mash, Russell A. Barkley|isbn=978-1-4625-4363-2|edition=Fifth edition|location=New York, NY|oclc=1130319849}}</ref> |10 - 15 minutes<ref name=":9" /> |A<ref name=":0">{{Cite book|url=https://www.worldcat.org/oclc/314222270|title=A guide to assessments that work|author=Hunsley, John|author2=Mash, Eric J.|date=2008|publisher=Oxford University Press|isbn=9780195310641|location=New York|oclc=314222270}}</ref> |E<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://store.aseba.org/School-Age-6-18-Materials/departments/11/ Purchase] |- |ASEBA: Teacher Report Form (TRF) |Teacher report |6-18 <ref name=":9" /> |10 - 15 minutes<ref name=":9" /> |A<ref name=":0" /> |E<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://store.aseba.org/School-Age-6-18-Materials/departments/11/ Purchase] |- |ASEBA: Youth Self-Report (YSR) |Youth self-report |11-18 <ref name=":9" /> |10 - 15 minutes<ref name=":9" /> |A<ref name=":0" /> |E<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://store.aseba.org/School-Age-6-18-Materials/departments/11/ Purchase] |- |Behavior Assessment for Children, Third Edition (BASC-3): Parent Rating Scale |Parent report |2-21 |10 - 20 minutes |A<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |E<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://www.pearsonassessments.com/store/usassessments/en/Store/Professional-Assessments/Behavior/Comprehensive/Behavior-Assessment-System-for-Children-%7C-Third-Edition-/p/100001402.html#:~:text=The%20BASC%E2%84%A2%20holds%20an,or%20adolescent's%20behavior%20and%20emotions. Purchase] |- |BASC-3: Teacher Rating Scale |Teacher report |2-21 |10 - 20 minutes |A<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |E<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://www.pearsonassessments.com/store/usassessments/en/Store/Professional-Assessments/Behavior/Comprehensive/Behavior-Assessment-System-for-Children-%7C-Third-Edition-/p/100001402.html#:~:text=The%20BASC%E2%84%A2%20holds%20an,or%20adolescent's%20behavior%20and%20emotions. Purchase] |- |BASC-3: Self-Report of Personality |Youth self-report |6 - college age |30 minutes |A<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |E<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://www.pearsonassessments.com/store/usassessments/en/Store/Professional-Assessments/Behavior/Comprehensive/Behavior-Assessment-System-for-Children-%7C-Third-Edition-/p/100001402.html#:~:text=The%20BASC%E2%84%A2%20holds%20an,or%20adolescent's%20behavior%20and%20emotions. Purchase] |- |Eyberg Child Behavior Inventory (ECBI) |Parent report |2-16 |5 minutes |A<ref name=":0" /> |G<ref name=":0" /> |E<ref name=":0" /> |E<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://www.parinc.com/products/pkey/97 Purchase] |- |Sutter-Eyberg Student Behavior Inventory - Revised (SESBI-R) |Teacher report |2-16 |5 minutes |A<ref name=":0" /> |G<ref name=":0" /> |E<ref name=":0" /> |E<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://www.parinc.com/products/pkey/97 Purchase] |- |Child and Adolescent Behavior Inventory (CABI) |Parent Report |3 - 18 |5 - 10 minutes | |A<ref>{{Cite journal|last=Burns|first=G. Leonard|last2=Preszler|first2=Jonathan|last3=Becker|first3=Stephen P.|date=2022-07-04|title=Psychometric and Normative Information on the Child and Adolescent Behavior Inventory in a Nationally Representative Sample of United States Children|url=https://doi.org/10.1080/15374416.2020.1852943|journal=Journal of Clinical Child & Adolescent Psychology|volume=51|issue=4|pages=443–452|doi=10.1080/15374416.2020.1852943|issn=1537-4416|pmc=PMC8272731|pmid=33428463}}</ref> | | | |PDF |- |Strengths and Difficulties Questionnaire (SDQ) |Parent report |3 - 16 |3 - 5 minutes |E<ref name=":10">{{Cite journal|last=Goodman|first=Robert|date=2001-11-01|title=Psychometric Properties of the Strengths and Difficulties Questionnaire|url=https://www.jaacap.org/article/S0890-8567(09)60543-8/abstract|journal=Journal of the American Academy of Child & Adolescent Psychiatry|language=English|volume=40|issue=11|pages=1337–1345|doi=10.1097/00004583-200111000-00015|issn=0890-8567|pmid=11699809}}</ref> |G<ref name=":10" /> | | | |[https://www.sdqinfo.org/a0.html SDQ Homepage][https://www.sdqinfo.org/py/sdqinfo/b0.py PDFs] |- |Disruptive Behavior Disorder Rating Scale |Parent report, teacher report |5 - 17 |5 - 10 minutes | | | | | |[https://web.archive.org/web/20151123022653/http://ccf.buffalo.edu/pdf/DBD_rating_scale.pdf PDF] |} '''Note:''' L = Less than adequate; A = Adequate; G = Good; E = Excellent; U = Unavailable; NA = Not applicable === Likelihood ratios and AUCs of screening instruments for ODD === * '''''For a list of the likelihood ratios for more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&wteswitched=1#Likelihood_ratios_and_AUCs_of_common_screening_instruments click here.]''''' {| class="wikitable sortable" border="1" ! Screening Measure (Primary Reference) ! Area Under Curve (AUC) ! LR+ (Score) ! LR- (Score) ! Clinical Generalizability !Where to Access |- |CBCL DSM-Oriented Scales<ref name=":7">Achenbach, T. M. (1991a). ''Manual for the Child Behavior Checklist/4–18 and 1991 Profile''. Burlington , VT : University of Vermont Department of Psychiatry.</ref> |.71 (N=475)<ref name=":3">Ebesutani, C., Bernstein, A., Nakamura, B. J., Chorpita, B. F., Higa-McMillan, C. K., & Weisz, J. R. (2010). Concurrent validity of the child behavior checklist DSM-oriented scales: Correspondence with DSM diagnoses and comparison to syndrome scales. ''Journal of Psychopathology and Behavioral Assessment, 32''(3), 373-384.</ref> | 2.80 (60+ to 70+) <ref name=":4">Warnick, E. M., Bracken, M. B., & Kasl, S. (2008). Screening efficiency of the Child Behavior Checklist and Strengths and Difficulties Questionnaire: a systematic review. ''Child and Adolescent Mental Health, 13''(3), 140-147.</ref> | .52 (60- to 70-)*<ref name=":4" /> |Youth aged 5 - 18 seeking out patient treatment across a variety of settings<ref name=":4" /> |[https://store.aseba.org/School-Age-6-18-Materials/departments/11/ Purchase] |- |SDQ<ref name=":6">{{Cite journal|last=Shabani|first=Amir|last2=Masoumian|first2=Samira|last3=Zamirinejad|first3=Somayeh|last4=Hejri|first4=Maryam|last5=Pirmorad|first5=Tahereh|last6=Yaghmaeezadeh|first6=Hooman|date=2021-05|title=Psychometric properties of Structured Clinical Interview for DSM‐5 Disorders‐Clinician Version (SCID‐5‐CV)|url=https://onlinelibrary.wiley.com/doi/10.1002/brb3.1894|journal=Brain and Behavior|language=en|volume=11|issue=5|doi=10.1002/brb3.1894|issn=2162-3279|pmc=PMC8119811|pmid=33729681}}</ref> | .81 -.88 (N = 18,416)<ref>{{Cite journal|last=Algorta|first=Guillermo Perez|last2=Dodd|first2=Alyson Lamont|last3=Stringaris|first3=Argyris|last4=Youngstrom|first4=Eric A.|date=2016-09|title=Diagnostic efficiency of the SDQ for parents to identify ADHD in the UK: a ROC analysis|url=http://link.springer.com/10.1007/s00787-015-0815-0|journal=European Child & Adolescent Psychiatry|language=en|volume=25|issue=9|pages=949–957|doi=10.1007/s00787-015-0815-0|issn=1018-8827|pmc=PMC4990620|pmid=26762184}}</ref> |7.00 <ref name=":4" /> |.55<ref name=":4" /> |Youth aged 5 - 18 seeking out patient treatment across a variety of settings<ref name=":4" /> |[https://www.sdqinfo.org/a0.html SDQ Homepage][https://www.sdqinfo.org/py/sdqinfo/b0.py PDFs] |- |SDQ - DSM -IV Conduct and ODD | |7.55 (3+)<ref name=":2">{{Cite journal|last=Goodman|first=Robert|date=2001-11-01|title=Psychometric Properties of the Strengths and Difficulties Questionnaire|url=https://www.jaacap.org/article/S0890-8567(09)60543-8/abstract|journal=Journal of the American Academy of Child & Adolescent Psychiatry|language=English|volume=40|issue=11|pages=1337–1345|doi=10.1097/00004583-200111000-00015|issn=0890-8567|pmid=11699809}}</ref> |.35 (3-)<ref name=":2" /> |Surveyed youth aged 5 - 15 in the UK <ref name=":2" /> |[https://www.sdqinfo.org/a0.html SDQ Homepage][https://www.sdqinfo.org/py/sdqinfo/b0.py PDFs] |- |ECBI- Intensity Scale<ref name=":8">Eyberg, S. M., & Robinson, E. A. (1983). Conduct problem behavior: Standardization of a behavioral rating scale with adolescents. Journal of Clinical Child Psychology, 12 (3), 347-354.</ref> | |6.92 (131+)<ref name=":5">{{Cite journal|last=LYNEHAM|first=HEIDI J.|last2=ABBOTT|first2=MAREE J.|last3=RAPEE|first3=RONALD M.|date=2007-06|title=Interrater Reliability of the Anxiety Disorders Interview Schedule for DSM-IV: Child and Parent Version|url=https://doi.org/10.1097/chi.0b013e3180465a09|journal=Journal of the American Academy of Child &amp; Adolescent Psychiatry|volume=46|issue=6|pages=731–736|doi=10.1097/chi.0b013e3180465a09|issn=0890-8567}}</ref> |.11 (131-)<ref name=":5" /> |Youth aged 7-16 had responses compared to diagnosis<ref name=":5" /> |[https://www.parinc.com/products/pkey/97 Purchase] |- |BASC-2 PRS - Aggression |.76 (N =156)<ref name=":12">{{Cite journal|last=Doyle|first=Alysa|last2=Ostrander|first2=Rick|last3=Skare|first3=Stacy|last4=Crosby|first4=Ross D.|last5=August|first5=Gerald J.|date=1997-09-01|title=Convergent and criterion-related validity of the behavior assessment system for children-parent rating scale|url=https://doi.org/10.1207/s15374424jccp2603_6|journal=Journal of Clinical Child Psychology|volume=26|issue=3|pages=276–284|doi=10.1207/s15374424jccp2603_6|issn=0047-228X|pmid=9292385}}</ref> |3.62 (65+)<ref name=":12" /> |.60 (65-)<ref name=":12" /> |Youth from first through fourth grade who were at risk for CD<ref name=":12" /> |[https://www.pearsonassessments.com/store/usassessments/en/Store/Professional-Assessments/Behavior/Comprehensive/Behavior-Assessment-System-for-Children-%7C-Third-Edition-/p/100001402.html#:~:text=The%20BASC%E2%84%A2%20holds%20an,or%20adolescent's%20behavior%20and%20emotions. Purchase] |} '''Note:''' “LR+” refers to the change in likelihood ratio associated with a positive test score, and “LR-” is the likelihood ratio for a low score. Likelihood ratios of 1 indicate that the test result did not change impressions at all. LRs larger than 10 or smaller than .10 are frequently clinically decisive; 5 or .20 are helpful, and between 2.0 and .5 are small enough that they rarely result in clinically meaningful changes of formulation<ref>Sackett, D. L., Straus, S. E., Richardson, W. S., Rosenberg, W., & Haynes, R. B. (2000). Evidence-based medicine: How to practice and teach EBM. Edinburgh: Churchill Livingstone.</ref>. '''Search terms:''' [Oppositional Defiant Disorder] AND [sensitivity OR specificity] in GoogleScholar and PsychINFO; === Interpreting ODD screening measure scores === * For information on interpreting screening measure scores, click [[Evidence based assessment/Prediction phase#Interpreting screening measure scores|here.]] == [[Evidence based assessment/Prescription phase|'''Prescription phase''']] == === Gold standard diagnostic interviews === * For a list of broad reaching diagnostic interviews sortable by disorder with PDFs (if applicable), [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prescription_phase&wteswitched=1#Common_Diagnostic_Interviews click here.] === Recommended diagnostic instruments for ODD === {| class="wikitable sortable" ! colspan="10" |Diagnostic instruments for ODD |- !Measure !Format (Reporter) !Age Range !Administration/ Completion Time !Interrater Reliability !Test-Retest Reliability !Construct Validity !Content Validity !Highly Recommended !Where to Access |- |Kiddie Schedule for Affective Disorders and Schizophrenia Present and Lifetime Version (KSADS-PL) |Structured interview |6-28 |45-75 minutes |E<ref name=":11">{{Cite journal|last=KAUFMAN|first=JOAN|last2=BIRMAHER|first2=BORIS|last3=BRENT|first3=DAVID|last4=RAO|first4=UMA|last5=FLYNN|first5=CYNTHIA|last6=MORECI|first6=PAULA|last7=WILLIAMSON|first7=DOUGLAS|last8=RYAN|first8=NEAL|date=1997-07|title=Schedule for Affective Disorders and Schizophrenia for School-Age Children-Present and Lifetime Version (K-SADS-PL): Initial Reliability and Validity Data|url=https://doi.org/10.1097/00004583-199707000-00021|journal=Journal of the American Academy of Child &amp; Adolescent Psychiatry|volume=36|issue=7|pages=980–988|doi=10.1097/00004583-199707000-00021|issn=0890-8567}}</ref> |G<ref name=":11" /> | | | |[https://www.pediatricbipolar.pitt.edu/resources/instruments Website to access] |- |Diagnostic Interview Schedule for Children (DISC-5) |Structured Interview (Self report and parent) |6-17 | | | | | |<ref name=":0" />[[File:Light green check.svg|center|frameless|36x36px]] |[https://telesage.com/netdisc-5/# Coming soon] |} '''Note:''' L = Less than adequate; A = Adequate; G = Good; E = Excellent; U = Unavailable; NA = Not applicable == [[Evidence based assessment/Process phase|'''Process phase''']] == The following section contains a list of process and outcome measures for oppositional defiant disorder. The section includes benchmarks based on published norms for several outcome and severity measures, as well as information about commonly used process measures. Process and outcome measures are used as part of the [[Evidence based assessment/Process phase|process phase]] of assessment. For more information of differences between process and outcome measures, see the page on the [[Evidence based assessment/Process phase|process phase of assessment]]. === Outcome and severity measures === * This table includes clinically significant benchmarks for generalized anxiety disorder specific outcome measures * Information on how to interpret this table can be [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase found here]. * Additionally, these [[Evidence based assessment/Vignettes|vignettes]] might be helpful resources for understanding appropriate adaptation of outcome measures in practice. * ''<u>For clinically significant change benchmarks for the CBCL, YSR, and TRF total, externalizing, internalizing, and attention benchmarks,</u>'' [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase&wteswitched=1#Clinically_significant_change_benchmarks_for_widely-used_outcome_measures see here.] {| class="wikitable sortable" border="1" |- | rowspan="1&quot;" style="text-align:center;font-size:130%;" | <b> Measure</b> | style="text-align:center;font-size:130%;" | <b> Subscale</b> | colspan="3" style="text-align:center;font-size:130%" width="300" | <b> Cut-off scores</b> | colspan="3" style="text-align:center;font-size:120%" | <b> Critical Change <br> (unstandardized scores)</b> |- | colspan="8" style="font-size:110%; text-align:center;" span | <b> Benchmarks Based on Published Norms</b> |- | colspan="2" | | style="text-align:center;font-size:110%" | <b> A</b> | style="text-align:center;font-size:110%" | <b> B</b> | style="text-align:center;font-size:110%" | <b> C</b> | style="text-align:center;font-size:110%" | <b> 95%</b> | style="text-align:center;font-size:110%" | <b> 90%</b> | style="text-align:center;font-size:110%" | <b> SE_difference</b> |- | rowspan="1" style="text-align:center;" | <b> CBCL T-scores <br> (2001 Norms)</b> | style="text-align:right;" | <i> Externalizing</i> | style="text-align:center;" | 49 | style="text-align:center;" | 70 | style="text-align:center;" | 58 | style="text-align:center;" | 7 | style="text-align:center;" | 6 | style="text-align:center;" | 3.4 |- | rowspan="1" style="text-align:center;" | <b> ECBI Scaled Scores <br> (1983 Norms)</b> | style="text-align:right;" | <i> Intensity</i> | style="text-align:center;" | 80.1 | style="text-align:center;" | 169.5 | style="text-align:center;" | 112.9 | style="text-align:center;" | 9.5 | style="text-align:center;" | 8 | style="text-align:center;" | 4.8 |- | rowspan="1" style="text-align:center;" | <b> ECBI Scaled Scores <br> (1983 Norms)</b> | style="text-align:right;" | <i> Problem</i> | style="text-align:center;" | 3.9 | style="text-align:center;" | 17.7 | style="text-align:center;" | 11.5 | style="text-align:center;" | 2.1 | style="text-align:center;" | 1.8 | style="text-align:center;" | 1.1 |} '''Note:''' “A” = Away from the clinical range, “B” = Back into the nonclinical range, “C” = Closer to the nonclinical than clinical mean. === Process measures=== See Section 1.1 for overview of evidence-based measures to use depending on etiology and symptomatology of Oppositional Defiant Disorder. ==Treatment== ===Behavioral parent training=== Behavioral Parent Training is considered the most effective treatment for childhood disruptive behavior disorders (e.g., Oppositional Defiant Disorder), especially for younger children (i.e., 3-8 year-olds). See http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/rule-breaking-defiance-and-acting-out/ a website sponsored by The Society for Child and Adolescent Psychology (APA, Division 53) and the Association for Behavioral and Cognitive Therapies (ABCT), for current summary of evidence-based treatments. ===Overview of recommendations for assessment and treatment=== See the [https://pathways.nice.org.uk/pathways/antisocial-behaviour-and-conduct-disorders-in-children-and-young-people#content=view-info-category%3Aview-about-menu National Institute for Health and Care Excellence (NICE) Practice Guidelines for Childhood Conduct Disorders], for an overview of recommendations for both assessment and treatment of Oppositional Defiant Disorder. == External Links == *[https://sccap53.org Society of Clinical Child and Adolescent Psychology] *http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/rule-breaking-defiance-and-acting-out/ == References == {{collapse top|Click here for references}} {{Reflist|30em}} [[Category:Psychological disorder portfolios|{{SUBPAGENAME}}]] {{collapse bottom}} [[Category:Psychological disorder portfolios|{{SUBPAGENAME}}]] arc1p24ateauis7vfgef2e87tjeohku 2410317 2410314 2022-07-29T22:20:01Z Aherman012 2943941 /* What is a "portfolio"? */ Setting up extended page wikitext text/x-wiki <noinclude>{{Helping Give Away Psychological Science Banner}}</noinclude> {{medical disclaimer}} {{:{{BASEPAGENAME}}/Sidebar}} ==[[Evidence based assessment/Portfolio template/What is a "portfolio"|'''What is a "portfolio"?''']]== * For background information on what assessment portfolios are, click the link in the heading above. * Want even 'more' information about this topic? There's an extended version of this page here. ==[[Evidence based assessment/Preparation phase|'''Preparation phase''']]== === Diagnostic criteria for oppositional defiant disorder=== {{blockquotetop}} <big>ICD-11 Diagnostic Criteria</big><br> <br> '''General Description:''' Oppositional defiant disorder is a persistent pattern (e.g., 6 months or more) of markedly defiant, disobedient, provocative or spiteful behaviour that occurs more frequently than is typically observed in individuals of comparable age and developmental level and that is not restricted to interaction with siblings. Oppositional defiant disorder may be manifest in prevailing, persistent angry or irritable mood, often accompanied by severe temper outbursts or in headstrong, argumentative and defiant behaviour. The behavior pattern is of sufficient severity to result in significant impairment in personal, family, social, educational, occupational or other important areas of functioning <br> <br> '''Oppositional Defiant Disorder With Chronic Irritability-Anger:''' All definitional requirements for oppositional defiant disorder are met. This form of oppositional defiant disorder is characterized by prevailing, persistent angry or irritable mood that may be present independent of any apparent provocation. The negative mood is often accompanied by regularly occurring severe temper outbursts that are grossly out of proportion in intensity or duration to the provocation. Chronic irritability and anger are characteristic of the individual’s functioning nearly every day, are observable across multiple settings or domains of functioning (e.g., home, school, social relationships), and are not restricted to the individual’s relationship with his/her parents or guardians. The pattern of chronic irritability and anger is not limited to occasional episodes (e.g., developmentally typical irritability) or discrete periods (e.g., irritable mood in the context of manic or depressive episodes). *Note: The ICD-11 lists 3 additional subcategories of oppositional defiant disorder with chronic irritability-anger (i.e., with limited prosocial emotions, with typical prosocial emotions, and unspecified). They can be found [https://icd.who.int/browse11/l-m/en#/http%3a%2f%2fid.who.int%2ficd%2fentity%2f818792113 here]. <br> '''Oppositional Defiant Disorder Without Chronic Irritability-Anger:''' Meets all definitional requirements for oppositional defiant disorder. This form of oppositional defiant disorder is not characterized by prevailing, persistent, angry or irritable mood, but does feature headstrong, argumentative, and defiant behavior. *Note: The ICD-11 lists 3 additional subcategories of oppositional defiant disorder without chronic irritability-anger (i.e., with limited prosocial emotions, with typical prosocial emotions, and unspecified). They can be found [https://icd.who.int/browse11/l-m/en#/http%3a%2f%2fid.who.int%2ficd%2fentity%2f736540987 here]. <br> <big>Changes in DSM-5</big> <br> The diagnostic criteria for ADHD changed slightly from DSM-IV to DSM-5. See the changes [https://www.ncbi.nlm.nih.gov/books/NBK519712/table/ch3.t14/ here]. {{blockquotebottom}}<ref>{{Cite web|url=https://icd.who.int/browse11/l-m/en#/http://id.who.int/icd/entity/1545599508|title=ICD-11 for Mortality and Morbidity Statistics|website=icd.who.int|access-date=2022-07-11}}</ref> ===Base rates of ODD in different clinical settings=== This section describes the demographic setting of the population(s) sampled, base rates of diagnosis, country/region sampled and the diagnostic method that was used. Using this information, clinicians will be able to anchor the rate of ODD that they are likely to see in their clinical practice. * '''''To see prevalence rates across multiple disorders, [[Evidence-based assessment/Preparation phase|click here.]]''''' {|class="wikitable sortable" border="1" |- ! Demography ! Setting !! Base Rate !! Diagnostic Method |- || Various locations across USA | Meta-analysis of 38 studies<ref>Canino G, Polanczyk G, Bauermeister JJ, et al. (2010) Does the prevalence of CD and ODD vary across cultures? Social Psychiatry Epidemiology, 45(7):695–704.</ref> || 3.3% || Varied |- || All of the U.S. | Nationally representative large-scale study (N = 3,119) <ref>Nock, M. K., Kazdin, A. E., Hiripi, E., & Kessler, R. C. (2007). Lifetime prevalence, correlates, and persistence of oppositional defiant disorder: results from the National Comorbidity Survey Replication. ''Journal of Child Psychology and Psychiatry, 48''(7), 703-713.</ref> || 10.2% (overall) 11.2% (males) 9.2% (females) || World Health Organization (WHO) Composite International Diagnostic Interview (CIDI)^r |- |Suburban and urban Colorado |Project to Learn about Youth-Mental health, school-based study for children from kindergarten to high-school (n = 236)<ref name=":1">{{Cite journal|last=Danielson|first=Melissa L.|last2=Bitsko|first2=Rebecca H.|last3=Holbrook|first3=Joseph R.|last4=Charania|first4=Sana N.|last5=Claussen|first5=Angelika H.|last6=McKeown|first6=Robert E.|last7=Cuffe|first7=Steven P.|last8=Owens|first8=Julie Sarno|last9=Evans|first9=Steven W.|date=2021-06-01|title=Community-Based Prevalence of Externalizing and Internalizing Disorders among School-Aged Children and Adolescents in Four Geographically Dispersed School Districts in the United States|url=https://doi.org/10.1007/s10578-020-01027-z|journal=Child Psychiatry & Human Development|language=en|volume=52|issue=3|pages=500–514|doi=10.1007/s10578-020-01027-z|issn=1573-3327|pmc=PMC8016018|pmid=32734339}}</ref> |6.8% |Diagnostic Interview Schedule for Children (DISC) |- |Urban and suburban Florida |Project to Learn about Youth-Mental health, school-based study for children from kindergarten to high-school (n = 289)<ref name=":1" /> |6.9% |Diagnostic Interview Schedule for Children (DISC) |- |Rural and suburban Ohio |Project to Learn about Youth-Mental health, school-based study for children from elementary to high-school (n = 152)<ref name=":1" /> |17.3% |Diagnostic Interview Schedule for Children (DISC) |- |Suburban and rural South Carolina |Project to Learn about Youth-Mental health, school-based study for children from elementary to high-school (n = 270)<ref name=":1" /> |5.7% |Diagnostic Interview Schedule for Children (DISC) |- || Semi-rural North Carolina | Preschool-aged children from pediatric practices (N = 306; age 2 - 5 years old)<ref>Egger, H.L., & Angold, A. (2006). Common emotional and behavioral disorders in preschool children: Presentation, nosology, and epidemiology. ''Journal of Child Psychiatry and Psychology, 47'', 313–337.</ref> || 6.6% || Preschool Age Psychiatric Assessment (PAPA)^p |- || Western North Carolina | The Great Smoky Mountains Study - longitudinal, population-based study of community sample<ref>Costello, E. J., Mustillo, S., Erkanli, A., Keeler, G., & Angold, A. (2003) Prevalence and development of psychiatric disorders in adolescence. ''Arch Gen Psychiatry, 60'', 837-844.</ref> || 2.33% (overall) 3.16% (males) 2.75% (females) || Child and Adolescent Psychiatric Assessment (CAPA)^{p, y} |- || Chicago | Preschool-aged children from inner city schools and pediatric practices (N = 796; age 2 - 5 years old)<ref>Lavigne, J. V., LeBailly, S. A., Hopkins, J., Gouze, K. R., & Binns, H. J. (2009). The prevalence of ADHD, ODD, depression, and anxiety in a community sample of 4-year-olds. ''Journal Of Clinical Child And Adolescent Psychology, 38''(3), 315-328. doi:10.1080/15374410902851382</ref> || 8.3% || Diagnostic Interview Schedule for Children–Parent Scale–Young Child Version (DISC-YC) ^p |- |Germany (Saarbrücken County) |All school-aged children were examined during a routine school-entry medical examination (N = 1676, mean age = 5.7)<ref>{{Cite journal|last=Niemczyk|first=Justine|last2=Equit|first2=Monika|last3=Braun-Bither|first3=Katrin|last4=Klein|first4=Anna-Maria|last5=von Gontard|first5=Alexander|date=2015-07-01|title=Prevalence of incontinence, attention deficit/hyperactivity disorder and oppositional defiant disorder in preschool children|url=https://doi.org/10.1007/s00787-014-0628-6|journal=European Child & Adolescent Psychiatry|language=en|volume=24|issue=7|pages=837–843|doi=10.1007/s00787-014-0628-6|issn=1435-165X}}</ref> |7.3% (males) 5.1%(females) |DISYPS-II |- |South Korea (Seoul) |Children were randomly surveyed across 6 school districts in Seoul (N = 1645, age 6 - 12 years old) |5.8% (males) 4.1%(females) |Diagnostic Interview Schedule for Children–Parent Scale IV (DISC-IV) |} ^p Parent interviewed as part of diagnostic assessment; ^y youth interviewed as part of diagnostic assessment, ^r adult interviewed for retrospective report as part of diagnostic assessment '''Note:''' Mash and Barkley note that prevalence rates of ODD must be qualified, because the definition of ODD has changed at a fast rate, the rates adolescents meeting criteria in any cross-sectional evaluation may be misleading because of the developmental progressions with and between ODD and Conduct Disorder, and categorical definitions of aggressive patterns may reflect arbitrary numbers of constituent estimates. These factors may lead to misleading prevalence rates. In addition, few studies have investigated the prevalence of ODD in preschool-aged children, and early onset of these behaviors is associated with more severe and stable impairment.<ref>Mash, E., & Barkley, R. (Eds.). (2003). ''Child Psychopathology''. 2nd Edition. New York: Guilford Press.</ref> ==[[Evidence based assessment/Prediction phase|'''Prediction phase''']]== The following section contains a list of screening and diagnostic instruments for ODD. The section includes administration information, psychometric data, and PDFs or links to the screenings. * Screenings are used as part of the [[Evidence based assessment/Prediction phase|prediction phase]] of assessment; for more information on interpretation of this data, or how screenings fit in to the assessment process, click [[Evidence based assessment/Prediction phase|here]]. * '''''For a list of more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&wteswitched=1#Psychometric_properties_of_common_screening_instruments click here.]''''' {| class="wikitable sortable" border="1" ! colspan="10" |Screening measures for ODD |- ! Measure ! Format (Reporter) ! Age Range ! Administration/ Completion Time ! Interrater Reliability ! Test-Retest Reliability ! Construct Validity ! Content Validity ! Highly Recommended !Where to access |- |Achenbach System of Empirically Based Assessments (ASEBA): Child Behavior Checklist (CBCL) |Parent report |6-18 <ref name=":9">{{Cite book|url=https://www.worldcat.org/oclc/1130319849|title=Assessment of disorders in childhood and adolescence|date=2020|others=Eric Arden Youngstrom, Mitchell J. Prinstein, Eric J. Mash, Russell A. Barkley|isbn=978-1-4625-4363-2|edition=Fifth edition|location=New York, NY|oclc=1130319849}}</ref> |10 - 15 minutes<ref name=":9" /> |A<ref name=":0">{{Cite book|url=https://www.worldcat.org/oclc/314222270|title=A guide to assessments that work|author=Hunsley, John|author2=Mash, Eric J.|date=2008|publisher=Oxford University Press|isbn=9780195310641|location=New York|oclc=314222270}}</ref> |E<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://store.aseba.org/School-Age-6-18-Materials/departments/11/ Purchase] |- |ASEBA: Teacher Report Form (TRF) |Teacher report |6-18 <ref name=":9" /> |10 - 15 minutes<ref name=":9" /> |A<ref name=":0" /> |E<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://store.aseba.org/School-Age-6-18-Materials/departments/11/ Purchase] |- |ASEBA: Youth Self-Report (YSR) |Youth self-report |11-18 <ref name=":9" /> |10 - 15 minutes<ref name=":9" /> |A<ref name=":0" /> |E<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://store.aseba.org/School-Age-6-18-Materials/departments/11/ Purchase] |- |Behavior Assessment for Children, Third Edition (BASC-3): Parent Rating Scale |Parent report |2-21 |10 - 20 minutes |A<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |E<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://www.pearsonassessments.com/store/usassessments/en/Store/Professional-Assessments/Behavior/Comprehensive/Behavior-Assessment-System-for-Children-%7C-Third-Edition-/p/100001402.html#:~:text=The%20BASC%E2%84%A2%20holds%20an,or%20adolescent's%20behavior%20and%20emotions. Purchase] |- |BASC-3: Teacher Rating Scale |Teacher report |2-21 |10 - 20 minutes |A<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |E<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://www.pearsonassessments.com/store/usassessments/en/Store/Professional-Assessments/Behavior/Comprehensive/Behavior-Assessment-System-for-Children-%7C-Third-Edition-/p/100001402.html#:~:text=The%20BASC%E2%84%A2%20holds%20an,or%20adolescent's%20behavior%20and%20emotions. Purchase] |- |BASC-3: Self-Report of Personality |Youth self-report |6 - college age |30 minutes |A<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |E<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://www.pearsonassessments.com/store/usassessments/en/Store/Professional-Assessments/Behavior/Comprehensive/Behavior-Assessment-System-for-Children-%7C-Third-Edition-/p/100001402.html#:~:text=The%20BASC%E2%84%A2%20holds%20an,or%20adolescent's%20behavior%20and%20emotions. Purchase] |- |Eyberg Child Behavior Inventory (ECBI) |Parent report |2-16 |5 minutes |A<ref name=":0" /> |G<ref name=":0" /> |E<ref name=":0" /> |E<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://www.parinc.com/products/pkey/97 Purchase] |- |Sutter-Eyberg Student Behavior Inventory - Revised (SESBI-R) |Teacher report |2-16 |5 minutes |A<ref name=":0" /> |G<ref name=":0" /> |E<ref name=":0" /> |E<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://www.parinc.com/products/pkey/97 Purchase] |- |Child and Adolescent Behavior Inventory (CABI) |Parent Report |3 - 18 |5 - 10 minutes | |A<ref>{{Cite journal|last=Burns|first=G. Leonard|last2=Preszler|first2=Jonathan|last3=Becker|first3=Stephen P.|date=2022-07-04|title=Psychometric and Normative Information on the Child and Adolescent Behavior Inventory in a Nationally Representative Sample of United States Children|url=https://doi.org/10.1080/15374416.2020.1852943|journal=Journal of Clinical Child & Adolescent Psychology|volume=51|issue=4|pages=443–452|doi=10.1080/15374416.2020.1852943|issn=1537-4416|pmc=PMC8272731|pmid=33428463}}</ref> | | | |PDF |- |Strengths and Difficulties Questionnaire (SDQ) |Parent report |3 - 16 |3 - 5 minutes |E<ref name=":10">{{Cite journal|last=Goodman|first=Robert|date=2001-11-01|title=Psychometric Properties of the Strengths and Difficulties Questionnaire|url=https://www.jaacap.org/article/S0890-8567(09)60543-8/abstract|journal=Journal of the American Academy of Child & Adolescent Psychiatry|language=English|volume=40|issue=11|pages=1337–1345|doi=10.1097/00004583-200111000-00015|issn=0890-8567|pmid=11699809}}</ref> |G<ref name=":10" /> | | | |[https://www.sdqinfo.org/a0.html SDQ Homepage][https://www.sdqinfo.org/py/sdqinfo/b0.py PDFs] |- |Disruptive Behavior Disorder Rating Scale |Parent report, teacher report |5 - 17 |5 - 10 minutes | | | | | |[https://web.archive.org/web/20151123022653/http://ccf.buffalo.edu/pdf/DBD_rating_scale.pdf PDF] |} '''Note:''' L = Less than adequate; A = Adequate; G = Good; E = Excellent; U = Unavailable; NA = Not applicable === Likelihood ratios and AUCs of screening instruments for ODD === * '''''For a list of the likelihood ratios for more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&wteswitched=1#Likelihood_ratios_and_AUCs_of_common_screening_instruments click here.]''''' {| class="wikitable sortable" border="1" ! Screening Measure (Primary Reference) ! Area Under Curve (AUC) ! LR+ (Score) ! LR- (Score) ! Clinical Generalizability !Where to Access |- |CBCL DSM-Oriented Scales<ref name=":7">Achenbach, T. M. (1991a). ''Manual for the Child Behavior Checklist/4–18 and 1991 Profile''. Burlington , VT : University of Vermont Department of Psychiatry.</ref> |.71 (N=475)<ref name=":3">Ebesutani, C., Bernstein, A., Nakamura, B. J., Chorpita, B. F., Higa-McMillan, C. K., & Weisz, J. R. (2010). Concurrent validity of the child behavior checklist DSM-oriented scales: Correspondence with DSM diagnoses and comparison to syndrome scales. ''Journal of Psychopathology and Behavioral Assessment, 32''(3), 373-384.</ref> | 2.80 (60+ to 70+) <ref name=":4">Warnick, E. M., Bracken, M. B., & Kasl, S. (2008). Screening efficiency of the Child Behavior Checklist and Strengths and Difficulties Questionnaire: a systematic review. ''Child and Adolescent Mental Health, 13''(3), 140-147.</ref> | .52 (60- to 70-)*<ref name=":4" /> |Youth aged 5 - 18 seeking out patient treatment across a variety of settings<ref name=":4" /> |[https://store.aseba.org/School-Age-6-18-Materials/departments/11/ Purchase] |- |SDQ<ref name=":6">{{Cite journal|last=Shabani|first=Amir|last2=Masoumian|first2=Samira|last3=Zamirinejad|first3=Somayeh|last4=Hejri|first4=Maryam|last5=Pirmorad|first5=Tahereh|last6=Yaghmaeezadeh|first6=Hooman|date=2021-05|title=Psychometric properties of Structured Clinical Interview for DSM‐5 Disorders‐Clinician Version (SCID‐5‐CV)|url=https://onlinelibrary.wiley.com/doi/10.1002/brb3.1894|journal=Brain and Behavior|language=en|volume=11|issue=5|doi=10.1002/brb3.1894|issn=2162-3279|pmc=PMC8119811|pmid=33729681}}</ref> | .81 -.88 (N = 18,416)<ref>{{Cite journal|last=Algorta|first=Guillermo Perez|last2=Dodd|first2=Alyson Lamont|last3=Stringaris|first3=Argyris|last4=Youngstrom|first4=Eric A.|date=2016-09|title=Diagnostic efficiency of the SDQ for parents to identify ADHD in the UK: a ROC analysis|url=http://link.springer.com/10.1007/s00787-015-0815-0|journal=European Child & Adolescent Psychiatry|language=en|volume=25|issue=9|pages=949–957|doi=10.1007/s00787-015-0815-0|issn=1018-8827|pmc=PMC4990620|pmid=26762184}}</ref> |7.00 <ref name=":4" /> |.55<ref name=":4" /> |Youth aged 5 - 18 seeking out patient treatment across a variety of settings<ref name=":4" /> |[https://www.sdqinfo.org/a0.html SDQ Homepage][https://www.sdqinfo.org/py/sdqinfo/b0.py PDFs] |- |SDQ - DSM -IV Conduct and ODD | |7.55 (3+)<ref name=":2">{{Cite journal|last=Goodman|first=Robert|date=2001-11-01|title=Psychometric Properties of the Strengths and Difficulties Questionnaire|url=https://www.jaacap.org/article/S0890-8567(09)60543-8/abstract|journal=Journal of the American Academy of Child & Adolescent Psychiatry|language=English|volume=40|issue=11|pages=1337–1345|doi=10.1097/00004583-200111000-00015|issn=0890-8567|pmid=11699809}}</ref> |.35 (3-)<ref name=":2" /> |Surveyed youth aged 5 - 15 in the UK <ref name=":2" /> |[https://www.sdqinfo.org/a0.html SDQ Homepage][https://www.sdqinfo.org/py/sdqinfo/b0.py PDFs] |- |ECBI- Intensity Scale<ref name=":8">Eyberg, S. M., & Robinson, E. A. (1983). Conduct problem behavior: Standardization of a behavioral rating scale with adolescents. Journal of Clinical Child Psychology, 12 (3), 347-354.</ref> | |6.92 (131+)<ref name=":5">{{Cite journal|last=LYNEHAM|first=HEIDI J.|last2=ABBOTT|first2=MAREE J.|last3=RAPEE|first3=RONALD M.|date=2007-06|title=Interrater Reliability of the Anxiety Disorders Interview Schedule for DSM-IV: Child and Parent Version|url=https://doi.org/10.1097/chi.0b013e3180465a09|journal=Journal of the American Academy of Child &amp; Adolescent Psychiatry|volume=46|issue=6|pages=731–736|doi=10.1097/chi.0b013e3180465a09|issn=0890-8567}}</ref> |.11 (131-)<ref name=":5" /> |Youth aged 7-16 had responses compared to diagnosis<ref name=":5" /> |[https://www.parinc.com/products/pkey/97 Purchase] |- |BASC-2 PRS - Aggression |.76 (N =156)<ref name=":12">{{Cite journal|last=Doyle|first=Alysa|last2=Ostrander|first2=Rick|last3=Skare|first3=Stacy|last4=Crosby|first4=Ross D.|last5=August|first5=Gerald J.|date=1997-09-01|title=Convergent and criterion-related validity of the behavior assessment system for children-parent rating scale|url=https://doi.org/10.1207/s15374424jccp2603_6|journal=Journal of Clinical Child Psychology|volume=26|issue=3|pages=276–284|doi=10.1207/s15374424jccp2603_6|issn=0047-228X|pmid=9292385}}</ref> |3.62 (65+)<ref name=":12" /> |.60 (65-)<ref name=":12" /> |Youth from first through fourth grade who were at risk for CD<ref name=":12" /> |[https://www.pearsonassessments.com/store/usassessments/en/Store/Professional-Assessments/Behavior/Comprehensive/Behavior-Assessment-System-for-Children-%7C-Third-Edition-/p/100001402.html#:~:text=The%20BASC%E2%84%A2%20holds%20an,or%20adolescent's%20behavior%20and%20emotions. Purchase] |} '''Note:''' “LR+” refers to the change in likelihood ratio associated with a positive test score, and “LR-” is the likelihood ratio for a low score. Likelihood ratios of 1 indicate that the test result did not change impressions at all. LRs larger than 10 or smaller than .10 are frequently clinically decisive; 5 or .20 are helpful, and between 2.0 and .5 are small enough that they rarely result in clinically meaningful changes of formulation<ref>Sackett, D. L., Straus, S. E., Richardson, W. S., Rosenberg, W., & Haynes, R. B. (2000). Evidence-based medicine: How to practice and teach EBM. Edinburgh: Churchill Livingstone.</ref>. '''Search terms:''' [Oppositional Defiant Disorder] AND [sensitivity OR specificity] in GoogleScholar and PsychINFO; === Interpreting ODD screening measure scores === * For information on interpreting screening measure scores, click [[Evidence based assessment/Prediction phase#Interpreting screening measure scores|here.]] == [[Evidence based assessment/Prescription phase|'''Prescription phase''']] == === Gold standard diagnostic interviews === * For a list of broad reaching diagnostic interviews sortable by disorder with PDFs (if applicable), [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prescription_phase&wteswitched=1#Common_Diagnostic_Interviews click here.] === Recommended diagnostic instruments for ODD === {| class="wikitable sortable" ! colspan="10" |Diagnostic instruments for ODD |- !Measure !Format (Reporter) !Age Range !Administration/ Completion Time !Interrater Reliability !Test-Retest Reliability !Construct Validity !Content Validity !Highly Recommended !Where to Access |- |Kiddie Schedule for Affective Disorders and Schizophrenia Present and Lifetime Version (KSADS-PL) |Structured interview |6-28 |45-75 minutes |E<ref name=":11">{{Cite journal|last=KAUFMAN|first=JOAN|last2=BIRMAHER|first2=BORIS|last3=BRENT|first3=DAVID|last4=RAO|first4=UMA|last5=FLYNN|first5=CYNTHIA|last6=MORECI|first6=PAULA|last7=WILLIAMSON|first7=DOUGLAS|last8=RYAN|first8=NEAL|date=1997-07|title=Schedule for Affective Disorders and Schizophrenia for School-Age Children-Present and Lifetime Version (K-SADS-PL): Initial Reliability and Validity Data|url=https://doi.org/10.1097/00004583-199707000-00021|journal=Journal of the American Academy of Child &amp; Adolescent Psychiatry|volume=36|issue=7|pages=980–988|doi=10.1097/00004583-199707000-00021|issn=0890-8567}}</ref> |G<ref name=":11" /> | | | |[https://www.pediatricbipolar.pitt.edu/resources/instruments Website to access] |- |Diagnostic Interview Schedule for Children (DISC-5) |Structured Interview (Self report and parent) |6-17 | | | | | |<ref name=":0" />[[File:Light green check.svg|center|frameless|36x36px]] |[https://telesage.com/netdisc-5/# Coming soon] |} '''Note:''' L = Less than adequate; A = Adequate; G = Good; E = Excellent; U = Unavailable; NA = Not applicable == [[Evidence based assessment/Process phase|'''Process phase''']] == The following section contains a list of process and outcome measures for oppositional defiant disorder. The section includes benchmarks based on published norms for several outcome and severity measures, as well as information about commonly used process measures. Process and outcome measures are used as part of the [[Evidence based assessment/Process phase|process phase]] of assessment. For more information of differences between process and outcome measures, see the page on the [[Evidence based assessment/Process phase|process phase of assessment]]. === Outcome and severity measures === * This table includes clinically significant benchmarks for generalized anxiety disorder specific outcome measures * Information on how to interpret this table can be [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase found here]. * Additionally, these [[Evidence based assessment/Vignettes|vignettes]] might be helpful resources for understanding appropriate adaptation of outcome measures in practice. * ''<u>For clinically significant change benchmarks for the CBCL, YSR, and TRF total, externalizing, internalizing, and attention benchmarks,</u>'' [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase&wteswitched=1#Clinically_significant_change_benchmarks_for_widely-used_outcome_measures see here.] {| class="wikitable sortable" border="1" |- | rowspan="1&quot;" style="text-align:center;font-size:130%;" | <b> Measure</b> | style="text-align:center;font-size:130%;" | <b> Subscale</b> | colspan="3" style="text-align:center;font-size:130%" width="300" | <b> Cut-off scores</b> | colspan="3" style="text-align:center;font-size:120%" | <b> Critical Change <br> (unstandardized scores)</b> |- | colspan="8" style="font-size:110%; text-align:center;" span | <b> Benchmarks Based on Published Norms</b> |- | colspan="2" | | style="text-align:center;font-size:110%" | <b> A</b> | style="text-align:center;font-size:110%" | <b> B</b> | style="text-align:center;font-size:110%" | <b> C</b> | style="text-align:center;font-size:110%" | <b> 95%</b> | style="text-align:center;font-size:110%" | <b> 90%</b> | style="text-align:center;font-size:110%" | <b> SE_difference</b> |- | rowspan="1" style="text-align:center;" | <b> CBCL T-scores <br> (2001 Norms)</b> | style="text-align:right;" | <i> Externalizing</i> | style="text-align:center;" | 49 | style="text-align:center;" | 70 | style="text-align:center;" | 58 | style="text-align:center;" | 7 | style="text-align:center;" | 6 | style="text-align:center;" | 3.4 |- | rowspan="1" style="text-align:center;" | <b> ECBI Scaled Scores <br> (1983 Norms)</b> | style="text-align:right;" | <i> Intensity</i> | style="text-align:center;" | 80.1 | style="text-align:center;" | 169.5 | style="text-align:center;" | 112.9 | style="text-align:center;" | 9.5 | style="text-align:center;" | 8 | style="text-align:center;" | 4.8 |- | rowspan="1" style="text-align:center;" | <b> ECBI Scaled Scores <br> (1983 Norms)</b> | style="text-align:right;" | <i> Problem</i> | style="text-align:center;" | 3.9 | style="text-align:center;" | 17.7 | style="text-align:center;" | 11.5 | style="text-align:center;" | 2.1 | style="text-align:center;" | 1.8 | style="text-align:center;" | 1.1 |} '''Note:''' “A” = Away from the clinical range, “B” = Back into the nonclinical range, “C” = Closer to the nonclinical than clinical mean. === Process measures=== See Section 1.1 for overview of evidence-based measures to use depending on etiology and symptomatology of Oppositional Defiant Disorder. ==Treatment== ===Behavioral parent training=== Behavioral Parent Training is considered the most effective treatment for childhood disruptive behavior disorders (e.g., Oppositional Defiant Disorder), especially for younger children (i.e., 3-8 year-olds). See http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/rule-breaking-defiance-and-acting-out/ a website sponsored by The Society for Child and Adolescent Psychology (APA, Division 53) and the Association for Behavioral and Cognitive Therapies (ABCT), for current summary of evidence-based treatments. ===Overview of recommendations for assessment and treatment=== See the [https://pathways.nice.org.uk/pathways/antisocial-behaviour-and-conduct-disorders-in-children-and-young-people#content=view-info-category%3Aview-about-menu National Institute for Health and Care Excellence (NICE) Practice Guidelines for Childhood Conduct Disorders], for an overview of recommendations for both assessment and treatment of Oppositional Defiant Disorder. == External Links == *[https://sccap53.org Society of Clinical Child and Adolescent Psychology] *http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/rule-breaking-defiance-and-acting-out/ == References == {{collapse top|Click here for references}} {{Reflist|30em}} [[Category:Psychological disorder portfolios|{{SUBPAGENAME}}]] {{collapse bottom}} [[Category:Psychological disorder portfolios|{{SUBPAGENAME}}]] sfapa8arauejypk4e8s5wcvuwrl8z05 2410319 2410317 2022-07-29T22:20:48Z Aherman012 2943941 /* What is a "portfolio"? */ added link to "extended" page wikitext text/x-wiki <noinclude>{{Helping Give Away Psychological Science Banner}}</noinclude> {{medical disclaimer}} {{:{{BASEPAGENAME}}/Sidebar}} ==[[Evidence based assessment/Portfolio template/What is a "portfolio"|'''What is a "portfolio"?''']]== * For background information on what assessment portfolios are, click the link in the heading above. * Want even 'more' information about this topic? There's an extended version of this page [[Evidence-based assessment/Oppositional defiant disorder (assessment portfolio)/extended version|here]]. ==[[Evidence based assessment/Preparation phase|'''Preparation phase''']]== === Diagnostic criteria for oppositional defiant disorder=== {{blockquotetop}} <big>ICD-11 Diagnostic Criteria</big><br> <br> '''General Description:''' Oppositional defiant disorder is a persistent pattern (e.g., 6 months or more) of markedly defiant, disobedient, provocative or spiteful behaviour that occurs more frequently than is typically observed in individuals of comparable age and developmental level and that is not restricted to interaction with siblings. Oppositional defiant disorder may be manifest in prevailing, persistent angry or irritable mood, often accompanied by severe temper outbursts or in headstrong, argumentative and defiant behaviour. The behavior pattern is of sufficient severity to result in significant impairment in personal, family, social, educational, occupational or other important areas of functioning <br> <br> '''Oppositional Defiant Disorder With Chronic Irritability-Anger:''' All definitional requirements for oppositional defiant disorder are met. This form of oppositional defiant disorder is characterized by prevailing, persistent angry or irritable mood that may be present independent of any apparent provocation. The negative mood is often accompanied by regularly occurring severe temper outbursts that are grossly out of proportion in intensity or duration to the provocation. Chronic irritability and anger are characteristic of the individual’s functioning nearly every day, are observable across multiple settings or domains of functioning (e.g., home, school, social relationships), and are not restricted to the individual’s relationship with his/her parents or guardians. The pattern of chronic irritability and anger is not limited to occasional episodes (e.g., developmentally typical irritability) or discrete periods (e.g., irritable mood in the context of manic or depressive episodes). *Note: The ICD-11 lists 3 additional subcategories of oppositional defiant disorder with chronic irritability-anger (i.e., with limited prosocial emotions, with typical prosocial emotions, and unspecified). They can be found [https://icd.who.int/browse11/l-m/en#/http%3a%2f%2fid.who.int%2ficd%2fentity%2f818792113 here]. <br> '''Oppositional Defiant Disorder Without Chronic Irritability-Anger:''' Meets all definitional requirements for oppositional defiant disorder. This form of oppositional defiant disorder is not characterized by prevailing, persistent, angry or irritable mood, but does feature headstrong, argumentative, and defiant behavior. *Note: The ICD-11 lists 3 additional subcategories of oppositional defiant disorder without chronic irritability-anger (i.e., with limited prosocial emotions, with typical prosocial emotions, and unspecified). They can be found [https://icd.who.int/browse11/l-m/en#/http%3a%2f%2fid.who.int%2ficd%2fentity%2f736540987 here]. <br> <big>Changes in DSM-5</big> <br> The diagnostic criteria for ADHD changed slightly from DSM-IV to DSM-5. See the changes [https://www.ncbi.nlm.nih.gov/books/NBK519712/table/ch3.t14/ here]. {{blockquotebottom}}<ref>{{Cite web|url=https://icd.who.int/browse11/l-m/en#/http://id.who.int/icd/entity/1545599508|title=ICD-11 for Mortality and Morbidity Statistics|website=icd.who.int|access-date=2022-07-11}}</ref> ===Base rates of ODD in different clinical settings=== This section describes the demographic setting of the population(s) sampled, base rates of diagnosis, country/region sampled and the diagnostic method that was used. Using this information, clinicians will be able to anchor the rate of ODD that they are likely to see in their clinical practice. * '''''To see prevalence rates across multiple disorders, [[Evidence-based assessment/Preparation phase|click here.]]''''' {|class="wikitable sortable" border="1" |- ! Demography ! Setting !! Base Rate !! Diagnostic Method |- || Various locations across USA | Meta-analysis of 38 studies<ref>Canino G, Polanczyk G, Bauermeister JJ, et al. (2010) Does the prevalence of CD and ODD vary across cultures? Social Psychiatry Epidemiology, 45(7):695–704.</ref> || 3.3% || Varied |- || All of the U.S. | Nationally representative large-scale study (N = 3,119) <ref>Nock, M. K., Kazdin, A. E., Hiripi, E., & Kessler, R. C. (2007). Lifetime prevalence, correlates, and persistence of oppositional defiant disorder: results from the National Comorbidity Survey Replication. ''Journal of Child Psychology and Psychiatry, 48''(7), 703-713.</ref> || 10.2% (overall) 11.2% (males) 9.2% (females) || World Health Organization (WHO) Composite International Diagnostic Interview (CIDI)^r |- |Suburban and urban Colorado |Project to Learn about Youth-Mental health, school-based study for children from kindergarten to high-school (n = 236)<ref name=":1">{{Cite journal|last=Danielson|first=Melissa L.|last2=Bitsko|first2=Rebecca H.|last3=Holbrook|first3=Joseph R.|last4=Charania|first4=Sana N.|last5=Claussen|first5=Angelika H.|last6=McKeown|first6=Robert E.|last7=Cuffe|first7=Steven P.|last8=Owens|first8=Julie Sarno|last9=Evans|first9=Steven W.|date=2021-06-01|title=Community-Based Prevalence of Externalizing and Internalizing Disorders among School-Aged Children and Adolescents in Four Geographically Dispersed School Districts in the United States|url=https://doi.org/10.1007/s10578-020-01027-z|journal=Child Psychiatry & Human Development|language=en|volume=52|issue=3|pages=500–514|doi=10.1007/s10578-020-01027-z|issn=1573-3327|pmc=PMC8016018|pmid=32734339}}</ref> |6.8% |Diagnostic Interview Schedule for Children (DISC) |- |Urban and suburban Florida |Project to Learn about Youth-Mental health, school-based study for children from kindergarten to high-school (n = 289)<ref name=":1" /> |6.9% |Diagnostic Interview Schedule for Children (DISC) |- |Rural and suburban Ohio |Project to Learn about Youth-Mental health, school-based study for children from elementary to high-school (n = 152)<ref name=":1" /> |17.3% |Diagnostic Interview Schedule for Children (DISC) |- |Suburban and rural South Carolina |Project to Learn about Youth-Mental health, school-based study for children from elementary to high-school (n = 270)<ref name=":1" /> |5.7% |Diagnostic Interview Schedule for Children (DISC) |- || Semi-rural North Carolina | Preschool-aged children from pediatric practices (N = 306; age 2 - 5 years old)<ref>Egger, H.L., & Angold, A. (2006). Common emotional and behavioral disorders in preschool children: Presentation, nosology, and epidemiology. ''Journal of Child Psychiatry and Psychology, 47'', 313–337.</ref> || 6.6% || Preschool Age Psychiatric Assessment (PAPA)^p |- || Western North Carolina | The Great Smoky Mountains Study - longitudinal, population-based study of community sample<ref>Costello, E. J., Mustillo, S., Erkanli, A., Keeler, G., & Angold, A. (2003) Prevalence and development of psychiatric disorders in adolescence. ''Arch Gen Psychiatry, 60'', 837-844.</ref> || 2.33% (overall) 3.16% (males) 2.75% (females) || Child and Adolescent Psychiatric Assessment (CAPA)^{p, y} |- || Chicago | Preschool-aged children from inner city schools and pediatric practices (N = 796; age 2 - 5 years old)<ref>Lavigne, J. V., LeBailly, S. A., Hopkins, J., Gouze, K. R., & Binns, H. J. (2009). The prevalence of ADHD, ODD, depression, and anxiety in a community sample of 4-year-olds. ''Journal Of Clinical Child And Adolescent Psychology, 38''(3), 315-328. doi:10.1080/15374410902851382</ref> || 8.3% || Diagnostic Interview Schedule for Children–Parent Scale–Young Child Version (DISC-YC) ^p |- |Germany (Saarbrücken County) |All school-aged children were examined during a routine school-entry medical examination (N = 1676, mean age = 5.7)<ref>{{Cite journal|last=Niemczyk|first=Justine|last2=Equit|first2=Monika|last3=Braun-Bither|first3=Katrin|last4=Klein|first4=Anna-Maria|last5=von Gontard|first5=Alexander|date=2015-07-01|title=Prevalence of incontinence, attention deficit/hyperactivity disorder and oppositional defiant disorder in preschool children|url=https://doi.org/10.1007/s00787-014-0628-6|journal=European Child & Adolescent Psychiatry|language=en|volume=24|issue=7|pages=837–843|doi=10.1007/s00787-014-0628-6|issn=1435-165X}}</ref> |7.3% (males) 5.1%(females) |DISYPS-II |- |South Korea (Seoul) |Children were randomly surveyed across 6 school districts in Seoul (N = 1645, age 6 - 12 years old) |5.8% (males) 4.1%(females) |Diagnostic Interview Schedule for Children–Parent Scale IV (DISC-IV) |} ^p Parent interviewed as part of diagnostic assessment; ^y youth interviewed as part of diagnostic assessment, ^r adult interviewed for retrospective report as part of diagnostic assessment '''Note:''' Mash and Barkley note that prevalence rates of ODD must be qualified, because the definition of ODD has changed at a fast rate, the rates adolescents meeting criteria in any cross-sectional evaluation may be misleading because of the developmental progressions with and between ODD and Conduct Disorder, and categorical definitions of aggressive patterns may reflect arbitrary numbers of constituent estimates. These factors may lead to misleading prevalence rates. In addition, few studies have investigated the prevalence of ODD in preschool-aged children, and early onset of these behaviors is associated with more severe and stable impairment.<ref>Mash, E., & Barkley, R. (Eds.). (2003). ''Child Psychopathology''. 2nd Edition. New York: Guilford Press.</ref> ==[[Evidence based assessment/Prediction phase|'''Prediction phase''']]== The following section contains a list of screening and diagnostic instruments for ODD. The section includes administration information, psychometric data, and PDFs or links to the screenings. * Screenings are used as part of the [[Evidence based assessment/Prediction phase|prediction phase]] of assessment; for more information on interpretation of this data, or how screenings fit in to the assessment process, click [[Evidence based assessment/Prediction phase|here]]. * '''''For a list of more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&wteswitched=1#Psychometric_properties_of_common_screening_instruments click here.]''''' {| class="wikitable sortable" border="1" ! colspan="10" |Screening measures for ODD |- ! Measure ! Format (Reporter) ! Age Range ! Administration/ Completion Time ! Interrater Reliability ! Test-Retest Reliability ! Construct Validity ! Content Validity ! Highly Recommended !Where to access |- |Achenbach System of Empirically Based Assessments (ASEBA): Child Behavior Checklist (CBCL) |Parent report |6-18 <ref name=":9">{{Cite book|url=https://www.worldcat.org/oclc/1130319849|title=Assessment of disorders in childhood and adolescence|date=2020|others=Eric Arden Youngstrom, Mitchell J. Prinstein, Eric J. Mash, Russell A. Barkley|isbn=978-1-4625-4363-2|edition=Fifth edition|location=New York, NY|oclc=1130319849}}</ref> |10 - 15 minutes<ref name=":9" /> |A<ref name=":0">{{Cite book|url=https://www.worldcat.org/oclc/314222270|title=A guide to assessments that work|author=Hunsley, John|author2=Mash, Eric J.|date=2008|publisher=Oxford University Press|isbn=9780195310641|location=New York|oclc=314222270}}</ref> |E<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://store.aseba.org/School-Age-6-18-Materials/departments/11/ Purchase] |- |ASEBA: Teacher Report Form (TRF) |Teacher report |6-18 <ref name=":9" /> |10 - 15 minutes<ref name=":9" /> |A<ref name=":0" /> |E<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://store.aseba.org/School-Age-6-18-Materials/departments/11/ Purchase] |- |ASEBA: Youth Self-Report (YSR) |Youth self-report |11-18 <ref name=":9" /> |10 - 15 minutes<ref name=":9" /> |A<ref name=":0" /> |E<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://store.aseba.org/School-Age-6-18-Materials/departments/11/ Purchase] |- |Behavior Assessment for Children, Third Edition (BASC-3): Parent Rating Scale |Parent report |2-21 |10 - 20 minutes |A<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |E<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://www.pearsonassessments.com/store/usassessments/en/Store/Professional-Assessments/Behavior/Comprehensive/Behavior-Assessment-System-for-Children-%7C-Third-Edition-/p/100001402.html#:~:text=The%20BASC%E2%84%A2%20holds%20an,or%20adolescent's%20behavior%20and%20emotions. Purchase] |- |BASC-3: Teacher Rating Scale |Teacher report |2-21 |10 - 20 minutes |A<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |E<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://www.pearsonassessments.com/store/usassessments/en/Store/Professional-Assessments/Behavior/Comprehensive/Behavior-Assessment-System-for-Children-%7C-Third-Edition-/p/100001402.html#:~:text=The%20BASC%E2%84%A2%20holds%20an,or%20adolescent's%20behavior%20and%20emotions. Purchase] |- |BASC-3: Self-Report of Personality |Youth self-report |6 - college age |30 minutes |A<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |E<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://www.pearsonassessments.com/store/usassessments/en/Store/Professional-Assessments/Behavior/Comprehensive/Behavior-Assessment-System-for-Children-%7C-Third-Edition-/p/100001402.html#:~:text=The%20BASC%E2%84%A2%20holds%20an,or%20adolescent's%20behavior%20and%20emotions. Purchase] |- |Eyberg Child Behavior Inventory (ECBI) |Parent report |2-16 |5 minutes |A<ref name=":0" /> |G<ref name=":0" /> |E<ref name=":0" /> |E<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://www.parinc.com/products/pkey/97 Purchase] |- |Sutter-Eyberg Student Behavior Inventory - Revised (SESBI-R) |Teacher report |2-16 |5 minutes |A<ref name=":0" /> |G<ref name=":0" /> |E<ref name=":0" /> |E<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://www.parinc.com/products/pkey/97 Purchase] |- |Child and Adolescent Behavior Inventory (CABI) |Parent Report |3 - 18 |5 - 10 minutes | |A<ref>{{Cite journal|last=Burns|first=G. Leonard|last2=Preszler|first2=Jonathan|last3=Becker|first3=Stephen P.|date=2022-07-04|title=Psychometric and Normative Information on the Child and Adolescent Behavior Inventory in a Nationally Representative Sample of United States Children|url=https://doi.org/10.1080/15374416.2020.1852943|journal=Journal of Clinical Child & Adolescent Psychology|volume=51|issue=4|pages=443–452|doi=10.1080/15374416.2020.1852943|issn=1537-4416|pmc=PMC8272731|pmid=33428463}}</ref> | | | |PDF |- |Strengths and Difficulties Questionnaire (SDQ) |Parent report |3 - 16 |3 - 5 minutes |E<ref name=":10">{{Cite journal|last=Goodman|first=Robert|date=2001-11-01|title=Psychometric Properties of the Strengths and Difficulties Questionnaire|url=https://www.jaacap.org/article/S0890-8567(09)60543-8/abstract|journal=Journal of the American Academy of Child & Adolescent Psychiatry|language=English|volume=40|issue=11|pages=1337–1345|doi=10.1097/00004583-200111000-00015|issn=0890-8567|pmid=11699809}}</ref> |G<ref name=":10" /> | | | |[https://www.sdqinfo.org/a0.html SDQ Homepage][https://www.sdqinfo.org/py/sdqinfo/b0.py PDFs] |- |Disruptive Behavior Disorder Rating Scale |Parent report, teacher report |5 - 17 |5 - 10 minutes | | | | | |[https://web.archive.org/web/20151123022653/http://ccf.buffalo.edu/pdf/DBD_rating_scale.pdf PDF] |} '''Note:''' L = Less than adequate; A = Adequate; G = Good; E = Excellent; U = Unavailable; NA = Not applicable === Likelihood ratios and AUCs of screening instruments for ODD === * '''''For a list of the likelihood ratios for more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&wteswitched=1#Likelihood_ratios_and_AUCs_of_common_screening_instruments click here.]''''' {| class="wikitable sortable" border="1" ! Screening Measure (Primary Reference) ! Area Under Curve (AUC) ! LR+ (Score) ! LR- (Score) ! Clinical Generalizability !Where to Access |- |CBCL DSM-Oriented Scales<ref name=":7">Achenbach, T. M. (1991a). ''Manual for the Child Behavior Checklist/4–18 and 1991 Profile''. Burlington , VT : University of Vermont Department of Psychiatry.</ref> |.71 (N=475)<ref name=":3">Ebesutani, C., Bernstein, A., Nakamura, B. J., Chorpita, B. F., Higa-McMillan, C. K., & Weisz, J. R. (2010). Concurrent validity of the child behavior checklist DSM-oriented scales: Correspondence with DSM diagnoses and comparison to syndrome scales. ''Journal of Psychopathology and Behavioral Assessment, 32''(3), 373-384.</ref> | 2.80 (60+ to 70+) <ref name=":4">Warnick, E. M., Bracken, M. B., & Kasl, S. (2008). Screening efficiency of the Child Behavior Checklist and Strengths and Difficulties Questionnaire: a systematic review. ''Child and Adolescent Mental Health, 13''(3), 140-147.</ref> | .52 (60- to 70-)*<ref name=":4" /> |Youth aged 5 - 18 seeking out patient treatment across a variety of settings<ref name=":4" /> |[https://store.aseba.org/School-Age-6-18-Materials/departments/11/ Purchase] |- |SDQ<ref name=":6">{{Cite journal|last=Shabani|first=Amir|last2=Masoumian|first2=Samira|last3=Zamirinejad|first3=Somayeh|last4=Hejri|first4=Maryam|last5=Pirmorad|first5=Tahereh|last6=Yaghmaeezadeh|first6=Hooman|date=2021-05|title=Psychometric properties of Structured Clinical Interview for DSM‐5 Disorders‐Clinician Version (SCID‐5‐CV)|url=https://onlinelibrary.wiley.com/doi/10.1002/brb3.1894|journal=Brain and Behavior|language=en|volume=11|issue=5|doi=10.1002/brb3.1894|issn=2162-3279|pmc=PMC8119811|pmid=33729681}}</ref> | .81 -.88 (N = 18,416)<ref>{{Cite journal|last=Algorta|first=Guillermo Perez|last2=Dodd|first2=Alyson Lamont|last3=Stringaris|first3=Argyris|last4=Youngstrom|first4=Eric A.|date=2016-09|title=Diagnostic efficiency of the SDQ for parents to identify ADHD in the UK: a ROC analysis|url=http://link.springer.com/10.1007/s00787-015-0815-0|journal=European Child & Adolescent Psychiatry|language=en|volume=25|issue=9|pages=949–957|doi=10.1007/s00787-015-0815-0|issn=1018-8827|pmc=PMC4990620|pmid=26762184}}</ref> |7.00 <ref name=":4" /> |.55<ref name=":4" /> |Youth aged 5 - 18 seeking out patient treatment across a variety of settings<ref name=":4" /> |[https://www.sdqinfo.org/a0.html SDQ Homepage][https://www.sdqinfo.org/py/sdqinfo/b0.py PDFs] |- |SDQ - DSM -IV Conduct and ODD | |7.55 (3+)<ref name=":2">{{Cite journal|last=Goodman|first=Robert|date=2001-11-01|title=Psychometric Properties of the Strengths and Difficulties Questionnaire|url=https://www.jaacap.org/article/S0890-8567(09)60543-8/abstract|journal=Journal of the American Academy of Child & Adolescent Psychiatry|language=English|volume=40|issue=11|pages=1337–1345|doi=10.1097/00004583-200111000-00015|issn=0890-8567|pmid=11699809}}</ref> |.35 (3-)<ref name=":2" /> |Surveyed youth aged 5 - 15 in the UK <ref name=":2" /> |[https://www.sdqinfo.org/a0.html SDQ Homepage][https://www.sdqinfo.org/py/sdqinfo/b0.py PDFs] |- |ECBI- Intensity Scale<ref name=":8">Eyberg, S. M., & Robinson, E. A. (1983). Conduct problem behavior: Standardization of a behavioral rating scale with adolescents. Journal of Clinical Child Psychology, 12 (3), 347-354.</ref> | |6.92 (131+)<ref name=":5">{{Cite journal|last=LYNEHAM|first=HEIDI J.|last2=ABBOTT|first2=MAREE J.|last3=RAPEE|first3=RONALD M.|date=2007-06|title=Interrater Reliability of the Anxiety Disorders Interview Schedule for DSM-IV: Child and Parent Version|url=https://doi.org/10.1097/chi.0b013e3180465a09|journal=Journal of the American Academy of Child &amp; Adolescent Psychiatry|volume=46|issue=6|pages=731–736|doi=10.1097/chi.0b013e3180465a09|issn=0890-8567}}</ref> |.11 (131-)<ref name=":5" /> |Youth aged 7-16 had responses compared to diagnosis<ref name=":5" /> |[https://www.parinc.com/products/pkey/97 Purchase] |- |BASC-2 PRS - Aggression |.76 (N =156)<ref name=":12">{{Cite journal|last=Doyle|first=Alysa|last2=Ostrander|first2=Rick|last3=Skare|first3=Stacy|last4=Crosby|first4=Ross D.|last5=August|first5=Gerald J.|date=1997-09-01|title=Convergent and criterion-related validity of the behavior assessment system for children-parent rating scale|url=https://doi.org/10.1207/s15374424jccp2603_6|journal=Journal of Clinical Child Psychology|volume=26|issue=3|pages=276–284|doi=10.1207/s15374424jccp2603_6|issn=0047-228X|pmid=9292385}}</ref> |3.62 (65+)<ref name=":12" /> |.60 (65-)<ref name=":12" /> |Youth from first through fourth grade who were at risk for CD<ref name=":12" /> |[https://www.pearsonassessments.com/store/usassessments/en/Store/Professional-Assessments/Behavior/Comprehensive/Behavior-Assessment-System-for-Children-%7C-Third-Edition-/p/100001402.html#:~:text=The%20BASC%E2%84%A2%20holds%20an,or%20adolescent's%20behavior%20and%20emotions. Purchase] |} '''Note:''' “LR+” refers to the change in likelihood ratio associated with a positive test score, and “LR-” is the likelihood ratio for a low score. Likelihood ratios of 1 indicate that the test result did not change impressions at all. LRs larger than 10 or smaller than .10 are frequently clinically decisive; 5 or .20 are helpful, and between 2.0 and .5 are small enough that they rarely result in clinically meaningful changes of formulation<ref>Sackett, D. L., Straus, S. E., Richardson, W. S., Rosenberg, W., & Haynes, R. B. (2000). Evidence-based medicine: How to practice and teach EBM. Edinburgh: Churchill Livingstone.</ref>. '''Search terms:''' [Oppositional Defiant Disorder] AND [sensitivity OR specificity] in GoogleScholar and PsychINFO; === Interpreting ODD screening measure scores === * For information on interpreting screening measure scores, click [[Evidence based assessment/Prediction phase#Interpreting screening measure scores|here.]] == [[Evidence based assessment/Prescription phase|'''Prescription phase''']] == === Gold standard diagnostic interviews === * For a list of broad reaching diagnostic interviews sortable by disorder with PDFs (if applicable), [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prescription_phase&wteswitched=1#Common_Diagnostic_Interviews click here.] === Recommended diagnostic instruments for ODD === {| class="wikitable sortable" ! colspan="10" |Diagnostic instruments for ODD |- !Measure !Format (Reporter) !Age Range !Administration/ Completion Time !Interrater Reliability !Test-Retest Reliability !Construct Validity !Content Validity !Highly Recommended !Where to Access |- |Kiddie Schedule for Affective Disorders and Schizophrenia Present and Lifetime Version (KSADS-PL) |Structured interview |6-28 |45-75 minutes |E<ref name=":11">{{Cite journal|last=KAUFMAN|first=JOAN|last2=BIRMAHER|first2=BORIS|last3=BRENT|first3=DAVID|last4=RAO|first4=UMA|last5=FLYNN|first5=CYNTHIA|last6=MORECI|first6=PAULA|last7=WILLIAMSON|first7=DOUGLAS|last8=RYAN|first8=NEAL|date=1997-07|title=Schedule for Affective Disorders and Schizophrenia for School-Age Children-Present and Lifetime Version (K-SADS-PL): Initial Reliability and Validity Data|url=https://doi.org/10.1097/00004583-199707000-00021|journal=Journal of the American Academy of Child &amp; Adolescent Psychiatry|volume=36|issue=7|pages=980–988|doi=10.1097/00004583-199707000-00021|issn=0890-8567}}</ref> |G<ref name=":11" /> | | | |[https://www.pediatricbipolar.pitt.edu/resources/instruments Website to access] |- |Diagnostic Interview Schedule for Children (DISC-5) |Structured Interview (Self report and parent) |6-17 | | | | | |<ref name=":0" />[[File:Light green check.svg|center|frameless|36x36px]] |[https://telesage.com/netdisc-5/# Coming soon] |} '''Note:''' L = Less than adequate; A = Adequate; G = Good; E = Excellent; U = Unavailable; NA = Not applicable == [[Evidence based assessment/Process phase|'''Process phase''']] == The following section contains a list of process and outcome measures for oppositional defiant disorder. The section includes benchmarks based on published norms for several outcome and severity measures, as well as information about commonly used process measures. Process and outcome measures are used as part of the [[Evidence based assessment/Process phase|process phase]] of assessment. For more information of differences between process and outcome measures, see the page on the [[Evidence based assessment/Process phase|process phase of assessment]]. === Outcome and severity measures === * This table includes clinically significant benchmarks for generalized anxiety disorder specific outcome measures * Information on how to interpret this table can be [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase found here]. * Additionally, these [[Evidence based assessment/Vignettes|vignettes]] might be helpful resources for understanding appropriate adaptation of outcome measures in practice. * ''<u>For clinically significant change benchmarks for the CBCL, YSR, and TRF total, externalizing, internalizing, and attention benchmarks,</u>'' [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase&wteswitched=1#Clinically_significant_change_benchmarks_for_widely-used_outcome_measures see here.] {| class="wikitable sortable" border="1" |- | rowspan="1&quot;" style="text-align:center;font-size:130%;" | <b> Measure</b> | style="text-align:center;font-size:130%;" | <b> Subscale</b> | colspan="3" style="text-align:center;font-size:130%" width="300" | <b> Cut-off scores</b> | colspan="3" style="text-align:center;font-size:120%" | <b> Critical Change <br> (unstandardized scores)</b> |- | colspan="8" style="font-size:110%; text-align:center;" span | <b> Benchmarks Based on Published Norms</b> |- | colspan="2" | | style="text-align:center;font-size:110%" | <b> A</b> | style="text-align:center;font-size:110%" | <b> B</b> | style="text-align:center;font-size:110%" | <b> C</b> | style="text-align:center;font-size:110%" | <b> 95%</b> | style="text-align:center;font-size:110%" | <b> 90%</b> | style="text-align:center;font-size:110%" | <b> SE_difference</b> |- | rowspan="1" style="text-align:center;" | <b> CBCL T-scores <br> (2001 Norms)</b> | style="text-align:right;" | <i> Externalizing</i> | style="text-align:center;" | 49 | style="text-align:center;" | 70 | style="text-align:center;" | 58 | style="text-align:center;" | 7 | style="text-align:center;" | 6 | style="text-align:center;" | 3.4 |- | rowspan="1" style="text-align:center;" | <b> ECBI Scaled Scores <br> (1983 Norms)</b> | style="text-align:right;" | <i> Intensity</i> | style="text-align:center;" | 80.1 | style="text-align:center;" | 169.5 | style="text-align:center;" | 112.9 | style="text-align:center;" | 9.5 | style="text-align:center;" | 8 | style="text-align:center;" | 4.8 |- | rowspan="1" style="text-align:center;" | <b> ECBI Scaled Scores <br> (1983 Norms)</b> | style="text-align:right;" | <i> Problem</i> | style="text-align:center;" | 3.9 | style="text-align:center;" | 17.7 | style="text-align:center;" | 11.5 | style="text-align:center;" | 2.1 | style="text-align:center;" | 1.8 | style="text-align:center;" | 1.1 |} '''Note:''' “A” = Away from the clinical range, “B” = Back into the nonclinical range, “C” = Closer to the nonclinical than clinical mean. === Process measures=== See Section 1.1 for overview of evidence-based measures to use depending on etiology and symptomatology of Oppositional Defiant Disorder. ==Treatment== ===Behavioral parent training=== Behavioral Parent Training is considered the most effective treatment for childhood disruptive behavior disorders (e.g., Oppositional Defiant Disorder), especially for younger children (i.e., 3-8 year-olds). See http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/rule-breaking-defiance-and-acting-out/ a website sponsored by The Society for Child and Adolescent Psychology (APA, Division 53) and the Association for Behavioral and Cognitive Therapies (ABCT), for current summary of evidence-based treatments. ===Overview of recommendations for assessment and treatment=== See the [https://pathways.nice.org.uk/pathways/antisocial-behaviour-and-conduct-disorders-in-children-and-young-people#content=view-info-category%3Aview-about-menu National Institute for Health and Care Excellence (NICE) Practice Guidelines for Childhood Conduct Disorders], for an overview of recommendations for both assessment and treatment of Oppositional Defiant Disorder. == External Links == *[https://sccap53.org Society of Clinical Child and Adolescent Psychology] *http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/rule-breaking-defiance-and-acting-out/ == References == {{collapse top|Click here for references}} {{Reflist|30em}} [[Category:Psychological disorder portfolios|{{SUBPAGENAME}}]] {{collapse bottom}} [[Category:Psychological disorder portfolios|{{SUBPAGENAME}}]] oy9lh467ddh8ujhbzz7r3j5ym01xk45 2410325 2410319 2022-07-29T22:28:57Z Aherman012 2943941 /* Prediction phase */ Condensed the page wikitext text/x-wiki <noinclude>{{Helping Give Away Psychological Science Banner}}</noinclude> {{medical disclaimer}} {{:{{BASEPAGENAME}}/Sidebar}} ==[[Evidence based assessment/Portfolio template/What is a "portfolio"|'''What is a "portfolio"?''']]== * For background information on what assessment portfolios are, click the link in the heading above. * Want even 'more' information about this topic? There's an extended version of this page [[Evidence-based assessment/Oppositional defiant disorder (assessment portfolio)/extended version|here]]. ==[[Evidence based assessment/Preparation phase|'''Preparation phase''']]== === Diagnostic criteria for oppositional defiant disorder=== {{blockquotetop}} <big>ICD-11 Diagnostic Criteria</big><br> <br> '''General Description:''' Oppositional defiant disorder is a persistent pattern (e.g., 6 months or more) of markedly defiant, disobedient, provocative or spiteful behaviour that occurs more frequently than is typically observed in individuals of comparable age and developmental level and that is not restricted to interaction with siblings. Oppositional defiant disorder may be manifest in prevailing, persistent angry or irritable mood, often accompanied by severe temper outbursts or in headstrong, argumentative and defiant behaviour. The behavior pattern is of sufficient severity to result in significant impairment in personal, family, social, educational, occupational or other important areas of functioning <br> <br> '''Oppositional Defiant Disorder With Chronic Irritability-Anger:''' All definitional requirements for oppositional defiant disorder are met. This form of oppositional defiant disorder is characterized by prevailing, persistent angry or irritable mood that may be present independent of any apparent provocation. The negative mood is often accompanied by regularly occurring severe temper outbursts that are grossly out of proportion in intensity or duration to the provocation. Chronic irritability and anger are characteristic of the individual’s functioning nearly every day, are observable across multiple settings or domains of functioning (e.g., home, school, social relationships), and are not restricted to the individual’s relationship with his/her parents or guardians. The pattern of chronic irritability and anger is not limited to occasional episodes (e.g., developmentally typical irritability) or discrete periods (e.g., irritable mood in the context of manic or depressive episodes). *Note: The ICD-11 lists 3 additional subcategories of oppositional defiant disorder with chronic irritability-anger (i.e., with limited prosocial emotions, with typical prosocial emotions, and unspecified). They can be found [https://icd.who.int/browse11/l-m/en#/http%3a%2f%2fid.who.int%2ficd%2fentity%2f818792113 here]. <br> '''Oppositional Defiant Disorder Without Chronic Irritability-Anger:''' Meets all definitional requirements for oppositional defiant disorder. This form of oppositional defiant disorder is not characterized by prevailing, persistent, angry or irritable mood, but does feature headstrong, argumentative, and defiant behavior. *Note: The ICD-11 lists 3 additional subcategories of oppositional defiant disorder without chronic irritability-anger (i.e., with limited prosocial emotions, with typical prosocial emotions, and unspecified). They can be found [https://icd.who.int/browse11/l-m/en#/http%3a%2f%2fid.who.int%2ficd%2fentity%2f736540987 here]. <br> <big>Changes in DSM-5</big> <br> The diagnostic criteria for ADHD changed slightly from DSM-IV to DSM-5. See the changes [https://www.ncbi.nlm.nih.gov/books/NBK519712/table/ch3.t14/ here]. {{blockquotebottom}}<ref>{{Cite web|url=https://icd.who.int/browse11/l-m/en#/http://id.who.int/icd/entity/1545599508|title=ICD-11 for Mortality and Morbidity Statistics|website=icd.who.int|access-date=2022-07-11}}</ref> ===Base rates of ODD in different clinical settings=== This section describes the demographic setting of the population(s) sampled, base rates of diagnosis, country/region sampled and the diagnostic method that was used. Using this information, clinicians will be able to anchor the rate of ODD that they are likely to see in their clinical practice. * '''''To see prevalence rates across multiple disorders, [[Evidence-based assessment/Preparation phase|click here.]]''''' {|class="wikitable sortable" border="1" |- ! Demography ! Setting !! Base Rate !! Diagnostic Method |- || Various locations across USA | Meta-analysis of 38 studies<ref>Canino G, Polanczyk G, Bauermeister JJ, et al. (2010) Does the prevalence of CD and ODD vary across cultures? Social Psychiatry Epidemiology, 45(7):695–704.</ref> || 3.3% || Varied |- || All of the U.S. | Nationally representative large-scale study (N = 3,119) <ref>Nock, M. K., Kazdin, A. E., Hiripi, E., & Kessler, R. C. (2007). Lifetime prevalence, correlates, and persistence of oppositional defiant disorder: results from the National Comorbidity Survey Replication. ''Journal of Child Psychology and Psychiatry, 48''(7), 703-713.</ref> || 10.2% (overall) 11.2% (males) 9.2% (females) || World Health Organization (WHO) Composite International Diagnostic Interview (CIDI)^r |- |Suburban and urban Colorado |Project to Learn about Youth-Mental health, school-based study for children from kindergarten to high-school (n = 236)<ref name=":1">{{Cite journal|last=Danielson|first=Melissa L.|last2=Bitsko|first2=Rebecca H.|last3=Holbrook|first3=Joseph R.|last4=Charania|first4=Sana N.|last5=Claussen|first5=Angelika H.|last6=McKeown|first6=Robert E.|last7=Cuffe|first7=Steven P.|last8=Owens|first8=Julie Sarno|last9=Evans|first9=Steven W.|date=2021-06-01|title=Community-Based Prevalence of Externalizing and Internalizing Disorders among School-Aged Children and Adolescents in Four Geographically Dispersed School Districts in the United States|url=https://doi.org/10.1007/s10578-020-01027-z|journal=Child Psychiatry & Human Development|language=en|volume=52|issue=3|pages=500–514|doi=10.1007/s10578-020-01027-z|issn=1573-3327|pmc=PMC8016018|pmid=32734339}}</ref> |6.8% |Diagnostic Interview Schedule for Children (DISC) |- |Urban and suburban Florida |Project to Learn about Youth-Mental health, school-based study for children from kindergarten to high-school (n = 289)<ref name=":1" /> |6.9% |Diagnostic Interview Schedule for Children (DISC) |- |Rural and suburban Ohio |Project to Learn about Youth-Mental health, school-based study for children from elementary to high-school (n = 152)<ref name=":1" /> |17.3% |Diagnostic Interview Schedule for Children (DISC) |- |Suburban and rural South Carolina |Project to Learn about Youth-Mental health, school-based study for children from elementary to high-school (n = 270)<ref name=":1" /> |5.7% |Diagnostic Interview Schedule for Children (DISC) |- || Semi-rural North Carolina | Preschool-aged children from pediatric practices (N = 306; age 2 - 5 years old)<ref>Egger, H.L., & Angold, A. (2006). Common emotional and behavioral disorders in preschool children: Presentation, nosology, and epidemiology. ''Journal of Child Psychiatry and Psychology, 47'', 313–337.</ref> || 6.6% || Preschool Age Psychiatric Assessment (PAPA)^p |- || Western North Carolina | The Great Smoky Mountains Study - longitudinal, population-based study of community sample<ref>Costello, E. J., Mustillo, S., Erkanli, A., Keeler, G., & Angold, A. (2003) Prevalence and development of psychiatric disorders in adolescence. ''Arch Gen Psychiatry, 60'', 837-844.</ref> || 2.33% (overall) 3.16% (males) 2.75% (females) || Child and Adolescent Psychiatric Assessment (CAPA)^{p, y} |- || Chicago | Preschool-aged children from inner city schools and pediatric practices (N = 796; age 2 - 5 years old)<ref>Lavigne, J. V., LeBailly, S. A., Hopkins, J., Gouze, K. R., & Binns, H. J. (2009). The prevalence of ADHD, ODD, depression, and anxiety in a community sample of 4-year-olds. ''Journal Of Clinical Child And Adolescent Psychology, 38''(3), 315-328. doi:10.1080/15374410902851382</ref> || 8.3% || Diagnostic Interview Schedule for Children–Parent Scale–Young Child Version (DISC-YC) ^p |- |Germany (Saarbrücken County) |All school-aged children were examined during a routine school-entry medical examination (N = 1676, mean age = 5.7)<ref>{{Cite journal|last=Niemczyk|first=Justine|last2=Equit|first2=Monika|last3=Braun-Bither|first3=Katrin|last4=Klein|first4=Anna-Maria|last5=von Gontard|first5=Alexander|date=2015-07-01|title=Prevalence of incontinence, attention deficit/hyperactivity disorder and oppositional defiant disorder in preschool children|url=https://doi.org/10.1007/s00787-014-0628-6|journal=European Child & Adolescent Psychiatry|language=en|volume=24|issue=7|pages=837–843|doi=10.1007/s00787-014-0628-6|issn=1435-165X}}</ref> |7.3% (males) 5.1%(females) |DISYPS-II |- |South Korea (Seoul) |Children were randomly surveyed across 6 school districts in Seoul (N = 1645, age 6 - 12 years old) |5.8% (males) 4.1%(females) |Diagnostic Interview Schedule for Children–Parent Scale IV (DISC-IV) |} ^p Parent interviewed as part of diagnostic assessment; ^y youth interviewed as part of diagnostic assessment, ^r adult interviewed for retrospective report as part of diagnostic assessment '''Note:''' Mash and Barkley note that prevalence rates of ODD must be qualified, because the definition of ODD has changed at a fast rate, the rates adolescents meeting criteria in any cross-sectional evaluation may be misleading because of the developmental progressions with and between ODD and Conduct Disorder, and categorical definitions of aggressive patterns may reflect arbitrary numbers of constituent estimates. These factors may lead to misleading prevalence rates. In addition, few studies have investigated the prevalence of ODD in preschool-aged children, and early onset of these behaviors is associated with more severe and stable impairment.<ref>Mash, E., & Barkley, R. (Eds.). (2003). ''Child Psychopathology''. 2nd Edition. New York: Guilford Press.</ref> ==[[Evidence based assessment/Prediction phase|'''Prediction phase''']]== The following section contains a list of screening and diagnostic instruments for ODD. The section includes administration information, psychometric data, and PDFs or links to the screenings. * Screenings are used as part of the [[Evidence based assessment/Prediction phase|prediction phase]] of assessment; for more information on interpretation of this data, or how screenings fit in to the assessment process, click [[Evidence based assessment/Prediction phase|here]]. * '''''For a list of more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&wteswitched=1#Psychometric_properties_of_common_screening_instruments click here.]''''' {| class="wikitable sortable" border="1" ! colspan="5" |Screening measures for ODD |- ! Measure ! Format (Reporter) ! Age Range ! Administration/ Completion Time !Where to access |- |Achenbach System of Empirically Based Assessments (ASEBA): Child Behavior Checklist (CBCL), Youth Self-Report (YSR) |Parent report (CBCL) Youth self-report (YSR) |6-18 (CBCL) 11-18 (YSR) <ref name=":9">{{Cite book|url=https://www.worldcat.org/oclc/1130319849|title=Assessment of disorders in childhood and adolescence|date=2020|others=Eric Arden Youngstrom, Mitchell J. Prinstein, Eric J. Mash, Russell A. Barkley|isbn=978-1-4625-4363-2|edition=Fifth edition|location=New York, NY|oclc=1130319849}}</ref> |10 - 15 minutes<ref name=":9" /> |[https://store.aseba.org/School-Age-6-18-Materials/departments/11/ Purchase] |- |Eyberg Child Behavior Inventory (ECBI) |Parent report |2-16 |5 minutes |[https://www.parinc.com/products/pkey/97 Purchase] |- |Strengths and Difficulties Questionnaire (SDQ) |Parent report |3 - 16 |3 - 5 minutes |[https://www.sdqinfo.org/a0.html SDQ Homepage][https://www.sdqinfo.org/py/sdqinfo/b0.py PDFs] |} '''Note:''' L = Less than adequate; A = Adequate; G = Good; E = Excellent; U = Unavailable; NA = Not applicable === Likelihood ratios and AUCs of screening instruments for ODD === * '''''For a list of the likelihood ratios for more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&wteswitched=1#Likelihood_ratios_and_AUCs_of_common_screening_instruments click here.]''''' {| class="wikitable sortable" border="1" ! Screening Measure (Primary Reference) ! Area Under Curve (AUC) ! LR+ (Score) ! LR- (Score) ! Clinical Generalizability !Where to Access |- |CBCL DSM-Oriented Scales<ref name=":7">Achenbach, T. M. (1991a). ''Manual for the Child Behavior Checklist/4–18 and 1991 Profile''. Burlington , VT : University of Vermont Department of Psychiatry.</ref> |.71 (N=475)<ref name=":3">Ebesutani, C., Bernstein, A., Nakamura, B. J., Chorpita, B. F., Higa-McMillan, C. K., & Weisz, J. R. (2010). Concurrent validity of the child behavior checklist DSM-oriented scales: Correspondence with DSM diagnoses and comparison to syndrome scales. ''Journal of Psychopathology and Behavioral Assessment, 32''(3), 373-384.</ref> | 2.80 (60+ to 70+) <ref name=":4">Warnick, E. M., Bracken, M. B., & Kasl, S. (2008). Screening efficiency of the Child Behavior Checklist and Strengths and Difficulties Questionnaire: a systematic review. ''Child and Adolescent Mental Health, 13''(3), 140-147.</ref> | .52 (60- to 70-)*<ref name=":4" /> |Youth aged 5 - 18 seeking out patient treatment across a variety of settings<ref name=":4" /> |[https://store.aseba.org/School-Age-6-18-Materials/departments/11/ Purchase] |- |SDQ<ref name=":6">{{Cite journal|last=Shabani|first=Amir|last2=Masoumian|first2=Samira|last3=Zamirinejad|first3=Somayeh|last4=Hejri|first4=Maryam|last5=Pirmorad|first5=Tahereh|last6=Yaghmaeezadeh|first6=Hooman|date=2021-05|title=Psychometric properties of Structured Clinical Interview for DSM‐5 Disorders‐Clinician Version (SCID‐5‐CV)|url=https://onlinelibrary.wiley.com/doi/10.1002/brb3.1894|journal=Brain and Behavior|language=en|volume=11|issue=5|doi=10.1002/brb3.1894|issn=2162-3279|pmc=PMC8119811|pmid=33729681}}</ref> | .81 -.88 (N = 18,416)<ref>{{Cite journal|last=Algorta|first=Guillermo Perez|last2=Dodd|first2=Alyson Lamont|last3=Stringaris|first3=Argyris|last4=Youngstrom|first4=Eric A.|date=2016-09|title=Diagnostic efficiency of the SDQ for parents to identify ADHD in the UK: a ROC analysis|url=http://link.springer.com/10.1007/s00787-015-0815-0|journal=European Child & Adolescent Psychiatry|language=en|volume=25|issue=9|pages=949–957|doi=10.1007/s00787-015-0815-0|issn=1018-8827|pmc=PMC4990620|pmid=26762184}}</ref> |7.55 (3+)<ref name=":2">{{Cite journal|last=Goodman|first=Robert|date=2001-11-01|title=Psychometric Properties of the Strengths and Difficulties Questionnaire|url=https://www.jaacap.org/article/S0890-8567(09)60543-8/abstract|journal=Journal of the American Academy of Child & Adolescent Psychiatry|language=English|volume=40|issue=11|pages=1337–1345|doi=10.1097/00004583-200111000-00015|issn=0890-8567|pmid=11699809}}</ref> |.35 (3-)<ref name=":2" /> |Surveyed youth aged 5 - 15 in the UK <ref name=":2" /> |[https://www.sdqinfo.org/a0.html SDQ Homepage][https://www.sdqinfo.org/py/sdqinfo/b0.py PDFs] |- |ECBI- Intensity Scale<ref name=":8">Eyberg, S. M., & Robinson, E. A. (1983). Conduct problem behavior: Standardization of a behavioral rating scale with adolescents. Journal of Clinical Child Psychology, 12 (3), 347-354.</ref> | |6.92 (131+)<ref name=":5">{{Cite journal|last=LYNEHAM|first=HEIDI J.|last2=ABBOTT|first2=MAREE J.|last3=RAPEE|first3=RONALD M.|date=2007-06|title=Interrater Reliability of the Anxiety Disorders Interview Schedule for DSM-IV: Child and Parent Version|url=https://doi.org/10.1097/chi.0b013e3180465a09|journal=Journal of the American Academy of Child &amp; Adolescent Psychiatry|volume=46|issue=6|pages=731–736|doi=10.1097/chi.0b013e3180465a09|issn=0890-8567}}</ref> |.11 (131-)<ref name=":5" /> |Youth aged 7-16 had responses compared to diagnosis<ref name=":5" /> |[https://www.parinc.com/products/pkey/97 Purchase] |} '''Note:''' “LR+” refers to the change in likelihood ratio associated with a positive test score, and “LR-” is the likelihood ratio for a low score. Likelihood ratios of 1 indicate that the test result did not change impressions at all. LRs larger than 10 or smaller than .10 are frequently clinically decisive; 5 or .20 are helpful, and between 2.0 and .5 are small enough that they rarely result in clinically meaningful changes of formulation<ref>Sackett, D. L., Straus, S. E., Richardson, W. S., Rosenberg, W., & Haynes, R. B. (2000). Evidence-based medicine: How to practice and teach EBM. Edinburgh: Churchill Livingstone.</ref>. '''Search terms:''' [Oppositional Defiant Disorder] AND [sensitivity OR specificity] in GoogleScholar and PsychINFO; === Interpreting ODD screening measure scores === * For information on interpreting screening measure scores, click [[Evidence based assessment/Prediction phase#Interpreting screening measure scores|here.]] == [[Evidence based assessment/Prescription phase|'''Prescription phase''']] == === Gold standard diagnostic interviews === * For a list of broad reaching diagnostic interviews sortable by disorder with PDFs (if applicable), [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prescription_phase&wteswitched=1#Common_Diagnostic_Interviews click here.] === Recommended diagnostic instruments for ODD === {| class="wikitable sortable" ! colspan="5" |Diagnostic instruments for ODD |- !Measure !Format (Reporter) !Age Range !Administration/ Completion Time !Where to Access |- |Kiddie Schedule for Affective Disorders and Schizophrenia Present and Lifetime Version (KSADS-PL) |Structured interview |6-28 |45-75 minutes |[https://www.pediatricbipolar.pitt.edu/resources/instruments Website to access] |- |Diagnostic Interview Schedule for Children (DISC-5) |Structured Interview (Self report and parent) |6-17 | |[https://telesage.com/netdisc-5/# Coming soon] |} '''Note:''' L = Less than adequate; A = Adequate; G = Good; E = Excellent; U = Unavailable; NA = Not applicable == [[Evidence based assessment/Process phase|'''Process phase''']] == The following section contains a list of process and outcome measures for oppositional defiant disorder. The section includes benchmarks based on published norms for several outcome and severity measures, as well as information about commonly used process measures. Process and outcome measures are used as part of the [[Evidence based assessment/Process phase|process phase]] of assessment. For more information of differences between process and outcome measures, see the page on the [[Evidence based assessment/Process phase|process phase of assessment]]. === Outcome and severity measures === * This table includes clinically significant benchmarks for generalized anxiety disorder specific outcome measures * Information on how to interpret this table can be [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase found here]. * Additionally, these [[Evidence based assessment/Vignettes|vignettes]] might be helpful resources for understanding appropriate adaptation of outcome measures in practice. * ''<u>For clinically significant change benchmarks for the CBCL, YSR, and TRF total, externalizing, internalizing, and attention benchmarks,</u>'' [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase&wteswitched=1#Clinically_significant_change_benchmarks_for_widely-used_outcome_measures see here.] {| class="wikitable sortable" border="1" |- | rowspan="1&quot;" style="text-align:center;font-size:130%;" | <b> Measure</b> | style="text-align:center;font-size:130%;" | <b> Subscale</b> | colspan="3" style="text-align:center;font-size:130%" width="300" | <b> Cut-off scores</b> | colspan="3" style="text-align:center;font-size:120%" | <b> Critical Change <br> (unstandardized scores)</b> |- | colspan="8" style="font-size:110%; text-align:center;" span | <b> Benchmarks Based on Published Norms</b> |- | colspan="2" | | style="text-align:center;font-size:110%" | <b> A</b> | style="text-align:center;font-size:110%" | <b> B</b> | style="text-align:center;font-size:110%" | <b> C</b> | style="text-align:center;font-size:110%" | <b> 95%</b> | style="text-align:center;font-size:110%" | <b> 90%</b> | style="text-align:center;font-size:110%" | <b> SE_difference</b> |- | rowspan="1" style="text-align:center;" | <b> CBCL T-scores <br> (2001 Norms)</b> | style="text-align:right;" | <i> Externalizing</i> | style="text-align:center;" | 49 | style="text-align:center;" | 70 | style="text-align:center;" | 58 | style="text-align:center;" | 7 | style="text-align:center;" | 6 | style="text-align:center;" | 3.4 |- | rowspan="1" style="text-align:center;" | <b> ECBI Scaled Scores <br> (1983 Norms)</b> | style="text-align:right;" | <i> Intensity</i> | style="text-align:center;" | 80.1 | style="text-align:center;" | 169.5 | style="text-align:center;" | 112.9 | style="text-align:center;" | 9.5 | style="text-align:center;" | 8 | style="text-align:center;" | 4.8 |- | rowspan="1" style="text-align:center;" | <b> ECBI Scaled Scores <br> (1983 Norms)</b> | style="text-align:right;" | <i> Problem</i> | style="text-align:center;" | 3.9 | style="text-align:center;" | 17.7 | style="text-align:center;" | 11.5 | style="text-align:center;" | 2.1 | style="text-align:center;" | 1.8 | style="text-align:center;" | 1.1 |} '''Note:''' “A” = Away from the clinical range, “B” = Back into the nonclinical range, “C” = Closer to the nonclinical than clinical mean. === Process measures=== See Section 1.1 for overview of evidence-based measures to use depending on etiology and symptomatology of Oppositional Defiant Disorder. ==Treatment== ===Behavioral parent training=== Behavioral Parent Training is considered the most effective treatment for childhood disruptive behavior disorders (e.g., Oppositional Defiant Disorder), especially for younger children (i.e., 3-8 year-olds). See http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/rule-breaking-defiance-and-acting-out/ a website sponsored by The Society for Child and Adolescent Psychology (APA, Division 53) and the Association for Behavioral and Cognitive Therapies (ABCT), for current summary of evidence-based treatments. ===Overview of recommendations for assessment and treatment=== See the [https://pathways.nice.org.uk/pathways/antisocial-behaviour-and-conduct-disorders-in-children-and-young-people#content=view-info-category%3Aview-about-menu National Institute for Health and Care Excellence (NICE) Practice Guidelines for Childhood Conduct Disorders], for an overview of recommendations for both assessment and treatment of Oppositional Defiant Disorder. == External Links == *[https://sccap53.org Society of Clinical Child and Adolescent Psychology] *http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/rule-breaking-defiance-and-acting-out/ == References == {{collapse top|Click here for references}} {{Reflist|30em}} [[Category:Psychological disorder portfolios|{{SUBPAGENAME}}]] {{collapse bottom}} [[Category:Psychological disorder portfolios|{{SUBPAGENAME}}]] bxoq3oirrpbrtb44lwjqhl93xmfgcad 0 207113 2410380 2410050 2022-07-30T02:54:20Z Maddiegray11 2936309 /* Recommended diagnostic interviews for substance use disorder */ took out ICD as a diagnostic interview wikitext text/x-wiki <noinclude>{{Helping Give Away Psychological Science Banner}}</noinclude> {{medical disclaimer}} {{:{{BASEPAGENAME}}/Sidebar}} ==[[Evidence based assessment/Portfolio template/What is a "portfolio"|'''What is a "portfolio?"''']]== For background information on what assessment portfolios are, click the link in the heading above. Want even 'more' information about this topic? There's an extended version of this page [[Evidence-based assessment/Substance use disorder (disorder portfolio)/extended version|here]]. == [[Evidence-based assessment/Preparation phase|'''Preparation Phase''']] == {{blockquotetop}} <big>'''Diagnostic Criteria for Substance Use Disorder'''</big> <big>'''ICD-11 Diagnostic Criteria for Substance Use Disorder'''</big> *Disorders due to substance use and addictive behaviours are mental and behavioural disorders that develop as a result of the use of predominantly psychoactive substances, including medications, or specific repetitive rewarding and reinforcing behaviours. **Note: The ICD-11 lists 20 additional subcategories of Substance Use Disorder. They can be found [https://icd.who.int/browse11/l-m/en#/http://id.who.int/icd/entity/1602669465 here]. <big>'''DSM Diagnostic Criteria for Substance Use Disorder'''</big> *Substance use disorder is a DSM disorder in the Substance-Related and Addictive Disorders chapter. It is characterized by the use of substances in a manner that leads to clinically significant impairment or distress. * The diagnostic criteria for Substance Use Disorder disorder changed slightly from [[DSM-IV]] to [[w:Diagnostic_and_Statistical_Manual_of_Mental_Disorders#DSM-IV-TR_.282000.29|DSM-5]]. Summaries are available [http://www.dsm5.org/Documents/changes%20from%20dsm-iv-tr%20to%20dsm-5.pdf here]. {{blockquotebottom}} === Base rates of SUD in different populations and clinical settings === {| class="wikitable sortable" border="1" |- ! Setting ! Base Rate ! Demography ! Diagnostic Method |- | General population of North Carolina, aged 12 or older | 6.7% | North Carolina | National Survey on Drug Use and Health (NSDUH), 2009 to 2013 |- | 43,093 individuals, 18+ years old collected between 2001 and 2002<ref>Hasin DS, Grant BF. The National Epidemiologic Survey on Alcohol and Related Conditions (NESARC) Waves 1 and 2: review and summary of findings. Soc Psychiatry Psychiatr Epidemiol. 2015 Nov;50(11):1609-40. doi: 10.1007/s00127-015-1088-0. Epub 2015 Jul 26. PMID: 26210739.</ref> | 17.8 (0.5) Alcohol Abuse; 12.5 (0.4) Alcohol Dependence; 7.7 (0.2) Drug Abuse; 2.6 (0.1) Drug Dependence | United States General Adult Population: National Epidemiologic Survey on Alcohol and Related Conditions (NESARC) (Grant et al., 2007) | National Institute on Alcohol Abuse and Alcoholism Alcohol Use Disorder and Associated Disabilities Interview Schedule- DSM IV Version (AUDADIS-IV) |- | 9,282 adults, 18+ years old ; collected between 2001 and 2003<ref>{{cite journal|last1=Kessler|first1=RC|last2=Green|first2=JG|last3=Gruber|first3=MJ|last4=Sampson|first4=NA|last5=Bromet|first5=E|last6=Cuitan|first6=M|last7=Furukawa|first7=TA|last8=Gureje|first8=O|last9=Hinkov|first9=H|date=June 2010|title=Screening for serious mental illness in the general population with the K6 screening scale: results from the WHO World Mental Health (WMH) survey initiative.|journal=International journal of methods in psychiatric research|volume=19 Suppl 1|pages=4-22|pmid=20527002|last10=Hu|first10=CY|last11=Lara|first11=C|last12=Lee|first12=S|last13=Mneimneh|first13=Z|last14=Myer|first14=L|last15=Oakley-Browne|first15=M|last16=Posada-Villa|first16=J|last17=Sagar|first17=R|last18=Viana|first18=MC|last19=Zaslavsky|first19=AM}}</ref> | 13.2 (0.6) Alcohol Abuse; 5.4 (0.3) Alcohol Dependence; 7.9 (0.4) Drug Abuse; 3.0 (0.2) Drug Dependence | United States General Adult Population: National Comorbidity Survey Replication (NCS-R) | World Mental Health Survey Initiative Version of the World Health Organization Composite International Diagnostic Interview (WMH-CIDI) which generates DSM-IV and International Classification of Diseases, 10th revision diagnoses |- | Urban General Medicine Practice, low-income primary care patients, 75% Hispanic<ref>{{cite journal|last1=Olfson|first1=M|last2=Shea|first2=S|last3=Feder|first3=A|last4=Fuentes|first4=M|last5=Nomura|first5=Y|last6=Gameroff|first6=M|last7=Weissman|first7=MM|date=NaN|title=Prevalence of anxiety, depression, and substance use disorders in an urban general medicine practice.|journal=Archives of family medicine|volume=9|issue=9|pages=876-83|pmid=11031395}}</ref> | 7.9% | New York | Patient Health Questionnaire |- | Incarcerated females<ref>{{cite journal|last1=Teplin|first1=LA|last2=Abram|first2=KM|last3=McClelland|first3=GM|date=June 1996|title=Prevalence of psychiatric disorders among incarcerated women. I. Pretrial jail detainees.|journal=Archives of general psychiatry|volume=53|issue=6|pages=505-12|pmid=8639033}}</ref> | 70.2% | Chicago prison - 40 % African American, 33% White, 25 % Hispanic | National Institute of Mental Health Diagnostic Interview Schedule Version 11I-R (NIMH DIS-III-R) |- |Incarcerated females (updated) (Proctor 2012) | 70% dependent | Minnesota State Prison System- 801 females, 18-58 years old, 57.7% Caucasian, 21.5% African American, 13.2% Native American |Substance Use Disorder Diagnostic Schedule-IV (SUDDS-IV) |- | Incarcerated male youths<ref>{{cite journal|last1=Wasserman|first1=GA|last2=McReynolds|first2=LS|last3=Lucas|first3=CP|last4=Fisher|first4=P|last5=Santos|first5=L|date=March 2002|title=The voice DISC-IV with incarcerated male youths: prevalence of disorder.|journal=Journal of the American Academy of Child and Adolescent Psychiatry|volume=41|issue=3|pages=314-21|pmid=11886026}}</ref> | 56.4% | Texas state prison – 45 % African American, 33% White, 20% Hispanic |Structured Clinical Interview for DSM IV – Substance Use Disorders Module |- | Individuals with schizophrenia across settings<ref>{{cite journal|last1=Regier|first1=DA|last2=Farmer|first2=ME|last3=Rae|first3=DS|last4=Locke|first4=BZ|last5=Keith|first5=SJ|last6=Judd|first6=LL|last7=Goodwin|first7=FK|date=21 November 1990|title=Comorbidity of mental disorders with alcohol and other drug abuse. Results from the Epidemiologic Catchment Area (ECA) Study.|journal=JAMA|volume=264|issue=19|pages=2511-8|pmid=2232018}}</ref> | 47% | New Haven, CT; Baltimore, MD; St. Louis, MO; Durham, NC; Los Angeles, CA | National Institute of Mental Health (NIMH) Diagnostic Interview Schedule |- | HIV+ men in community health clinics<ref>{{cite journal|last1=Dew|first1=MA|last2=Becker|first2=JT|last3=Sanchez|first3=J|last4=Caldararo|first4=R|last5=Lopez|first5=OL|last6=Wess|first6=J|last7=Dorst|first7=SK|last8=Banks|first8=G|date=March 1997|title=Prevalence and predictors of depressive, anxiety and substance use disorders in HIV-infected and uninfected men: a longitudinal evaluation.|journal=Psychological medicine|volume=27|issue=2|pages=395-409|pmid=9089832}}</ref> | 24.4% | Alleghany County, PA | Structured Clinical Interview for DSM-III-R |- | Internal medicine inpatients<ref>{{cite journal|last1=Hansen|first1=MS|last2=Fink|first2=P|last3=Frydenberg|first3=M|last4=Oxhøj|first4=M|last5=Søndergaard|first5=L|last6=Munk-Jørgensen|first6=P|date=April 2001|title=Mental disorders among internal medical inpatients: prevalence, detection, and treatment status.|journal=Journal of psychosomatic research|volume=50|issue=4|pages=199-204|pmid=11369025}}</ref> | 10.9% | Denmark | Symptom Check List (SCL-8) |} == [[Evidence based assessment/Prediction phase|'''Prediction phase''']] == === Psychometric properties of screening measures for Substance Use Disorder === The following section contains a list of screening and diagnostic instruments for Substance Use Disorder. === Screening instruments and diagnostic interviews === {| class="wikitable sortable" |- ! Measure !Format (Reporter) !Age Range !Administration/ Completion Time !Where to Access |- |Structured Clinical International Diagnostic Interview (SCID-I) |Interview |Adults |30 minutes-3 hours |https://osf.io/x9smc |- |[https://osf.io/h2n9j/?view_only=ec71313a9e844abb977e241b5443f0db Substance Dependence Severity Scale (SDSS)] |Interview |16-adult |30-45 minutes | |- |[https://osf.io/7dh4s Global Appraisal of Individual Needs- Initial (GAIN-I) ($1.00 license fee per project for use of Beta version)] |Interview |12-adult |1.5-2.5 hours | |} '''Note:''' Reliability and validity are included in the extended version [[Evidence-based assessment/Substance use disorder (disorder portfolio)/extended version|here]]. This table includes measures with Good or Excellent ratings. === Likelihood ratios and AUCs of screening measures for '''(insert portfolio name)''' === * '''''For a list of the likelihood ratios for more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&wteswitched=1#Likelihood_ratios_and_AUCs_of_common_screening_instruments click here.]''''' {| class="wikitable sortable" border="1" |- ! Screening Measure (Primary Reference) ! AUC ! LR+ (Score) ! LR- (Score) ! Clinical Generalizability |- | [https://osf.io/gq86m/?view_only=628c7440cb5e4cbead514916f342a8cf Kessler 6 Screening Scale (K6)] <ref>{{cite journal|last1=Kessler|first1=RC|last2=Green|first2=JG|last3=Gruber|first3=MJ|last4=Sampson|first4=NA|last5=Bromet|first5=E|last6=Cuitan|first6=M|last7=Furukawa|first7=TA|last8=Gureje|first8=O|last9=Hinkov|first9=H|last10=Hu|first10=CY|last11=Lara|first11=C|last12=Lee|first12=S|last13=Mneimneh|first13=Z|last14=Myer|first14=L|last15=Oakley-Browne|first15=M|last16=Posada-Villa|first16=J|last17=Sagar|first17=R|last18=Viana|first18=MC|last19=Zaslavsky|first19=AM|title=Screening for serious mental illness in the general population with the K6 screening scale: results from the WHO World Mental Health (WMH) survey initiative.|journal=International journal of methods in psychiatric research|date=June 2010|volume=19 Suppl 1|pages=4-22|pmid=20527002}}</ref><ref>Swartz, J. A., & Lurigio, A. J. (2006). Screening for serious mental illness in populations with co-occurring substance use disorders: Performance of the K6 scale. Journal of substance abuse treatment, 31(3), 287-296</ref> | 0.84 (N=41,770) | 3.96 (13+) | 0.296 (0-12) | High: The sample of 41,770 was drawn from initial surveys that were carried out in 14 countries. |- | [https://osf.io/czhgd Alcohol, Smoking and Substance Involvement Screening Test (ASSIST)]<ref>{{cite journal|last1=Humeniuk|first1=R|last2=Ali|first2=R|last3=Babor|first3=TF|last4=Farrell|first4=M|last5=Formigoni|first5=ML|last6=Jittiwutikarn|first6=J|last7=de Lacerda|first7=RB|last8=Ling|first8=W|last9=Marsden|first9=J|last10=Monteiro|first10=M|last11=Nhiwatiwa|first11=S|last12=Pal|first12=H|last13=Poznyak|first13=V|last14=Simon|first14=S|title=Validation of the Alcohol, Smoking And Substance Involvement Screening Test (ASSIST).|journal=Addiction (Abingdon, England)|date=June 2008|volume=103|issue=6|pages=1039-47|pmid=18373724}}</ref> | 0.84 (N=1,047) | 2.76 | 0.28 | High: The sample of 1,047 participants was drawn from drug treatment and primary health care settings in Australia, Brazil, India, Thailand, the United Kingdom, the U.S. and Zimbabwe. |- | [https://osf.io/nk5vx/?view_only=348d7f1ee5e741f8a6657233403dda66 Drug Use Disorders Identification Test (DUDIT)]<ref>{{Cite journal|last=Voluse|first=Andrew C.|last2=Gioia|first2=Christopher J.|last3=Sobell|first3=Linda Carter|last4=Dum|first4=Mariam|last5=Sobell|first5=Mark B.|last6=Simco|first6=Edward R.|title=Psychometric properties of the Drug Use Disorders Identification Test (DUDIT) with substance abusers in outpatient and residential treatment|url=https://doi.org/10.1016/j.addbeh.2011.07.030|journal=Addictive Behaviors|volume=37|issue=1|pages=36–41|doi=10.1016/j.addbeh.2011.07.030}}</ref> |0.95 (N=153) | 6 |0.12 | High: 153 participants from outpatient and residential substance use treatment programs |} '''Note:''' “LR+” refers to the change in likelihood ratio associated with a positive test score, and “LR-” is the likelihood ratio for a low score. Likelihood ratios of 1 indicate that the test result did not change impressions at all. LRs larger than 10 or smaller than 0.1 are frequently clinically decisive; 5 or 0.2 are helpful, and between 2.0 and 0.5 are small enough that they rarely result in clinically meaningful changes of formulation (Sackett et al., 2000). ''' Search terms''': [substance use OR substance use disorders] AND [sensitivity OR specificity] in Google Scholar and PsycINFO =='''[[Evidence-based assessment/Prescription phase|Prescription phase]]'''== ===Gold standard diagnostic interviews=== * For a list of broad reaching diagnostic interviews sortable by disorder with PDFs (if applicable), [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prescription_phase&wteswitched=1#Common_Diagnostic_Interviews click here.] ===Recommended diagnostic interviews for substance use disorder=== {| class="wikitable sortable" border="1" ! colspan="5" |Diagnostic instruments for '''(insert portfolio name)''' |- ! Measure ! Format (Reporter) ! Age Range ! Administration/ Completion Time !Where to Access |- | Diagnostic Interview Schedule- V (DIS) | | | | |- | Structured Clinical Interview for DSM-V (SCID) | | | | |- | The Psychiatric Research Interview for Substance and Mental Disorders | | | | |- |The Mini International Psychiatric Interview (M.I.N.I)<ref name="Sheehan1998" /> | | | | |} '''Note:''' Reliability and validity are included in the extended version (link). This table includes measures with Good or Excellent ratings. == '''[[Evidence-based assessment/Process phase|Process phase]]''' == === Outcome and severity measures === This table includes clinically significant benchmarks for '''(insert portfolio name here)''' specific outcome measures * Information on how to interpret this table can be [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase found here]. * Additionally, these [[Evidence based assessment/Vignettes|vignettes]] might be helpful resources for understanding appropriate adaptation of outcome measures in practice. *''<u>For clinically significant change benchmarks for the CBCL, YSR, and TRF total, externalizing, internalizing, and attention benchmarks,</u>'' [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase&wteswitched=1#Clinically_significant_change_benchmarks_for_widely-used_outcome_measures see here.] Clinically significant change benchmarks with common instruments and mood rating scales {| class="wikitable sortable" border="1" | colspan="7" | ==== '''Clinically significant change benchmarks with common instruments and mood rating scales''' ==== |- | style="text-align:center;font-size:120%" width="300" | | colspan="3" style="text-align:center;font-size:120%" width="300" | <b> Cut* Scores</b> | colspan="3" style="text-align:center;font-size:120%" | <b> Critical Change <br> (Unstandardized Scores)</b> |- | style="text-align:center;font-size:130%;" | <b> Measure</b> | style="text-align:center;font-size:110%" | <b> A</b> | style="text-align:center;font-size:110%" | <b> B</b> | style="text-align:center;font-size:110%" | <b> C</b> | style="text-align:center;font-size:110%" | <b> 95%</b> | style="text-align:center;font-size:110%" | <b> 90%</b> | style="text-align:center;font-size:110%" | <b> SE_difference</b> |- | colspan="7" style="font-size:110%; text-align:center;" span | <b> Benchmarks Based on Published Norms</b> |- | style="text-align:center;font-size:100%;" | ''[https://osf.io/wn3rb Rutgers Alcohol Problem Index]''<ref name="Roberts">{{cite journal|last1=Roberts|first1=LJ|last2=Neal|first2=DJ|last3=Kivlahan|first3=DR|last4=Baer|first4=JS|last5=Marlatt|first5=GA|title=Individual drinking changes following a brief intervention among college students: clinical significance in an indicated preventive context.|journal=Journal of consulting and clinical psychology|date=June 2000|volume=68|issue=3|pages=500-5|pmid=10883566}}</ref> <ref name="White1989">{{cite journal|last1=White|first1=HR|last2=Labouvie|first2=EW|title=Towards the assessment of adolescent problem drinking.|journal=Journal of studies on alcohol|date=January 1989|volume=50|issue=1|pages=30-7|pmid=2927120}}</ref> | style="text-align:center;font-size:100%;" span |0.8 | style="text-align:center;font-size:100%;" span | 4.9 | style="text-align:center;font-size:100%;" span | 4.0 | style="text-align:center;font-size:100%;" span | 4.1 | style="text-align:center;font-size:100%;" span | 3.5 | style="text-align:center;font-size:100%;" span | 2.1 |- | style="text-align:center;font-size:100%;" | ''[https://osf.io/vudep Alcohol Dependence Scale (ADS)]''<ref name="Roberts" /> [https://osf.io/vudep (copyrighted)] | style="text-align:center;font-size:100%;" span | 1.2 | style="text-align:center;font-size:100%;" span | 9.9 | style="text-align:center;font-size:100%;" span | 7.8 | style="text-align:center;font-size:100%;" span | 1.4 | style="text-align:center;font-size:100%;" span | 1.2 | style="text-align:center;font-size:100%;" span | 0.7 |- | style="text-align:center;font-size:100%;" | ''[https://osf.io/23mwt/?view_only=1924fe3e90334b11bb1e099a691b2ffb Drug Abuse Screening Test (DAST)]'' <ref>{{cite journal|last1=Skinner|first1=HA|title=The drug abuse screening test.|journal=Addictive behaviors|date=1982|volume=7|issue=4|pages=363-71|pmid=7183189}}</ref> | style="text-align:center;font-size:100%;" span | 0.1 | style="text-align:center;font-size:100%;" span | 2.6 | style="text-align:center;font-size:100%;" span | 1.8 | style="text-align:center;font-size:100%;" span | 1.6 | style="text-align:center;font-size:100%;" span | 1.3 | style="text-align:center;font-size:100%;" span | 0.8 |} '''Note:''' "A" = Away from the clinical range, "B" = Back into the nonclinical range, "C" = Closer to the nonclinical than clinical mean. '''Search terms''': [substance use OR substance use disorder] AND [clinical significance OR outcomes] in Google Scholar and PsycINFO === Treatment === In the United States, according to SAMHSA, of the 8.9 million adults with a dual diagnosis, 44% received some form of treatment in the past year. Given the frequent co-occurrence of mood disorders and substance use disorders, the recommended first step in treatment is for clinicians to deliver a comprehensive screening evaluation that will inform their treatment approach. There are a host of empirically supported treatments for substance use disorders, though medication interventions and psychotherapy are most common. ==== Medication ==== Specifically, medications have been shown to be most effective in the treatment of alcohol and opioid dependence. Naltrexone (50 mg/day) administered for 12 weeks has been shown to decrease cravings for alcohol and the number of days in which alcohol was consumed.<ref>{{cite journal|last1=Volpicelli|first1=JR|last2=Alterman|first2=AI|last3=Hayashida|first3=M|last4=O'Brien|first4=CP|date=November 1992|title=Naltrexone in the treatment of alcohol dependence.|journal=Archives of general psychiatry|volume=49|issue=11|pages=876-80|pmid=1345133}}</ref> Disulfiram (250 mg/day), administered for one year, has been shown to help reduce drinking frequency after relapse.<ref>{{cite journal|last1=Chick|first1=J|last2=Gough|first2=K|last3=Falkowski|first3=W|last4=Kershaw|first4=P|last5=Hore|first5=B|last6=Mehta|first6=B|last7=Ritson|first7=B|last8=Ropner|first8=R|last9=Torley|first9=D|date=July 1992|title=Disulfiram treatment of alcoholism.|journal=The British journal of psychiatry : the journal of mental science|volume=161|pages=84-9|pmid=1638335}}</ref><ref name="Fuller1986">{{cite journal|last1=Fuller|first1=RK|last2=Branchey|first2=L|last3=Brightwell|first3=DR|last4=Derman|first4=RM|last5=Emrick|first5=CD|last6=Iber|first6=FL|last7=James|first7=KE|last8=Lacoursiere|first8=RB|last9=Lee|first9=KK|date=19 September 1986|title=Disulfiram treatment of alcoholism. A Veterans Administration cooperative study.|journal=JAMA|volume=256|issue=11|pages=1449-55|pmid=3528541|last10=Lowenstam|first10=I}}</ref> In the context of opioid dependence, Methadone has been the gold standard medication treatment for over 30 years. According to numerous studies, patients on higher doses of methadone (>50mg/day) report less illicit opioid use, as well as increased retention rates in treatment.<ref>{{cite journal|last1=Farrell|first1=M|last2=Ward|first2=J|last3=Mattick|first3=R|last4=Hall|first4=W|last5=Stimson|first5=G|last6=Des Jarlais|first6=D|last7=Gossop|first7=M|last8=Strang|first8=J|date=1994.|title=Methadone maintenance treatment in opiate dependence: a review.|journal=British Medical Journal|volume=309|issue=6960|page=997}}</ref> Buprenorphine is an alternative to Methadone to treat opioid dependence and research similarly supports its clinical efficacy. Buprenorphine (60 mg/day) has been shown to bring about improved retention rates, as well as reduced illicit opioid use. ==== Therapy ==== * Cognitive Behavioral Therapies ** While medication serves as an effective intervention for some with drug dependence, behavioral interventions are also empirically supported. A number of studies suggest that CBT is an effective intervention for substance use. In a 2010 review, McHugh, Hearon and Otto<ref>{{cite journal|last1=McHugh|first1=RK|last2=Hearon|first2=BA|last3=Otto|first3=MW|date=September 2010|title=Cognitive behavioral therapy for substance use disorders.|journal=The Psychiatric clinics of North America|volume=33|issue=3|pages=511-25|pmid=20599130}}</ref> found that CBT for substance use, which synthesizes cognitive and motivational elements, as well as skills-building interventions, is effective both as a stand-alone treatment and when combined with other treatments. Acceptance and Commitment Therapy (ACT) has also been used to treat substance-using populations with encouraging results. Specifically, Lanza and Menéndez<ref>{{cite journal|last1=Villagrá Lanza|first1=P|last2=González Menéndez|first2=A|date=2013|title=Acceptance and Commitment Therapy for drug abuse in incarcerated women.|journal=Psicothema|volume=25|issue=3|pages=307-12|pmid=23910743}}</ref> employed a 16-session ACT in the treatment of incarcerated females. In this population, abstinence rates, as well as anxiety sensitivity and other comorbid psychopathology showed improvement. Another behavior intervention that has been successfully implemented is Mindfulness Therapy for Substance Use.<ref>{{cite journal|last1=Marcus|first1=Marianne T.|last2=Zgierska|first2=Aleksandra|date=27 October 2009|title=Mindfulness-Based Therapies for Substance Use Disorders: Part 1|journal=Substance Abuse|volume=30|issue=4|pages=263–265|doi=10.1080/08897070903250027}}</ref> Research indicates that this modality is effective across a range of populations through use of methods that help patients to develop nonreactive, acceptance behaviors. * Contingency Management ** One common technique implemented as a treatment method across psychiatric disorders is contingency management, wherein the problematic behavior of the individual is closely monitored and reinforcers are delivered contingent upon detection of a target behavior.<ref>{{cite book|title=Contingency management for adolescent substance abuse : a practitioner's guide|last1=al.]|first1=Scott W. Henggeler ... [et|date=2012|publisher=Guilford Press|isbn=1462502474|location=New York, NY}}</ref> In the case of substance use, abstinence is monitored via urine screens or other objective methods and the patient is rewarded for abstinence through prizes or vouchers, which are conversely withheld in the event that the patient does not remain abstinent as determined by urine screening or related methods. Recent work has demonstrated that contingency management can be an effective method for delaying time to first use after treatment and achieving short-term sobriety in individuals with substance use disorders. However, the effects of this intervention seem to be contingent upon the magnitude of the reward, and effects are not evident long-term.<ref>{{cite journal|last1=Petry|first1=NM|last2=Alessi|first2=SM|last3=Barry|first3=D|last4=Carroll|first4=KM|date=June 2015|title=Standard magnitude prize reinforcers can be as efficacious as larger magnitude reinforcers in cocaine-dependent methadone patients.|journal=Journal of consulting and clinical psychology|volume=83|issue=3|pages=464-72|pmid=25198284}}</ref> * Motivational Interviewing ** Motivational Interviewing is a treatment option that seems to be particularly useful for individuals who are ambivalent about changing behavior. This type of intervention requires the therapist to build a collaborative relationship with the patient, using empathic and non-confrontational approaches to help the patient enhance personal motivation to change.<ref>{{cite book|title=Motivational interviewing : helping people change|last1=Miller|first1=William R.|last2=Rollnick|first2=Stephen|date=2013|publisher=Guilford Press|isbn=1609182278|edition=3rd|location=New York, NY}}</ref> Finally, behavioral activation therapy attempts to address comorbid diagnoses commonly occurring with substance use disorders (i.e., depression) in an attempt to improve outcomes for individuals who may be harder to treat. This technique aims to increase positive reinforcers and decrease intensity and occurrences of negative consequences and life events. This treatment approach has been effective in reducing severity of depression and anxiety symptoms, and increasing enjoyment and reward value of posttreatment activities as compared to treatment as usual.<ref>{{cite journal|last1=Daughters|first1=SB|last2=Braun|first2=AR|last3=Sargeant|first3=MN|last4=Reynolds|first4=EK|last5=Hopko|first5=DR|last6=Blanco|first6=C|last7=Lejuez|first7=CW|date=January 2008|title=Effectiveness of a brief behavioral treatment for inner-city illicit drug users with elevated depressive symptoms: the life enhancement treatment for substance use (LETS Act!).|journal=The Journal of clinical psychiatry|volume=69|issue=1|pages=122-9|pmid=18312046}}</ref> ==Web based resources== * '''[http://www.drugabuse.gov National Institute on Drug Abuse]''' * '''[http://www.samhsa.gov/treatment/ Substance Abuse and Mental Health Services Administration (SAMHSA)]''' * '''[http://www.addictionrecoveryguide.org/ The Addiction Recovery Guide]''' * [http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/drug-and-alcohol-abuse/ EffectiveChildTherapy.Org information on Substance Abuse] * [https://sccap53.org Society of Clinical Child and Adolescent Psychology] ==References== {{collapse top|Click Expand for references}} {{reflist|2}} # {{collapse bottom}} [[Category:Psychological disorder portfolios|{{SUBPAGENAME}}]] epmw4r0k1f7mxhptfniisbqwyntv0in 2410381 2410380 2022-07-30T02:57:51Z Maddiegray11 2936309 /* Recommended diagnostic interviews for substance use disorder */removed invalid citation wikitext text/x-wiki <noinclude>{{Helping Give Away Psychological Science Banner}}</noinclude> {{medical disclaimer}} {{:{{BASEPAGENAME}}/Sidebar}} ==[[Evidence based assessment/Portfolio template/What is a "portfolio"|'''What is a "portfolio?"''']]== For background information on what assessment portfolios are, click the link in the heading above. Want even 'more' information about this topic? There's an extended version of this page [[Evidence-based assessment/Substance use disorder (disorder portfolio)/extended version|here]]. == [[Evidence-based assessment/Preparation phase|'''Preparation Phase''']] == {{blockquotetop}} <big>'''Diagnostic Criteria for Substance Use Disorder'''</big> <big>'''ICD-11 Diagnostic Criteria for Substance Use Disorder'''</big> *Disorders due to substance use and addictive behaviours are mental and behavioural disorders that develop as a result of the use of predominantly psychoactive substances, including medications, or specific repetitive rewarding and reinforcing behaviours. **Note: The ICD-11 lists 20 additional subcategories of Substance Use Disorder. They can be found [https://icd.who.int/browse11/l-m/en#/http://id.who.int/icd/entity/1602669465 here]. <big>'''DSM Diagnostic Criteria for Substance Use Disorder'''</big> *Substance use disorder is a DSM disorder in the Substance-Related and Addictive Disorders chapter. It is characterized by the use of substances in a manner that leads to clinically significant impairment or distress. * The diagnostic criteria for Substance Use Disorder disorder changed slightly from [[DSM-IV]] to [[w:Diagnostic_and_Statistical_Manual_of_Mental_Disorders#DSM-IV-TR_.282000.29|DSM-5]]. Summaries are available [http://www.dsm5.org/Documents/changes%20from%20dsm-iv-tr%20to%20dsm-5.pdf here]. {{blockquotebottom}} === Base rates of SUD in different populations and clinical settings === {| class="wikitable sortable" border="1" |- ! Setting ! Base Rate ! Demography ! Diagnostic Method |- | General population of North Carolina, aged 12 or older | 6.7% | North Carolina | National Survey on Drug Use and Health (NSDUH), 2009 to 2013 |- | 43,093 individuals, 18+ years old collected between 2001 and 2002<ref>Hasin DS, Grant BF. The National Epidemiologic Survey on Alcohol and Related Conditions (NESARC) Waves 1 and 2: review and summary of findings. Soc Psychiatry Psychiatr Epidemiol. 2015 Nov;50(11):1609-40. doi: 10.1007/s00127-015-1088-0. Epub 2015 Jul 26. PMID: 26210739.</ref> | 17.8 (0.5) Alcohol Abuse; 12.5 (0.4) Alcohol Dependence; 7.7 (0.2) Drug Abuse; 2.6 (0.1) Drug Dependence | United States General Adult Population: National Epidemiologic Survey on Alcohol and Related Conditions (NESARC) (Grant et al., 2007) | National Institute on Alcohol Abuse and Alcoholism Alcohol Use Disorder and Associated Disabilities Interview Schedule- DSM IV Version (AUDADIS-IV) |- | 9,282 adults, 18+ years old ; collected between 2001 and 2003<ref>{{cite journal|last1=Kessler|first1=RC|last2=Green|first2=JG|last3=Gruber|first3=MJ|last4=Sampson|first4=NA|last5=Bromet|first5=E|last6=Cuitan|first6=M|last7=Furukawa|first7=TA|last8=Gureje|first8=O|last9=Hinkov|first9=H|date=June 2010|title=Screening for serious mental illness in the general population with the K6 screening scale: results from the WHO World Mental Health (WMH) survey initiative.|journal=International journal of methods in psychiatric research|volume=19 Suppl 1|pages=4-22|pmid=20527002|last10=Hu|first10=CY|last11=Lara|first11=C|last12=Lee|first12=S|last13=Mneimneh|first13=Z|last14=Myer|first14=L|last15=Oakley-Browne|first15=M|last16=Posada-Villa|first16=J|last17=Sagar|first17=R|last18=Viana|first18=MC|last19=Zaslavsky|first19=AM}}</ref> | 13.2 (0.6) Alcohol Abuse; 5.4 (0.3) Alcohol Dependence; 7.9 (0.4) Drug Abuse; 3.0 (0.2) Drug Dependence | United States General Adult Population: National Comorbidity Survey Replication (NCS-R) | World Mental Health Survey Initiative Version of the World Health Organization Composite International Diagnostic Interview (WMH-CIDI) which generates DSM-IV and International Classification of Diseases, 10th revision diagnoses |- | Urban General Medicine Practice, low-income primary care patients, 75% Hispanic<ref>{{cite journal|last1=Olfson|first1=M|last2=Shea|first2=S|last3=Feder|first3=A|last4=Fuentes|first4=M|last5=Nomura|first5=Y|last6=Gameroff|first6=M|last7=Weissman|first7=MM|date=NaN|title=Prevalence of anxiety, depression, and substance use disorders in an urban general medicine practice.|journal=Archives of family medicine|volume=9|issue=9|pages=876-83|pmid=11031395}}</ref> | 7.9% | New York | Patient Health Questionnaire |- | Incarcerated females<ref>{{cite journal|last1=Teplin|first1=LA|last2=Abram|first2=KM|last3=McClelland|first3=GM|date=June 1996|title=Prevalence of psychiatric disorders among incarcerated women. I. Pretrial jail detainees.|journal=Archives of general psychiatry|volume=53|issue=6|pages=505-12|pmid=8639033}}</ref> | 70.2% | Chicago prison - 40 % African American, 33% White, 25 % Hispanic | National Institute of Mental Health Diagnostic Interview Schedule Version 11I-R (NIMH DIS-III-R) |- |Incarcerated females (updated) (Proctor 2012) | 70% dependent | Minnesota State Prison System- 801 females, 18-58 years old, 57.7% Caucasian, 21.5% African American, 13.2% Native American |Substance Use Disorder Diagnostic Schedule-IV (SUDDS-IV) |- | Incarcerated male youths<ref>{{cite journal|last1=Wasserman|first1=GA|last2=McReynolds|first2=LS|last3=Lucas|first3=CP|last4=Fisher|first4=P|last5=Santos|first5=L|date=March 2002|title=The voice DISC-IV with incarcerated male youths: prevalence of disorder.|journal=Journal of the American Academy of Child and Adolescent Psychiatry|volume=41|issue=3|pages=314-21|pmid=11886026}}</ref> | 56.4% | Texas state prison – 45 % African American, 33% White, 20% Hispanic |Structured Clinical Interview for DSM IV – Substance Use Disorders Module |- | Individuals with schizophrenia across settings<ref>{{cite journal|last1=Regier|first1=DA|last2=Farmer|first2=ME|last3=Rae|first3=DS|last4=Locke|first4=BZ|last5=Keith|first5=SJ|last6=Judd|first6=LL|last7=Goodwin|first7=FK|date=21 November 1990|title=Comorbidity of mental disorders with alcohol and other drug abuse. Results from the Epidemiologic Catchment Area (ECA) Study.|journal=JAMA|volume=264|issue=19|pages=2511-8|pmid=2232018}}</ref> | 47% | New Haven, CT; Baltimore, MD; St. Louis, MO; Durham, NC; Los Angeles, CA | National Institute of Mental Health (NIMH) Diagnostic Interview Schedule |- | HIV+ men in community health clinics<ref>{{cite journal|last1=Dew|first1=MA|last2=Becker|first2=JT|last3=Sanchez|first3=J|last4=Caldararo|first4=R|last5=Lopez|first5=OL|last6=Wess|first6=J|last7=Dorst|first7=SK|last8=Banks|first8=G|date=March 1997|title=Prevalence and predictors of depressive, anxiety and substance use disorders in HIV-infected and uninfected men: a longitudinal evaluation.|journal=Psychological medicine|volume=27|issue=2|pages=395-409|pmid=9089832}}</ref> | 24.4% | Alleghany County, PA | Structured Clinical Interview for DSM-III-R |- | Internal medicine inpatients<ref>{{cite journal|last1=Hansen|first1=MS|last2=Fink|first2=P|last3=Frydenberg|first3=M|last4=Oxhøj|first4=M|last5=Søndergaard|first5=L|last6=Munk-Jørgensen|first6=P|date=April 2001|title=Mental disorders among internal medical inpatients: prevalence, detection, and treatment status.|journal=Journal of psychosomatic research|volume=50|issue=4|pages=199-204|pmid=11369025}}</ref> | 10.9% | Denmark | Symptom Check List (SCL-8) |} == [[Evidence based assessment/Prediction phase|'''Prediction phase''']] == === Psychometric properties of screening measures for Substance Use Disorder === The following section contains a list of screening and diagnostic instruments for Substance Use Disorder. === Screening instruments and diagnostic interviews === {| class="wikitable sortable" |- ! Measure !Format (Reporter) !Age Range !Administration/ Completion Time !Where to Access |- |Structured Clinical International Diagnostic Interview (SCID-I) |Interview |Adults |30 minutes-3 hours |https://osf.io/x9smc |- |[https://osf.io/h2n9j/?view_only=ec71313a9e844abb977e241b5443f0db Substance Dependence Severity Scale (SDSS)] |Interview |16-adult |30-45 minutes | |- |[https://osf.io/7dh4s Global Appraisal of Individual Needs- Initial (GAIN-I) ($1.00 license fee per project for use of Beta version)] |Interview |12-adult |1.5-2.5 hours | |} '''Note:''' Reliability and validity are included in the extended version [[Evidence-based assessment/Substance use disorder (disorder portfolio)/extended version|here]]. This table includes measures with Good or Excellent ratings. === Likelihood ratios and AUCs of screening measures for '''(insert portfolio name)''' === * '''''For a list of the likelihood ratios for more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&wteswitched=1#Likelihood_ratios_and_AUCs_of_common_screening_instruments click here.]''''' {| class="wikitable sortable" border="1" |- ! Screening Measure (Primary Reference) ! AUC ! LR+ (Score) ! LR- (Score) ! Clinical Generalizability |- | [https://osf.io/gq86m/?view_only=628c7440cb5e4cbead514916f342a8cf Kessler 6 Screening Scale (K6)] <ref>{{cite journal|last1=Kessler|first1=RC|last2=Green|first2=JG|last3=Gruber|first3=MJ|last4=Sampson|first4=NA|last5=Bromet|first5=E|last6=Cuitan|first6=M|last7=Furukawa|first7=TA|last8=Gureje|first8=O|last9=Hinkov|first9=H|last10=Hu|first10=CY|last11=Lara|first11=C|last12=Lee|first12=S|last13=Mneimneh|first13=Z|last14=Myer|first14=L|last15=Oakley-Browne|first15=M|last16=Posada-Villa|first16=J|last17=Sagar|first17=R|last18=Viana|first18=MC|last19=Zaslavsky|first19=AM|title=Screening for serious mental illness in the general population with the K6 screening scale: results from the WHO World Mental Health (WMH) survey initiative.|journal=International journal of methods in psychiatric research|date=June 2010|volume=19 Suppl 1|pages=4-22|pmid=20527002}}</ref><ref>Swartz, J. A., & Lurigio, A. J. (2006). Screening for serious mental illness in populations with co-occurring substance use disorders: Performance of the K6 scale. Journal of substance abuse treatment, 31(3), 287-296</ref> | 0.84 (N=41,770) | 3.96 (13+) | 0.296 (0-12) | High: The sample of 41,770 was drawn from initial surveys that were carried out in 14 countries. |- | [https://osf.io/czhgd Alcohol, Smoking and Substance Involvement Screening Test (ASSIST)]<ref>{{cite journal|last1=Humeniuk|first1=R|last2=Ali|first2=R|last3=Babor|first3=TF|last4=Farrell|first4=M|last5=Formigoni|first5=ML|last6=Jittiwutikarn|first6=J|last7=de Lacerda|first7=RB|last8=Ling|first8=W|last9=Marsden|first9=J|last10=Monteiro|first10=M|last11=Nhiwatiwa|first11=S|last12=Pal|first12=H|last13=Poznyak|first13=V|last14=Simon|first14=S|title=Validation of the Alcohol, Smoking And Substance Involvement Screening Test (ASSIST).|journal=Addiction (Abingdon, England)|date=June 2008|volume=103|issue=6|pages=1039-47|pmid=18373724}}</ref> | 0.84 (N=1,047) | 2.76 | 0.28 | High: The sample of 1,047 participants was drawn from drug treatment and primary health care settings in Australia, Brazil, India, Thailand, the United Kingdom, the U.S. and Zimbabwe. |- | [https://osf.io/nk5vx/?view_only=348d7f1ee5e741f8a6657233403dda66 Drug Use Disorders Identification Test (DUDIT)]<ref>{{Cite journal|last=Voluse|first=Andrew C.|last2=Gioia|first2=Christopher J.|last3=Sobell|first3=Linda Carter|last4=Dum|first4=Mariam|last5=Sobell|first5=Mark B.|last6=Simco|first6=Edward R.|title=Psychometric properties of the Drug Use Disorders Identification Test (DUDIT) with substance abusers in outpatient and residential treatment|url=https://doi.org/10.1016/j.addbeh.2011.07.030|journal=Addictive Behaviors|volume=37|issue=1|pages=36–41|doi=10.1016/j.addbeh.2011.07.030}}</ref> |0.95 (N=153) | 6 |0.12 | High: 153 participants from outpatient and residential substance use treatment programs |} '''Note:''' “LR+” refers to the change in likelihood ratio associated with a positive test score, and “LR-” is the likelihood ratio for a low score. Likelihood ratios of 1 indicate that the test result did not change impressions at all. LRs larger than 10 or smaller than 0.1 are frequently clinically decisive; 5 or 0.2 are helpful, and between 2.0 and 0.5 are small enough that they rarely result in clinically meaningful changes of formulation (Sackett et al., 2000). ''' Search terms''': [substance use OR substance use disorders] AND [sensitivity OR specificity] in Google Scholar and PsycINFO =='''[[Evidence-based assessment/Prescription phase|Prescription phase]]'''== ===Gold standard diagnostic interviews=== * For a list of broad reaching diagnostic interviews sortable by disorder with PDFs (if applicable), [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prescription_phase&wteswitched=1#Common_Diagnostic_Interviews click here.] ===Recommended diagnostic interviews for substance use disorder=== {| class="wikitable sortable" border="1" ! colspan="5" |Diagnostic instruments for '''(insert portfolio name)''' |- ! Measure ! Format (Reporter) ! Age Range ! Administration/ Completion Time !Where to Access |- | Diagnostic Interview Schedule- V (DIS) | | | | |- | Structured Clinical Interview for DSM-V (SCID) | | | | |- | The Psychiatric Research Interview for Substance and Mental Disorders | | | | |- |The Mini International Psychiatric Interview (M.I.N.I) | | | | |} '''Note:''' Reliability and validity are included in the extended version (link). This table includes measures with Good or Excellent ratings. == '''[[Evidence-based assessment/Process phase|Process phase]]''' == === Outcome and severity measures === This table includes clinically significant benchmarks for '''(insert portfolio name here)''' specific outcome measures * Information on how to interpret this table can be [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase found here]. * Additionally, these [[Evidence based assessment/Vignettes|vignettes]] might be helpful resources for understanding appropriate adaptation of outcome measures in practice. *''<u>For clinically significant change benchmarks for the CBCL, YSR, and TRF total, externalizing, internalizing, and attention benchmarks,</u>'' [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase&wteswitched=1#Clinically_significant_change_benchmarks_for_widely-used_outcome_measures see here.] Clinically significant change benchmarks with common instruments and mood rating scales {| class="wikitable sortable" border="1" | colspan="7" | ==== '''Clinically significant change benchmarks with common instruments and mood rating scales''' ==== |- | style="text-align:center;font-size:120%" width="300" | | colspan="3" style="text-align:center;font-size:120%" width="300" | <b> Cut* Scores</b> | colspan="3" style="text-align:center;font-size:120%" | <b> Critical Change <br> (Unstandardized Scores)</b> |- | style="text-align:center;font-size:130%;" | <b> Measure</b> | style="text-align:center;font-size:110%" | <b> A</b> | style="text-align:center;font-size:110%" | <b> B</b> | style="text-align:center;font-size:110%" | <b> C</b> | style="text-align:center;font-size:110%" | <b> 95%</b> | style="text-align:center;font-size:110%" | <b> 90%</b> | style="text-align:center;font-size:110%" | <b> SE_difference</b> |- | colspan="7" style="font-size:110%; text-align:center;" span | <b> Benchmarks Based on Published Norms</b> |- | style="text-align:center;font-size:100%;" | ''[https://osf.io/wn3rb Rutgers Alcohol Problem Index]''<ref name="Roberts">{{cite journal|last1=Roberts|first1=LJ|last2=Neal|first2=DJ|last3=Kivlahan|first3=DR|last4=Baer|first4=JS|last5=Marlatt|first5=GA|title=Individual drinking changes following a brief intervention among college students: clinical significance in an indicated preventive context.|journal=Journal of consulting and clinical psychology|date=June 2000|volume=68|issue=3|pages=500-5|pmid=10883566}}</ref> <ref name="White1989">{{cite journal|last1=White|first1=HR|last2=Labouvie|first2=EW|title=Towards the assessment of adolescent problem drinking.|journal=Journal of studies on alcohol|date=January 1989|volume=50|issue=1|pages=30-7|pmid=2927120}}</ref> | style="text-align:center;font-size:100%;" span |0.8 | style="text-align:center;font-size:100%;" span | 4.9 | style="text-align:center;font-size:100%;" span | 4.0 | style="text-align:center;font-size:100%;" span | 4.1 | style="text-align:center;font-size:100%;" span | 3.5 | style="text-align:center;font-size:100%;" span | 2.1 |- | style="text-align:center;font-size:100%;" | ''[https://osf.io/vudep Alcohol Dependence Scale (ADS)]''<ref name="Roberts" /> [https://osf.io/vudep (copyrighted)] | style="text-align:center;font-size:100%;" span | 1.2 | style="text-align:center;font-size:100%;" span | 9.9 | style="text-align:center;font-size:100%;" span | 7.8 | style="text-align:center;font-size:100%;" span | 1.4 | style="text-align:center;font-size:100%;" span | 1.2 | style="text-align:center;font-size:100%;" span | 0.7 |- | style="text-align:center;font-size:100%;" | ''[https://osf.io/23mwt/?view_only=1924fe3e90334b11bb1e099a691b2ffb Drug Abuse Screening Test (DAST)]'' <ref>{{cite journal|last1=Skinner|first1=HA|title=The drug abuse screening test.|journal=Addictive behaviors|date=1982|volume=7|issue=4|pages=363-71|pmid=7183189}}</ref> | style="text-align:center;font-size:100%;" span | 0.1 | style="text-align:center;font-size:100%;" span | 2.6 | style="text-align:center;font-size:100%;" span | 1.8 | style="text-align:center;font-size:100%;" span | 1.6 | style="text-align:center;font-size:100%;" span | 1.3 | style="text-align:center;font-size:100%;" span | 0.8 |} '''Note:''' "A" = Away from the clinical range, "B" = Back into the nonclinical range, "C" = Closer to the nonclinical than clinical mean. '''Search terms''': [substance use OR substance use disorder] AND [clinical significance OR outcomes] in Google Scholar and PsycINFO === Treatment === In the United States, according to SAMHSA, of the 8.9 million adults with a dual diagnosis, 44% received some form of treatment in the past year. Given the frequent co-occurrence of mood disorders and substance use disorders, the recommended first step in treatment is for clinicians to deliver a comprehensive screening evaluation that will inform their treatment approach. There are a host of empirically supported treatments for substance use disorders, though medication interventions and psychotherapy are most common. ==== Medication ==== Specifically, medications have been shown to be most effective in the treatment of alcohol and opioid dependence. Naltrexone (50 mg/day) administered for 12 weeks has been shown to decrease cravings for alcohol and the number of days in which alcohol was consumed.<ref>{{cite journal|last1=Volpicelli|first1=JR|last2=Alterman|first2=AI|last3=Hayashida|first3=M|last4=O'Brien|first4=CP|date=November 1992|title=Naltrexone in the treatment of alcohol dependence.|journal=Archives of general psychiatry|volume=49|issue=11|pages=876-80|pmid=1345133}}</ref> Disulfiram (250 mg/day), administered for one year, has been shown to help reduce drinking frequency after relapse.<ref>{{cite journal|last1=Chick|first1=J|last2=Gough|first2=K|last3=Falkowski|first3=W|last4=Kershaw|first4=P|last5=Hore|first5=B|last6=Mehta|first6=B|last7=Ritson|first7=B|last8=Ropner|first8=R|last9=Torley|first9=D|date=July 1992|title=Disulfiram treatment of alcoholism.|journal=The British journal of psychiatry : the journal of mental science|volume=161|pages=84-9|pmid=1638335}}</ref><ref name="Fuller1986">{{cite journal|last1=Fuller|first1=RK|last2=Branchey|first2=L|last3=Brightwell|first3=DR|last4=Derman|first4=RM|last5=Emrick|first5=CD|last6=Iber|first6=FL|last7=James|first7=KE|last8=Lacoursiere|first8=RB|last9=Lee|first9=KK|date=19 September 1986|title=Disulfiram treatment of alcoholism. A Veterans Administration cooperative study.|journal=JAMA|volume=256|issue=11|pages=1449-55|pmid=3528541|last10=Lowenstam|first10=I}}</ref> In the context of opioid dependence, Methadone has been the gold standard medication treatment for over 30 years. According to numerous studies, patients on higher doses of methadone (>50mg/day) report less illicit opioid use, as well as increased retention rates in treatment.<ref>{{cite journal|last1=Farrell|first1=M|last2=Ward|first2=J|last3=Mattick|first3=R|last4=Hall|first4=W|last5=Stimson|first5=G|last6=Des Jarlais|first6=D|last7=Gossop|first7=M|last8=Strang|first8=J|date=1994.|title=Methadone maintenance treatment in opiate dependence: a review.|journal=British Medical Journal|volume=309|issue=6960|page=997}}</ref> Buprenorphine is an alternative to Methadone to treat opioid dependence and research similarly supports its clinical efficacy. Buprenorphine (60 mg/day) has been shown to bring about improved retention rates, as well as reduced illicit opioid use. ==== Therapy ==== * Cognitive Behavioral Therapies ** While medication serves as an effective intervention for some with drug dependence, behavioral interventions are also empirically supported. A number of studies suggest that CBT is an effective intervention for substance use. In a 2010 review, McHugh, Hearon and Otto<ref>{{cite journal|last1=McHugh|first1=RK|last2=Hearon|first2=BA|last3=Otto|first3=MW|date=September 2010|title=Cognitive behavioral therapy for substance use disorders.|journal=The Psychiatric clinics of North America|volume=33|issue=3|pages=511-25|pmid=20599130}}</ref> found that CBT for substance use, which synthesizes cognitive and motivational elements, as well as skills-building interventions, is effective both as a stand-alone treatment and when combined with other treatments. Acceptance and Commitment Therapy (ACT) has also been used to treat substance-using populations with encouraging results. Specifically, Lanza and Menéndez<ref>{{cite journal|last1=Villagrá Lanza|first1=P|last2=González Menéndez|first2=A|date=2013|title=Acceptance and Commitment Therapy for drug abuse in incarcerated women.|journal=Psicothema|volume=25|issue=3|pages=307-12|pmid=23910743}}</ref> employed a 16-session ACT in the treatment of incarcerated females. In this population, abstinence rates, as well as anxiety sensitivity and other comorbid psychopathology showed improvement. Another behavior intervention that has been successfully implemented is Mindfulness Therapy for Substance Use.<ref>{{cite journal|last1=Marcus|first1=Marianne T.|last2=Zgierska|first2=Aleksandra|date=27 October 2009|title=Mindfulness-Based Therapies for Substance Use Disorders: Part 1|journal=Substance Abuse|volume=30|issue=4|pages=263–265|doi=10.1080/08897070903250027}}</ref> Research indicates that this modality is effective across a range of populations through use of methods that help patients to develop nonreactive, acceptance behaviors. * Contingency Management ** One common technique implemented as a treatment method across psychiatric disorders is contingency management, wherein the problematic behavior of the individual is closely monitored and reinforcers are delivered contingent upon detection of a target behavior.<ref>{{cite book|title=Contingency management for adolescent substance abuse : a practitioner's guide|last1=al.]|first1=Scott W. Henggeler ... [et|date=2012|publisher=Guilford Press|isbn=1462502474|location=New York, NY}}</ref> In the case of substance use, abstinence is monitored via urine screens or other objective methods and the patient is rewarded for abstinence through prizes or vouchers, which are conversely withheld in the event that the patient does not remain abstinent as determined by urine screening or related methods. Recent work has demonstrated that contingency management can be an effective method for delaying time to first use after treatment and achieving short-term sobriety in individuals with substance use disorders. However, the effects of this intervention seem to be contingent upon the magnitude of the reward, and effects are not evident long-term.<ref>{{cite journal|last1=Petry|first1=NM|last2=Alessi|first2=SM|last3=Barry|first3=D|last4=Carroll|first4=KM|date=June 2015|title=Standard magnitude prize reinforcers can be as efficacious as larger magnitude reinforcers in cocaine-dependent methadone patients.|journal=Journal of consulting and clinical psychology|volume=83|issue=3|pages=464-72|pmid=25198284}}</ref> * Motivational Interviewing ** Motivational Interviewing is a treatment option that seems to be particularly useful for individuals who are ambivalent about changing behavior. This type of intervention requires the therapist to build a collaborative relationship with the patient, using empathic and non-confrontational approaches to help the patient enhance personal motivation to change.<ref>{{cite book|title=Motivational interviewing : helping people change|last1=Miller|first1=William R.|last2=Rollnick|first2=Stephen|date=2013|publisher=Guilford Press|isbn=1609182278|edition=3rd|location=New York, NY}}</ref> Finally, behavioral activation therapy attempts to address comorbid diagnoses commonly occurring with substance use disorders (i.e., depression) in an attempt to improve outcomes for individuals who may be harder to treat. This technique aims to increase positive reinforcers and decrease intensity and occurrences of negative consequences and life events. This treatment approach has been effective in reducing severity of depression and anxiety symptoms, and increasing enjoyment and reward value of posttreatment activities as compared to treatment as usual.<ref>{{cite journal|last1=Daughters|first1=SB|last2=Braun|first2=AR|last3=Sargeant|first3=MN|last4=Reynolds|first4=EK|last5=Hopko|first5=DR|last6=Blanco|first6=C|last7=Lejuez|first7=CW|date=January 2008|title=Effectiveness of a brief behavioral treatment for inner-city illicit drug users with elevated depressive symptoms: the life enhancement treatment for substance use (LETS Act!).|journal=The Journal of clinical psychiatry|volume=69|issue=1|pages=122-9|pmid=18312046}}</ref> ==Web based resources== * '''[http://www.drugabuse.gov National Institute on Drug Abuse]''' * '''[http://www.samhsa.gov/treatment/ Substance Abuse and Mental Health Services Administration (SAMHSA)]''' * '''[http://www.addictionrecoveryguide.org/ The Addiction Recovery Guide]''' * [http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/drug-and-alcohol-abuse/ EffectiveChildTherapy.Org information on Substance Abuse] * [https://sccap53.org Society of Clinical Child and Adolescent Psychology] ==References== {{collapse top|Click Expand for references}} {{reflist|2}} # {{collapse bottom}} [[Category:Psychological disorder portfolios|{{SUBPAGENAME}}]] jg67sq5yolvkchrjivxb4kyhsgg821b 2 211310 2410385 2409295 2022-07-30T08:26:04Z Guy vandegrift 813252 wikitext text/x-wiki {| class="wikitable" style="" |- ! round ! p ! q ! ratio ! cents |- | 294.13 | 32 | 27 | 1.185185185 | 294.1349974 |- | 300 | -1 | 3 | 1.189207115 | 300 |- | 301.85 | 25 | 21 | 1.19047619 | 301.8465204 |- | 315.64 | 6 | 5 | 1.2 | 315.641287 |- | 386.31 | 5 | 4 | 1.25 | 386.3137139 |- | 400 | -1 | 4 | 1.25992105 | 400 |- | 407.82 | 81 | 64 | 1.265625 | 407.8200035 |- | 427.37 | 32 | 25 | 1.28 | 427.3725723 |- | 435.08 | 9 | 7 | 1.285714286 | 435.0840953 |- | 498.04 | 4 | 3 | 1.333333333 | 498.0449991 |- | 500 | -1 | 5 | 1.334839854 | 500 |- | 519.55 | 27 | 20 | 1.35 | 519.5512887 |- | 568.72 | 25 | 18 | 1.388888889 | 568.717426 |- | 582.51 | 7 | 5 | 1.4 | 582.5121926 |- | 590.22 | 45 | 32 | 1.40625 | 590.2237156 |- | 600 | -1 | 6 | 1.414213562 | 600 |- | 609.78 | 64 | 45 | 1.422222222 | 609.7762844 |- | 617.49 | 10 | 7 | 1.428571429 | 617.4878074 |- | 631.28 | 36 | 25 | 1.44 | 631.282574 |- | 680.45 | 40 | 27 | 1.481481481 | 680.4487113 |- | 700 | -1 | 7 | 1.498307077 | 700 |- | 701.96 | 3 | 2 | 1.5 | 701.9550009 |- | 764.92 | 14 | 9 | 1.555555556 | 764.9159047 |- | 772.63 | 25 | 16 | 1.5625 | 772.6274277 |- | 800 | -1 | 8 | 1.587401052 | 800 |- | 813.69 | 8 | 5 | 1.6 | 813.6862861 |- | 835.19 | 81 | 50 | 1.62 | 835.1925757 |- | 884.36 | 5 | 3 | 1.666666667 | 884.358713 |- | 900 | -1 | 9 | 1.681792831 | 900 |- | 905.87 | 27 | 16 | 1.6875 | 905.8650026 |- | 933.13 | 12 | 7 | 1.714285714 | 933.1290944 |- | 968.83 | 7 | 4 | 1.75 | 968.8259065 |- | 996.09 | 16 | 9 | 1.777777778 | 996.0899983 |- | 1000 | -1 | 10 | 1.781797436 | 1000 |- | 1017.6 | 9 | 5 | 1.8 | 1017.596288 |} gqjh3q41mbph5h10quuk1k58kpegduh 0 214229 2410336 2410225 2022-07-30T00:09:32Z Dave Braunschweig 426084 Reverted edits by [[Special:Contributions/2401:4900:41F6:580B:26D2:D88B:2E17:F841|2401:4900:41F6:580B:26D2:D88B:2E17:F841]] ([[User_talk:2401:4900:41F6:580B:26D2:D88B:2E17:F841|talk]]) to last version by [[User:Dave Braunschweig|Dave Braunschweig]] using [[Wikiversity:Rollback|rollback]] wikitext text/x-wiki Made by ANTARACentral_Board_of_Secondary_Education | CBSE]]. These syllabus are periodically reviewed and revised. The NCERT book for a particular subject is divided into various chapters and every chapter has a set of questions following the chapter. This section provides answers to the questions at the end of each chapter in the '''Geography''' book, '''Our Environment''', for [[NCERT/Textbook_Solutions#Solutions_to_Textbooks_of_Class_VII |'''Class-VII''']]. ==Chapter 01 -Environment Geography == The Questions with Answers of this chapter are provided below:- '''Question 01:''' Answer the following questions:- '''(i)''' What is an ecosystem? '''Answer:Ecosystem is a community of living organisms in conjunction with the nonliving components of their environment (things like air, water and mineral soil), interacting as a system.''' '''(ii)''' What do you mean by natural environment? '''Answer: The environment which is created by nature comprises of land, water, air, plants and animals. This is known as natural environment ''' '''(iii)''' Which are the major components of the environment? '''Answer:''' The components of the environment are NATURAL(land,air , water.living things) HUMAN-MADE (building,parks bridge, roads, industry, monument,etc...)Human( individuals, families, community, religion, educational, economic , political situation, '''(iv)''' Give four examples of human made environment. '''Answer:''' Few examples of human made environment are: # Parks # Buildings # Roads # Vehicles # Bridges # Industries #monuments '''(v)''' What is lithosphere? '''Answer:''' [[Wikipedia:Lithosphere| Lithosphere]] is the solid crust or the hard top layer of the earth. It includes the crust and the uppermost mantle, which constitute the hard and rigid outer layer of the Earth. '''(vi)''' Which are the two major components of biotic environment? '''Answer:''' Two major components of biotic environment are Plants and Animals. '''(vii)''' What is biosphere? '''Answer:''' Biosphere is a narrow zone of the earth where land, water and air interact with each other to support life. It consists of plant and animal kingdom together. It is a global sum of all ecosystems. '''Question 2.''' Choose the correct answer. '''(i)''' Which is not a natural ecosystem? (a) Desert (b)Aquariumum (c) Forest '''Answer:''' (b) Aquarium '''(ii)''' Which is not a component of human environment? (a) Land (b) Religion (c) Community '''Answer:''' (a) Land '''(iii)''' Which is a human made environment? (a) Mountain (b) Sea (c) Road '''Answer:''' (c) Road '''(iv)''' Which is a threat to environment? (a) Growing plant (b) Growing population (c) Growing crops '''Answer:''' (b) Growing population '''Question 3.''' Match the following. {| style="width:100%;" class="wikitable" border="0" cellpadding="2" cellspacing="2" ! scope="col" width=20%| Col-1 ! scope="col" width=20%| Col-2 |- ! | Biosphere | blanket of air which surrounds the earth |- ! scope="row" | Atmosphere | domain of water |- ! scope="row" | Hydrosphere | gravitational force of the earth |- ! scope="row" | Environment | our surroundings |- ! scope="row" | ----- | narrow zone where land water and air interact |- ! scope="row" | ----- | relation between the organisms and their surroundings |- |} '''Answer:''' {| style="width:100%;" class="wikitable" border="0" cellpadding="2" cellspacing="2" ! scope="col" width=20%| Col-1 ! scope="col" width=20%| Col-2 |- ! | Biosphere | narrow zone where land water and air interact |- ! scope="row" | Atmosphere | blanket of air which surrounds the earth |- ! scope="row" | Hydrosphere | domain of water |- ! scope="row" | Environment | our surroundings |- |} '''Question 4.''' Give reasons. (i) Man modifies his environment (ii) Plants and animals depend on each other '''Answer:''' (i) Man modifies his environment because of his growing needs. He is capable of modifying it according to his need to live a comfortable life. Humans learn new ways to use and change environment and as a result invented many things. Industrial revolution enabled large scale production of goods. Transportation became faster and more comfortable. Information revolution made communication easier and faster across the world. (ii) Plants and animals depend on each other for their sustainability. Animals consume plants for their living and also takes oxygen from them. Few carnivorous animals inturn eat other animals. Plants are dependent on animals as they give out carbon dioxide which is important for photosynthesis. Also, dead remains of animals provide nutrients to the plants. ==Chapter 02 Inside Our Earth Geography== The Questions with Answers of this chapter are provided below:- '''Question 1.''' Answer the following questions. '''(i)''' What are the three layers of the earth? '''Answer''' The three layers of the Earth are the crust, the mantle and the core. '''(ii)''' What is a rock? '''Answer''' Any natural mass of solid mineral matter that makes up the Earth’s crust is called a rock. '''(iii)''' Name three types of rocks. '''Answer''' The three types of rocks are igneous rocks, sedimentary rocks and metamorphic rocks. '''(iv)''' How are extrusive and intrusive rocks formed? '''Answer''' Extrusive rocks are formed by the molten lava which comes on the earth’s surface and rapidly cools down to becomes solid. When the molten magma cools down deep inside the earth’s crust then the solid rocks so formed are called intrusive rocks. '''(v)''' What do you mean by a rock cycle? '''Answer''' When one type of rock changes to another type under certain conditions in a cyclic manner then this process of transformation of the rock from one to another is known as the rock cycle. '''(vi)''' What are the uses of rocks? '''Answer''' The uses of the rocks are as follows : * Hard rocks are used in construction of buildings and roads. * Some rocks are shiny and precious therefore used for making jewellery. * Rocks are made up of different minerals and are very important to humankind. * Some are used as fuels. For example, coal, natural gas and petroleum. * Soft rocks are used for making talcum powder, chalks etc. '''(vii)''' What are metamorphic rocks? '''Answer''' The rocks which are formed due to conversion of igneous and sedimentary rocks under great heat and pressure is called metamorphic rocks. '''Question 2.''' Tick the correct answer. '''(i)''' The rock which is made up of molten magma is (a) Igneous (b) Sedimentary (c) Metamorphic '''Answer''' (a) Igneous '''(ii)''' The innermost layer of the earth is (a) Crust (b) Core (c) Mantle '''Answer''' (b) Core '''(iii)''' Gold, petroleum and coal are examples of (a) Rocks (b) Minerals (c) Fossils '''Answer''' (b) Minerals '''(iv)''' Rocks which contain fossils are (a) Sedimentary rocks (b) Metamorphic rocks (c) Igneous rocks '''Answer''' (a) Sedimentary rocks '''(v)''' The thinnest layer of the earth is (a) Crust (b) Mantle (c) Core '''Answer''' (a) Crust '''Question 3.''' Match the following. {| style="width:100%;" class="wikitable" border="0" cellpadding="2" cellspacing="2" ! scope="col" width=20%| Col-1 ! scope="col" width=20%| Col-2 |- ! | Core | Earth’s surface |- ! scope="row" | Minerals | Used for roads and buildings |- ! scope="row" | Rocks | Made of silicon and alumina |- ! scope="row" | Clay | Has definite chemical composition |- ! scope="row" | Sial | Innermost layer |- ! scope="row" | ----- | Changes into slate |- ! scope="row" | ----- | Process of transformation of the rock |- |} '''Answer''' (i) Cores (e) Innermost layers (ii) Minerals (d) Has definite chemical composition (iii) Rocks (b) Used for roads and buildings (iv) Clay (f) Changes into slate (v) Sial (c) Made of silicon and alumina '''Question 4.''' Give reasons. '''(i)''' We cannot go to the centre of the earth. '''Answer''' We cannot go to the centre of the earth because the it has very high temperature and pressure and lies 6000 km below the ocean floor. We will not able to survive there because there is no oxygen or<graph>{ "version": 2, "width": 400, "height": 200, "data": [ { "name": "table", "values": [ { "x": 0, "y": 1 }, { "x": 1, "y": 3 }, { "x": 2, "y": 2 }, { "x": 3, "y": 4 } ] } ], "scales": [ { "name": "x", "type": "ordinal", "range": "width", "zero": false, "domain": { "data": "table", "field": "x" } }, { "name": "y", "type": "linear", "range": "height", "nice": true, "domain": { "data": "table", "field": "y" } } ], "axes": [ { "type": "x", "scale": "x" }, { "type": "y", "scale": "y" } ], "marks": [ { "type": "rect", "from": { "data": "table" }, "properties": { "enter": { "x": { "scale": "x", "field": "x" }, "y": { "scale": "y", "field": "y" }, "y2": { "scale": "y", "value": 0 }, "fill": { "value": "steelblue" }, "width": { "scale": "x", "band": "true", "offset": -1 } } } } ] }</graph>conditions. favourable '''(ii)''' Sedimentary rocks are formed from sediments. '''Answer''' Sedimentary rocks are formed from sediments because of extreme compression and hardening of the particles of sediment which are transported and deposited by wind, water etc. '''(iii)''' Limestone is changed into marble. '''Answer''' Limestone is changed into marble because of extreme heat and pressure as it is a sedimentary rock. ==Chapter 03 Our Changing Earth== ==Chapter 04 Air== ==Chapter 05 Water== ==Chapter 06 Natural Vegetation and Wildlife== ==Chapter 07 Human Environment–Settlement, Transport and Communication== ==Chapter 08 Human Environment Interactions (The Tropical and the Subtropical Region)== ==Chapter 09 Life in the Temperate Grasslands== ==Chapter 10 Life in the Deserts== ==See Also== [[NCERT/Textbook Solutions]] [[NCERT/Textbook Solutions/Class VII/History]] [[NCERT/Textbook Solutions/Class VII/Civics]] ==External Links== * [http://www.ncert.nic.in/index.html Official website of NCERT] * [http://epathshala.nic.in/e-pathshala-4/flipbook/ NCERT textbooks available online] ==References== {{Reflist}} nz4mqo23rfjazdo4t99ztid36cjz4x2 0 232469 2410310 2410002 2022-07-29T20:28:50Z Young1lim 21186 /* ARM Assembly Programming (II) */ wikitext text/x-wiki == '''Background''' == '''Combinational and Sequential Circuits''' * [[Media:DD2.B.4..Adder.20131007.pdf |Adder]] * [[Media:DD3.A.1.LatchFF.20160308.pdf |Latches and Flipflops]] '''FSM''' * [[Media:DD3.A.3.FSM.20131030.pdf |FSM]] * [[Media:CArch.2.A.Bubble.20131021.pdf |FSM Example]] '''Tiny CPU Example''' * [[Media:CDsgn6.TinyCPU.2.A.ISA.20160511.pdf |Instruction Set]] * [[Media:CDsgn6.TinyCPU.2.B.DPath.20160502.pdf |Data Path]] * [[Media:CDsgn6.TinyCPU.2.C.CPath.20160427.pdf |Control Path]] * [[Media:CDsgn6.TinyCPU.2.D.Implement.20160513.pdf |FPGA Implementation]] </br> == '''Microprocessor Architecture''' == * ARM Architecture : - Programmer's Model ([[Media:ARM.1Arch.1A.Model.20180321.pdf |pdf]]) : - Pipelined Architecture ([[Media:ARM.1Arch.2A.Pipeline.20180419.pdf |pdf]]) * ARM Organization * ARM Cortex-M Processor Architecture * ARM Processor Cores </br> == '''Instruction Set Architecture''' == * ARM Instruction Set : - Overview ([[Media:ARM.2ISA.1A.Overview.20190611.pdf |pdf]]) : - Addressing Modes ([[Media:ARM.2ISA.2A.AddrMode.20191108.pdf |pdf]]) : - Multiple Transfer ([[Media:ARM.2ISA.3A.MTransfer.20190903.pdf |pdf]]) : - Assembler Format :: - Data Processing ([[Media:ARM.2ISA.4A.Proc.Format.20200204.pdf |pdf]]) :: - Data Transfer ([[Media:ARM.2ISA.4B.Trans.Format.20200205.pdf |pdf]]) :: - Coprocessor ([[Media:ARM.2ISA.4C.CoProc.Format.20191214.pdf |pdf]]) :: - Summary ([[Media:ARM.2ISA.4D.Summary.Format.20200205.pdf |pdf]]) : - Binary Encoding ([[Media:ARM.2ISA.5A.Encoding.201901105.pdf |pdf]]) * Thumb Instruction Set </br> == '''Assembly Programming''' == === ARM Assembly Programming (I) === * 1. Overview ([[Media:ARM.2ASM.1A.Overview.20200101.pdf |pdf]]) * 2. Example Programs ([[Media:ARM.2ASM.2A.Program.20200108.pdf |pdf]]) * 3. Addressing Modes ([[Media:ARM.2ASM.3A.Address.20200127.pdf |pdf]]) * 4. Data Transfer ([[Media:ARM.2ASM.4A.DTransfer.20200206.pdf |pdf]]) * 5. Data Processing ([[Media:ARM.2ASM.5A.DProcess.20200208.pdf |pdf]]) * 6. Control ([[Media:ARM.2ASM.6A.Control.20200215.pdf |pdf]]) * 7. Arrays ([[Media:ARM.2ASM.7A.Array.20200311.pdf |pdf]]) * 8. Data Structures ([[Media:ARM.2ASM.8A.DataStruct.20200718.pdf |pdf]]) * 9. Finite State Machines ([[Media:ARM.2ASM.9A.FSM.20200417.pdf |pdf]]) * 10. Functions ([[Media:ARM.2ASM.10A.Function.20210115.pdf |pdf]]) * 11. Parameter Passing ([[Media:ARM.2ASM.11A.Parameter.20210106.pdf |pdf]]) * 12. Stack Frames ([[Media:ARM.2ASM.12A.StackFrame.20210611.pdf |pdf]]) :: :: === ARM Assembly Programming (II) === :: * 1. Thumb instruction programming ([[Media:ARM.2ASM.Thumb.20210612.pdf |pdf]]) * 2. Exceptions ([[Media:ARM.2ASM.Exception.20220722.pdf |pdf]]) * 3. Exception Programming ([[Media:ARM.2ASM.ExceptionProg.20220311.pdf |pdf]]) * 4. Exception Handlers ([[Media:ARM.2ASM.ExceptionHandler.20220131.pdf |pdf]]) * 5. Interrupt Programming ([[Media:ARM.2ASM.InterruptProg.20211030.pdf |pdf]]) * 6. Interrupt Handlers ([[Media:ARM.2ASM.InterruptHandler.20211030.pdf |pdf]]) * 7. Vector Interrupt Controllers ([[Media:ARM.2ASM.VIC.20220728.pdf |pdf]]) </br> * ARM Assembly Exercises ([[Media:ESys.3.A.ARM-ASM-Exercise.20160608.pdf |A.pdf]], [[Media:ESys.3.B.Assembly.20160716.pdf |B.pdf]]) :: === ARM Assembly Programming (III) === * 1. Fixed point arithmetic (integer division) * 2. Floating point arithmetic * 3. Matrix multiply === ARM Linking === * arm link ([[Media:arm_link.20211208.pdf |pdf]]) </br> === ARM Microcontroller Programming === * 1. Input / Output * 2. Serial / Parallel Port Interfacing * 3. Analog I/O Interfacing * 4. Communication </br> == '''Architectural Support''' == </br> '''ARM Architectural Support''' * High Level Languages * System Development * Operating Systems </br> == '''Memory and Peripheral Architecture''' == </br> == '''System and Peripheral Buses''' == </br> == '''Serial Bus''' == </br> == '''Interrupts and Exceptions ''' == </br> == '''Timers ''' == </br> == '''Synchrnoization'''== </br> === H/W and S/W Synchronization === * busy wait synchronization * handshake interface </br> === Interrupt Synchronization === * interrupt synchronization * reentrant programming * buffered IO * periodic interrupt * periodic polling </br> ==''' Interfacing '''== </br> === Time Interfacing === * input capture * output compare </br> === Serial Interfacing === * Programming UART * Programming SPI * Programming I2C * Programming USB </br> === Analog Interfacing === * OP Amp * Filters * ADC * DAC </br>== '''Instruction Set Architecture''' == * ARM Instruction Set :: - Overview ([[Media:ARM.2ISA.1A.Overview.20180528.pdf |pdf]]) :: - Binary Encoding ([[Media:ARM.2ISA.2A.Encoding.20180528.pdf |pdf]]) :: - Assembler Format ([[Media:ARM.2ISA.3A.Format.20180528.pdf |pdf]]) * Thumb Instruction Set * ARM Assembly Language ([[Media:ESys3.1A.Assembly.20160608.pdf |pdf]]) * ARM Machine Language ([[Media:ESys3.2A.Machine.20160615.pdf |pdf]]) </br> </br> go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] biu26kcvbirpxbftxnilpyr0njg9y5k 3 250233 2410333 2409826 2022-07-30T00:05:01Z Dave Braunschweig 426084 /* Activity Review */ Reply wikitext text/x-wiki {{Robelbox|theme=9|title=Welcome!|width=100%}} <div style="{{Robelbox/pad}}"> '''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] Faendalimas!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Marshallsumter|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOUI JS signature icon LTR.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity. To [[Wikiversity:Introduction|get started]], you may <!-- The Left column --> <div style="width:50.0%; float:left"> * [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]]. * Visit a (kind of) [[Wikiversity:Random|random project]]. * [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]]. * Find out about [[Wikiversity:Research|research]] activities on Wikiversity. * [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left. * Enable VisualEditor under [[Special:Preferences#mw-prefsection-betafeatures|Beta]] settings to make article editing easier. </div> <!-- The Right column --> <div style="width:50.0%; float:left"> * Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity. * Give [[Wikiversity:Feedback|feedback]] about your initial observations * Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]]. * [[Wikiversity:Chat|Chat]] with other Wikiversitans on [irc://irc.freenode.net/wikiversity-en <kbd>#wikiversity-en</kbd>]. * Follow Wikiversity on [[twitter]] (http://twitter.com/Wikiversity) and [[identi.ca]] (http://identi.ca/group/wikiversity). </div> <br clear="both"/> You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 22:06, 9 June 2019 (UTC)</div> {{Robelbox/close}} == Suggestions == Links to your [https://orcid.org/0000-0003-1279-2722 ORCID page] and to your list of publications. [[User:Sylvain Ribault|Sylvain Ribault]] ([[User talk:Sylvain Ribault|discuss]] • [[Special:Contributions/Sylvain Ribault|contribs]]) 21:13, 10 June 2019 (UTC) :thank you, I have added the Orchid Page, Research Gate and the Author Page about me from Wikispecies. cheers [[User:Faendalimas|<span style="color: #004730">Scott Thomson</span>]] (<small class="nickname">Faendalimas</small>) ^{[[User talk:Faendalimas|<span style="color: maroon">talk</span>]]} 21:25, 10 June 2019 (UTC) == Custodian == Congratulations! You are now a Wikiversity custodian. I'm sure you're familiar with the tools. Please add yourself to [[Wikiversity:Staff]] and regularly monitor [[Wikiversity:Request custodian action]] and [[Wikiversity:Notices for custodians]]. Let us know whenever you have any questions. Thanks! -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 13:46, 2 November 2019 (UTC) Thank you for moving the discussion of Location Hypotheses of Atlantis. When, where and how may I start with moving the Location Hypotheses of Atlantis out of Draft? thanks again! [[User:RAYLEIGH22|RAYLEIGH22]] ([[User talk:RAYLEIGH22|discuss]] • [[Special:Contributions/RAYLEIGH22|contribs]]) 17:40, 5 November 2019 (UTC) Faendalimas, Congratulations on your achieving custodian status. I look forward to working with you! [[User:RAYLEIGH22|RAYLEIGH22]] ([[User talk:RAYLEIGH22|discuss]] • [[Special:Contributions/RAYLEIGH22|contribs]]) 13:54, 6 November 2019 (UTC) == thankless job == Helping to develop policy is often a tedious and underappreciated effot. I would like to thank you for your work on the CU policy and encourage you to continue these helpful contributions to our project. We greatly appreciate your efforts to improve our site and we want you to know that we value the thought and care that you have demonstrated in our community discussions. Congratulations on your recent custodianship! I'm confident that your participation will improve wikiversity. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 06:52, 4 November 2019 (UTC) :Thankyou its all cool, it is pretty much the same everywhere. How long do you wish to leave this policy up? It seems to have been accepted. Cheers [[User:Faendalimas|<span style="color: #004730">Scott Thomson</span>]] (<small class="nickname">Faendalimas</small>) ^{[[User talk:Faendalimas|<span style="color: maroon">talk</span>]]} 06:56, 4 November 2019 (UTC) ::Usually I would close a discussion like this as SNOW as it is obvious that there is very strong support and little chance of substantive objection. Given that CU is a more serious consideration than most routine proposals that we discuss I'd like to leave it open for a little longer than we typically would. This is merely to ensure that the process is not rushed and to provide time for infrequent contributors to participate, should they choose. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 07:24, 4 November 2019 (UTC) :::Two weeks is generally a good amount of time for such proposals. This would be about 7 November to hit that point, it could go a little longer though. I would suggest that if you plan on seeing if we can get local checkusers it may be worth applying for those as soon as the policy is in place, while its fresh. Yourself and Dave are going to need to mass mail the members, carefully of course, however if Justin and I apply as some seem to want us to we cannot be involved in that as it would be seen as canvassing. Do not be surpised if some Stewards and Global Sysops cast votes in a vote like that. Cheers [[User:Faendalimas|<span style="color: #004730">Scott Thomson</span>]] (<small class="nickname">Faendalimas</small>) ^{[[User talk:Faendalimas|<span style="color: maroon">talk</span>]]} 07:28, 4 November 2019 (UTC) I must change this for ALL of you. A sincere thank you to everyone who is working to make Wikiversity possible. May this site provide a research home for all of those who need one and may the research create breakthroughs and advancement in many subjects that would not have been possible without Wikiversity. Thank you all once again! [[User:RAYLEIGH22|RAYLEIGH22]] ([[User talk:RAYLEIGH22|discuss]] • [[Special:Contributions/RAYLEIGH22|contribs]]) 13:58, 6 November 2019 (UTC) *I closed the CU discussion because the rate of new comments had slowed down considerably. It didn't look like we'd reached the criteria in a reasonable amount of time. Thank you again for working on the policy. That framework will be important as the community grows. Perhaps we'll reach a threshold where participation in these discussions is sufficient to reach the requirements. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 19:41, 29 January 2020 (UTC) **Yes {{ping|Mu301}} perfectly reasonable to close at this point. I am not worried about it. You have a policy in place now thats the important thing. Local CU´s can wait for now. Cheers [[User:Faendalimas|<span style="color: #004730">Scott Thomson</span>]] (<small class="nickname">Faendalimas</small>) ^{[[User talk:Faendalimas|<span style="color: maroon">talk</span>]]} 14:57, 1 February 2020 (UTC) == Wikiversity:Requests for CheckUser == Please review: [[Wikiversity:Requests for CheckUser]]. Do you have any suggestions for adding instructions to this page? Not that we need to anytime soon, as it will likely take quite a bit of time before we need it. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 05:38, 27 November 2019 (UTC) :{{ping|Mu301}} yeah there is quite a bit needed in that page, it works better if its a form. This is the Wikispecies one [[species:Wikispecies:Requests_for_checkuser|here]] it has to explain the various definitions etc we use for everyone. I will put it together if we end up needing it if you like. I figured it would be easier to determine if we will have local checkusers first. Cheers [[User:Faendalimas|<span style="color: #004730">Scott Thomson</span>]] (<small class="nickname">Faendalimas</small>) ^{[[User talk:Faendalimas|<span style="color: maroon">talk</span>]]} 06:29, 27 November 2019 (UTC) ::Yes, I'd wait until there are about a dozen or so supports. Also, I archived the old page to the talk. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 21:17, 27 November 2019 (UTC) == How we will see unregistered users == <section begin=content/> Hi! You get this message because you are an admin on a Wikimedia wiki. When someone edits a Wikimedia wiki without being logged in today, we show their IP address. As you may already know, we will not be able to do this in the future. This is a decision by the Wikimedia Foundation Legal department, because norms and regulations for privacy online have changed. Instead of the IP we will show a masked identity. You as an admin '''will still be able to access the IP'''. There will also be a new user right for those who need to see the full IPs of unregistered users to fight vandalism, harassment and spam without being admins. Patrollers will also see part of the IP even without this user right. We are also working on [[m:IP Editing: Privacy Enhancement and Abuse Mitigation/Improving tools|better tools]] to help. If you have not seen it before, you can [[m:IP Editing: Privacy Enhancement and Abuse Mitigation|read more on Meta]]. If you want to make sure you don’t miss technical changes on the Wikimedia wikis, you can [[m:Global message delivery/Targets/Tech ambassadors|subscribe]] to [[m:Tech/News|the weekly technical newsletter]]. We have [[m:IP Editing: Privacy Enhancement and Abuse Mitigation#IP Masking Implementation Approaches (FAQ)|two suggested ways]] this identity could work. '''We would appreciate your feedback''' on which way you think would work best for you and your wiki, now and in the future. You can [[m:Talk:IP Editing: Privacy Enhancement and Abuse Mitigation|let us know on the talk page]]. You can write in your language. The suggestions were posted in October and we will decide after 17 January. Thank you. /[[m:User:Johan (WMF)|Johan (WMF)]]<section end=content/> 18:14, 4 January 2022 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=User:Johan_(WMF)/Target_lists/Admins2022(3)&oldid=22532499 --> == Activity Review == Wikiversity has adopted a [[Wikiversity:Request_custodian_action#Reviews_for_Inactivity|policy]] that expects users with administrative rights to be active participants and to regularly use those rights (at least five edits and five actions within the last 12 months). It appears that you are no longer active at Wikiversity and do not currently have a need for administrative rights. If you would prefer to retain your administrative role, please rejoin us with your contributions and support. If your account remains inactive, we will need to request that stewards remove your administrative rights. Let me know if you have any questions, and thank you for your previous efforts on our behalf. [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 13:53, 25 July 2022 (UTC) :Hi {{ping|Dave Braunschweig}}, sorry I have been very busy and somewhat limited in my involvement in local wikis as I have spent the last 2 years on the Ombuds Commission for WMF. This year I am Chair of the OC. If you feel it necessary to remove these rights this is fair enough. I shall leave it to you. I have watched the pages but am as I said both limited to a degree and very busy for direct involvement. Thanks for the message. Cheers [[User:Faendalimas|<span style="color: #004730">Scott Thomson</span>]] (<small class="nickname">Faendalimas</small>) ^{[[User talk:Faendalimas|<span style="color: maroon">talk</span>]]} 15:51, 27 July 2022 (UTC) ::Thanks for the update. We have run into issues where we claim to be a small wiki in need of SWMT support, but they see our staff list and ask why. ::It would be without prejudice. You are welcome to request administrative rights again when you have time and use for them. ::Thanks! ::[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 00:05, 30 July 2022 (UTC) gbsf13dk0o096k1gfuwswcjdkzcd34i 2 257428 2410376 2410142 2022-07-30T02:22:19Z Archie97305 2915204 wikitext text/x-wiki https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/Sandbox https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/plenary https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/inKind https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/specialdelivery meritorium . meritorious : merit .or.iou.us {{User alternative account|VeronicaJeanAnderson}} ∨⚡\🗲↯/ϟ∧ ✮☆⚝⛤🟊✰✭▲◂◁◀◢⍟◶✪⚪⬤🔥◍⚫⨁⚉⨂❂✧✷✸✡✵ http://slither.io/ https://www.thescienceofpsychotherapy.com/behaviour-affection-and-emotional-control/ {| class="wikitable" style="text-align: center;" |+ ⚞🧿⚟_◞◜↷◝◟_◞◜⚞🧿⚟🧿⚞🧿⚟◝◟_◞◜↶◝◟_⚞🧿⚟ |- || ✪⚪⬤🔥◍⚫⨁ || [https://www.twitch.tv/archie97305 👀] | style="background:pink;" | <span style="color:#808080"> ≡ odd → +1 </span> | style="background:pink;" | <span style="color:#808080"> [ { ( East </span> | style="background:black;" | <span style="color:white"> ⚫🔴⚪○💮⭕</span> | style="background:#808080;" | <span style="color:pink"> West ) } ] </span> | style="background:#808080;" | <span style="color:pink"> iff even ⇒ ÷2 </span> |- || Primary || [https://www.amnesty.org/en/ 1] |- || Secondary || [https://philosophy.stackexchange.com/questions/31029/why-was-the-horseshoe-symbol-%E2%8A%83-selected-for-material-implication 2] |- || Tertiary || [https://en.wikiversity.org/wiki/Wikiversity:Main_Page 3] |- || Quaternary || [http://localhost:8080/ 4] |- || Quinary || [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson 5] |- || Senary || [https://en.wikiversity.org/wiki/User:Archie97305 6] |- || Septenary || [https://maritimearchaeological.org/beeswax-wreck/ 7] |- || Octenary || [https://www.youtube.com/freecodecamp 8] |- || nonary || [https://www.freecodecamp.org/ 9] |- ||Base Name ||[https://wordsmith.org/board/ubbthreads.php?ubb=showflat&Number=84101 `] |- || binary || 2 |- ||ternary ||3 |- ||quaternary ||4 |- ||quinary ||5 |- ||senary ||6 |- ||septenary ||7 |- ||octal ||8 |- ||nonary |- ||decimal |- ||undenary |- ||duodecimal |- ||hexadecimal ||16 |- ||vigesimal ||20 |- ||sexagesimal ||60 |} How do you want your water served when you get here? https://pubs.usgs.gov/sir/2005/5168/pdf/sir2005-5168.pdf Robert Lee Stinson %VOX "tautology club says hi" w 11am "Naturalist Society for the Humane Treatment of Monsters" from dnd game on twitter [https://www.youtube.com/watch?v=n7uNA5fO1iI rice ex in CA] https://www.oregonwild.org/about/blog/oregon-grizzly-country https://therevelator.org/yellowstone-grizzlies-unbearable-divides/ https://developer.mozilla.org/en-US/docs/Web/CSS/color_value/hsl https://www.researchgate.net/about Amare, Nicole & Manning, A.. (2012). Seeing typeface personality: Emotional responses to form as tone. IEEE International Professional Communication Conference. 1-9. 10.1109/IPCC.2012.6408605. Various studies have correlated specific visual characteristics of typefaces with specific overall emotional effects: curvilinear forms and open letter shapes generally feel “friendly” but also “formal” or “informal,” depending on other factors; large contrasts in stroke widths, cap height, and aspect ratio generally feel “interesting,” but also “attractive” or “aggressive,” depending on other factors; low-variety and low-contrast forms generally feel “professional” but also “reliable” or “boring.” Although the current findings on typeface personality are useful, they have not indicated a systematic explanation for why specific physical typeface forms have the specific emotion effects that they do. This paper will report results of an empirical study in which 102 participants indicated their immediate emotional responses to each of 36 distinct typeface designs. Results support correlation between specific typeface features (variety vs. contrast vs. pattern) and specific emotional parameters (amusement vs. agitation vs. focus), explaining findings of previous studies, suggesting various classroom approaches to purpose-driven typeface selection. {{User alternative account|VeronicaJeanAnderson}} {| class="wikitable" style="text-align: center;" |+ ᐪgenki-ness; +, -tachi . . . |- | style="background:black;" | <span style="color:white"> [ { ( A B E ) } ] </span> | style="background:black;" | <span style="color:white"> [ { ( [https://www.twitch.tv/archie97305 👀] ) } ]</span> |- || Primary | style="background:#FFFFE6;" | <span style="color:black"> index.html</span> || notepad/atom (atom is deprecated) |- || Secondary | style="background:#FFF2E6;" | <span style="color:black"> vue </span> || [https://www.vim.org/ vim] [https://github.com/vim/vim-win32-installer/releases installer] |- || Tertiary | style="background:white;" | <span style="color:black"> css </span> || global css @ || gg css @ || NPC css @ |- || Quaternary | style="background:#FFE6FB;" | <span style="color:black"> pug </span> |- || Quinary | style="background:#F9F9F9;" | <span style="color:pink"> (direct object) </span> |- || Senary || b |} https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/sandbox trying to create a 1 -> 2 -> 3 -> 4 -> 5 -> 6 system in the apartment here that can be copied from site to site using artistic threads to help a Nice And Proper NAP-er navigate between properties with ease while maintaining adequate supportive care that we all require to enable us to focus on whatever catches our fancy. sun; natural light; breathe; BGs carbs; hygiene; laundry away bedroom; needles; blood; garbage out kitchen/nutritional/study social/outreach/linking worlds back porch 0 -- Computer Science, information and general works {| class="wikitable" style="text-align: center;" |+ ᐪgenki-ness; +, -tachi . . . |- | style="background:black;" | <span style="color:white"> [ { ( T O P ) } ] </span> | style="background:black;" | <span style="color:white"> [ ℳ ] </span> | style="background:white;" | <span style="color:black"> { ¢ } </span> | style="background:#F9F9F9;" | <span style="color:pink"> ( ৳ ) </span> | style="background:black;" | <span style="color:white"> [ { ( I.n C.ase of E.mergency ) } ] </span> | style="background:teal;" | <span style="color:lime"> ᐪ l i p s c h i t z </span> || [https://www.youtube.com/watch?v=qrrz54UtkCc ᐪ] |- || Primary | style="background:#FFFFE6;" | <span style="color:black"> physical</span> | style="background:#FFE6E6;" | <span style="color:black"> emotional</span> | style="background:#E6EAFF;" | <span style="color:black"> social</span> || This reflects health enough to communicate with people intimately enough to address real immediate issues | style="background:#FFFFE6;" | <span style="color:teal"> ^ torikomu </span> ||[https://www.youtube.com/watch?v=YxvBPH4sArQ ^] |- || Secondary | style="background:#FFF2E6;" | <span style="color:black"> occupational</span> | style="background:#F2E6FF;" | <span style="color:black"> intellectual</span> | style="background:#E6FFEA;" | <span style="color:black"> environmental</span> || This reflects living somewhere promoting healthy reasoning | style="background:#FFE6E6;" | <span style="color: teal"> | kaizen | </span> || | |- || Tertiary | style="background:white;" | <span style="color:black"> spiritual</span> | style="background:#BFBFBF;" | <span style="color:white"> factual </span> | style="background:#F2F2F2;" | <span style="color:black"> nutritional</span> || This reflects healthy mindful every habits | style="background:#E6EAFF;" | <span style="color:teal"> . genkiness . .</span> || . |- || Quaternary | style="background:#FFE6FB;" | <span style="color:black"> generational</span> | style="background:#E6FFFF;" | <span style="color:black"> miscellaneal</span> | style="background:#F2E0CE;" | <span style="color:black"> punctuational</span> || This reflects having it all together enough to enjoy the holidays | style="background:#FFF2E6;" | <span style="color:lime"> # goblin </span> || # |- || Quinary | style="background:#F9F9F9;" | <span style="color:pink"> (direct object) </span> | style="background:white;" | <span style="color:black"> {verb} </span> | style="background:black;" | <span style="color:white"> [noun] </span> || This reflects deliberate professional progress | style="background:#F2E6FF;" | <span style="color:lime"> / tsugu /</span> || / |- || Senary || b || 〇 || x || This reflects influencing others | style="background:#E6FFEA;" | <span style="color:lime"> @ g @ g @ </span> || [https://www.youtube.com/watch?v=SYnVYJDxu2Q @] |} == 100 -- Philosophy and psychology == How can I use color to manipulate behavior and improve communication? ===named=== ==== Re⋮Beccaδ#639 ==== ===== rebeccapurple :: #663399 ===== https://meyerweb.com/eric/thoughts/2014/06/19/rebeccapurple/ ====black==== ====white==== ====græy==== ====pink==== ====indigo==== ====midnightblue==== ===hex=== ====#fff==== ====#fff==== {| class="wikitable" style="text-align: center;" |+ |- | style="background:black;" | <span style="color:white"> [ white { on black ⚞🧿⚟ #fff on #000 ⚞🧿⚟ } ] </span> |- | style="background:#808080;" | <span style="color:pink"> [ pink { on 50% grey ⚞🧿⚟ #ffc0cb on #808080 ⚞🧿⚟ } ] </span> |- | style="background:#808080;" | <span style="color:#191970"> [ midnightblue { on 50% grey ⚞🧿⚟ #191970 on #808080 ⚞🧿⚟ } ] </span> |- | style="background:#808080;" | <span style="color:#4b0082"> [ indigo { on 50% grey ⚞🧿⚟ #4b0082 on #808080 ⚞🧿⚟ } ] </span> |} ===cmyk=== https://colordesigner.io/convert/cmyktohex ====gg on cmyk(0,0,0,33.3) w|materializecss.com==== https://materializecss.com/color.html {| class="wikitable" style="text-align: center;" |+ |- | style="background:#ababab" | <span style="color:#fff9c4"> [ gg_yellow { on cmyk(0,0,0,33.3) ⚞🧿⚟ #fff9c4 on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#ffe0b2"> [ gg_orange { on cmyk(0,0,0,33.3) ⚞🧿⚟ #ffe0b2 on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#ffcdd2"> [ gg_red { on cmyk(0,0,0,33.3) ⚞🧿⚟ #ffcdd2 on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#e1bee7"> [ gg_purple { on cmyk(0,0,0,33.3) ⚞🧿⚟ #e1bee7 on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#bbdefb"> [ gg_blue { on cmyk(0,0,0,33.3) ⚞🧿⚟ #bbdefb on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#c8e6c9"> [ gg_green { on cmyk(0,0,0,33.3) ⚞🧿⚟ #c8e6c9 on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#efefef"> [ gg_white { on cmyk(0,0,0,33.3) ⚞🧿⚟ #efefef on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#111"> [ gg_black { on cmyk(0,0,0,33.3) ⚞🧿⚟ #111 on #ababab ⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#808080"> [ gg_grey { on cmyk(0,0,0,33.3) ⚞🧿⚟ #808080 on #ababab ⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#f8bbd0"> [ gg_pink { on cmyk(0,0,0,33.3) ⚞🧿⚟ #f8bbd0 on #ababab ⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#b2ebf2"> [ gg_cyan { on cmyk(0,0,0,33.3) ⚞🧿⚟ #b2ebf2 on #ababab ⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#d7ccc8"> [ gg_brown { on cmyk(0,0,0,33.3) ⚞🧿⚟ #d7ccc8 on #ababab )⚞🧿⚟} ] </span> |} ===rgba=== === TrumPutin-ism === Trump has demonstrably alienated the USA from allies both foreign and domestic. While Oregon's AG works on Epstein and Weinstein, contemporaneous crimes go unabated and have created a new problem where otherwise law abiding folk find themselves on the wrong side of the law. Oregon doesn't have enough public defenders to fight violent crime, yet children are alienated from their church and families to hide atrocities they don't even know about. == 200 -- Religion == Royal We 1000 things I did 1992-2022 other than lie my way onto the supreme court to overturn Roe v Wade {| |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ 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|| ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |} == 300 -- Social sciences == https://wattention.com/traditional-rice-harvesting-in-japan/ https://www.wwoofjapan.com/home/index.php?lang=en == 400 -- Language == === Programming === ==== .png ==== ==== Esperanto ==== ==== HTML ==== ==== PUG ==== == 500 -- Pure Science == == 600 -- Technology == === local hosts=== [http://localhost:8080/ 8080] file:///D:/index.html === Roland SP 404MKII === https://www.roland.com/global/products/sp-404mk2/ https://www.roland.com/global/support/by_product/sp-404mk2/owners_manuals/ @https://static.roland.com/manuals/sp-404mk2_app/eng/19610757.html === VIM === https://vim-adventures.com/ == 700 -- Arts and recreation == == 800 -- Literature == == 900 -- History and geography == https://geology.com/stories/13/bear-areas/ === pre-2030 === 2022 "booked" by Hillsboro Police for sending email addressing "Christian Hate" and "Spiritual War" along with "exorcisms" and "Halloween Hysteria" in Marion County, OR where Salem Police Department abdicated from protecting some children in Salem from 2016-2021. 2021 Kaiser Permanente promised cash settlement to mitigate their abdication in Marion County. KP lawyer with intimate details about my vagina: terrence .j . loeber@kp.org 2012 "unliked" on FB by some Nazarene peers after openly questioning Alex Jones' allegation that Sandy Hook didn't happen and asking for compassion for parents who were called actors while they grieved publicly through no choice of their own. 2011 Lupron given by KP for menorrhagia as alternative to b/c pills first rxd in 86. How many women who have "mostly" been on b/c pills from 87-11 are obese? Why no menorrhagia while immersed in Japan? How close to a traditional Japanese diet can I get in the Willamette Valley and how close to no meat will my body allow? === TrumPutish War Against Humanity === Trump has demonstrably alienated the USA from allies both foreign and domestic. While Oregon's AG works on Epstein and Weinstein, contemporaneous crimes go unabated and have created a new problem where otherwise law abiding folk find themselves on the wrong side of the law. Oregon doesn't have enough public defenders to fight violent crime, yet children are alienated from their church and families to hide atrocities they don't even know about. === Ring of Fire === === Post "Roe v Wade" === Who did Roe v Wade protect? Why would a Nazarene raised pro-life support an "underground" network post Roe v Wade? === Our Contemporary "Underground Railroad" needs a submarine? === Why did Portland, OR close the Shanghai Tunnels recently? Human Trafficking through Astoria, OR has been going on "forever". How do we align an "underground railroad" with contemporary supports? == 10 -- A & + == == 11 -- B * x == == 12 -- C f(◯) == == 13 -- D Δ δ ƍ ≜ 𝜟 𝝳 == == 14 -- E 🐘 𓃰 == == 15 -- F == == 16 -- G == == 17 -- H == == 18 -- I == == 19 -- J == == 20 -- K == == 21 -- L == == 22 -- M == == 23 -- N == == 24 -- O == == 25 -- P == == 26 -- Q == == 27 -- R == == 28 -- S == == 29 -- T == == 30 -- U == == 31 -- V == == 32 -- W == == 33 -- X == == 34 -- Y == == 35 -- Z == le47fffi9zqzuk5btj6a47yauqevnar 2410382 2410376 2022-07-30T03:00:00Z A020f0ff 2928078 wikitext text/x-wiki ○ Ya (hiragana: や, katakana: ヤ) https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/Sandbox https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/plenary https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/inKind https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/specialdelivery meritorium . meritorious : merit .or.iou.us {{User alternative account|VeronicaJeanAnderson}} ∨⚡\🗲↯/ϟ∧ ✮☆⚝⛤🟊✰✭▲◂◁◀◢⍟◶✪⚪⬤🔥◍⚫⨁⚉⨂❂✧✷✸✡✵ http://slither.io/ https://www.thescienceofpsychotherapy.com/behaviour-affection-and-emotional-control/ {| class="wikitable" style="text-align: center;" |+ ⚞🧿⚟_◞◜↷◝◟_◞◜⚞🧿⚟🧿⚞🧿⚟◝◟_◞◜↶◝◟_⚞🧿⚟ |- || ✪⚪⬤🔥◍⚫⨁ || [https://www.twitch.tv/archie97305 👀] | style="background:pink;" | <span style="color:#808080"> ≡ odd → +1 </span> | style="background:pink;" | <span style="color:#808080"> [ { ( East </span> | style="background:black;" | <span style="color:white"> ⚫🔴⚪○💮⭕</span> | style="background:#808080;" | <span style="color:pink"> West ) } ] </span> | style="background:#808080;" | <span style="color:pink"> iff even ⇒ ÷2 </span> |- || Primary || [https://www.amnesty.org/en/ 1] |- || Secondary || [https://philosophy.stackexchange.com/questions/31029/why-was-the-horseshoe-symbol-%E2%8A%83-selected-for-material-implication 2] |- || Tertiary || [https://en.wikiversity.org/wiki/Wikiversity:Main_Page 3] |- || Quaternary || [http://localhost:8080/ 4] |- || Quinary || [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson 5] |- || Senary || [https://en.wikiversity.org/wiki/User:Archie97305 6] |- || Septenary || [https://maritimearchaeological.org/beeswax-wreck/ 7] |- || Octenary || [https://www.youtube.com/freecodecamp 8] |- || nonary || [https://www.freecodecamp.org/ 9] |- ||Base Name ||[https://wordsmith.org/board/ubbthreads.php?ubb=showflat&Number=84101 `] |- || binary || 2 |- ||ternary ||3 |- ||quaternary ||4 |- ||quinary ||5 |- ||senary ||6 |- ||septenary ||7 |- ||octal ||8 |- ||nonary |- ||decimal |- ||undenary |- ||duodecimal |- ||hexadecimal ||16 |- ||vigesimal ||20 |- ||sexagesimal ||60 |} How do you want your water served when you get here? https://pubs.usgs.gov/sir/2005/5168/pdf/sir2005-5168.pdf Robert Lee Stinson %VOX "tautology club says hi" w 11am "Naturalist Society for the Humane Treatment of Monsters" from dnd game on twitter [https://www.youtube.com/watch?v=n7uNA5fO1iI rice ex in CA] https://www.oregonwild.org/about/blog/oregon-grizzly-country https://therevelator.org/yellowstone-grizzlies-unbearable-divides/ https://developer.mozilla.org/en-US/docs/Web/CSS/color_value/hsl https://www.researchgate.net/about Amare, Nicole & Manning, A.. (2012). Seeing typeface personality: Emotional responses to form as tone. IEEE International Professional Communication Conference. 1-9. 10.1109/IPCC.2012.6408605. Various studies have correlated specific visual characteristics of typefaces with specific overall emotional effects: curvilinear forms and open letter shapes generally feel “friendly” but also “formal” or “informal,” depending on other factors; large contrasts in stroke widths, cap height, and aspect ratio generally feel “interesting,” but also “attractive” or “aggressive,” depending on other factors; low-variety and low-contrast forms generally feel “professional” but also “reliable” or “boring.” Although the current findings on typeface personality are useful, they have not indicated a systematic explanation for why specific physical typeface forms have the specific emotion effects that they do. This paper will report results of an empirical study in which 102 participants indicated their immediate emotional responses to each of 36 distinct typeface designs. Results support correlation between specific typeface features (variety vs. contrast vs. pattern) and specific emotional parameters (amusement vs. agitation vs. focus), explaining findings of previous studies, suggesting various classroom approaches to purpose-driven typeface selection. {{User alternative account|VeronicaJeanAnderson}} {| class="wikitable" style="text-align: center;" |+ ᐪgenki-ness; +, -tachi . . . |- | style="background:black;" | <span style="color:white"> [ { ( A B E ) } ] </span> | style="background:black;" | <span style="color:white"> [ { ( [https://www.twitch.tv/archie97305 👀] ) } ]</span> |- || Primary | style="background:#FFFFE6;" | <span style="color:black"> index.html</span> || notepad/atom (atom is deprecated) |- || Secondary | style="background:#FFF2E6;" | <span style="color:black"> vue </span> || [https://www.vim.org/ vim] [https://github.com/vim/vim-win32-installer/releases installer] |- || Tertiary | style="background:white;" | <span style="color:black"> css </span> || global css @ || gg css @ || NPC css @ |- || Quaternary | style="background:#FFE6FB;" | <span style="color:black"> pug </span> |- || Quinary | style="background:#F9F9F9;" | <span style="color:pink"> (direct object) </span> |- || Senary || b |} https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/sandbox trying to create a 1 -> 2 -> 3 -> 4 -> 5 -> 6 system in the apartment here that can be copied from site to site using artistic threads to help a Nice And Proper NAP-er navigate between properties with ease while maintaining adequate supportive care that we all require to enable us to focus on whatever catches our fancy. sun; natural light; breathe; BGs carbs; hygiene; laundry away bedroom; needles; blood; garbage out kitchen/nutritional/study social/outreach/linking worlds back porch 0 -- Computer Science, information and general works {| class="wikitable" style="text-align: center;" |+ ᐪgenki-ness; +, -tachi . . . |- | style="background:black;" | <span style="color:white"> [ { ( T O P ) } ] </span> | style="background:black;" | <span style="color:white"> [ ℳ ] </span> | style="background:white;" | <span style="color:black"> { ¢ } </span> | style="background:#F9F9F9;" | <span style="color:pink"> ( ৳ ) </span> | style="background:black;" | <span style="color:white"> [ { ( I.n C.ase of E.mergency ) } ] </span> | style="background:teal;" | <span style="color:lime"> ᐪ l i p s c h i t z </span> || [https://www.youtube.com/watch?v=qrrz54UtkCc ᐪ] |- || Primary | style="background:#FFFFE6;" | <span style="color:black"> physical</span> | style="background:#FFE6E6;" | <span style="color:black"> emotional</span> | style="background:#E6EAFF;" | <span style="color:black"> social</span> || This reflects health enough to communicate with people intimately enough to address real immediate issues | style="background:#FFFFE6;" | <span style="color:teal"> ^ torikomu </span> ||[https://www.youtube.com/watch?v=YxvBPH4sArQ ^] |- || Secondary | style="background:#FFF2E6;" | <span style="color:black"> occupational</span> | style="background:#F2E6FF;" | <span style="color:black"> intellectual</span> | style="background:#E6FFEA;" | <span style="color:black"> environmental</span> || This reflects living somewhere promoting healthy reasoning | style="background:#FFE6E6;" | <span style="color: teal"> | kaizen | </span> || | |- || Tertiary | style="background:white;" | <span style="color:black"> spiritual</span> | style="background:#BFBFBF;" | <span style="color:white"> factual </span> | style="background:#F2F2F2;" | <span style="color:black"> nutritional</span> || This reflects healthy mindful every habits | style="background:#E6EAFF;" | <span style="color:teal"> . genkiness . .</span> || . |- || Quaternary | style="background:#FFE6FB;" | <span style="color:black"> generational</span> | style="background:#E6FFFF;" | <span style="color:black"> miscellaneal</span> | style="background:#F2E0CE;" | <span style="color:black"> punctuational</span> || This reflects having it all together enough to enjoy the holidays | style="background:#FFF2E6;" | <span style="color:lime"> # goblin </span> || # |- || Quinary | style="background:#F9F9F9;" | <span style="color:pink"> (direct object) </span> | style="background:white;" | <span style="color:black"> {verb} </span> | style="background:black;" | <span style="color:white"> [noun] </span> || This reflects deliberate professional progress | style="background:#F2E6FF;" | <span style="color:lime"> / tsugu /</span> || / |- || Senary || b || 〇 || x || This reflects influencing others | style="background:#E6FFEA;" | <span style="color:lime"> @ g @ g @ </span> || [https://www.youtube.com/watch?v=SYnVYJDxu2Q @] |} == 100 -- Philosophy and psychology == How can I use color to manipulate behavior and improve communication? ===named=== ==== Re⋮Beccaδ#639 ==== ===== rebeccapurple :: #663399 ===== https://meyerweb.com/eric/thoughts/2014/06/19/rebeccapurple/ ====black==== ====white==== ====græy==== ====pink==== ====indigo==== ====midnightblue==== ===hex=== ====#fff==== ====#fff==== {| class="wikitable" style="text-align: center;" |+ |- | style="background:black;" | <span style="color:white"> [ white { on black ⚞🧿⚟ #fff on #000 ⚞🧿⚟ } ] </span> |- | style="background:#808080;" | <span style="color:pink"> [ pink { on 50% grey ⚞🧿⚟ #ffc0cb on #808080 ⚞🧿⚟ } ] </span> |- | style="background:#808080;" | <span style="color:#191970"> [ midnightblue { on 50% grey ⚞🧿⚟ #191970 on #808080 ⚞🧿⚟ } ] </span> |- | style="background:#808080;" | <span style="color:#4b0082"> [ indigo { on 50% grey ⚞🧿⚟ #4b0082 on #808080 ⚞🧿⚟ } ] </span> |} ===cmyk=== https://colordesigner.io/convert/cmyktohex ====gg on cmyk(0,0,0,33.3) w|materializecss.com==== https://materializecss.com/color.html {| class="wikitable" style="text-align: center;" |+ |- | style="background:#ababab" | <span style="color:#fff9c4"> [ gg_yellow { on cmyk(0,0,0,33.3) ⚞🧿⚟ #fff9c4 on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#ffe0b2"> [ gg_orange { on cmyk(0,0,0,33.3) ⚞🧿⚟ #ffe0b2 on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#ffcdd2"> [ gg_red { on cmyk(0,0,0,33.3) ⚞🧿⚟ #ffcdd2 on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#e1bee7"> [ gg_purple { on cmyk(0,0,0,33.3) ⚞🧿⚟ #e1bee7 on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#bbdefb"> [ gg_blue { on cmyk(0,0,0,33.3) ⚞🧿⚟ #bbdefb on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#c8e6c9"> [ gg_green { on cmyk(0,0,0,33.3) ⚞🧿⚟ #c8e6c9 on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#efefef"> [ gg_white { on cmyk(0,0,0,33.3) ⚞🧿⚟ #efefef on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#111"> [ gg_black { on cmyk(0,0,0,33.3) ⚞🧿⚟ #111 on #ababab ⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#808080"> [ gg_grey { on cmyk(0,0,0,33.3) ⚞🧿⚟ #808080 on #ababab ⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#f8bbd0"> [ gg_pink { on cmyk(0,0,0,33.3) ⚞🧿⚟ #f8bbd0 on #ababab ⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#b2ebf2"> [ gg_cyan { on cmyk(0,0,0,33.3) ⚞🧿⚟ #b2ebf2 on #ababab ⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#d7ccc8"> [ gg_brown { on cmyk(0,0,0,33.3) ⚞🧿⚟ #d7ccc8 on #ababab )⚞🧿⚟} ] </span> |} ===rgba=== === TrumPutin-ism === Trump has demonstrably alienated the USA from allies both foreign and domestic. While Oregon's AG works on Epstein and Weinstein, contemporaneous crimes go unabated and have created a new problem where otherwise law abiding folk find themselves on the wrong side of the law. Oregon doesn't have enough public defenders to fight violent crime, yet children are alienated from their church and families to hide atrocities they don't even know about. == 200 -- Religion == Royal We 1000 things I did 1992-2022 other than lie my way onto the supreme court to overturn Roe v Wade {| |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |} == 300 -- Social sciences == https://wattention.com/traditional-rice-harvesting-in-japan/ https://www.wwoofjapan.com/home/index.php?lang=en == 400 -- Language == === Programming === ==== .png ==== ==== Esperanto ==== ==== HTML ==== ==== PUG ==== == 500 -- Pure Science == == 600 -- Technology == === local hosts=== [http://localhost:8080/ 8080] file:///D:/index.html === Roland SP 404MKII === https://www.roland.com/global/products/sp-404mk2/ https://www.roland.com/global/support/by_product/sp-404mk2/owners_manuals/ @https://static.roland.com/manuals/sp-404mk2_app/eng/19610757.html === VIM === https://vim-adventures.com/ == 700 -- Arts and recreation == == 800 -- Literature == == 900 -- History and geography == https://geology.com/stories/13/bear-areas/ === pre-2030 === 2022 "booked" by Hillsboro Police for sending email addressing "Christian Hate" and "Spiritual War" along with "exorcisms" and "Halloween Hysteria" in Marion County, OR where Salem Police Department abdicated from protecting some children in Salem from 2016-2021. 2021 Kaiser Permanente promised cash settlement to mitigate their abdication in Marion County. KP lawyer with intimate details about my vagina: terrence .j . loeber@kp.org 2012 "unliked" on FB by some Nazarene peers after openly questioning Alex Jones' allegation that Sandy Hook didn't happen and asking for compassion for parents who were called actors while they grieved publicly through no choice of their own. 2011 Lupron given by KP for menorrhagia as alternative to b/c pills first rxd in 86. How many women who have "mostly" been on b/c pills from 87-11 are obese? Why no menorrhagia while immersed in Japan? How close to a traditional Japanese diet can I get in the Willamette Valley and how close to no meat will my body allow? === TrumPutish War Against Humanity === Trump has demonstrably alienated the USA from allies both foreign and domestic. While Oregon's AG works on Epstein and Weinstein, contemporaneous crimes go unabated and have created a new problem where otherwise law abiding folk find themselves on the wrong side of the law. Oregon doesn't have enough public defenders to fight violent crime, yet children are alienated from their church and families to hide atrocities they don't even know about. === Ring of Fire === === Post "Roe v Wade" === Who did Roe v Wade protect? Why would a Nazarene raised pro-life support an "underground" network post Roe v Wade? === Our Contemporary "Underground Railroad" needs a submarine? === Why did Portland, OR close the Shanghai Tunnels recently? Human Trafficking through Astoria, OR has been going on "forever". How do we align an "underground railroad" with contemporary supports? == 10 -- A & + == == 11 -- B * x == == 12 -- C f(◯) == == 13 -- D Δ δ ƍ ≜ 𝜟 𝝳 == == 14 -- E 🐘 𓃰 == == 15 -- F == == 16 -- G == == 17 -- H == == 18 -- I == == 19 -- J == == 20 -- K == == 21 -- L == == 22 -- M == == 23 -- N == == 24 -- O == == 25 -- P == == 26 -- Q == == 27 -- R == == 28 -- S == == 29 -- T == == 30 -- U == == 31 -- V == == 32 -- W == == 33 -- X == == 34 -- Y == == 35 -- Z == t8bxjcl5tdohoo7kkc3qusjgqk1ictx 2410383 2410382 2022-07-30T03:01:47Z A020f0ff 2928078 wikitext text/x-wiki ○ Ya (hiragana: や, katakana: ヤ) ∘ ヤフー https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/Sandbox https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/plenary https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/inKind https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/specialdelivery meritorium . meritorious : merit .or.iou.us {{User alternative account|VeronicaJeanAnderson}} ∨⚡\🗲↯/ϟ∧ ✮☆⚝⛤🟊✰✭▲◂◁◀◢⍟◶✪⚪⬤🔥◍⚫⨁⚉⨂❂✧✷✸✡✵ http://slither.io/ https://www.thescienceofpsychotherapy.com/behaviour-affection-and-emotional-control/ {| class="wikitable" style="text-align: center;" |+ ⚞🧿⚟_◞◜↷◝◟_◞◜⚞🧿⚟🧿⚞🧿⚟◝◟_◞◜↶◝◟_⚞🧿⚟ |- || ✪⚪⬤🔥◍⚫⨁ || [https://www.twitch.tv/archie97305 👀] | style="background:pink;" | <span style="color:#808080"> ≡ odd → +1 </span> | style="background:pink;" | <span style="color:#808080"> [ { ( East </span> | style="background:black;" | <span style="color:white"> ⚫🔴⚪○💮⭕</span> | style="background:#808080;" | <span style="color:pink"> West ) } ] </span> | style="background:#808080;" | <span style="color:pink"> iff even ⇒ ÷2 </span> |- || Primary || [https://www.amnesty.org/en/ 1] |- || Secondary || [https://philosophy.stackexchange.com/questions/31029/why-was-the-horseshoe-symbol-%E2%8A%83-selected-for-material-implication 2] |- || Tertiary || [https://en.wikiversity.org/wiki/Wikiversity:Main_Page 3] |- || Quaternary || [http://localhost:8080/ 4] |- || Quinary || [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson 5] |- || Senary || [https://en.wikiversity.org/wiki/User:Archie97305 6] |- || Septenary || [https://maritimearchaeological.org/beeswax-wreck/ 7] |- || Octenary || [https://www.youtube.com/freecodecamp 8] |- || nonary || [https://www.freecodecamp.org/ 9] |- ||Base Name ||[https://wordsmith.org/board/ubbthreads.php?ubb=showflat&Number=84101 `] |- || binary || 2 |- ||ternary ||3 |- ||quaternary ||4 |- ||quinary ||5 |- ||senary ||6 |- ||septenary ||7 |- ||octal ||8 |- ||nonary |- ||decimal |- ||undenary |- ||duodecimal |- ||hexadecimal ||16 |- ||vigesimal ||20 |- ||sexagesimal ||60 |} How do you want your water served when you get here? https://pubs.usgs.gov/sir/2005/5168/pdf/sir2005-5168.pdf Robert Lee Stinson %VOX "tautology club says hi" w 11am "Naturalist Society for the Humane Treatment of Monsters" from dnd game on twitter [https://www.youtube.com/watch?v=n7uNA5fO1iI rice ex in CA] https://www.oregonwild.org/about/blog/oregon-grizzly-country https://therevelator.org/yellowstone-grizzlies-unbearable-divides/ https://developer.mozilla.org/en-US/docs/Web/CSS/color_value/hsl https://www.researchgate.net/about Amare, Nicole & Manning, A.. (2012). Seeing typeface personality: Emotional responses to form as tone. IEEE International Professional Communication Conference. 1-9. 10.1109/IPCC.2012.6408605. Various studies have correlated specific visual characteristics of typefaces with specific overall emotional effects: curvilinear forms and open letter shapes generally feel “friendly” but also “formal” or “informal,” depending on other factors; large contrasts in stroke widths, cap height, and aspect ratio generally feel “interesting,” but also “attractive” or “aggressive,” depending on other factors; low-variety and low-contrast forms generally feel “professional” but also “reliable” or “boring.” Although the current findings on typeface personality are useful, they have not indicated a systematic explanation for why specific physical typeface forms have the specific emotion effects that they do. This paper will report results of an empirical study in which 102 participants indicated their immediate emotional responses to each of 36 distinct typeface designs. Results support correlation between specific typeface features (variety vs. contrast vs. pattern) and specific emotional parameters (amusement vs. agitation vs. focus), explaining findings of previous studies, suggesting various classroom approaches to purpose-driven typeface selection. {{User alternative account|VeronicaJeanAnderson}} {| class="wikitable" style="text-align: center;" |+ ᐪgenki-ness; +, -tachi . . . |- | style="background:black;" | <span style="color:white"> [ { ( A B E ) } ] </span> | style="background:black;" | <span style="color:white"> [ { ( [https://www.twitch.tv/archie97305 👀] ) } ]</span> |- || Primary | style="background:#FFFFE6;" | <span style="color:black"> index.html</span> || notepad/atom (atom is deprecated) |- || Secondary | style="background:#FFF2E6;" | <span style="color:black"> vue </span> || [https://www.vim.org/ vim] [https://github.com/vim/vim-win32-installer/releases installer] |- || Tertiary | style="background:white;" | <span style="color:black"> css </span> || global css @ || gg css @ || NPC css @ |- || Quaternary | style="background:#FFE6FB;" | <span style="color:black"> pug </span> |- || Quinary | style="background:#F9F9F9;" | <span style="color:pink"> (direct object) </span> |- || Senary || b |} https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/sandbox trying to create a 1 -> 2 -> 3 -> 4 -> 5 -> 6 system in the apartment here that can be copied from site to site using artistic threads to help a Nice And Proper NAP-er navigate between properties with ease while maintaining adequate supportive care that we all require to enable us to focus on whatever catches our fancy. sun; natural light; breathe; BGs carbs; hygiene; laundry away bedroom; needles; blood; garbage out kitchen/nutritional/study social/outreach/linking worlds back porch 0 -- Computer Science, information and general works {| class="wikitable" style="text-align: center;" |+ ᐪgenki-ness; +, -tachi . . . |- | style="background:black;" | <span style="color:white"> [ { ( T O P ) } ] </span> | style="background:black;" | <span style="color:white"> [ ℳ ] </span> | style="background:white;" | <span style="color:black"> { ¢ } </span> | style="background:#F9F9F9;" | <span style="color:pink"> ( ৳ ) </span> | style="background:black;" | <span style="color:white"> [ { ( I.n C.ase of E.mergency ) } ] </span> | style="background:teal;" | <span style="color:lime"> ᐪ l i p s c h i t z </span> || [https://www.youtube.com/watch?v=qrrz54UtkCc ᐪ] |- || Primary | style="background:#FFFFE6;" | <span style="color:black"> physical</span> | style="background:#FFE6E6;" | <span style="color:black"> emotional</span> | style="background:#E6EAFF;" | <span style="color:black"> social</span> || This reflects health enough to communicate with people intimately enough to address real immediate issues | style="background:#FFFFE6;" | <span style="color:teal"> ^ torikomu </span> ||[https://www.youtube.com/watch?v=YxvBPH4sArQ ^] |- || Secondary | style="background:#FFF2E6;" | <span style="color:black"> occupational</span> | style="background:#F2E6FF;" | <span style="color:black"> intellectual</span> | style="background:#E6FFEA;" | <span style="color:black"> environmental</span> || This reflects living somewhere promoting healthy reasoning | style="background:#FFE6E6;" | <span style="color: teal"> | kaizen | </span> || | |- || Tertiary | style="background:white;" | <span style="color:black"> spiritual</span> | style="background:#BFBFBF;" | <span style="color:white"> factual </span> | style="background:#F2F2F2;" | <span style="color:black"> nutritional</span> || This reflects healthy mindful every habits | style="background:#E6EAFF;" | <span style="color:teal"> . genkiness . .</span> || . |- || Quaternary | style="background:#FFE6FB;" | <span style="color:black"> generational</span> | style="background:#E6FFFF;" | <span style="color:black"> miscellaneal</span> | style="background:#F2E0CE;" | <span style="color:black"> punctuational</span> || This reflects having it all together enough to enjoy the holidays | style="background:#FFF2E6;" | <span style="color:lime"> # goblin </span> || # |- || Quinary | style="background:#F9F9F9;" | <span style="color:pink"> (direct object) </span> | style="background:white;" | <span style="color:black"> {verb} </span> | style="background:black;" | <span style="color:white"> [noun] </span> || This reflects deliberate professional progress | style="background:#F2E6FF;" | <span style="color:lime"> / tsugu /</span> || / |- || Senary || b || 〇 || x || This reflects influencing others | style="background:#E6FFEA;" | <span style="color:lime"> @ g @ g @ </span> || [https://www.youtube.com/watch?v=SYnVYJDxu2Q @] |} == 100 -- Philosophy and psychology == How can I use color to manipulate behavior and improve communication? ===named=== ==== Re⋮Beccaδ#639 ==== ===== rebeccapurple :: #663399 ===== https://meyerweb.com/eric/thoughts/2014/06/19/rebeccapurple/ ====black==== ====white==== ====græy==== ====pink==== ====indigo==== ====midnightblue==== ===hex=== ====#fff==== ====#fff==== {| class="wikitable" style="text-align: center;" |+ |- | style="background:black;" | <span style="color:white"> [ white { on black ⚞🧿⚟ #fff on #000 ⚞🧿⚟ } ] </span> |- | style="background:#808080;" | <span style="color:pink"> [ pink { on 50% grey ⚞🧿⚟ #ffc0cb on #808080 ⚞🧿⚟ } ] </span> |- | style="background:#808080;" | <span style="color:#191970"> [ midnightblue { on 50% grey ⚞🧿⚟ #191970 on #808080 ⚞🧿⚟ } ] </span> |- | style="background:#808080;" | <span style="color:#4b0082"> [ indigo { on 50% grey ⚞🧿⚟ #4b0082 on #808080 ⚞🧿⚟ } ] </span> |} ===cmyk=== https://colordesigner.io/convert/cmyktohex ====gg on cmyk(0,0,0,33.3) w|materializecss.com==== https://materializecss.com/color.html {| class="wikitable" style="text-align: center;" |+ |- | style="background:#ababab" | <span style="color:#fff9c4"> [ gg_yellow { on cmyk(0,0,0,33.3) ⚞🧿⚟ #fff9c4 on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#ffe0b2"> [ gg_orange { on cmyk(0,0,0,33.3) ⚞🧿⚟ #ffe0b2 on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#ffcdd2"> [ gg_red { on cmyk(0,0,0,33.3) ⚞🧿⚟ #ffcdd2 on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#e1bee7"> [ gg_purple { on cmyk(0,0,0,33.3) ⚞🧿⚟ #e1bee7 on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#bbdefb"> [ gg_blue { on cmyk(0,0,0,33.3) ⚞🧿⚟ #bbdefb on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#c8e6c9"> [ gg_green { on cmyk(0,0,0,33.3) ⚞🧿⚟ #c8e6c9 on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#efefef"> [ gg_white { on cmyk(0,0,0,33.3) ⚞🧿⚟ #efefef on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#111"> [ gg_black { on cmyk(0,0,0,33.3) ⚞🧿⚟ #111 on #ababab ⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#808080"> [ gg_grey { on cmyk(0,0,0,33.3) ⚞🧿⚟ #808080 on #ababab ⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#f8bbd0"> [ gg_pink { on cmyk(0,0,0,33.3) ⚞🧿⚟ #f8bbd0 on #ababab ⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#b2ebf2"> [ gg_cyan { on cmyk(0,0,0,33.3) ⚞🧿⚟ #b2ebf2 on #ababab ⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#d7ccc8"> [ gg_brown { on cmyk(0,0,0,33.3) ⚞🧿⚟ #d7ccc8 on #ababab )⚞🧿⚟} ] </span> |} ===rgba=== === TrumPutin-ism === Trump has demonstrably alienated the USA from allies both foreign and domestic. While Oregon's AG works on Epstein and Weinstein, contemporaneous crimes go unabated and have created a new problem where otherwise law abiding folk find themselves on the wrong side of the law. Oregon doesn't have enough public defenders to fight violent crime, yet children are alienated from their church and families to hide atrocities they don't even know about. == 200 -- Religion == Royal We 1000 things I did 1992-2022 other than lie my way onto the supreme court to overturn Roe v Wade {| |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |} == 300 -- Social sciences == https://wattention.com/traditional-rice-harvesting-in-japan/ https://www.wwoofjapan.com/home/index.php?lang=en == 400 -- Language == === Programming === ==== .png ==== ==== Esperanto ==== ==== HTML ==== ==== PUG ==== == 500 -- Pure Science == == 600 -- Technology == === local hosts=== [http://localhost:8080/ 8080] file:///D:/index.html === Roland SP 404MKII === https://www.roland.com/global/products/sp-404mk2/ https://www.roland.com/global/support/by_product/sp-404mk2/owners_manuals/ @https://static.roland.com/manuals/sp-404mk2_app/eng/19610757.html === VIM === https://vim-adventures.com/ == 700 -- Arts and recreation == == 800 -- Literature == == 900 -- History and geography == https://geology.com/stories/13/bear-areas/ === pre-2030 === 2022 "booked" by Hillsboro Police for sending email addressing "Christian Hate" and "Spiritual War" along with "exorcisms" and "Halloween Hysteria" in Marion County, OR where Salem Police Department abdicated from protecting some children in Salem from 2016-2021. 2021 Kaiser Permanente promised cash settlement to mitigate their abdication in Marion County. KP lawyer with intimate details about my vagina: terrence .j . loeber@kp.org 2012 "unliked" on FB by some Nazarene peers after openly questioning Alex Jones' allegation that Sandy Hook didn't happen and asking for compassion for parents who were called actors while they grieved publicly through no choice of their own. 2011 Lupron given by KP for menorrhagia as alternative to b/c pills first rxd in 86. How many women who have "mostly" been on b/c pills from 87-11 are obese? Why no menorrhagia while immersed in Japan? How close to a traditional Japanese diet can I get in the Willamette Valley and how close to no meat will my body allow? === TrumPutish War Against Humanity === Trump has demonstrably alienated the USA from allies both foreign and domestic. While Oregon's AG works on Epstein and Weinstein, contemporaneous crimes go unabated and have created a new problem where otherwise law abiding folk find themselves on the wrong side of the law. Oregon doesn't have enough public defenders to fight violent crime, yet children are alienated from their church and families to hide atrocities they don't even know about. === Ring of Fire === === Post "Roe v Wade" === Who did Roe v Wade protect? Why would a Nazarene raised pro-life support an "underground" network post Roe v Wade? === Our Contemporary "Underground Railroad" needs a submarine? === Why did Portland, OR close the Shanghai Tunnels recently? Human Trafficking through Astoria, OR has been going on "forever". How do we align an "underground railroad" with contemporary supports? == 10 -- A & + == == 11 -- B * x == == 12 -- C f(◯) == == 13 -- D Δ δ ƍ ≜ 𝜟 𝝳 == == 14 -- E 🐘 𓃰 == == 15 -- F == == 16 -- G == == 17 -- H == == 18 -- I == == 19 -- J == == 20 -- K == == 21 -- L == == 22 -- M == == 23 -- N == == 24 -- O == == 25 -- P == == 26 -- Q == == 27 -- R == == 28 -- S == == 29 -- T == == 30 -- U == == 31 -- V == == 32 -- W == == 33 -- X == == 34 -- Y == == 35 -- Z == ke36hpiafm9q9rrhcvoc6h82xpionhj 2410384 2410383 2022-07-30T03:06:30Z A020f0ff 2928078 wikitext text/x-wiki ○ ∘ Ya (hiragana: や, katakana: ヤ) ヤフー 屋 - Wiktionaryhttps://en.wiktionary.org › wiki › 屋 Semantic compound of 尸+至. 尸 does not represent the radical for death, but is a pictogram depicting a cloth draped. 至 means "dead end". Home (家) A home, or domicile, is a space used as a permanent or semi-permanent residence for one or many humans. It is a fully or semi sheltered space and can have both interior and exterior aspects to it. Ya (や) Kana Ya is one of the Japanese kana, each of which represents one mora. The hiragana is written in three strokes, while the katakana is written in two. Both represent. Their shapes have origins in the character 也. Wikipedia hiragana origin: 也 spelling kana: 大和のヤ Yamato no "ya" transliteration: ya unicode: U+3084, U+30E4 What is the pronunciation of Ya line? In historical kana orthography, it is written as "yau", "say", and "eu", and read as "yo", "you", and "yo", respectively. Even in modern times, "saying" and "going" are sometimes pronounced as "yu" and "yuku . " From the above, it can be said that Ya line is the yoon of that line . https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/Sandbox https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/plenary https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/inKind https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/specialdelivery meritorium . meritorious : merit .or.iou.us {{User alternative account|VeronicaJeanAnderson}} ∨⚡\🗲↯/ϟ∧ ✮☆⚝⛤🟊✰✭▲◂◁◀◢⍟◶✪⚪⬤🔥◍⚫⨁⚉⨂❂✧✷✸✡✵ http://slither.io/ https://www.thescienceofpsychotherapy.com/behaviour-affection-and-emotional-control/ {| class="wikitable" style="text-align: center;" |+ ⚞🧿⚟_◞◜↷◝◟_◞◜⚞🧿⚟🧿⚞🧿⚟◝◟_◞◜↶◝◟_⚞🧿⚟ |- || ✪⚪⬤🔥◍⚫⨁ || [https://www.twitch.tv/archie97305 👀] | style="background:pink;" | <span style="color:#808080"> ≡ odd → +1 </span> | style="background:pink;" | <span style="color:#808080"> [ { ( East </span> | style="background:black;" | <span style="color:white"> ⚫🔴⚪○💮⭕</span> | style="background:#808080;" | <span style="color:pink"> West ) } ] </span> | style="background:#808080;" | <span style="color:pink"> iff even ⇒ ÷2 </span> |- || Primary || [https://www.amnesty.org/en/ 1] |- || Secondary || [https://philosophy.stackexchange.com/questions/31029/why-was-the-horseshoe-symbol-%E2%8A%83-selected-for-material-implication 2] |- || Tertiary || [https://en.wikiversity.org/wiki/Wikiversity:Main_Page 3] |- || Quaternary || [http://localhost:8080/ 4] |- || Quinary || [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson 5] |- || Senary || [https://en.wikiversity.org/wiki/User:Archie97305 6] |- || Septenary || [https://maritimearchaeological.org/beeswax-wreck/ 7] |- || Octenary || [https://www.youtube.com/freecodecamp 8] |- || nonary || [https://www.freecodecamp.org/ 9] |- ||Base Name ||[https://wordsmith.org/board/ubbthreads.php?ubb=showflat&Number=84101 `] |- || binary || 2 |- ||ternary ||3 |- ||quaternary ||4 |- ||quinary ||5 |- ||senary ||6 |- ||septenary ||7 |- ||octal ||8 |- ||nonary |- ||decimal |- ||undenary |- ||duodecimal |- ||hexadecimal ||16 |- ||vigesimal ||20 |- ||sexagesimal ||60 |} How do you want your water served when you get here? https://pubs.usgs.gov/sir/2005/5168/pdf/sir2005-5168.pdf Robert Lee Stinson %VOX "tautology club says hi" w 11am "Naturalist Society for the Humane Treatment of Monsters" from dnd game on twitter [https://www.youtube.com/watch?v=n7uNA5fO1iI rice ex in CA] https://www.oregonwild.org/about/blog/oregon-grizzly-country https://therevelator.org/yellowstone-grizzlies-unbearable-divides/ https://developer.mozilla.org/en-US/docs/Web/CSS/color_value/hsl https://www.researchgate.net/about Amare, Nicole & Manning, A.. (2012). Seeing typeface personality: Emotional responses to form as tone. IEEE International Professional Communication Conference. 1-9. 10.1109/IPCC.2012.6408605. Various studies have correlated specific visual characteristics of typefaces with specific overall emotional effects: curvilinear forms and open letter shapes generally feel “friendly” but also “formal” or “informal,” depending on other factors; large contrasts in stroke widths, cap height, and aspect ratio generally feel “interesting,” but also “attractive” or “aggressive,” depending on other factors; low-variety and low-contrast forms generally feel “professional” but also “reliable” or “boring.” Although the current findings on typeface personality are useful, they have not indicated a systematic explanation for why specific physical typeface forms have the specific emotion effects that they do. This paper will report results of an empirical study in which 102 participants indicated their immediate emotional responses to each of 36 distinct typeface designs. Results support correlation between specific typeface features (variety vs. contrast vs. pattern) and specific emotional parameters (amusement vs. agitation vs. focus), explaining findings of previous studies, suggesting various classroom approaches to purpose-driven typeface selection. {{User alternative account|VeronicaJeanAnderson}} {| class="wikitable" style="text-align: center;" |+ ᐪgenki-ness; +, -tachi . . . |- | style="background:black;" | <span style="color:white"> [ { ( A B E ) } ] </span> | style="background:black;" | <span style="color:white"> [ { ( [https://www.twitch.tv/archie97305 👀] ) } ]</span> |- || Primary | style="background:#FFFFE6;" | <span style="color:black"> index.html</span> || notepad/atom (atom is deprecated) |- || Secondary | style="background:#FFF2E6;" | <span style="color:black"> vue </span> || [https://www.vim.org/ vim] [https://github.com/vim/vim-win32-installer/releases installer] |- || Tertiary | style="background:white;" | <span style="color:black"> css </span> || global css @ || gg css @ || NPC css @ |- || Quaternary | style="background:#FFE6FB;" | <span style="color:black"> pug </span> |- || Quinary | style="background:#F9F9F9;" | <span style="color:pink"> (direct object) </span> |- || Senary || b |} https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/sandbox trying to create a 1 -> 2 -> 3 -> 4 -> 5 -> 6 system in the apartment here that can be copied from site to site using artistic threads to help a Nice And Proper NAP-er navigate between properties with ease while maintaining adequate supportive care that we all require to enable us to focus on whatever catches our fancy. sun; natural light; breathe; BGs carbs; hygiene; laundry away bedroom; needles; blood; garbage out kitchen/nutritional/study social/outreach/linking worlds back porch 0 -- Computer Science, information and general works {| class="wikitable" style="text-align: center;" |+ ᐪgenki-ness; +, -tachi . . . |- | style="background:black;" | <span style="color:white"> [ { ( T O P ) } ] </span> | style="background:black;" | <span style="color:white"> [ ℳ ] </span> | style="background:white;" | <span style="color:black"> { ¢ } </span> | style="background:#F9F9F9;" | <span style="color:pink"> ( ৳ ) </span> | style="background:black;" | <span style="color:white"> [ { ( I.n C.ase of E.mergency ) } ] </span> | style="background:teal;" | <span style="color:lime"> ᐪ l i p s c h i t z </span> || [https://www.youtube.com/watch?v=qrrz54UtkCc ᐪ] |- || Primary | style="background:#FFFFE6;" | <span style="color:black"> physical</span> | style="background:#FFE6E6;" | <span style="color:black"> emotional</span> | style="background:#E6EAFF;" | <span style="color:black"> social</span> || This reflects health enough to communicate with people intimately enough to address real immediate issues | style="background:#FFFFE6;" | <span style="color:teal"> ^ torikomu </span> ||[https://www.youtube.com/watch?v=YxvBPH4sArQ ^] |- || Secondary | style="background:#FFF2E6;" | <span style="color:black"> occupational</span> | style="background:#F2E6FF;" | <span style="color:black"> intellectual</span> | style="background:#E6FFEA;" | <span style="color:black"> environmental</span> || This reflects living somewhere promoting healthy reasoning | style="background:#FFE6E6;" | <span style="color: teal"> | kaizen | </span> || | |- || Tertiary | style="background:white;" | <span style="color:black"> spiritual</span> | style="background:#BFBFBF;" | <span style="color:white"> factual </span> | style="background:#F2F2F2;" | <span style="color:black"> nutritional</span> || This reflects healthy mindful every habits | style="background:#E6EAFF;" | <span style="color:teal"> . genkiness . .</span> || . |- || Quaternary | style="background:#FFE6FB;" | <span style="color:black"> generational</span> | style="background:#E6FFFF;" | <span style="color:black"> miscellaneal</span> | style="background:#F2E0CE;" | <span style="color:black"> punctuational</span> || This reflects having it all together enough to enjoy the holidays | style="background:#FFF2E6;" | <span style="color:lime"> # goblin </span> || # |- || Quinary | style="background:#F9F9F9;" | <span style="color:pink"> (direct object) </span> | style="background:white;" | <span style="color:black"> {verb} </span> | style="background:black;" | <span style="color:white"> [noun] </span> || This reflects deliberate professional progress | style="background:#F2E6FF;" | <span style="color:lime"> / tsugu /</span> || / |- || Senary || b || 〇 || x || This reflects influencing others | style="background:#E6FFEA;" | <span style="color:lime"> @ g @ g @ </span> || [https://www.youtube.com/watch?v=SYnVYJDxu2Q @] |} == 100 -- Philosophy and psychology == How can I use color to manipulate behavior and improve communication? ===named=== ==== Re⋮Beccaδ#639 ==== ===== rebeccapurple :: #663399 ===== https://meyerweb.com/eric/thoughts/2014/06/19/rebeccapurple/ ====black==== ====white==== ====græy==== ====pink==== ====indigo==== ====midnightblue==== ===hex=== ====#fff==== ====#fff==== {| class="wikitable" style="text-align: center;" |+ |- | style="background:black;" | <span style="color:white"> [ white { on black ⚞🧿⚟ #fff on #000 ⚞🧿⚟ } ] </span> |- | style="background:#808080;" | <span style="color:pink"> [ pink { on 50% grey ⚞🧿⚟ #ffc0cb on #808080 ⚞🧿⚟ } ] </span> |- | style="background:#808080;" | <span style="color:#191970"> [ midnightblue { on 50% grey ⚞🧿⚟ #191970 on #808080 ⚞🧿⚟ } ] </span> |- | style="background:#808080;" | <span style="color:#4b0082"> [ indigo { on 50% grey ⚞🧿⚟ #4b0082 on #808080 ⚞🧿⚟ } ] </span> |} ===cmyk=== https://colordesigner.io/convert/cmyktohex ====gg on cmyk(0,0,0,33.3) w|materializecss.com==== https://materializecss.com/color.html {| class="wikitable" style="text-align: center;" |+ |- | style="background:#ababab" | <span style="color:#fff9c4"> [ gg_yellow { on cmyk(0,0,0,33.3) ⚞🧿⚟ #fff9c4 on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#ffe0b2"> [ gg_orange { on cmyk(0,0,0,33.3) ⚞🧿⚟ #ffe0b2 on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#ffcdd2"> [ gg_red { on cmyk(0,0,0,33.3) ⚞🧿⚟ #ffcdd2 on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#e1bee7"> [ gg_purple { on cmyk(0,0,0,33.3) ⚞🧿⚟ #e1bee7 on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#bbdefb"> [ gg_blue { on cmyk(0,0,0,33.3) ⚞🧿⚟ #bbdefb on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#c8e6c9"> [ gg_green { on cmyk(0,0,0,33.3) ⚞🧿⚟ #c8e6c9 on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#efefef"> [ gg_white { on cmyk(0,0,0,33.3) ⚞🧿⚟ #efefef on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#111"> [ gg_black { on cmyk(0,0,0,33.3) ⚞🧿⚟ #111 on #ababab ⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#808080"> [ gg_grey { on cmyk(0,0,0,33.3) ⚞🧿⚟ #808080 on #ababab ⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#f8bbd0"> [ gg_pink { on cmyk(0,0,0,33.3) ⚞🧿⚟ #f8bbd0 on #ababab ⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#b2ebf2"> [ gg_cyan { on cmyk(0,0,0,33.3) ⚞🧿⚟ #b2ebf2 on #ababab ⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#d7ccc8"> [ gg_brown { on cmyk(0,0,0,33.3) ⚞🧿⚟ #d7ccc8 on #ababab )⚞🧿⚟} ] </span> |} ===rgba=== === TrumPutin-ism === Trump has demonstrably alienated the USA from allies both foreign and domestic. While Oregon's AG works on Epstein and Weinstein, contemporaneous crimes go unabated and have created a new problem where otherwise law abiding folk find themselves on the wrong side of the law. Oregon doesn't have enough public defenders to fight violent crime, yet children are alienated from their church and families to hide atrocities they don't even know about. == 200 -- Religion == Royal We 1000 things I did 1992-2022 other than lie my way onto the supreme court to overturn Roe v Wade {| |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |} == 300 -- Social sciences == https://wattention.com/traditional-rice-harvesting-in-japan/ https://www.wwoofjapan.com/home/index.php?lang=en == 400 -- Language == === Programming === ==== .png ==== ==== Esperanto ==== ==== HTML ==== ==== PUG ==== == 500 -- Pure Science == == 600 -- Technology == === local hosts=== [http://localhost:8080/ 8080] file:///D:/index.html === Roland SP 404MKII === https://www.roland.com/global/products/sp-404mk2/ https://www.roland.com/global/support/by_product/sp-404mk2/owners_manuals/ @https://static.roland.com/manuals/sp-404mk2_app/eng/19610757.html === VIM === https://vim-adventures.com/ == 700 -- Arts and recreation == == 800 -- Literature == == 900 -- History and geography == https://geology.com/stories/13/bear-areas/ === pre-2030 === 2022 "booked" by Hillsboro Police for sending email addressing "Christian Hate" and "Spiritual War" along with "exorcisms" and "Halloween Hysteria" in Marion County, OR where Salem Police Department abdicated from protecting some children in Salem from 2016-2021. 2021 Kaiser Permanente promised cash settlement to mitigate their abdication in Marion County. KP lawyer with intimate details about my vagina: terrence .j . loeber@kp.org 2012 "unliked" on FB by some Nazarene peers after openly questioning Alex Jones' allegation that Sandy Hook didn't happen and asking for compassion for parents who were called actors while they grieved publicly through no choice of their own. 2011 Lupron given by KP for menorrhagia as alternative to b/c pills first rxd in 86. How many women who have "mostly" been on b/c pills from 87-11 are obese? Why no menorrhagia while immersed in Japan? How close to a traditional Japanese diet can I get in the Willamette Valley and how close to no meat will my body allow? === TrumPutish War Against Humanity === Trump has demonstrably alienated the USA from allies both foreign and domestic. While Oregon's AG works on Epstein and Weinstein, contemporaneous crimes go unabated and have created a new problem where otherwise law abiding folk find themselves on the wrong side of the law. Oregon doesn't have enough public defenders to fight violent crime, yet children are alienated from their church and families to hide atrocities they don't even know about. === Ring of Fire === === Post "Roe v Wade" === Who did Roe v Wade protect? Why would a Nazarene raised pro-life support an "underground" network post Roe v Wade? === Our Contemporary "Underground Railroad" needs a submarine? === Why did Portland, OR close the Shanghai Tunnels recently? Human Trafficking through Astoria, OR has been going on "forever". How do we align an "underground railroad" with contemporary supports? == 10 -- A & + == == 11 -- B * x == == 12 -- C f(◯) == == 13 -- D Δ δ ƍ ≜ 𝜟 𝝳 == == 14 -- E 🐘 𓃰 == == 15 -- F == == 16 -- G == == 17 -- H == == 18 -- I == == 19 -- J == == 20 -- K == == 21 -- L == == 22 -- M == == 23 -- N == == 24 -- O == == 25 -- P == == 26 -- Q == == 27 -- R == == 28 -- S == == 29 -- T == == 30 -- U == == 31 -- V == == 32 -- W == == 33 -- X == == 34 -- Y == == 35 -- Z == es08o907gicpgvpa2asomqxut80z6b2 0 262457 2410242 2410023 2022-07-29T14:50:30Z Ncharamut 2824970 /* Description */ updated training RAs wikitext text/x-wiki == '''Addresses''' == === '''''Future Directions Address 1: Father Inclusion, Engagement, Retention, and Positive Outcomes in Child and Adolescent Research''''' === '''Presented by Dr. Greg Fabiano, Ph.D.''' ====Description==== In this address, Dr. Greg Fabiano outlines future directions in the next generation of father-focused studies in the child and adolescent psychology literature, with an emphasis on improving the study of the parameters of inclusion, engagement, retention, and measurement of outcomes. Watch the YouTube video recording of the address enter link [https://www.youtube.com/watch?v=EogVEgi3n80&feature=youtu.be here] === '''''Future Directions Address 2: Research and Intervention with Youths in Poverty''''' === '''Presented by Dr. Martha Wadsworth, Ph.D.''' ====Description==== In this address, Dr. Martha Wadsworth integrates theory and empirical findings about understanding and fostering the process of resilience and adaptation in children and families who live in poverty. Watch the YouTube video recording of the address enter link [https://www.youtube.com/watch?v=Mj3bgzgXI6g here] === '''''Future Directions Address 3: Examination of Brain Networks in Neurodevelopmental Disorders''''' === '''Presented by Dr. Lucina Uddin, Ph.D.''' ====Description==== In this address, Dr. Lucina Uddin discusses future directions for neuroscience researchers examining brain networks in neurodevelopmental disorders, highlighting gaps in the current literature. Watch the YouTube video recording of the address enter link [https://www.youtube.com/watch?v=CoSM8C2ucRE here] === '''''Future Directions Address 4: the Treatment of Youth Mental Health''''' === '''Presented by Dr. Bruce Chorpita, Ph.D.''' ====Description==== In this address, Dr. Bruce Chorpita discusses mental health care systems and presents ideas and examples of methods that may preserve the strengths of the two major paradigms in children’s mental health, evidence-based treatments, and individualized care models, but that also have the potential to extend their applicability and impact. Watch the YouTube video recording of the address enter link [https://www.youtube.com/watch?v=bnKkc2iyeLo here]. == '''Workshops''' == === '''''Selecting Mentors when Applying to Doctoral Programs''''' === '''Presented by Dr. Andres De Los Reyes, Ph.D.''' ===== Description ===== Applying to doctoral programs marks an important life milestone for you and other undergraduate majors and post-baccalaureate trainees. Importantly, some of the considerations for choosing where to receive undergraduate training (e.g., faculty-to-student ratio, quality of institution) take a "back seat" to the key factor in doctoral training that most impacts your career: Identifying the person who will serve as your mentor. Undergraduate programs rarely offer formal instruction in choosing doctoral mentors, and some of the factors you might consider could vary from year-to-year and by mentor. Overall, when selecting a mentor you should consider who fits your needs and learning style. Because the quality of your doctoral training is mostly impacted by your mentor, you should think about who is going to fit best with your goals. When searching for mentors to apply to, you should first identify researchers who study topics similar to your interests. You can do this by reading research articles and looking for ones that align with your "burning question". Other factors you should consider when selecting mentors include: the size of their lab, the time they devote to their mentees, the area in which their research encompasses, and the resources they have available. To determine what to look for in these factors, it is important to know your values and learning style. Watch the YouTube recording of the workshop [https://www.youtube.com/watch?v=OXpO0FqAY9E here]. === '''''Responding to Peer Review Commentary''''' === '''Presented by''' '''Dr. Andres De Los Reyes, Ph.D.''' ===== Description ===== Publishing articles involves submitting scholarly manuscripts to peer-reviewed journals. A key component of publishing manuscripts involves receiving commentary about your work from peers in your field, and satisfactorily responding to such commentary. Yet, researchers rarely receive formal training on responding to peer review commentary. The first thing to know when submitting an article for peer review is that it can take several months to get feedback and the feedback among reviewers hardly converges. However, the good news is, typically if you receive a revise and resubmit, revise accordingly and resubmit the paper, the finished product will likely be of higher quality than when you first submitted for publication. When submitting a paper for publication, you should consider which journal you believe will give you a fair review and you should submit 2-4 reviewers in your cover letter that you think have the expertise to review your work. Once you get your decision from peer-review, you should wait a few days before working on revisions and/or responding. Next you should itemize the decision letter creating a to-do list of the revisions (this will become a template for the cover letter you send in response). If there are suggested revisions you cannot do, you need a solid reason as to why you are unable to and you should cite this if possible. If you need help, it is okay to reach out to the editor and/or colleagues for support and advice. If reviewers give conflicting advice, you should give a reason as to why you went with one reviewer's suggestion over the others. Lastly, you can sometimes get a reviewer who is particularly mean, in that case you should let the editor know about the review and if you consistently have a hard time with a reviewer, you can let the editor know in your cover letter that you wish for the person to not review your paper. Watch the YouTube recording of the workshop [https://www.youtube.com/watch?v=cJCIRuzD5sY here]. === '''''Strategies for Developing a Research Program''''' === '''Presented by Dr. Andres De Los Reyes, Ph.D.''' ===== Description ===== Research isn’t all elegant study designs, accurate data collection, and sophisticated equations. Researchers must also communicate their ideas and findings with scholarly audiences, and do so effectively. These audiences are no different from those found at your local theater: They understand each paper you write or talk you deliver insofar as it tells a compelling story. Yet, your storytelling doesn’t stop with a single paper or talk. Scholarly records span years and multiple pieces of work. Successful researchers learn to synthesize their records to tell a larger story: a research program. Finding your "burning question" will help when developing a research program. This can then be used to build the theoretical framework to which you begin to answer your "burning question". The goal of developing a research program should be similar to that of making a film, you want to elicit positive emotion in your audience and make them think. The first three papers you write in your lab should have a role in building your research program and connect back to your "burning question". Each paper should propel your next paper in this "trilogy" and the third paper should bring you back to the beginning one. Tailored to the lives of early career researchers, these tools reveal keen insights into nailing the job talk that launches your career. Watch the YouTube recording of the workshop [https://www.youtube.com/watch?v=HRRvMJiI7zs here]. === '''''New Models of Collaboration and Dissemination''''' === '''Presented by Dr. Eric Youngstrom, Ph.D.''' ===== Description ===== Wikipedia and Wikiversity offer powerful tools for disseminating knowledge to diverse audiences, including scientists and other key stakeholders (e.g., parents and policy makers). These tools greatly increase in utility if scientists receive training on how to leverage these tools for disseminating knowledge. In this workshop, Dr. Eric Youngstrom provides attendees with the know-how for using Wikipedia and Wikiversity, with a focus on how these tools help advance the mission of the open science movement. He begins by outlining copyright for open source information including a description of creative commons (CC) licensing that Wiki platforms use. There are several ways to use a CC license one of which is more high tech and involves registering your work on the CC website and the other which involves you placing CC BY 4.0 NAME on your work. Next Dr. Youngstrom describes how to use OSF to promote open access science. Housing materials on OSF allows others to easily access the work you are doing without having to go through paywalls. Lastly, Dr. Youngstrom describes the non-profit, Helping Give Away Psychological Science (HGAPS) and the work that HGAPS is doing to promote open science. This includes utilizing free platforms such as OSF and Wiki platforms to disseminate information. He concludes with a discussion on the HGAPS Assessment Center and the free assessment resources housed there. Watch the YouTube recording of the video [https://www.youtube.com/watch?v=9hIrVr1RaOc here]. === '''''Tools For Lab Building: Training Undergraduate Research Assistants''''' === '''Presented by Dr. Sarah Racz, Ph.D. and Dr. Yo Jackson, Ph.D.''' ===== Description ===== For many research teams, undergraduate research assistants (RA) form a core component of their personnel. A key challenge involves not only the varying motivations of these personnel and their ultimate career goals, but also their relative inexperience with research generally. Often, we found ourselves immersing these students in their first research experiences. In this workshop, Dr. Racz and Jackson discuss concrete strategies for providing standardized research training experiences for undergraduates, with a focus on developing personnel to assist in accurate data collection and creating a hospitable work environment for students, post-doctoral fellows, staff, and faculty. There are many benefits of having undergraduate research assistants in your lab such as showing you are invested in training students, building your lab, and bringing in new ideas to the lab. When recruiting undergrad RAs, your university may have systems in place to assign these students to your lab or you can have open recruitment usually by advertisement or word of mouth. You should also have an application for students to apply, screening criteria for applicants, and an interview process. Once RAs have been hired, you should set clear expectations from the beginning and they should be outlined in a contract that the RA signs. You should have documents outlining the lab processes that RAs can go through and you can also utilize advanced RAs to help train new ones. It is good to set a hours per week expectation for RAs and to have a policy for when they miss scheduled lab time. Give specific tasks to RAs and have regular meetings to discuss tasks and lab duties. It is good to provide a range of skills and experiences to RAs and to have a benchmark so they have something to work towards. You may want to require a specific commitment to the lab such as 2 semesters to ensure retention in the lab as well as reward good performance with more responsibilities. You should also have clear guidelines for publishing with undergrad RAs and these opportunities should be reserved for RAs who have been in the lab for a while and have shown good performance. Watch the YouTube Recording [https://www.youtube.com/watch?v=pN1Jpg3_5do here]. === '''''Building and Maintaining Research Partnerships with Schools''''' === '''Presented by Dr. Tim Cavell, Ph.D. and Dr. Elizabeth Talbott, Ph.D.''' ===== Description ===== A key component of research embedded in primary and secondary schools involves building long-term partnerships with key stakeholders in the school system. These stakeholders include administrators, teachers, classroom aids, school staff, and parents. In this workshop, we provide concrete advice on how to build lasting partnerships with school systems in an effort to conduct research with meaningful impacts on these systems. Watch the YouTube recording of the workshop [https://www.youtube.com/watch?v=7LxkQ6X0aqQ here]. === '''''Getting Your First Grant''''' === '''Presented by Dr. Deborah Drabick, Ph.D. and Dr. Katie Ehrlich, Ph.D.''' ===== Description ===== Submitting your first grant as a Ph.D. can appear on the surface to be a daunting task, with many expectations, requirements, and complicated forms. In this workshop, we leverage years of experience with extramural funding to explain the grant submission process, and provide attendees with concrete tools for submitting successful applications via multiple post-Ph.D. mechanisms, including project grants and K Series applications. Watch the YouTube recording of the workshop [https://www.youtube.com/watch?v=KcYjR9IznGo here]. === '''''Demystifying Academic Job Interviewing''''' === '''Presented by Dr. Kathryn Humphreys and Dr. Jessica Schleider''' ===== Description ===== The academic job interview factors prominently into faculty hiring decisions. It represents a public sample of your program of research and your style of teaching, as well as your critical thinking, responsiveness to feedback, and a whole range of non-specific variables, like your "accessibility," "collegiality," or "likeability." In this workshop, we provide a detailed overview of a winning formula for crafting an outstanding job interview experience, in an effort to minimize the anxiety and maximize the impact associated with your interview visit. Watch the YouTube Recording of the Workshop [https://www.youtube.com/watch?v=8iZLmNOOIDQ here]. === '''''Preparing a Training Grant: Overview''''' === '''Presented by Dr. Stephen Becker, Ph.D. and Dr. Meghan Miller, Ph.D.''' ===== Description ===== Submitting a training grant involves considering multiple factors that focus on not only a proposed study but also a concrete plan for developing the skills needed to execute this study. By construction, these applications carry many expectations, requirements, and complicated forms. In this workshop, we leverage our years of experience with extramural funding to clarify the process of submitting a training grant, and provide attendees with concrete tools for submitting successful training grant applications. Watch the YouTube recording of the workshop [https://www.youtube.com/watch?v=7qbPoHiAmFk here]. == '''Ceremony for the ''Future Directions Launch Award''''' == === Jessie Greenlee === * Award Winner in the area of Autism Spectrum Disorder * Postdoctoral Fellow at the University of Wisonsin-Madson ==== About the award recipient ==== Jessie is a recipient of the 2020 Future Directions Launch Award in Autism. Jessie completed a Postdoctoral Fellow at the Waisman Center at the University of Wisconsin-Madison. She received her Ph.D. in Developmental Psychology from Virginia Commonwealth University in 2019. Jessie’s research investigates the mechanisms through which individual and contextual factors are associated with mental and physical health disparities in vulnerable populations. She is particularly interested in understanding how families promote healthy social and emotional development in individuals with special healthcare needs. Jessie is currently working on several projects aimed at understanding how different sub-system family processes (e.g., marital conflict, co-parenting, parent-child relationship quality) impact outcomes for children and youth with autism spectrum disorder. Jessie is currently hold a position as an Assistant Professor of Psychology at Lafayette College in PA. Learn more about Jessie's work here: [https://www.researchgate.net/profile/Jessica&#x20;Greenlee www.researchgate.net/profile/Jessica_Greenlee] Watch the YouTube recording of the remarks [https://www.youtube.com/watch?v=BRggTSAhIW0 here]. === Tyler McFayden === * Award Winner in the area of Autism Spectrum Disorder and Language Development * Received Ph.D. at Virginia Tech ==== About the award recipient ==== Tyler received the 2020 Future Directions Launch Award in Autism. She is a current predoctoral clinical intern at the University of North Carolina- Chapel Hill’s (UNC-CH) Carolina Institute for Developmental Disabilities. Tyler plans to remain at UNC-CH for her postdoctoral training to participate in the NIMH-funded T32 Postdoctoral Research Training Program focusing on intellectual and developmental disabilities. Tyler attended Virginia Tech’s Clinical and Developmental Psychological doctoral program under the mentorship of Dr. Thomas Ollendick, where she worked in a typically developing infant lab, studying early language development, and an autism lab investigating endophenotypes of social communication. Tyler is particularly interested in how language develops in early infancy and in groups without spoken language (e.g., Deaf/Hard of Hearing and minimally-verbal/partially speaking autistic youth) to best inform social communication interventions. Learn more about Tyler’s work here: [https://www.researchgate.net/profile/Tyler&#x20;Mcfayden www.researchgate.net/profile/Tyler_Mcfayden] Watch the YouTube recording of the remarks [https://www.youtube.com/watch?v=L_ebFjTEJXk here]. rqfzaatoyaz97wrp0paizg80x7am6le 2410251 2410242 2022-07-29T17:59:47Z Ncharamut 2824970 /* Description */ updated research in schools wikitext text/x-wiki == '''Addresses''' == === '''''Future Directions Address 1: Father Inclusion, Engagement, Retention, and Positive Outcomes in Child and Adolescent Research''''' === '''Presented by Dr. Greg Fabiano, Ph.D.''' ====Description==== In this address, Dr. Greg Fabiano outlines future directions in the next generation of father-focused studies in the child and adolescent psychology literature, with an emphasis on improving the study of the parameters of inclusion, engagement, retention, and measurement of outcomes. Watch the YouTube video recording of the address enter link [https://www.youtube.com/watch?v=EogVEgi3n80&feature=youtu.be here] === '''''Future Directions Address 2: Research and Intervention with Youths in Poverty''''' === '''Presented by Dr. Martha Wadsworth, Ph.D.''' ====Description==== In this address, Dr. Martha Wadsworth integrates theory and empirical findings about understanding and fostering the process of resilience and adaptation in children and families who live in poverty. Watch the YouTube video recording of the address enter link [https://www.youtube.com/watch?v=Mj3bgzgXI6g here] === '''''Future Directions Address 3: Examination of Brain Networks in Neurodevelopmental Disorders''''' === '''Presented by Dr. Lucina Uddin, Ph.D.''' ====Description==== In this address, Dr. Lucina Uddin discusses future directions for neuroscience researchers examining brain networks in neurodevelopmental disorders, highlighting gaps in the current literature. Watch the YouTube video recording of the address enter link [https://www.youtube.com/watch?v=CoSM8C2ucRE here] === '''''Future Directions Address 4: the Treatment of Youth Mental Health''''' === '''Presented by Dr. Bruce Chorpita, Ph.D.''' ====Description==== In this address, Dr. Bruce Chorpita discusses mental health care systems and presents ideas and examples of methods that may preserve the strengths of the two major paradigms in children’s mental health, evidence-based treatments, and individualized care models, but that also have the potential to extend their applicability and impact. Watch the YouTube video recording of the address enter link [https://www.youtube.com/watch?v=bnKkc2iyeLo here]. == '''Workshops''' == === '''''Selecting Mentors when Applying to Doctoral Programs''''' === '''Presented by Dr. Andres De Los Reyes, Ph.D.''' ===== Description ===== Applying to doctoral programs marks an important life milestone for you and other undergraduate majors and post-baccalaureate trainees. Importantly, some of the considerations for choosing where to receive undergraduate training (e.g., faculty-to-student ratio, quality of institution) take a "back seat" to the key factor in doctoral training that most impacts your career: Identifying the person who will serve as your mentor. Undergraduate programs rarely offer formal instruction in choosing doctoral mentors, and some of the factors you might consider could vary from year-to-year and by mentor. Overall, when selecting a mentor you should consider who fits your needs and learning style. Because the quality of your doctoral training is mostly impacted by your mentor, you should think about who is going to fit best with your goals. When searching for mentors to apply to, you should first identify researchers who study topics similar to your interests. You can do this by reading research articles and looking for ones that align with your "burning question". Other factors you should consider when selecting mentors include: the size of their lab, the time they devote to their mentees, the area in which their research encompasses, and the resources they have available. To determine what to look for in these factors, it is important to know your values and learning style. Watch the YouTube recording of the workshop [https://www.youtube.com/watch?v=OXpO0FqAY9E here]. === '''''Responding to Peer Review Commentary''''' === '''Presented by''' '''Dr. Andres De Los Reyes, Ph.D.''' ===== Description ===== Publishing articles involves submitting scholarly manuscripts to peer-reviewed journals. A key component of publishing manuscripts involves receiving commentary about your work from peers in your field, and satisfactorily responding to such commentary. Yet, researchers rarely receive formal training on responding to peer review commentary. The first thing to know when submitting an article for peer review is that it can take several months to get feedback and the feedback among reviewers hardly converges. However, the good news is, typically if you receive a revise and resubmit, revise accordingly and resubmit the paper, the finished product will likely be of higher quality than when you first submitted for publication. When submitting a paper for publication, you should consider which journal you believe will give you a fair review and you should submit 2-4 reviewers in your cover letter that you think have the expertise to review your work. Once you get your decision from peer-review, you should wait a few days before working on revisions and/or responding. Next you should itemize the decision letter creating a to-do list of the revisions (this will become a template for the cover letter you send in response). If there are suggested revisions you cannot do, you need a solid reason as to why you are unable to and you should cite this if possible. If you need help, it is okay to reach out to the editor and/or colleagues for support and advice. If reviewers give conflicting advice, you should give a reason as to why you went with one reviewer's suggestion over the others. Lastly, you can sometimes get a reviewer who is particularly mean, in that case you should let the editor know about the review and if you consistently have a hard time with a reviewer, you can let the editor know in your cover letter that you wish for the person to not review your paper. Watch the YouTube recording of the workshop [https://www.youtube.com/watch?v=cJCIRuzD5sY here]. === '''''Strategies for Developing a Research Program''''' === '''Presented by Dr. Andres De Los Reyes, Ph.D.''' ===== Description ===== Research isn’t all elegant study designs, accurate data collection, and sophisticated equations. Researchers must also communicate their ideas and findings with scholarly audiences, and do so effectively. These audiences are no different from those found at your local theater: They understand each paper you write or talk you deliver insofar as it tells a compelling story. Yet, your storytelling doesn’t stop with a single paper or talk. Scholarly records span years and multiple pieces of work. Successful researchers learn to synthesize their records to tell a larger story: a research program. Finding your "burning question" will help when developing a research program. This can then be used to build the theoretical framework to which you begin to answer your "burning question". The goal of developing a research program should be similar to that of making a film, you want to elicit positive emotion in your audience and make them think. The first three papers you write in your lab should have a role in building your research program and connect back to your "burning question". Each paper should propel your next paper in this "trilogy" and the third paper should bring you back to the beginning one. Tailored to the lives of early career researchers, these tools reveal keen insights into nailing the job talk that launches your career. Watch the YouTube recording of the workshop [https://www.youtube.com/watch?v=HRRvMJiI7zs here]. === '''''New Models of Collaboration and Dissemination''''' === '''Presented by Dr. Eric Youngstrom, Ph.D.''' ===== Description ===== Wikipedia and Wikiversity offer powerful tools for disseminating knowledge to diverse audiences, including scientists and other key stakeholders (e.g., parents and policy makers). These tools greatly increase in utility if scientists receive training on how to leverage these tools for disseminating knowledge. In this workshop, Dr. Eric Youngstrom provides attendees with the know-how for using Wikipedia and Wikiversity, with a focus on how these tools help advance the mission of the open science movement. He begins by outlining copyright for open source information including a description of creative commons (CC) licensing that Wiki platforms use. There are several ways to use a CC license one of which is more high tech and involves registering your work on the CC website and the other which involves you placing CC BY 4.0 NAME on your work. Next Dr. Youngstrom describes how to use OSF to promote open access science. Housing materials on OSF allows others to easily access the work you are doing without having to go through paywalls. Lastly, Dr. Youngstrom describes the non-profit, Helping Give Away Psychological Science (HGAPS) and the work that HGAPS is doing to promote open science. This includes utilizing free platforms such as OSF and Wiki platforms to disseminate information. He concludes with a discussion on the HGAPS Assessment Center and the free assessment resources housed there. Watch the YouTube recording of the video [https://www.youtube.com/watch?v=9hIrVr1RaOc here]. === '''''Tools For Lab Building: Training Undergraduate Research Assistants''''' === '''Presented by Dr. Sarah Racz, Ph.D. and Dr. Yo Jackson, Ph.D.''' ===== Description ===== For many research teams, undergraduate research assistants (RA) form a core component of their personnel. A key challenge involves not only the varying motivations of these personnel and their ultimate career goals, but also their relative inexperience with research generally. Often, we found ourselves immersing these students in their first research experiences. In this workshop, Dr. Racz and Jackson discuss concrete strategies for providing standardized research training experiences for undergraduates, with a focus on developing personnel to assist in accurate data collection and creating a hospitable work environment for students, post-doctoral fellows, staff, and faculty. There are many benefits of having undergraduate research assistants in your lab such as showing you are invested in training students, building your lab, and bringing in new ideas to the lab. When recruiting undergrad RAs, your university may have systems in place to assign these students to your lab or you can have open recruitment usually by advertisement or word of mouth. You should also have an application for students to apply, screening criteria for applicants, and an interview process. Once RAs have been hired, you should set clear expectations from the beginning and they should be outlined in a contract that the RA signs. You should have documents outlining the lab processes that RAs can go through and you can also utilize advanced RAs to help train new ones. It is good to set a hours per week expectation for RAs and to have a policy for when they miss scheduled lab time. Give specific tasks to RAs and have regular meetings to discuss tasks and lab duties. It is good to provide a range of skills and experiences to RAs and to have a benchmark so they have something to work towards. You may want to require a specific commitment to the lab such as 2 semesters to ensure retention in the lab as well as reward good performance with more responsibilities. You should also have clear guidelines for publishing with undergrad RAs and these opportunities should be reserved for RAs who have been in the lab for a while and have shown good performance. Watch the YouTube Recording [https://www.youtube.com/watch?v=pN1Jpg3_5do here]. === '''''Building and Maintaining Research Partnerships with Schools''''' === '''Presented by Dr. Tim Cavell, Ph.D. and Dr. Elizabeth Talbott, Ph.D.''' ===== Description ===== A key component of research embedded in primary and secondary schools involves building long-term partnerships with key stakeholders in the school system. These stakeholders include administrators, teachers, classroom aids, school staff, and parents. In this workshop, Drs. Cavell and Talbott provide concrete advice on how to build lasting partnerships with school systems in an effort to conduct research with meaningful impacts on these systems. Schools are important and useful settings to conduct research in since they serve all children from pre-k to 21 years old. When thinking about partnering with a school for research, it is important to consider the culture and history of the school, the current events in the district, the geography of the schools you would like to partner with, and the district's needs. Before trying to partner with a school, you should familiarize yourself with the relationship the school may already have with researchers and whether your project would be better served in another community such as schools that primarily serve underrepresented students. You should also determine who has the authority to greenlight or stop your project and should contact that individual. Sometimes it is helpful to get a referral from someone who has a good relationship with that school or district. It is usually best to start with a phone call when trying to make contact. You should have an initial ask prepared and be ready to play the long game. It can be helpful to have the school identify a point of contact for you so you know who to communicate with. When thinking about conducting research in schools you should consider some key aspects of what will be required of schools. Your research should try to require little if any work from school staff, have minimal disruption to instructional time, be valuable to the teachers and staff, require minimal space in the school for your research team to conduct their work, among others. Obtaining consent is a very important aspect to conducting research in schools and you should think about how to best obtain it. Think about how you want to collect the data in schools considering both personnel needed and the measures you will use. Lastly, prepare your graduate students to conduct research in schools using your knowledge from working with schools. Watch the YouTube recording of the workshop [https://www.youtube.com/watch?v=7LxkQ6X0aqQ here]. === '''''Getting Your First Grant''''' === '''Presented by Dr. Deborah Drabick, Ph.D. and Dr. Katie Ehrlich, Ph.D.''' ===== Description ===== Submitting your first grant as a Ph.D. can appear on the surface to be a daunting task, with many expectations, requirements, and complicated forms. In this workshop, we leverage years of experience with extramural funding to explain the grant submission process, and provide attendees with concrete tools for submitting successful applications via multiple post-Ph.D. mechanisms, including project grants and K Series applications. Watch the YouTube recording of the workshop [https://www.youtube.com/watch?v=KcYjR9IznGo here]. === '''''Demystifying Academic Job Interviewing''''' === '''Presented by Dr. Kathryn Humphreys and Dr. Jessica Schleider''' ===== Description ===== The academic job interview factors prominently into faculty hiring decisions. It represents a public sample of your program of research and your style of teaching, as well as your critical thinking, responsiveness to feedback, and a whole range of non-specific variables, like your "accessibility," "collegiality," or "likeability." In this workshop, we provide a detailed overview of a winning formula for crafting an outstanding job interview experience, in an effort to minimize the anxiety and maximize the impact associated with your interview visit. Watch the YouTube Recording of the Workshop [https://www.youtube.com/watch?v=8iZLmNOOIDQ here]. === '''''Preparing a Training Grant: Overview''''' === '''Presented by Dr. Stephen Becker, Ph.D. and Dr. Meghan Miller, Ph.D.''' ===== Description ===== Submitting a training grant involves considering multiple factors that focus on not only a proposed study but also a concrete plan for developing the skills needed to execute this study. By construction, these applications carry many expectations, requirements, and complicated forms. In this workshop, we leverage our years of experience with extramural funding to clarify the process of submitting a training grant, and provide attendees with concrete tools for submitting successful training grant applications. Watch the YouTube recording of the workshop [https://www.youtube.com/watch?v=7qbPoHiAmFk here]. == '''Ceremony for the ''Future Directions Launch Award''''' == === Jessie Greenlee === * Award Winner in the area of Autism Spectrum Disorder * Postdoctoral Fellow at the University of Wisonsin-Madson ==== About the award recipient ==== Jessie is a recipient of the 2020 Future Directions Launch Award in Autism. Jessie completed a Postdoctoral Fellow at the Waisman Center at the University of Wisconsin-Madison. She received her Ph.D. in Developmental Psychology from Virginia Commonwealth University in 2019. Jessie’s research investigates the mechanisms through which individual and contextual factors are associated with mental and physical health disparities in vulnerable populations. She is particularly interested in understanding how families promote healthy social and emotional development in individuals with special healthcare needs. Jessie is currently working on several projects aimed at understanding how different sub-system family processes (e.g., marital conflict, co-parenting, parent-child relationship quality) impact outcomes for children and youth with autism spectrum disorder. Jessie is currently hold a position as an Assistant Professor of Psychology at Lafayette College in PA. Learn more about Jessie's work here: [https://www.researchgate.net/profile/Jessica&#x20;Greenlee www.researchgate.net/profile/Jessica_Greenlee] Watch the YouTube recording of the remarks [https://www.youtube.com/watch?v=BRggTSAhIW0 here]. === Tyler McFayden === * Award Winner in the area of Autism Spectrum Disorder and Language Development * Received Ph.D. at Virginia Tech ==== About the award recipient ==== Tyler received the 2020 Future Directions Launch Award in Autism. She is a current predoctoral clinical intern at the University of North Carolina- Chapel Hill’s (UNC-CH) Carolina Institute for Developmental Disabilities. Tyler plans to remain at UNC-CH for her postdoctoral training to participate in the NIMH-funded T32 Postdoctoral Research Training Program focusing on intellectual and developmental disabilities. Tyler attended Virginia Tech’s Clinical and Developmental Psychological doctoral program under the mentorship of Dr. Thomas Ollendick, where she worked in a typically developing infant lab, studying early language development, and an autism lab investigating endophenotypes of social communication. Tyler is particularly interested in how language develops in early infancy and in groups without spoken language (e.g., Deaf/Hard of Hearing and minimally-verbal/partially speaking autistic youth) to best inform social communication interventions. Learn more about Tyler’s work here: [https://www.researchgate.net/profile/Tyler&#x20;Mcfayden www.researchgate.net/profile/Tyler_Mcfayden] Watch the YouTube recording of the remarks [https://www.youtube.com/watch?v=L_ebFjTEJXk here]. pj9wr8q9emvrhumh35hnf29dr18pzwk 0 263444 2410306 2410028 2022-07-29T20:16:03Z Scogdill 1331941 wikitext text/x-wiki == Also Known As == *Louise, Duchess of Devonshire *Louisa, Duchess of Manchester *Luise Friederike August Gräfin von Alten *Louisa Montagu *Louise Cavendish *The Double Duchess == Acquaintances, Friends and Enemies == === Friends === *[[Social Victorians/People/Albert Edward, Prince of Wales | Albert Edward, Prince of Wales]] (beginning about 1852) *[[Social Victorians/People/Spencer Compton Cavendish|Spencer Compton Cavendish]], Lord Hartington (later 8th Duke of Devonshire) *Daisy, Lady Warwick *Lady Mayoress, Mrs. Benjamin Samuel Faudel-Phillips, 2nd Baronet,<ref>{{Cite journal|date=2020-08-25|title=Faudel-Phillips baronets|url=https://en.wikipedia.org/w/index.php?title=Faudel-Phillips_baronets&oldid=974879290|journal=Wikipedia|language=en}}</ref> presented to Victoria by Louisa Cavendish at a Queen's Drawing-room on Wednesday, 24 February 1897 at Buckingham Palace.<ref name=":4">"The Queen's Drawing Room" ''Morning Post'' 25 February 1897 Thursday: 5 [of 10], Col. 5a–7b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970225/047/0005.</ref>{{rp|p. 5, Col. 6c}} *Mrs. J. E. Mellor, presented to Victoria by Louisa Cavendish at a Queen's Drawing-room on Wednesday, 24 February 1897 at Buckingham Palace.<ref name=":4" />{{rp|p. 5, Col. 6c}} === Enemies === * Consuelo, Duchess of Marlborough (at least, in 1901)<ref name=":1">Murphy, Sophia. ''The Duchess of Devonshire's Ball''. London: Sidgwick & Jackson, 1984.</ref>{{rp|pp. 31–32}} == Organizations == == Timeline == '''1852 July 22''', Luise Friederike Auguste Gräfin von Alten and William Drogo Montagu married.<ref name=":2">"Luise Friederike Auguste Gräfin von Alten." {{Cite web|url=http://www.thepeerage.com/p10947.htm#i109469|title=Person Page|website=www.thepeerage.com|access-date=2020-09-25}}</ref> '''1863, early, or late 1862''', Louise and Spencer Compton Cavendish began a relationship.<ref name=":1" />{{rp|p. 26}} '''1873 December 10''', Mary Louise Elizabeth Montagu (daughter) and William Douglas-Hamilton married. '''1876 May 22''', Consuelo Iznaga y Clement and George Victor Drogo Montagu (son) married in Grace Church, New York City.<ref>{{Cite journal|date=2020-08-24|title=George Montagu, 8th Duke of Manchester|url=https://en.wikipedia.org/w/index.php?title=George_Montagu,_8th_Duke_of_Manchester&oldid=974659520|journal=Wikipedia|language=en}}</ref><ref>{{Cite journal|date=2020-07-27|title=Consuelo Montagu, Duchess of Manchester|url=https://en.wikipedia.org/w/index.php?title=Consuelo_Montagu,_Duchess_of_Manchester&oldid=969888488|journal=Wikipedia|language=en}}</ref> '''1876 August 10''', Louisa Augusta Beatrice Montagu (daughter) and Archibald Acheson married. '''1889 January 5''', Alice Maude Olivia Montagu (daughter) and Edward Stanley married. '''1890 March 22''', William Drogo Montagu (7th Duke) died.<ref name=":3">"William Drogo Montagu, 7th Duke of Manchester." {{Cite web|url=http://www.thepeerage.com/p10128.htm#i101274|title=Person Page|website=www.thepeerage.com|access-date=2020-09-25}}</ref> '''1890 November 14''', William Angus Drogo Montagu (grandson) and Helena Zimmerman married secretly, in London.<ref>"Helena Zimmerman." {{Cite web|url=http://www.thepeerage.com/p34555.htm#i345545|title=Person Page|website=www.thepeerage.com|access-date=2020-09-25}}</ref> '''1892 August 16''', Louise Friederike Auguste Gräfin von Alten Montagu and Spencer Compton Cavendish, her second husband, married.<ref name=":2" /> '''1897 July 2, Friday''', Louise Cavendish (#18 on the list of attendees) hosted her famous [[Social Victorians/1897 Fancy Dress Ball| fancy-dress ball]] at Devonshire House in London. '''1897 July 20''', Mary Louise Elizabeth Montagu Douglas-Hamilton and Robert Carnaby Foster married. '''1900 November 14''', William Angus Drogo Montagu and Helena Zimmerman married.<ref>{{Cite journal|date=2020-07-17|title=Helena, Countess of Kintore|url=https://en.wikipedia.org/w/index.php?title=Helena,_Countess_of_Kintore&oldid=968067371|journal=Wikipedia|language=en}}</ref> '''1901 Spring''', Paris, Consuelo Spencer-Churchill, Duchess of Marlborough, describes a meeting with Louise Cavendish in the spring following Queen Victoria's death at the horse racetrack, Longchamps:<blockquote>A renowned character and virtually dictator of what was known as the fast set as opposed to the Victorian, Her Grace was a German aristocrat by birth. She had first been married to the impoverished Duke of Manchester, and when he died had improved her status by marriage to the rich Duke of Devonshire, who waged an undisputed influence in politics. Rumour had her beautiful, but when I knew her she was a raddled old woman, covering her wrinkles with paint and her pate with a brown wig. Her mouth was a red gash and from it, when she saw me, issued a stream of abuse. How could I, she complained, pointing to my white gloves, show so little respect to the memory of a great Queen? What a carefree world we must have lived in, that etiquette even in such small matters could assume so much importance?<ref>Balsan, Consuelo Vanderbilt. ''The Glitter and the Gold: The American Duchess — In Her Own Words''. New York: St. Martin's, 1953.</ref>{{rp|p. 115}}</blockquote> === Annual Events === Every year, as Duchess of Devonshire, Louise held a dance on the night after the Derby at Epsom Downs, which at this point was held on Wednesdays after Easter. == Costume at the Duchess of Devonshire's 2 July 1897 Fancy-dress Ball == [[File:Louise Frederica Augusta Cavendish (née von Alten), Duchess of Devonshire (formerly Duchess of Manchester) as Zenobia, Queen of Palmyra.jpg|thumb|Louise, Duchess of Devonshire as Zenobia, Queen of Palmyra|alt=Louise, Duchess of Devonshire in costume as Zenobia, Queen of Palmyra]] At their fancy-dress ball, Louisa, Duchess of Devonshire sat at Table 1 during the first seating for supper, escorted in to the table by the Prince of Wales.<ref name=":7">"Fancy Dress Ball at Devonshire House." ''Morning Post'' Saturday 3 July 1897: 7 [of 12], Col. 4a–8 Col. 2b. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970703/054/0007.</ref>{{rp|p. 7, Col. 4c}} Her costume was designed by M. Comelli (Attillo Giuseppe Comelli, 1858–1925, artist and costumier for opera, ballet and theatre in London as well as Europe and the U.S.<ref name=":13">{{Cite book|url=https://books.google.com/books?id=SZh2DwAAQBAJ&pg=PT207&lpg=PT207&dq=Attilio+Comelli&source=bl&ots=lFB0If7CwV&sig=ACfU3U1_Ost_lhmMvzMMs6NvuhK5SlRhJw&hl=en&sa=X&ved=2ahUKEwjKlsTw2sH3AhXYAp0JHVIxDWA4KBDoAXoECBAQAw#v=onepage&q=Attilio%20Comelli&f=false|title=Forgotten Designers Costume Designers of American Broadway Revues and Musicals From 1900-1930|last=Unruh|first=Delbert|date=2018-11-06|publisher=Page Publishing Inc|isbn=978-1-64082-758-5|language=en}} N.P.</ref>)<ref name=":5">“The Devonshire House Ball.” The ''Man of Ross'' 10 July 1897, Saturday: 2 [of 8], Col. 4b. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0001463/18970710/033/0002.</ref> <ref name=":8">"The Duchess of Devonshire's Fancy Dress Ball. Special Telegram." ''Belfast News-Letter'' Saturday 03 July 1897: 5 [of 8], Col. 9 [of 9]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/BL/0000038/18970703/015/0005.</ref>{{rp|p. 5, Col. 9a}} <ref name=":9">"By One Who Was There." “The Duchess’s Costume Ball.” ''Westminster Gazette'' 03 July 1897 Saturday: 5 [of 8], Cols. 1a–3b [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002947/18970703/035/0005.</ref> and constructed by the House of Worth. Comelli seems to have designed [[Social Victorians/People/Louisa Montagu Cavendish#The Duchess and Her Entourage|the costumes of her retinue as well]]. According to Russell Harris,<blockquote>For her costume, the Duchess commissioned Monsieur Comelli (1858-1925), a well-known designer of opera costumes for the London theatre and opera stage, and then had the design made up by Worth of Paris. ''Munsey’s Magazine'' noted “it is safe to say that the Queen of Palmyra never owned such a sumptuous costume in her lifetime.”<ref>Harris, Russell. {{Cite web|url=http://www.rvondeh.dircon.co.uk/incalmprose/devonshiredss.html|title=Louise, Duchess of Devonshire, née Countess von Alten of Hanover (1832-1911), as Zenobia, Queen of Palmyra|website=www.rvondeh.dircon.co.uk|access-date=2022-05-05}} ''Narrated in Calm Prose: Photographs from the V&A's Lafayette Archive of Guests in Costume at the Duchess of Devonshire's Diamond Jubilee Ball, July 1897''. http://www.rvondeh.dircon.co.uk/incalmprose/devonshiredss.html.</ref></blockquote>Lafayette's portrait of "Louise Frederica Augusta Cavendish (née von Alten), Duchess of Devonshire (formerly Duchess of Manchester)" in costume is photogravure #5 in the album presented to the Duchess of Devonshire and now in the National Portrait Gallery.<ref>"Devonshire House Fancy Dress Ball (1897): photogravures by Walker & Boutall after various photographers." 1899. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait-list.php?set=515.</ref> The printing on the portrait says, "The Duchess of Devonshire as Zenobia Queen of Palmyra," with a Long S in ''Duchess''.<ref>"Louise Frederica Augusta Cavendish (née von Alten), Duchess of Devonshire (formerly Duchess of Manchester) as Zenobia, Queen of Palmyra." Devonshire House Fancy Dress Ball Album. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw158357/Louise-Frederica-Augusta-Cavendish-ne-von-Alten-Duchess-of-Devonshire-formerly-Duchess-of-Manchester-as-Zenobia-Queen-of-Palmyra.</ref> Often, the V&A Lafayette Archive contains more than one portrait of a sitter for this ball, but the uncropped portrait (above right), which shows the unfinished end of the balustrade in front of the Duchess and the edge of the painted flat behind it, seems to have been the only portrait taken by Lafayette of the Duchess in costume. The copy owned by the National Portrait Gallery in London and the copy included in the album are cropped so that those unfinished edges do not show, but they appear to be from the same photograph. === Newspaper Descriptions of the Duchess's Costume === Newspaper articles about the Duchess's presence at the ball focused on her hosting, her costume, [[Social Victorians/People/Louisa Montagu Cavendish#The Duchess's Jewelry|her jewelry]], and [[Social Victorians/People/Louisa Montagu Cavendish#The Duchess's Entourage|her entourage]], often in the same story. These almost exactly identical descriptions suggest [[Social Victorians/1897 Fancy Dress Ball/anthology#Scissors-and-Paste Journalism|scissors-and-paste journalism]] or a shared primary source: * "The Duchess of Devonshire was a dazzling vision, dressed as 'Zenobia,' in a glistening gold gauze gown, elaborately ornamented with suns and discs, wrought in purple and green gems outlined with gold, and having a large diamond as centre. The space between was fluted with fine silver spangles. This robe was open in front over an under dress of white crépe de chine, delicately worked in crystals, and at each side of the opening on the gold robe were large fan-shaped groups of peacock feathers, worked in multicoloured jewels. The [[Social Victorians/Terminology#Corsage|corsage]] was to correspond, and had a magnificent girdle of jewels, the train of bright green velvet, hung like a fan, without folds, being fastened at each side of the shoulders by diamond brooches, and caught at the waist with a similar ornament. It was a mass of gorgeous embroidery, carried out in heliotrope velvet, lotus flowers studded with tinted gems, and other devices in terra-cotta and electric blue velvet — all enriched with gold, diamond, and jewelled embroidery — and lined with pale blue satin. The crown worn with this was high, and of filigree gold, surmounted with two horns, each tipped with a large diamond. It was encrusted with large diamonds, rubies, and emeralds, and long chains of pearls fell under the chin and about the head — one magnificent pear-shaped pearl resting on the forehead. Attending the hostess were four children, four fan-bearers, and four trumpeters, all magnificently arrayed in artistically embroidered Assyrian robes, helmets, and other accessories, correct in every detail."<ref>"Duchess of Devonshire's Fancy Ball. A Brilliant Spectacle. Some of the Dresses." London ''Daily News'' Saturday 3 July 1897: 5 [of 10], Col. 6a–6, Col. 1b. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000051/18970703/024/0005 and http://www.britishnewspaperarchive.co.uk/viewer/BL/0000051/18970703/024/0006.</ref>{{rp|p. 5, Col. 6a}} * "The Duchess of Devonshire, as Zenobia, Queen of Palmyra, wore a magnificent costume, supplied by Worth, of Paris. The skirt of gold tissue was embroidered all over in a star-like design in emeralds, sapphires, diamonds, and other jewels, outlined with gold, the corners where it opened in front being elaborately wrought in the same jewels and gold to represent peacocks' outspread tails. This opened to show an under-dress of cream crêpe de chine, delicately embroidered in silver, gold, and pearls, and sprinkled all over with diamonds. The train was attached to the shoulders by two slender points, and was fastened at the waist with a large diamond ornament. It was of green velvet of a lovely shade, and was superbly embroidered in Oriental designs, introducing the lotus flower in rubies, sapphires, amethysts, emeralds, and diamonds, in four borderings on contrasting grounds, separated with gold cord. The train was lined with turquoise satin. The bodice was composed of gold tissue to match the skirt, and the front was of crêpe de chine, hidden with a stomacher of real diamonds, rubies, and emeralds, and there was a jewelled belt. A gold crown encrusted with emeralds, diamonds, and rubies, with a diamond drop at each curved end, and two upstanding white ostrich feathers in the centre, and round the front were festoons of pearls, with a large pear-shaped pearl in the centre falling on the forehead."<ref>“The Ball at Devonshire House. Magnificent Spectacle. Description of the Dresses.” London ''Evening Standard'' 3 July 1897 Saturday: 3 [of 12], Cols. 1a–5b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000183/18970703/015/0004.</ref>{{rp|p. 3, Col. 2b}} *"The Duchess of Devonshire, as Zenobia, Queen of Palmyra, wore a magnificent costume. The skirt of gold tissue was embroidered all over in a star-like design in emeralds, sapphires, diamonds, and other jewels outlined with gold, the corners where it opened in front being elaborately wrought in the same jewels and gold to represent peacocks' outspread tails. This opened to show an under-dress of cream crepe de chine, delicately embroidered in silver, gold, and pearls, and sprinkled all over with diamonds. The train was attached to the shoulders by two slender points, and was fastened at the waist with a large diamond ornament. It was of green velvet of a lovely shade, and was superbly embroidered in Oriental designs, introducing the lotus flower in rubies, sapphires, amethysts, emeralds, and diamonds, in four borderings on contrasting grounds, separated with gold cord. The train was lined with turquoise satin. The bodice was composed of gold tissue to match the skirt, and the front was of crepe de chine, hidden with a stomacher of real diamonds, rubies, and emeralds, and there was a jeweled belt. A gold crown encrusted with emeralds, diamonds, and rubies with a diamond drop at each curved end and two upstanding white ostrich feathers in the centre, and round the front were festoons of pearls with a large pear-shaped pearl in the centre falling on the forehead."<ref name=":7" />{{rp|p. 7, Col. 7a}} *"The Duchess of Devonshire, as Zenobia, Queen of Palmyra, wore a magnificent costume. The skirt of gold tissue was embroidered all over in a star-like design in emeralds, sapphires, diamonds, and other jewels outlined with gold, the corners where it opened in front being elaborately wrought in the same jewels and gold to represent peacocks’ outspread tails. This opened to show an underdress of cream crêpe de chine, delicately embroidered in silver, gold, and pearls and sprinkled all over with diamonds. The train, which was attached to the shoulders by two slender points and was fastened at the waist with a large diamond ornament, was a green velvet of a lovely shade, and was superbly embroidered in Oriental designs introducing the lotus flower in rubies, sapphires, amethysts, emeralds, and diamonds, with four borderings on contrasting grounds, separated with gold cord. The train was lined with turquoise satin. The bodice was composed of gold tissue to match the skirt, and the front was of crêpe de chine hidden with a stomacher of real diamonds, rubies and emeralds. Jewelled belt. A gold crown incrusted with emeralds, diamonds, and rubies, with a diamond drop at each curved end and two upstanding white ostrich feathers in the middle, and round the front festoons of pearls with a large pear-shaped pearl in the centre falling on the forehead."<ref name=":6">"Ball at Devonshire House." The ''Times'' Saturday 3 July 1897: 12, Cols. 1A–4C ''The Times Digital Archive''. Web. 28 Nov. 2015.</ref>{{rp|p. 12, Col. 3b}} *According to the article in ''The Graphic'', written by [[Social Victorians/People/Lady Violet Greville|Lady Violet Greville]] though this caption to the Lafayette photograph seems to have been boilerplate and printed in other places, the Duchess of Devonshire wore a "Skirt of gold tissue, embroidered all over with emeralds, sapphires, diamonds, and other jewels outlined with gold. This opened to show an underdress of crème crêpe de chine, embroidered in silver, gold, and pearls, and sprinkled all over with diamonds. The train was green velvet, superbly embroidered in Oriental designs. The bodice was composed of gold tissue, and the front was of crêpe de chine hidden with a stomacher of diamonds, rubies, and emeralds. A gold crown encrusted with emeralds, diamonds, and rubies, with a diamond drop at each curved end and two upstanding white ostrich feathers in the middle, and round the front festoons of pearls with a large pear-shaped pearl in the centre."<ref name=":10">Greville, Violet, Lady. "Devonshire House Ball." The ''Graphic'' Saturday 10 July 1897: 15 [of 24]: Col. 1a–16, Col. 1c. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000057/18970710/019/0015.</ref>{{rp|p. 15, Col. 3b}} *The ''Guernsey Star'' describes first [[Social Victorians/People/Spencer Compton Cavendish|Spencer Compton Cavendish, Duke of Devonshire]] and then Louisa, Duchess: "The host himself personated Charles V. of Germany in a costume copied from a celebrated picture by Titian, while the hostess was attired with great Oriental magnificence as Zenobia. Her dress was tissue of silver in front [sic], wrought with jewels. The over-dress was cloth of gold magnificently wrought with jewels, and Her Grace wore a bandeau of gold round her head, studded with diamonds, turquoise, and emeralds, and surrounded by hanging chains of superb pearls."<ref>"Duchess of Devonshire's Fancy-Dress Ball. Brilliant Spectacle." The [Guernsey] ''Star'' 6 July 1897, Tuesday: 1 [of 4], Col. 1a–2b [of 7]. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000184/18970706/003/0001.</ref>{{rp|p. 1, Col. 2a}} == The Duchess's Jewelry == Gossipy newspaper reports before the ball reported on the jewelry associated with the costumes for the ball. For example, according to the Edinburgh ''Evening News'' on 21 June 1897, less than two weeks before the party, "The ball being a fancy dress one, men as well as women will be able in certain characters to wear jewels. The Duchess of Devonshire, who is to appear as Zenobia, is getting her jewels reset after the antique style."<ref>“The Duchess of Devonshire’s Ball.” Edinburgh ''Evening News'' 21 June 1897, Monday: 4 [of 6], Col. 5c [of 7]. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000452/18970621/079/0004.</ref> While almost all descriptions of her mention her jewels because they were sewn onto the costume itself, these emphasize her jewelry: * "The Duchess was attired with great Oriental magnificence as Zenobia. Her dress was a tissue of silver, embroidered with gold and jewels, an overmantle of cloth of gold embroidered in the same manner hung from the shoulders, and she wore a bandeau of gold studded with gems, and surrounded by hanging chains of pearls over her elaborate headdress; strings and ropes of jewels and pearls were worn round the neck, and hung down almost to the knees."<ref>“The Duchess of Devonshire’s Ball.” The ''Gentlewoman'' 10 July 1897 Saturday: 32–42 [of 76], Cols. 1a–3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970710/155/0032. </ref>{{rp|p. 32, Cols. 1c–2a}} * "A wonderfully beautiful dress was that which was worn by the Duchess of Devonshire as Zenobia, Queen of Palmyra. It was of golden tissue, sewn with silver paillettes, and jewelled with diamonds and other precious stones. In front there were silk embroideries, in many vivid shades of colour, and here the golden draperies opened to show a petticoat of white crêpe de chine, embroidered with pearls and gold. The short train was of brilliant green velvet, exquisitely embroidered. One of the Duchess of Devonshire’s beautiful diamond and emerald tiaras had been taken to pieces to form a stomacher, the effect of which was dazzling in its brilliancy. Long chains of pearls and other wonderful jewels were worn with this beautiful dress."<ref>“The Devonshire House Ball. A Brilliant Gathering.” The ''Pall Mall Gazette'' 3 July 1897, Saturday: 7 [of 10], Col. 2a–3a [of 3]. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000098/18970703/019/0007.</ref>{{rp|p. 7, Col. 2b}} * In the article about the ball in the ''Graphic'', [[Social Victorians/People/Lady Violet Greville|Lady Violet Greville]] says, "The Ducal hostess herself elected to appear as Zenobia, Queen of Palmyra, with lavish magnificence, and wearing a corruscation of jewels which must have eclipsed the state of even the all-subduing majesty the Duchess impersonated."<ref name=":10" />{{rp|p. 16, Col. 1a}} *The Duchess was dressed "as Zenobia, Queen of Palmyra, her dress a marvel of soft tissues and exquisite ornament, and her tiara a still greater marvel of the jeweller's art."<ref name=":6" />{{rp|p. 12, Col. 2a}} <ref>"The Duchess of Devonshire’s Historic Ball. Some of the Fancy Costumes." Supplement. The ''Leicester Chronicle and Leicestershire Mercury'' 10 July 1897, Saturday: 11 [of 12], Cols. 4a–b [of 7]. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000173/18970710/141/0011.</ref>{{rp|p. 11, 4a}} The Duchess's pearls, which were an important feature of her costume, occasioned a great deal of direct commentary in the newspaper accounts. === Goldsmith, Pearl & Diamond Merchant, & Silversmith === The Duchess's jewelry occasioned a great deal of reportage in the articles about the ball. '''It was reported that she had her jewels restrung to be used in the costume. stomacher and review of jewelry in more general articles''' An invoice and receipt in the Archives of the Duke of Devonshire (Devonshire Collections, Chatsworth) is from a concern whose preprinted stationery has a crown in the upper-left corner, suggesting that they had a royal warrant, and no name other than Goldsmith, Pearl & Diamond Merchant, & Silversmith. This document offers a unique view into the evolution of one necklace, at least, over the years. It lists what are apparently three restringing of some pearls of Louise, Duchess of Devonshire. The three restringings appear to be dated: The first necklace is a "Pearl Necklet in original 4 rows," dated 20 October 1892 (but the stationery was printed to assume the invoice would be used in the 1880s, so the 9 is written over the second 8, and the 2 has been added).<ref name=":14">Invoice and receipt. Goldsmith, Pearl & Diamond Merchant & Silversmith. Date of itemized invoices for restringing pearls: 20 October 1892, 1 March 1897, 1909. The Devonshire Collections, Chatsworth, Reference number FIS/4/1/2.</ref>(p. 1) The necklet contained a "Total [of] Total 224 large pearls": # 1st [row] 51 large pearls # 2nd 53 large pearls # 3rd 57 large pearls # 4th 63 large pearls The second necklace is a "Necklet as re-strung on October 15th 1892, with addition of small pearls supplied, now consists of 5 rows, containing" a total of "224 large pearls & 227 small" <ref name=":14" />(p. 1) # 1st 41 large pearls & 40 small # 2nd 42 large pearls & 42 small # 3rd 44 large pearls & 45 small # 4th 47 large pearls & 48 small # 5th 50 large pearls & 51 small The third necklace is a "Pearl Necklet as again re-strung with additional pearls supplied 1 March 1897, now consisting of 5 Rows containing" a total of "262 Large Pearls & 267 Small"<ref name=":14" />(p. 2): # 1st Row 45 Large Pearls & 44 Small # 2nd Row 48 large Pearls & 49 Small # 3rd Row 51 Large Pearls & 52 Small # 4th Row 56 Large Pearls & 65 small Possibly these pearls may have been restrung in 1909 into a cornet?<ref name=":14" />(p. 2) If the Duchess wore one of these stringings of her pearls for the ball, then it must have been the second necklet, strung in 1892, a 5-strand necklace. None of the newspaper accounts refer to a 5-strand pearl necklace, although her pearls are often mentioned. == The Duchess's Entourage == Besides the Duke of Devonshire, the retinue of Louise, Duchess of Devonshire as Zenobia, Queen of Palmyra, included her grandson, [[Social Victorians/People/William Angus Drogo Montagu|William Angus Drago Montagu, 9th Duke of Manchester]], dressed as a Georgian courtier. According to a single source, the Belfast ''News-Letter,''<ref name=":8" />{{rp|p. 5, Col. 9a}} the rest of her entourage — all in costume — seems to have been made up of the following: * Four children * Four trumpeters * Four fan-bearers Three newspapers — The Belfast ''News-Letter'', the ''Man of Ross'' and the ''Westminster Gazette'' — say that the Duchess's entourage included three groups: children, trumpeters and fan-bearers. Only the Belfast ''News-Letter'' says that each group had four members. These three sources describe the Duchess's retinue and how the people in it were dressed: *"The Duchess of Devonshire was dazzingly [sic] magnificent as 'Zenobia,' arrayed in the glistening fabrics and massive jewels in which artists have delighted to depict the Warrior Queen, the costume in this case being specially designed by the clever French artist, M. Comelli, who was also responsible for the splendid attire of the Queen's suite. This was composed of four children in white Assyrian robes, draped with pink shawls; four trumpeters in white cloth robes, embroidered in subdued tones of silks, with a purple shawl draped over, beautifully ornamented with embroidery, and wearing fringed steel helmets and leather cuirasses embossed in steel; and four fan-bearers attired in pale blue robes, with crimson shawls, enriched with gold and jewelled embroidery, adorned with jewelled diadems, and holding long-handled fans of white feathers, mounted in blue and gold — a gloriously magnificent pageant."<ref name=":8" />{{rp|p. 5, Col. 9a}} *"The duchess was dressed as Zenobia, in gold cloth, gorgeously embroidered in gold, brilliants, and coloured stones, and opening over an under dress of white crêpe de Chine, worked finely in brilliants. The train of light green velvet was lined with blue, and sumptuously embroidered in jewels and gold, the colouring being particularly artistic. With this dress were worn splendid jewels, and a large horn crown, encrusted with diamonds, emeralds, and rubies. The duchess was attended by a suite of children, trumpeters, and fan-bearers, all picturesquely attired in Assyian [sic] costumes — the whole group being specially designed by M. Comelli."<ref name=":5" /> *"The host was dressed as Charles V. of Germany, in black velvet, satin, and fur; and the Duchess made the most gorgeous of Zenobias, in a gown of gold gauze, and a green velvet train — both a mass of exquisite oriental embroidery. The crown and hanging ropes of pearls, the jewelled girdle, and the train of children, fan-bearers, and trumpeters — all in Babylonish garb — as designed by M. Comelli, made a gloriously imposing and picturesque group."<ref name=":9" /> === Details of the Costumes in the Entourage === The Archives of the Duke of Devonshire (Devonshire Collections, Chatsworth) has "receipts" or invoices that functioned as receipts for several commercial concerns that were involved in making costumes or accessories for costumes for this ball. They are the following: * [[Social Victorians/People/Louisa Montagu Cavendish#M. (Attillo Giuseppe) Comelli|M. (Attillo Giuseppe) Comelli]] * [[Social Victorians/People/Louisa Montagu Cavendish#B. Burnet & Co.|B. Burnet & Co.]] * [[Social Victorians/People/Louisa Montagu Cavendish#Arthur Millward, Theatrical Jeweller|Arthur Millward, Theatrical Jeweller]] * [[Social Victorians/People/Louisa Montagu Cavendish#Liberty & Co., Ltd.|Liberty & Co., Ltd.]] * [[Social Victorians/People/Louisa Montagu Cavendish#Lafayette, Ltd.|Lafayette, Ltd.]] * [[Social Victorians/People/Louisa Montagu Cavendish#Goldsmith, Pearl & Diamond Merchant, & Silversmith|Goldsmith, Pearl & Diamond Merchant, & Silversmith]] This list of commercial concerns almost certainly cannot be the complete list of all concerns that contributed to the costumes. These are the only receipts or invoices about expenses for the ball, however, that the Chatsworth Archive contains; similar documents were likely not even kept or were destroyed with other papers not retained at some point in time. The business concerns listed above were specialized and likely used for different elements of the costumes. As a theatrical designer, Comelli would have depended on the suppliers he knew and arranged with them for the construction of these costumes. The Chatsworth Archive calls these documents ''receipts'', which indeed they are because they were returned to Devonshire House as receipts for payment. From our perspective, though, they are invoices that contain specifics about what was used to make the costumes. The analysis of these invoices has led to an understanding of what the people who attended the Duchess in her entourage wore and a clearer sense, perhaps, of how many people walked in that entourage. This analysis is based on the items listed on the invoices and their pricing, most of which is included in the section for each invoice, below. While the Belfast ''News-Letter'' says that each group contained four members,<ref name=":8" />{{rp|p. 5, Col. 9a}} the invoices and receipts suggest that the newspaper, the single source for this information, was wrong about the number of people in each group. It is theoretically possible, of course, that suppliers other than the ones in the Chatsworth Archive made some of these costumes and that other invoices and receipts must have existed at that time. But the [[Social Victorians/People/Louisa Montagu Cavendish#M. (Attillo Giuseppe) Comelli|Comelli memo, below]], seems definitive: he designed and seems to have overseen the construction of the costumes, which numbered six rather than twelve. '''[Collier discussion?]''' Besides providing welcome detail about the costumes of the people in the Duchess's entourage, which is available nowhere else, these invoices also raise at least as many questions as they answer. ==== M. (Attillo Giuseppe) Comelli ==== Attillo Giuseppe Comelli was a designer for opera, ballet and theatre in Europe, the UK and North America.<ref name=":13" /> The receipt in the Chatsworth Archive was sent from Covent Garden. The invoice lists £4 for "Making six costumes," 3''s'' for "Extras" and 12s for "Cab fares for men paid by the request of M. Comelli."<ref>M. Comelli, Covent Garden, to Duchess of Devonshire. Date of invoice 13 July 1897; postmarked 25 October 1897, for receipt of payment(?). The Devonshire Collections, Chatsworth, Reference number L/109/4(3).</ref> Three other names are on this invoice and receipt: * L. L[?] Collier [?], written under and perhaps as part of the direction to the Duchess of Devonshire * Mr. Strong ("Forwarded to Mr Strong. [sic] by the instructions of M. Comelli," written in the same hand as wrote the majority of the memo) * Floyd [?] Collier [??] ("Received with Thanks," presumably thanking for the payment, in a different hand) ==== B. Burnet & Co. ==== An invoice and receipt from B. Burnet & Co., held in the Archives of the Duke of Devonshire, has specific information about some of the fabrics, trims and accessories purchased for the costumes of the Duchess's retinue.<ref name=":11">B. Burnet & Co. to Louise, Duchess of Devonshire. Date of invoice 2 and 6 July 1897; postmarked 25 October 1897, for receipt of payment(?). The Devonshire Collections, Chatsworth, Reference number L/109/4(3).</ref> Besides itemizing some costume or accessory elements that seem clearly to be for the groups, the invoice also lists items not easy to associate with particular costumes, like the following: * 12 yards of White silk fringe 8in deep<ref name=":11" />{{rp|back left}} * 12 1/2 yards of "wht cloth"<ref name=":11" />{{rp|back left}} * 9 yards of "[[Social Victorians/Terminology#Selesia|Selesia]]"<ref name=":11" />{{rp|back left}} * 2 yards of Canvas<ref name=":11" />{{rp|back right}} * 4 Tan Wool Tights<ref name=":11" />{{rp|back right}} * 2 Tan Boys Tights<ref name=":11" />{{rp|back right}} At this time, we are not sure which costumes these elements were used for. Possibly the white silk fringe and the white cloth would have been used to construct the robes for the children and trumpeters in the entourage. The number of tights suggests that the six costumes on this invoice all included tights. With other elements of the trumpeters' costumes, the Burnet invoice also lists "6 prs Assyrian Buskins." Probably, to a late Victorian, buskins would have been "defensive leggings"<ref>Demmin, Auguste. An illustrated History of Arms and Armour: From the Earliest Period to the Present Time. George Bell, 1894. Google Books https://books.google.com/books?id=ArRCAAAAYAAJ: 106.</ref> laced together and covering the lower leg and often feet of a soldier. To a clothing and military historian, buskins (or greaves) were worn by people in a number of cultures over millennia and varied widely in style and construction. Buskins appear in Assyrian art held at the time by the British Museum. Listing six pairs of buskins suggests that every costume in the Duchess's entrourage included buskins, possibly worn over the tan tights. The Burnet invoice lists "4 Broad Belts," which may have held "4 Skins Fleshers."<ref name=":11" />{{rp|p. 1, front of invoice}} (A skin flesher is a kind of knife used to separate the skin from the meat in animals.) If each group included only two members, then perhaps the belts and fleshers were worn not only by the trumpeters but also by the fan-bearers. The Millward invoice (specifics in the section on the Millward invoice below) lists "8 Doz 'Plaques' for Belts'" with a drawing of an upright rectangle with a circle in the middle, which might have been a jewel. Double lines around the rectangle suggest that the plaques were not flat or the metal was not thin. The drawing does not give any ideas about how the plaques were attached to the belts, if they were. It is impossible to tell if the plaques were attached to the "4 Broad Belts" (likely for the trumpeters and fan-bearers), but unless they were quite tiny, "8 Doz 'Plaques'" would be far too many for the belts of only the two children. A different hand, probably "[L.??] L. Collier," wrote the following sentence at the end of the invoice and receipt, above the postmark:<blockquote>All the above named articles were used for the six [?] dresses made for the Devonshire Ball.<ref name=":11" />(back right)</blockquote>This same hand, signing what is possibly "Floyd Collier," also signed the postmark of the Comelli invoice and receipt. On the Burnet document, this writer, possibly an assistant or employee of the Duchess of Devonshire, says that "six dresses" were made (if in fact, that word is ''six''). (No "Collier" is listed among the staff or servants of the Duke of Devonshire at the end of the 19th century.<ref>"Servants A-H." ''Historic Servants and Staff. Servants and Staff Database''. Retrieved 18 July 2022 https://www.chatsworth.org/media/11528/servants-a-h.pdf.</ref> The invoice appears to itemize materials used for six costumes: two children, two trumpeters and two fan-bearers. ==== Arthur Millward, Theatrical Jeweller ==== An invoice and request for payment from Arthur Millward, Theatrical Jeweller, held in the Archives of Chatsworth House, has more specifics about some of the fabrics, trims and accessories for the costumes of the Duchess's retinue.<ref name=":12">Memorandum. Arthur Millward, Theatrical Jeweller, to Louise, Duchess of Devonshire. Date of itemized invoice 1 July 1897; date of request for payment(?) 27 August 1897. The Devonshire Collections, Chatsworth, Reference number L/109/4(?).</ref> This invoice lists the following, which could have been used in any of the costumes for the entourage: * 8 Doz 'Plaques' for Belts [discussed with the belts in the section on the Burnet invoice, above] * 4 Large Armlets * 4 Bracelets * 8 Armlets<ref name=":12" />{{rp|p. 2, back}} Because Millward was a Theatrical Jeweller, it seems likely that most (if not all) of the items listed on the invoice were made of metal and the jewels mentioned were artificial, made of glass or paste. Other items on the invoice seem to belong to the costumes of the trumpeters, which the Belfast ''News-Letter'' says included helmets: * 2 Helmets * 2 Centre pieces The Millward invoice shows tiny line drawings next to the words ''2 Helmets'' and ''2 Centre pieces''. These drawings suggest that the Centre pieces were attached to the helmets rather than being anything that would have been put on a table as decoration. Other items seem to belong to the costumes of the fan-bearers: * 2 Pearl & Gold Headdresses * 2 Fan properties with Feathers The "Pearl & Gold Headdresses" were likely the "jewelled diadems" mentioned in the Belfast ''News-Letter''. The "Fan properties with Feathers" are likely to have been the "long-handled fans of white feathers, mounted in blue and gold" mentioned in the newspaper report.<ref name=":8" />{{rp|p. 5, Col. 9a}} At the end of the Millward invoice, a "reduction as agreed with M [Mr?] Commelli [sic]" of £1 10''s'' is subtracted from a total of £22 3''s''. No reason for this reduction is given.<ref name=":12" />{{rp|p. 2, back}} ==== Liberty & Co., Ltd. ==== One invoice and receipt from the Chatsworth Archive, dated 12 July, to the Duchess of Devonshire, lists "13 yds S&W Satin[?]," 7 yards of blue and 6 of purple.<ref>Invoice and receipt. Liberty & Co. Ltd. To Her Grace, The Duchess of Devonshire. Date of itemized invoice 12 July [1897]. The Devonshire Collections, Chatsworth, Reference number L/109/4(?).</ref> Because the fabric is satin and from Liberty, it is possible that it was not used in the costumes of the people in the entourage but perhaps for the costume of the Duchess herself??? ==== Lafayette, Ltd. ==== The invoice and receipt from Lafayette, Ltd., the photographer that set up a temporary studio in the garden to take portraits of people at the ball in their costumes, may not be related to the ball at all.<ref>Invoice and receipt. Lafayette, Ltd. To His Grace The Duke of Devonshire. Date of itemized invoice 12 April 1897; addressed to the Duke, 18 February 1898; receipt and thanks for payment, 7 April 1898. The Devonshire Collections, Chatsworth, Reference number L/109/4(?).</ref> Three dates are written on the preprinted stationery: # 18/2/98 (18 February 1898), under the direction to "His Grace The Duke of Devonshire" # 4/12/97 (4 December 1897), next to the single item on the invoice for which a charge is listed: "6 [??] £1.10.0" # 7/4/98 (7 April 1898), in a different hand, with "Recd by cheque 7/4/98 Lafayette Ltd pp[?] [??] thanks" At the bottom of the page, in the hand that wrote all of the invoice except the receipt and thanks, is "With Lafayette Ltds Compliments." ==== Details for the Children in the Entourage ==== According to the ''Belfast News-Letter'', four children were "in white Assyrian robes, draped with pink shawls."<ref name=":8" />{{rp|p. 5, Col. 9a}} According to the B. Burnet invoice, the following was purchased for "White Cloth Dresses":<ref name=":11" />{{rp|p. 2, back left of invoice}} * "2 Terra Gown draperies with Stars 200 in all" * "2 Cloth fronts embroidered with Square Medallions down centre" * "2 do do [ditto ditto, that is, cloth fronts] embroidered double border down front each side and collar" * "4 Sleeves embroidered Small Medallions" The Burnet & Co. invoice lists 6 yards of "Terra" Silk Fringe, which perhaps was used to trim the "terra draperies," or shawls, made from 3 1/4 yards of "Light Terra Satinette" worn by the children? ==== Details for the Trumpeters in the Entourage ==== According to the ''Belfast News-Letter'', four trumpeters were "in white cloth robes, embroidered in subdued tones of silks, with a purple shawl draped over, beautifully ornamented with embroidery, and wearing fringed steel helmets and leather cuirasses embossed in steel."<ref name=":8" />{{rp|p. 5, Col. 9a}} The trumpeters appear to have been dressed as soldiers or military men. According to the B. Burnet invoice, the following was purchased for the trumpeters' costumes:<ref name=":11" />{{rp|p. 1, front of invoice}} * 7 '''units (yards?)''' of purple silk [probably used for shawls?] * "2 skirt fronts with border alround [sic]" * "2 sets of Leather Cuarasses [sic] Embroidered front & back" * "4 Sleeves embroidered loop stitch" The Millward invoice lists * 2 Helmets * 2 Centre Pieces [probably for helmets rather than table decorations] ==== Details for the Fan-bearers in the Entourage ==== According to the ''Belfast News-Letter'', four fan-bearers were "attired in pale blue robes, with crimson shawls, enriched with gold and jewelled embroidery, adorned with jewelled diadems, and holding long-handled fans of white feathers, mounted in blue and gold."<ref name=":8" />{{rp|p. 5, Col. 9a}} According to the B. Burnet invoice, the following was purchased for the fan bearers's costumes:<ref name=":11" />{{rp|pp. 1–2, front and left-back of invoice}} * "Embroidering 2 Crimson draperies with Stars 334 in all" * "2 Top [?] fronts embroidered & round necks" * "4 Sleeves embroidered long stitch"The Millward invoice lists * 2 Pearl & Gold Headdresses * 2 Fan properties with Feathers<ref name=":12" />{{rp|p. 2, back}} The Burnet & Co. invoice lists 12 yards of "Red Silk Fringe," which perhaps was used to trim the "crimson shawls" or "Crimson draperies," which may have been made from the 5 yards of "Red Satinette." Again, this list suggests two rather than four costumes. === The Historical Zenobia === Zenobia (240 – c. 274) was queen of the Syrian Palmyrene Empire, ruling as regent for her son after her husband's assassination.<ref>{{Cite journal|date=2022-05-03|title=Zenobia|url=https://en.wikipedia.org/w/index.php?title=Zenobia&oldid=1086005949|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Zenobia.</ref> She was the subject of much art in the 19th century, including literature, opera, sculpture, and paintings. Middle-eastern traveller Lady Hester Stanhope (1776–1839) discussed Zenobia in her memoirs, published in 1847.<ref>{{Cite journal|date=2022-03-07|title=Lady Hester Stanhope|url=https://en.wikipedia.org/w/index.php?title=Lady_Hester_Stanhope&oldid=1075838273|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Lady_Hester_Stanhope.</ref> == Demographics == *Nationality: born in Hanover, in what is now Germany<ref name=":0">{{Cite journal|date=2020-07-27|title=Louisa Cavendish, Duchess of Devonshire|url=https://en.wikipedia.org/w/index.php?title=Louisa_Cavendish,_Duchess_of_Devonshire&oldid=969824214|journal=Wikipedia|language=en}}</ref> === Residences === ==== As Duchess of Manchester ==== *Kimbolton Castle, Huntingdonshire *Manchester House, London ==== As Duchess of Devonshire ==== *Devonshire House, London (mid-April until mid-July, for the Season) *Compton Place, Eastbourne (mid-July until 12 August<ref name=":1" />{{rp|p. 32}}) *Bolton Abbey, Yorkshire (12 August until the middle of September<ref name=":1" />{{rp|p. 32}}) *Chatsworth, Derbyshire (middle of September until early Spring<ref name=":1" />{{rp|p. 32}}) *Lismore Castle, County Waterford (early Spring until the middle of April<ref name=":1" />{{rp|p. 32}}) == Family == *Louisa (or Luise) Friederike Auguste Gräfin von Alten Montagu Cavendish (15 January 1832 – 15 November 1911)<ref name=":2" /><ref name=":0" /> *William Drogo Montagu, 7th Duke of Manchester (15 October 1823 – 22 March 1890)<ref name=":3" /><ref>{{Cite journal|date=2020-09-07|title=William Montagu, 7th Duke of Manchester|url=https://en.wikipedia.org/w/index.php?title=William_Montagu,_7th_Duke_of_Manchester&oldid=977197445|journal=Wikipedia|language=en}}</ref> #George Victor Drogo Montagu, 8th Duke of Manchester (17 June 1853 – 18 August 1892) #Mary Louise [Louisa?] Elizabeth Montagu Douglas-Hamilton Forster (27 December 1854 – 10 February 1934) #Louisa Augusta Beatrice Montagu Acheson (c. 1856 – 3 March 1944) #Charles William Augustus Montagu (23 November 1860 – 10 November 1939) #Alice Maude Olivia Montagu Stanley (15 August 1862 – 23 July 1957) *[[Social Victorians/People/Spencer Compton Cavendish|Spencer Compton Cavendish]], 8th Duke of Devonshire (23 July 1833 – 24 March 1908) == Notes and Questions == #As Duchess of Manchester Luise was not invited to the wedding between Bertie and Alix, Victoria's punishment for Luise's having gotten the Duke of Derby to promise her the position of Mistress of the Robes (and then exacting that promise).<ref>Leslie, Anita. ''The Marlborough House Set''. New York: Doubleday, 1973.</ref>{{rp|pp. 47–48}} #"As a young woman she was extremely beautiful; Princess Catherine Radziwill saw her at a reception given by the Empress of Germany and recalls on being introduced to her 'how she struck me as the loveliest creature I had ever set eyes upon. Indeed I have only met three women in my whole existence who could be compared to her.'"<ref name=":1" />{{rp|p. 21}} == Footnotes == {{reflist}} dy18j5tavw1ctsiax8t1hr74e87ii4i 2410318 2410306 2022-07-29T22:20:31Z Scogdill 1331941 wikitext text/x-wiki == Also Known As == *Louise, Duchess of Devonshire *Louisa, Duchess of Manchester *Luise Friederike August Gräfin von Alten *Louisa Montagu *Louise Cavendish *The Double Duchess == Acquaintances, Friends and Enemies == === Friends === *[[Social Victorians/People/Albert Edward, Prince of Wales | Albert Edward, Prince of Wales]] (beginning about 1852) *[[Social Victorians/People/Spencer Compton Cavendish|Spencer Compton Cavendish]], Lord Hartington (later 8th Duke of Devonshire) *Daisy, Lady Warwick *Lady Mayoress, Mrs. Benjamin Samuel Faudel-Phillips, 2nd Baronet,<ref>{{Cite journal|date=2020-08-25|title=Faudel-Phillips baronets|url=https://en.wikipedia.org/w/index.php?title=Faudel-Phillips_baronets&oldid=974879290|journal=Wikipedia|language=en}}</ref> presented to Victoria by Louisa Cavendish at a Queen's Drawing-room on Wednesday, 24 February 1897 at Buckingham Palace.<ref name=":4">"The Queen's Drawing Room" ''Morning Post'' 25 February 1897 Thursday: 5 [of 10], Col. 5a–7b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970225/047/0005.</ref>{{rp|p. 5, Col. 6c}} *Mrs. J. E. Mellor, presented to Victoria by Louisa Cavendish at a Queen's Drawing-room on Wednesday, 24 February 1897 at Buckingham Palace.<ref name=":4" />{{rp|p. 5, Col. 6c}} === Enemies === * Consuelo, Duchess of Marlborough (at least, in 1901)<ref name=":1">Murphy, Sophia. ''The Duchess of Devonshire's Ball''. London: Sidgwick & Jackson, 1984.</ref>{{rp|pp. 31–32}} == Organizations == == Timeline == '''1852 July 22''', Luise Friederike Auguste Gräfin von Alten and William Drogo Montagu married.<ref name=":2">"Luise Friederike Auguste Gräfin von Alten." {{Cite web|url=http://www.thepeerage.com/p10947.htm#i109469|title=Person Page|website=www.thepeerage.com|access-date=2020-09-25}}</ref> '''1863, early, or late 1862''', Louise and Spencer Compton Cavendish began a relationship.<ref name=":1" />{{rp|p. 26}} '''1873 December 10''', Mary Louise Elizabeth Montagu (daughter) and William Douglas-Hamilton married. '''1876 May 22''', Consuelo Iznaga y Clement and George Victor Drogo Montagu (son) married in Grace Church, New York City.<ref>{{Cite journal|date=2020-08-24|title=George Montagu, 8th Duke of Manchester|url=https://en.wikipedia.org/w/index.php?title=George_Montagu,_8th_Duke_of_Manchester&oldid=974659520|journal=Wikipedia|language=en}}</ref><ref>{{Cite journal|date=2020-07-27|title=Consuelo Montagu, Duchess of Manchester|url=https://en.wikipedia.org/w/index.php?title=Consuelo_Montagu,_Duchess_of_Manchester&oldid=969888488|journal=Wikipedia|language=en}}</ref> '''1876 August 10''', Louisa Augusta Beatrice Montagu (daughter) and Archibald Acheson married. '''1889 January 5''', Alice Maude Olivia Montagu (daughter) and Edward Stanley married. '''1890 March 22''', William Drogo Montagu (7th Duke) died.<ref name=":3">"William Drogo Montagu, 7th Duke of Manchester." {{Cite web|url=http://www.thepeerage.com/p10128.htm#i101274|title=Person Page|website=www.thepeerage.com|access-date=2020-09-25}}</ref> '''1890 November 14''', William Angus Drogo Montagu (grandson) and Helena Zimmerman married secretly, in London.<ref>"Helena Zimmerman." {{Cite web|url=http://www.thepeerage.com/p34555.htm#i345545|title=Person Page|website=www.thepeerage.com|access-date=2020-09-25}}</ref> '''1892 August 16''', Louise Friederike Auguste Gräfin von Alten Montagu and Spencer Compton Cavendish, her second husband, married.<ref name=":2" /> '''1897 July 2, Friday''', Louise Cavendish (#18 on the list of attendees) hosted her famous [[Social Victorians/1897 Fancy Dress Ball| fancy-dress ball]] at Devonshire House in London. '''1897 July 20''', Mary Louise Elizabeth Montagu Douglas-Hamilton and Robert Carnaby Foster married. '''1900 November 14''', William Angus Drogo Montagu and Helena Zimmerman married.<ref>{{Cite journal|date=2020-07-17|title=Helena, Countess of Kintore|url=https://en.wikipedia.org/w/index.php?title=Helena,_Countess_of_Kintore&oldid=968067371|journal=Wikipedia|language=en}}</ref> '''1901 Spring''', Paris, Consuelo Spencer-Churchill, Duchess of Marlborough, describes a meeting with Louise Cavendish in the spring following Queen Victoria's death at the horse racetrack, Longchamps:<blockquote>A renowned character and virtually dictator of what was known as the fast set as opposed to the Victorian, Her Grace was a German aristocrat by birth. She had first been married to the impoverished Duke of Manchester, and when he died had improved her status by marriage to the rich Duke of Devonshire, who waged an undisputed influence in politics. Rumour had her beautiful, but when I knew her she was a raddled old woman, covering her wrinkles with paint and her pate with a brown wig. Her mouth was a red gash and from it, when she saw me, issued a stream of abuse. How could I, she complained, pointing to my white gloves, show so little respect to the memory of a great Queen? What a carefree world we must have lived in, that etiquette even in such small matters could assume so much importance?<ref>Balsan, Consuelo Vanderbilt. ''The Glitter and the Gold: The American Duchess — In Her Own Words''. New York: St. Martin's, 1953.</ref>{{rp|p. 115}}</blockquote> === Annual Events === Every year, as Duchess of Devonshire, Louise held a dance on the night after the Derby at Epsom Downs, which at this point was held on Wednesdays after Easter. == Costume at the Duchess of Devonshire's 2 July 1897 Fancy-dress Ball == [[File:Louise Frederica Augusta Cavendish (née von Alten), Duchess of Devonshire (formerly Duchess of Manchester) as Zenobia, Queen of Palmyra.jpg|thumb|Louise, Duchess of Devonshire as Zenobia, Queen of Palmyra|alt=Louise, Duchess of Devonshire in costume as Zenobia, Queen of Palmyra]] At their fancy-dress ball, Louisa, Duchess of Devonshire sat at Table 1 during the first seating for supper, escorted in to the table by the Prince of Wales.<ref name=":7">"Fancy Dress Ball at Devonshire House." ''Morning Post'' Saturday 3 July 1897: 7 [of 12], Col. 4a–8 Col. 2b. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970703/054/0007.</ref>{{rp|p. 7, Col. 4c}} Her costume was designed by M. Comelli (Attillo Giuseppe Comelli, 1858–1925, artist and costumier for opera, ballet and theatre in London as well as Europe and the U.S.<ref name=":13">{{Cite book|url=https://books.google.com/books?id=SZh2DwAAQBAJ&pg=PT207&lpg=PT207&dq=Attilio+Comelli&source=bl&ots=lFB0If7CwV&sig=ACfU3U1_Ost_lhmMvzMMs6NvuhK5SlRhJw&hl=en&sa=X&ved=2ahUKEwjKlsTw2sH3AhXYAp0JHVIxDWA4KBDoAXoECBAQAw#v=onepage&q=Attilio%20Comelli&f=false|title=Forgotten Designers Costume Designers of American Broadway Revues and Musicals From 1900-1930|last=Unruh|first=Delbert|date=2018-11-06|publisher=Page Publishing Inc|isbn=978-1-64082-758-5|language=en}} N.P.</ref>)<ref name=":5">“The Devonshire House Ball.” The ''Man of Ross'' 10 July 1897, Saturday: 2 [of 8], Col. 4b. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0001463/18970710/033/0002.</ref> <ref name=":8">"The Duchess of Devonshire's Fancy Dress Ball. Special Telegram." ''Belfast News-Letter'' Saturday 03 July 1897: 5 [of 8], Col. 9 [of 9]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/BL/0000038/18970703/015/0005.</ref>{{rp|p. 5, Col. 9a}} <ref name=":9">"By One Who Was There." “The Duchess’s Costume Ball.” ''Westminster Gazette'' 03 July 1897 Saturday: 5 [of 8], Cols. 1a–3b [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002947/18970703/035/0005.</ref> and constructed by the House of Worth. Comelli seems to have designed [[Social Victorians/People/Louisa Montagu Cavendish#The Duchess and Her Entourage|the costumes of her retinue as well]]. According to Russell Harris,<blockquote>For her costume, the Duchess commissioned Monsieur Comelli (1858-1925), a well-known designer of opera costumes for the London theatre and opera stage, and then had the design made up by Worth of Paris. ''Munsey’s Magazine'' noted “it is safe to say that the Queen of Palmyra never owned such a sumptuous costume in her lifetime.”<ref>Harris, Russell. {{Cite web|url=http://www.rvondeh.dircon.co.uk/incalmprose/devonshiredss.html|title=Louise, Duchess of Devonshire, née Countess von Alten of Hanover (1832-1911), as Zenobia, Queen of Palmyra|website=www.rvondeh.dircon.co.uk|access-date=2022-05-05}} ''Narrated in Calm Prose: Photographs from the V&A's Lafayette Archive of Guests in Costume at the Duchess of Devonshire's Diamond Jubilee Ball, July 1897''. http://www.rvondeh.dircon.co.uk/incalmprose/devonshiredss.html.</ref></blockquote>Lafayette's portrait of "Louise Frederica Augusta Cavendish (née von Alten), Duchess of Devonshire (formerly Duchess of Manchester)" in costume is photogravure #5 in the album presented to the Duchess of Devonshire and now in the National Portrait Gallery.<ref>"Devonshire House Fancy Dress Ball (1897): photogravures by Walker & Boutall after various photographers." 1899. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait-list.php?set=515.</ref> The printing on the portrait says, "The Duchess of Devonshire as Zenobia Queen of Palmyra," with a Long S in ''Duchess''.<ref>"Louise Frederica Augusta Cavendish (née von Alten), Duchess of Devonshire (formerly Duchess of Manchester) as Zenobia, Queen of Palmyra." Devonshire House Fancy Dress Ball Album. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw158357/Louise-Frederica-Augusta-Cavendish-ne-von-Alten-Duchess-of-Devonshire-formerly-Duchess-of-Manchester-as-Zenobia-Queen-of-Palmyra.</ref> Often, the V&A Lafayette Archive contains more than one portrait of a sitter for this ball, but the uncropped portrait (above right), which shows the unfinished end of the balustrade in front of the Duchess and the edge of the painted flat behind it, seems to have been the only portrait taken by Lafayette of the Duchess in costume. The copy owned by the National Portrait Gallery in London and the copy included in the album are cropped so that those unfinished edges do not show, but they appear to be from the same photograph. === Newspaper Descriptions of the Duchess's Costume === Newspaper articles about the Duchess's presence at the ball focused on her hosting, her costume, [[Social Victorians/People/Louisa Montagu Cavendish#The Duchess's Jewelry|her jewelry]], and [[Social Victorians/People/Louisa Montagu Cavendish#The Duchess's Entourage|her entourage]], often in the same story. These almost exactly identical descriptions suggest [[Social Victorians/1897 Fancy Dress Ball/anthology#Scissors-and-Paste Journalism|scissors-and-paste journalism]] or a shared primary source: * "The Duchess of Devonshire was a dazzling vision, dressed as 'Zenobia,' in a glistening gold gauze gown, elaborately ornamented with suns and discs, wrought in purple and green gems outlined with gold, and having a large diamond as centre. The space between was fluted with fine silver spangles. This robe was open in front over an under dress of white crépe de chine, delicately worked in crystals, and at each side of the opening on the gold robe were large fan-shaped groups of peacock feathers, worked in multicoloured jewels. The [[Social Victorians/Terminology#Corsage|corsage]] was to correspond, and had a magnificent [[Social Victorians/Terminology#Girdle|girdle]] of jewels, the train of bright green velvet, hung like a fan, without folds, being fastened at each side of the shoulders by diamond brooches, and caught at the waist with a similar ornament. It was a mass of gorgeous embroidery, carried out in heliotrope velvet, lotus flowers studded with tinted gems, and other devices in terra-cotta and electric blue velvet — all enriched with gold, diamond, and jewelled embroidery — and lined with pale blue satin. The crown worn with this was high, and of filigree gold, surmounted with two horns, each tipped with a large diamond. It was encrusted with large diamonds, rubies, and emeralds, and long chains of pearls fell under the chin and about the head — one magnificent pear-shaped pearl resting on the forehead. Attending the hostess were four children, four fan-bearers, and four trumpeters, all magnificently arrayed in artistically embroidered Assyrian robes, helmets, and other accessories, correct in every detail."<ref name=":15">"Duchess of Devonshire's Fancy Ball. A Brilliant Spectacle. Some of the Dresses." London ''Daily News'' Saturday 3 July 1897: 5 [of 10], Col. 6a–6, Col. 1b. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000051/18970703/024/0005 and http://www.britishnewspaperarchive.co.uk/viewer/BL/0000051/18970703/024/0006.</ref>{{rp|p. 5, Col. 6a}} * "The Duchess of Devonshire, as Zenobia, Queen of Palmyra, wore a magnificent costume, supplied by Worth, of Paris. The skirt of gold tissue was embroidered all over in a star-like design in emeralds, sapphires, diamonds, and other jewels, outlined with gold, the corners where it opened in front being elaborately wrought in the same jewels and gold to represent peacocks' outspread tails. This opened to show an under-dress of cream crêpe de chine, delicately embroidered in silver, gold, and pearls, and sprinkled all over with diamonds. The train was attached to the shoulders by two slender points, and was fastened at the waist with a large diamond ornament. It was of green velvet of a lovely shade, and was superbly embroidered in Oriental designs, introducing the lotus flower in rubies, sapphires, amethysts, emeralds, and diamonds, in four borderings on contrasting grounds, separated with gold cord. The train was lined with turquoise satin. The bodice was composed of gold tissue to match the skirt, and the front was of crêpe de chine, hidden with a stomacher of real diamonds, rubies, and emeralds, and there was a jewelled belt. A gold crown encrusted with emeralds, diamonds, and rubies, with a diamond drop at each curved end, and two upstanding white ostrich feathers in the centre, and round the front were festoons of pearls, with a large pear-shaped pearl in the centre falling on the forehead."<ref name=":16">“The Ball at Devonshire House. Magnificent Spectacle. Description of the Dresses.” London ''Evening Standard'' 3 July 1897 Saturday: 3 [of 12], Cols. 1a–5b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000183/18970703/015/0004.</ref>{{rp|p. 3, Col. 2b}} *"The Duchess of Devonshire, as Zenobia, Queen of Palmyra, wore a magnificent costume. The skirt of gold tissue was embroidered all over in a star-like design in emeralds, sapphires, diamonds, and other jewels outlined with gold, the corners where it opened in front being elaborately wrought in the same jewels and gold to represent peacocks' outspread tails. This opened to show an under-dress of cream crepe de chine, delicately embroidered in silver, gold, and pearls, and sprinkled all over with diamonds. The train was attached to the shoulders by two slender points, and was fastened at the waist with a large diamond ornament. It was of green velvet of a lovely shade, and was superbly embroidered in Oriental designs, introducing the lotus flower in rubies, sapphires, amethysts, emeralds, and diamonds, in four borderings on contrasting grounds, separated with gold cord. The train was lined with turquoise satin. The bodice was composed of gold tissue to match the skirt, and the front was of crepe de chine, hidden with a stomacher of real diamonds, rubies, and emeralds, and there was a jeweled belt. A gold crown encrusted with emeralds, diamonds, and rubies with a diamond drop at each curved end and two upstanding white ostrich feathers in the centre, and round the front were festoons of pearls with a large pear-shaped pearl in the centre falling on the forehead."<ref name=":7" />{{rp|p. 7, Col. 7a}} *"The Duchess of Devonshire, as Zenobia, Queen of Palmyra, wore a magnificent costume. The skirt of gold tissue was embroidered all over in a star-like design in emeralds, sapphires, diamonds, and other jewels outlined with gold, the corners where it opened in front being elaborately wrought in the same jewels and gold to represent peacocks’ outspread tails. This opened to show an underdress of cream crêpe de chine, delicately embroidered in silver, gold, and pearls and sprinkled all over with diamonds. The train, which was attached to the shoulders by two slender points and was fastened at the waist with a large diamond ornament, was a green velvet of a lovely shade, and was superbly embroidered in Oriental designs introducing the lotus flower in rubies, sapphires, amethysts, emeralds, and diamonds, with four borderings on contrasting grounds, separated with gold cord. The train was lined with turquoise satin. The bodice was composed of gold tissue to match the skirt, and the front was of crêpe de chine hidden with a stomacher of real diamonds, rubies and emeralds. Jewelled belt. A gold crown incrusted with emeralds, diamonds, and rubies, with a diamond drop at each curved end and two upstanding white ostrich feathers in the middle, and round the front festoons of pearls with a large pear-shaped pearl in the centre falling on the forehead."<ref name=":6">"Ball at Devonshire House." The ''Times'' Saturday 3 July 1897: 12, Cols. 1A–4C ''The Times Digital Archive''. Web. 28 Nov. 2015.</ref>{{rp|p. 12, Col. 3b}} *According to the article in ''The Graphic'', written by [[Social Victorians/People/Lady Violet Greville|Lady Violet Greville]] though this caption to the Lafayette photograph seems to have been boilerplate and printed in other places, the Duchess of Devonshire wore a "Skirt of gold tissue, embroidered all over with emeralds, sapphires, diamonds, and other jewels outlined with gold. This opened to show an underdress of crème crêpe de chine, embroidered in silver, gold, and pearls, and sprinkled all over with diamonds. The train was green velvet, superbly embroidered in Oriental designs. The bodice was composed of gold tissue, and the front was of crêpe de chine hidden with a stomacher of diamonds, rubies, and emeralds. A gold crown encrusted with emeralds, diamonds, and rubies, with a diamond drop at each curved end and two upstanding white ostrich feathers in the middle, and round the front festoons of pearls with a large pear-shaped pearl in the centre."<ref name=":10">Greville, Violet, Lady. "Devonshire House Ball." The ''Graphic'' Saturday 10 July 1897: 15 [of 24]: Col. 1a–16, Col. 1c. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000057/18970710/019/0015.</ref>{{rp|p. 15, Col. 3b}} *The ''Guernsey Star'' describes first [[Social Victorians/People/Spencer Compton Cavendish|Spencer Compton Cavendish, Duke of Devonshire]] and then Louisa, Duchess: "The host himself personated Charles V. of Germany in a costume copied from a celebrated picture by Titian, while the hostess was attired with great Oriental magnificence as Zenobia. Her dress was tissue of silver in front [sic], wrought with jewels. The over-dress was cloth of gold magnificently wrought with jewels, and Her Grace wore a bandeau of gold round her head, studded with diamonds, turquoise, and emeralds, and surrounded by hanging chains of superb pearls."<ref name=":17">"Duchess of Devonshire's Fancy-Dress Ball. Brilliant Spectacle." The [Guernsey] ''Star'' 6 July 1897, Tuesday: 1 [of 4], Col. 1a–2b [of 7]. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000184/18970706/003/0001.</ref>{{rp|p. 1, Col. 2a}} == The Duchess's Jewelry == Gossipy newspaper reports before the ball reported on the jewelry associated with the costumes for the ball. For example, according to the Edinburgh ''Evening News'' on 21 June 1897, less than two weeks before the party, "The ball being a fancy dress one, men as well as women will be able in certain characters to wear jewels. The Duchess of Devonshire, who is to appear as Zenobia, is getting her jewels reset after the antique style."<ref>“The Duchess of Devonshire’s Ball.” Edinburgh ''Evening News'' 21 June 1897, Monday: 4 [of 6], Col. 5c [of 7]. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000452/18970621/079/0004.</ref> While almost all descriptions of her mention her jewels because they were sewn onto the costume itself, these emphasize her jewelry: * "The Duchess was attired with great Oriental magnificence as Zenobia. Her dress was a tissue of silver, embroidered with gold and jewels, an overmantle of cloth of gold embroidered in the same manner hung from the shoulders, and she wore a bandeau of gold studded with gems, and surrounded by hanging chains of pearls over her elaborate headdress; strings and ropes of jewels and pearls were worn round the neck, and hung down almost to the knees."<ref>“The Duchess of Devonshire’s Ball.” The ''Gentlewoman'' 10 July 1897 Saturday: 32–42 [of 76], Cols. 1a–3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970710/155/0032. </ref>{{rp|p. 32, Cols. 1c–2a}} * "A wonderfully beautiful dress was that which was worn by the Duchess of Devonshire as Zenobia, Queen of Palmyra. It was of golden tissue, sewn with silver paillettes, and jewelled with diamonds and other precious stones. In front there were silk embroideries, in many vivid shades of colour, and here the golden draperies opened to show a petticoat of white crêpe de chine, embroidered with pearls and gold. The short train was of brilliant green velvet, exquisitely embroidered. One of the Duchess of Devonshire’s beautiful diamond and emerald tiaras had been taken to pieces to form a stomacher, the effect of which was dazzling in its brilliancy. Long chains of pearls and other wonderful jewels were worn with this beautiful dress."<ref>“The Devonshire House Ball. A Brilliant Gathering.” The ''Pall Mall Gazette'' 3 July 1897, Saturday: 7 [of 10], Col. 2a–3a [of 3]. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000098/18970703/019/0007.</ref>{{rp|p. 7, Col. 2b}} * In the article about the ball in the ''Graphic'', [[Social Victorians/People/Lady Violet Greville|Lady Violet Greville]] says, "The Ducal hostess herself elected to appear as Zenobia, Queen of Palmyra, with lavish magnificence, and wearing a corruscation of jewels which must have eclipsed the state of even the all-subduing majesty the Duchess impersonated."<ref name=":10" />{{rp|p. 16, Col. 1a}} *The Duchess was dressed "as Zenobia, Queen of Palmyra, her dress a marvel of soft tissues and exquisite ornament, and her tiara a still greater marvel of the jeweller's art."<ref name=":6" />{{rp|p. 12, Col. 2a}} <ref>"The Duchess of Devonshire’s Historic Ball. Some of the Fancy Costumes." Supplement. The ''Leicester Chronicle and Leicestershire Mercury'' 10 July 1897, Saturday: 11 [of 12], Cols. 4a–b [of 7]. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000173/18970710/141/0011.</ref>{{rp|p. 11, 4a}} Points to make * The Duchess's pearls, which were an important feature of her costume, occasioned a great deal of direct commentary in the newspaper accounts. * Some accounts say that the jewels on the Duchess's costume were actual precious or semiprecious stones, but not all and this might not be right. Zenobia's Crown The crown that the Duchess wore as Zenobia is difficult to see clearly in the Lafayette photograph (above right). It was lavish, "encrusted" with jewels and featuring pearls: * The London ''Daily News'' says, "The crown worn with this was high, and of filigree gold, surmounted with two horns, each tipped with a large diamond. It was encrusted with large diamonds, rubies, and emeralds, and long chains of pearls fell under the chin and about the head — one magnificent pear-shaped pearl resting on the forehead."<ref name=":15" />(p. 5, Col. 6a) * The London ''Evening Standard'' says, "A gold crown encrusted with emeralds, diamonds, and rubies, with a diamond drop at each curved end, and two upstanding white ostrich feathers in the centre, and round the front were festoons of pearls, with a large pear-shaped pearl in the centre falling on the forehead."<ref name=":16" />{{rp|p. 3, Col. 2b}} * The London ''Morning Post'' says, "A gold crown encrusted with emeralds, diamonds, and rubies with a diamond drop at each curved end and two upstanding white ostrich feathers in the centre, and round the front were festoons of pearls with a large pear-shaped pearl in the centre falling on the forehead."<ref name=":7" />{{rp|p. 7, Col. 7a}} * The London ''Times'' says, "A gold crown incrusted with emeralds, diamonds, and rubies, with a diamond drop at each curved end and two upstanding white ostrich feathers in the middle, and round the front festoons of pearls with a large pear-shaped pearl in the centre falling on the forehead."<ref name=":6" />{{rp|p. 12, Col. 3b}} * The London ''Graphic'' says, "A gold crown encrusted with emeralds, diamonds, and rubies, with a diamond drop at each curved end and two upstanding white ostrich feathers in the middle, and round the front festoons of pearls with a large pear-shaped pearl in the centre."<ref name=":10" />{{rp|p. 15, Col. 3b}} * The Guernsey ''Star'' says, "Her Grace wore a bandeau of gold round her head, studded with diamonds, turquoise, and emeralds, and surrounded by hanging chains of superb pearls."<ref name=":17" />{{rp|p. 1, Col. 2a}} === Goldsmith, Pearl & Diamond Merchant, & Silversmith === The Duchess's jewelry occasioned a great deal of reportage in the articles about the ball. '''It was reported that she had her jewels restrung to be used in the costume. stomacher and review of jewelry in more general articles''' An invoice and receipt in the Archives of the Duke of Devonshire (Devonshire Collections, Chatsworth) is from a concern whose preprinted stationery has a crown in the upper-left corner, suggesting that they had a royal warrant, and no name other than Goldsmith, Pearl & Diamond Merchant, & Silversmith. This document offers a unique view into the evolution of one necklace, at least, over the years. It lists what are apparently three restringing of some pearls of Louise, Duchess of Devonshire. The three restringings appear to be dated: The first necklace is a "Pearl Necklet in original 4 rows," dated 20 October 1892 (but the stationery was printed to assume the invoice would be used in the 1880s, so the 9 is written over the second 8, and the 2 has been added).<ref name=":14">Invoice and receipt. Goldsmith, Pearl & Diamond Merchant & Silversmith. Date of itemized invoices for restringing pearls: 20 October 1892, 1 March 1897, 1909. The Devonshire Collections, Chatsworth, Reference number FIS/4/1/2.</ref>(p. 1) The necklet contained a "Total [of] Total 224 large pearls": # 1st [row] 51 large pearls # 2nd 53 large pearls # 3rd 57 large pearls # 4th 63 large pearls The second necklace is a "Necklet as re-strung on October 15th 1892, with addition of small pearls supplied, now consists of 5 rows, containing" a total of "224 large pearls & 227 small" <ref name=":14" />(p. 1) # 1st 41 large pearls & 40 small # 2nd 42 large pearls & 42 small # 3rd 44 large pearls & 45 small # 4th 47 large pearls & 48 small # 5th 50 large pearls & 51 small The third necklace is a "Pearl Necklet as again re-strung with additional pearls supplied 1 March 1897, now consisting of 5 Rows containing" a total of "262 Large Pearls & 267 Small"<ref name=":14" />(p. 2): # 1st Row 45 Large Pearls & 44 Small # 2nd Row 48 large Pearls & 49 Small # 3rd Row 51 Large Pearls & 52 Small # 4th Row 56 Large Pearls & 65 small Possibly these pearls may have been restrung in 1909 into a cornet?<ref name=":14" />(p. 2) If the Duchess wore one of these stringings of her pearls for the ball, then it must have been the second necklet, strung in 1892, a 5-strand necklace. None of the newspaper accounts refer to a 5-strand pearl necklace, although her pearls are often mentioned. == The Duchess's Entourage == Besides the Duke of Devonshire, the retinue of Louise, Duchess of Devonshire as Zenobia, Queen of Palmyra, included her grandson, [[Social Victorians/People/William Angus Drogo Montagu|William Angus Drago Montagu, 9th Duke of Manchester]], dressed as a Georgian courtier. Four newspapers say that the Duchess's entourage included three groups, all in costume: children, trumpeters and fan-bearers. According to two sources, the London ''Daily News''<ref name=":15" />(p. 5, Col. 6a) and the Belfast ''News-Letter,''<ref name=":8" />{{rp|p. 5, Col. 9a}} these groups each had four members. The London ''Daily News'' is likely the source (though not the only one) for the Belfast ''News-Letter'', which took part in [[Social Victorians/1897 Fancy Dress Ball/anthology#Scissors-and-Paste Journalism|scissors-and-paste journalism]], like so many other newspapers of the 19th century. ['''check this: which one was published earlier in the day, and on which day?'''] The ''Man of Ross'' and the ''Westminster Gazette'' do not address the number of members of the groups. These four sources describe the Duchess's retinue and how the people in it were dressed: *"Attending the hostess were four children, four fan-bearers, and four trumpeters, all magnificently arrayed in artistically embroidered Assyrian robes, helmets, and other accessories, correct in every detail."<ref name=":15" />{{rp|p. 5, Col. 6a}} *"The Duchess of Devonshire was dazzingly [sic] magnificent as 'Zenobia,' arrayed in the glistening fabrics and massive jewels in which artists have delighted to depict the Warrior Queen, the costume in this case being specially designed by the clever French artist, M. Comelli, who was also responsible for the splendid attire of the Queen's suite. This was composed of four children in white Assyrian robes, draped with pink shawls; four trumpeters in white cloth robes, embroidered in subdued tones of silks, with a purple shawl draped over, beautifully ornamented with embroidery, and wearing fringed steel helmets and leather cuirasses embossed in steel; and four fan-bearers attired in pale blue robes, with crimson shawls, enriched with gold and jewelled embroidery, adorned with jewelled diadems, and holding long-handled fans of white feathers, mounted in blue and gold — a gloriously magnificent pageant."<ref name=":8" />{{rp|p. 5, Col. 9a}} *"The duchess was dressed as Zenobia, in gold cloth, gorgeously embroidered in gold, brilliants, and coloured stones, and opening over an under dress of white crêpe de Chine, worked finely in brilliants. The train of light green velvet was lined with blue, and sumptuously embroidered in jewels and gold, the colouring being particularly artistic. With this dress were worn splendid jewels, and a large horn crown, encrusted with diamonds, emeralds, and rubies. The duchess was attended by a suite of children, trumpeters, and fan-bearers, all picturesquely attired in Assyian [sic] costumes — the whole group being specially designed by M. Comelli."<ref name=":5" /> *"The host was dressed as Charles V. of Germany, in black velvet, satin, and fur; and the Duchess made the most gorgeous of Zenobias, in a gown of gold gauze, and a green velvet train — both a mass of exquisite oriental embroidery. The crown and hanging ropes of pearls, the jewelled girdle, and the train of children, fan-bearers, and trumpeters — all in Babylonish garb — as designed by M. Comelli, made a gloriously imposing and picturesque group."<ref name=":9" /> === Details of the Costumes in the Entourage === The Archives of the Duke of Devonshire (Devonshire Collections, Chatsworth) has "receipts" or invoices that functioned as receipts for several commercial concerns that were involved in making costumes or accessories for costumes for this ball. They are the following: * [[Social Victorians/People/Louisa Montagu Cavendish#M. (Attillo Giuseppe) Comelli|M. (Attillo Giuseppe) Comelli]] * [[Social Victorians/People/Louisa Montagu Cavendish#B. Burnet & Co.|B. Burnet & Co.]] * [[Social Victorians/People/Louisa Montagu Cavendish#Arthur Millward, Theatrical Jeweller|Arthur Millward, Theatrical Jeweller]] * [[Social Victorians/People/Louisa Montagu Cavendish#Liberty & Co., Ltd.|Liberty & Co., Ltd.]] * [[Social Victorians/People/Louisa Montagu Cavendish#Lafayette, Ltd.|Lafayette, Ltd.]] * [[Social Victorians/People/Louisa Montagu Cavendish#Goldsmith, Pearl & Diamond Merchant, & Silversmith|Goldsmith, Pearl & Diamond Merchant, & Silversmith]] This list of commercial concerns almost certainly cannot be the complete list of all concerns that contributed to the costumes. These are the only receipts or invoices about expenses for the ball, however, that the Chatsworth Archive contains; similar documents were likely not even kept or were destroyed with other papers not retained at some point in time. The business concerns listed above were specialized and likely used for different elements of the costumes. As a theatrical designer, Comelli would have depended on the suppliers he knew and arranged with them for the construction of these costumes. The Chatsworth Archive calls these documents ''receipts'', which indeed they are because they were returned to Devonshire House as receipts for payment. From our perspective, though, they are invoices that contain specifics about what was used to make the costumes. The analysis of these invoices has led to an understanding of what the people who attended the Duchess in her entourage wore and a clearer sense, perhaps, of how many people walked in that entourage. This analysis is based on the items listed on the invoices and their pricing, most of which is included in the section for each invoice, below. While the Belfast ''News-Letter'' says that each group contained four members,<ref name=":8" />{{rp|p. 5, Col. 9a}} the invoices and receipts suggest that the newspaper, the single source for this information, was wrong about the number of people in each group. It is theoretically possible, of course, that suppliers other than the ones in the Chatsworth Archive made some of these costumes and that other invoices and receipts must have existed at that time. But the [[Social Victorians/People/Louisa Montagu Cavendish#M. (Attillo Giuseppe) Comelli|Comelli memo, below]], seems definitive: he designed and seems to have overseen the construction of the costumes, which numbered six rather than twelve. '''[Collier discussion?]''' Besides providing welcome detail about the costumes of the people in the Duchess's entourage, which is available nowhere else, these invoices also raise at least as many questions as they answer. ==== M. (Attillo Giuseppe) Comelli ==== Attillo Giuseppe Comelli was a designer for opera, ballet and theatre in Europe, the UK and North America.<ref name=":13" /> The receipt in the Chatsworth Archive was sent from Covent Garden. The invoice lists £4 for "Making six costumes," 3''s'' for "Extras" and 12s for "Cab fares for men paid by the request of M. Comelli."<ref>M. Comelli, Covent Garden, to Duchess of Devonshire. Date of invoice 13 July 1897; postmarked 25 October 1897, for receipt of payment(?). The Devonshire Collections, Chatsworth, Reference number L/109/4(3).</ref> Three other names are on this invoice and receipt: * L. L[?] Collier [?], written under and perhaps as part of the direction to the Duchess of Devonshire * Mr. Strong ("Forwarded to Mr Strong. [sic] by the instructions of M. Comelli," written in the same hand as wrote the majority of the memo) * Floyd [?] Collier [??] ("Received with Thanks," presumably thanking for the payment, in a different hand) ==== B. Burnet & Co. ==== An invoice and receipt from B. Burnet & Co., held in the Archives of the Duke of Devonshire, has specific information about some of the fabrics, trims and accessories purchased for the costumes of the Duchess's retinue.<ref name=":11">B. Burnet & Co. to Louise, Duchess of Devonshire. Date of invoice 2 and 6 July 1897; postmarked 25 October 1897, for receipt of payment(?). The Devonshire Collections, Chatsworth, Reference number L/109/4(3).</ref> Besides itemizing some costume or accessory elements that seem clearly to be for the groups, the invoice also lists items not easy to associate with particular costumes, like the following: * 12 yards of White silk fringe 8in deep<ref name=":11" />{{rp|back left}} * 12 1/2 yards of "wht cloth"<ref name=":11" />{{rp|back left}} * 9 yards of "[[Social Victorians/Terminology#Selesia|Selesia]]"<ref name=":11" />{{rp|back left}} * 2 yards of Canvas<ref name=":11" />{{rp|back right}} * 4 Tan Wool Tights<ref name=":11" />{{rp|back right}} * 2 Tan Boys Tights<ref name=":11" />{{rp|back right}} At this time, we are not sure which costumes these elements were used for. Possibly the white silk fringe and the white cloth would have been used to construct the robes for the children and trumpeters in the entourage. The number of tights suggests that the six costumes on this invoice all included tights. With other elements of the trumpeters' costumes, the Burnet invoice also lists "6 prs Assyrian Buskins." Probably, to a late Victorian, buskins would have been "defensive leggings"<ref>Demmin, Auguste. An illustrated History of Arms and Armour: From the Earliest Period to the Present Time. George Bell, 1894. Google Books https://books.google.com/books?id=ArRCAAAAYAAJ: 106.</ref> laced together and covering the lower leg and often feet of a soldier. To a clothing and military historian, buskins (or greaves) were worn by people in a number of cultures over millennia and varied widely in style and construction. Buskins appear in Assyrian art held at the time by the British Museum. Listing six pairs of buskins suggests that every costume in the Duchess's entrourage included buskins, possibly worn over the tan tights. The Burnet invoice lists "4 Broad Belts," which may have held "4 Skins Fleshers."<ref name=":11" />{{rp|p. 1, front of invoice}} (A skin flesher is a kind of knife used to separate the skin from the meat in animals.) If each group included only two members, then perhaps the belts and fleshers were worn not only by the trumpeters but also by the fan-bearers. The Millward invoice (specifics in the section on the Millward invoice below) lists "8 Doz 'Plaques' for Belts'" with a drawing of an upright rectangle with a circle in the middle, which might have been a jewel. Double lines around the rectangle suggest that the plaques were not flat or the metal was not thin. The drawing does not give any ideas about how the plaques were attached to the belts, if they were. It is impossible to tell if the plaques were attached to the "4 Broad Belts" (likely for the trumpeters and fan-bearers), but unless they were quite tiny, "8 Doz 'Plaques'" would be far too many for the belts of only the two children. A different hand, probably "[L.??] L. Collier," wrote the following sentence at the end of the invoice and receipt, above the postmark:<blockquote>All the above named articles were used for the six [?] dresses made for the Devonshire Ball.<ref name=":11" />(back right)</blockquote>This same hand, signing what is possibly "Floyd Collier," also signed the postmark of the Comelli invoice and receipt. On the Burnet document, this writer, possibly an assistant or employee of the Duchess of Devonshire, says that "six dresses" were made (if in fact, that word is ''six''). (No "Collier" is listed among the staff or servants of the Duke of Devonshire at the end of the 19th century.<ref>"Servants A-H." ''Historic Servants and Staff. Servants and Staff Database''. Retrieved 18 July 2022 https://www.chatsworth.org/media/11528/servants-a-h.pdf.</ref> The invoice appears to itemize materials used for six costumes: two children, two trumpeters and two fan-bearers. ==== Arthur Millward, Theatrical Jeweller ==== An invoice and request for payment from Arthur Millward, Theatrical Jeweller, held in the Archives of Chatsworth House, has more specifics about some of the fabrics, trims and accessories for the costumes of the Duchess's retinue.<ref name=":12">Memorandum. Arthur Millward, Theatrical Jeweller, to Louise, Duchess of Devonshire. Date of itemized invoice 1 July 1897; date of request for payment(?) 27 August 1897. The Devonshire Collections, Chatsworth, Reference number L/109/4(?).</ref> This invoice lists the following, which could have been used in any of the costumes for the entourage: * 8 Doz 'Plaques' for Belts [discussed with the belts in the section on the Burnet invoice, above] * 4 Large Armlets * 4 Bracelets * 8 Armlets<ref name=":12" />{{rp|p. 2, back}} Because Millward was a Theatrical Jeweller, it seems likely that most (if not all) of the items listed on the invoice were made of metal and the jewels mentioned were artificial, made of glass or paste. Other items on the invoice seem to belong to the costumes of the trumpeters, which the Belfast ''News-Letter'' says included helmets: * 2 Helmets * 2 Centre pieces The Millward invoice shows tiny line drawings next to the words ''2 Helmets'' and ''2 Centre pieces''. These drawings suggest that the Centre pieces were attached to the helmets rather than being anything that would have been put on a table as decoration. Other items seem to belong to the costumes of the fan-bearers: * 2 Pearl & Gold Headdresses * 2 Fan properties with Feathers The "Pearl & Gold Headdresses" were likely the "jewelled diadems" mentioned in the Belfast ''News-Letter''. The "Fan properties with Feathers" are likely to have been the "long-handled fans of white feathers, mounted in blue and gold" mentioned in the newspaper report.<ref name=":8" />{{rp|p. 5, Col. 9a}} At the end of the Millward invoice, a "reduction as agreed with M [Mr?] Commelli [sic]" of £1 10''s'' is subtracted from a total of £22 3''s''. No reason for this reduction is given.<ref name=":12" />{{rp|p. 2, back}} ==== Liberty & Co., Ltd. ==== One invoice and receipt from the Chatsworth Archive, dated 12 July, to the Duchess of Devonshire, lists "13 yds S&W Satin[?]," 7 yards of blue and 6 of purple.<ref>Invoice and receipt. Liberty & Co. Ltd. To Her Grace, The Duchess of Devonshire. Date of itemized invoice 12 July [1897]. The Devonshire Collections, Chatsworth, Reference number L/109/4(?).</ref> Because the fabric is satin and from Liberty, it is possible that it was not used in the costumes of the people in the entourage but perhaps for the costume of the Duchess herself??? ==== Lafayette, Ltd. ==== The invoice and receipt from Lafayette, Ltd., the photographer that set up a temporary studio in the garden to take portraits of people at the ball in their costumes, may not be related to the ball at all.<ref>Invoice and receipt. Lafayette, Ltd. To His Grace The Duke of Devonshire. Date of itemized invoice 12 April 1897; addressed to the Duke, 18 February 1898; receipt and thanks for payment, 7 April 1898. The Devonshire Collections, Chatsworth, Reference number L/109/4(?).</ref> Three dates are written on the preprinted stationery: # 18/2/98 (18 February 1898), under the direction to "His Grace The Duke of Devonshire" # 4/12/97 (4 December 1897), next to the single item on the invoice for which a charge is listed: "6 [??] £1.10.0" # 7/4/98 (7 April 1898), in a different hand, with "Recd by cheque 7/4/98 Lafayette Ltd pp[?] [??] thanks" At the bottom of the page, in the hand that wrote all of the invoice except the receipt and thanks, is "With Lafayette Ltds Compliments." ==== Details for the Children in the Entourage ==== According to the ''Belfast News-Letter'', four children were "in white Assyrian robes, draped with pink shawls."<ref name=":8" />{{rp|p. 5, Col. 9a}} According to the B. Burnet invoice, the following was purchased for "White Cloth Dresses":<ref name=":11" />{{rp|p. 2, back left of invoice}} * "2 Terra Gown draperies with Stars 200 in all" * "2 Cloth fronts embroidered with Square Medallions down centre" * "2 do do [ditto ditto, that is, cloth fronts] embroidered double border down front each side and collar" * "4 Sleeves embroidered Small Medallions" The Burnet & Co. invoice lists 6 yards of "Terra" Silk Fringe, which perhaps was used to trim the "terra draperies," or shawls, made from 3 1/4 yards of "Light Terra Satinette" worn by the children? ==== Details for the Trumpeters in the Entourage ==== According to the ''Belfast News-Letter'', four trumpeters were "in white cloth robes, embroidered in subdued tones of silks, with a purple shawl draped over, beautifully ornamented with embroidery, and wearing fringed steel helmets and leather cuirasses embossed in steel."<ref name=":8" />{{rp|p. 5, Col. 9a}} The trumpeters appear to have been dressed as soldiers or military men. According to the B. Burnet invoice, the following was purchased for the trumpeters' costumes:<ref name=":11" />{{rp|p. 1, front of invoice}} * 7 '''units (yards?)''' of purple silk [probably used for shawls?] * "2 skirt fronts with border alround [sic]" * "2 sets of Leather Cuarasses [sic] Embroidered front & back" * "4 Sleeves embroidered loop stitch" The Millward invoice lists * 2 Helmets * 2 Centre Pieces [probably for helmets rather than table decorations] ==== Details for the Fan-bearers in the Entourage ==== According to the ''Belfast News-Letter'', four fan-bearers were "attired in pale blue robes, with crimson shawls, enriched with gold and jewelled embroidery, adorned with jewelled diadems, and holding long-handled fans of white feathers, mounted in blue and gold."<ref name=":8" />{{rp|p. 5, Col. 9a}} According to the B. Burnet invoice, the following was purchased for the fan bearers's costumes:<ref name=":11" />{{rp|pp. 1–2, front and left-back of invoice}} * "Embroidering 2 Crimson draperies with Stars 334 in all" * "2 Top [?] fronts embroidered & round necks" * "4 Sleeves embroidered long stitch"The Millward invoice lists * 2 Pearl & Gold Headdresses * 2 Fan properties with Feathers<ref name=":12" />{{rp|p. 2, back}} The Burnet & Co. invoice lists 12 yards of "Red Silk Fringe," which perhaps was used to trim the "crimson shawls" or "Crimson draperies," which may have been made from the 5 yards of "Red Satinette." Again, this list suggests two rather than four costumes. === The Historical Zenobia === Zenobia (240 – c. 274) was queen of the Syrian Palmyrene Empire, ruling as regent for her son after her husband's assassination.<ref>{{Cite journal|date=2022-05-03|title=Zenobia|url=https://en.wikipedia.org/w/index.php?title=Zenobia&oldid=1086005949|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Zenobia.</ref> She was the subject of much art in the 19th century, including literature, opera, sculpture, and paintings. Middle-eastern traveller Lady Hester Stanhope (1776–1839) discussed Zenobia in her memoirs, published in 1847.<ref>{{Cite journal|date=2022-03-07|title=Lady Hester Stanhope|url=https://en.wikipedia.org/w/index.php?title=Lady_Hester_Stanhope&oldid=1075838273|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Lady_Hester_Stanhope.</ref> == Demographics == *Nationality: born in Hanover, in what is now Germany<ref name=":0">{{Cite journal|date=2020-07-27|title=Louisa Cavendish, Duchess of Devonshire|url=https://en.wikipedia.org/w/index.php?title=Louisa_Cavendish,_Duchess_of_Devonshire&oldid=969824214|journal=Wikipedia|language=en}}</ref> === Residences === ==== As Duchess of Manchester ==== *Kimbolton Castle, Huntingdonshire *Manchester House, London ==== As Duchess of Devonshire ==== *Devonshire House, London (mid-April until mid-July, for the Season) *Compton Place, Eastbourne (mid-July until 12 August<ref name=":1" />{{rp|p. 32}}) *Bolton Abbey, Yorkshire (12 August until the middle of September<ref name=":1" />{{rp|p. 32}}) *Chatsworth, Derbyshire (middle of September until early Spring<ref name=":1" />{{rp|p. 32}}) *Lismore Castle, County Waterford (early Spring until the middle of April<ref name=":1" />{{rp|p. 32}}) == Family == *Louisa (or Luise) Friederike Auguste Gräfin von Alten Montagu Cavendish (15 January 1832 – 15 November 1911)<ref name=":2" /><ref name=":0" /> *William Drogo Montagu, 7th Duke of Manchester (15 October 1823 – 22 March 1890)<ref name=":3" /><ref>{{Cite journal|date=2020-09-07|title=William Montagu, 7th Duke of Manchester|url=https://en.wikipedia.org/w/index.php?title=William_Montagu,_7th_Duke_of_Manchester&oldid=977197445|journal=Wikipedia|language=en}}</ref> #George Victor Drogo Montagu, 8th Duke of Manchester (17 June 1853 – 18 August 1892) #Mary Louise [Louisa?] Elizabeth Montagu Douglas-Hamilton Forster (27 December 1854 – 10 February 1934) #Louisa Augusta Beatrice Montagu Acheson (c. 1856 – 3 March 1944) #Charles William Augustus Montagu (23 November 1860 – 10 November 1939) #Alice Maude Olivia Montagu Stanley (15 August 1862 – 23 July 1957) *[[Social Victorians/People/Spencer Compton Cavendish|Spencer Compton Cavendish]], 8th Duke of Devonshire (23 July 1833 – 24 March 1908) == Notes and Questions == #As Duchess of Manchester Luise was not invited to the wedding between Bertie and Alix, Victoria's punishment for Luise's having gotten the Duke of Derby to promise her the position of Mistress of the Robes (and then exacting that promise).<ref>Leslie, Anita. ''The Marlborough House Set''. New York: Doubleday, 1973.</ref>{{rp|pp. 47–48}} #"As a young woman she was extremely beautiful; Princess Catherine Radziwill saw her at a reception given by the Empress of Germany and recalls on being introduced to her 'how she struck me as the loveliest creature I had ever set eyes upon. Indeed I have only met three women in my whole existence who could be compared to her.'"<ref name=":1" />{{rp|p. 21}} == Footnotes == {{reflist}} b5k1v2x4q9doh3xabbxw0qdpswd6rqe 0 271693 2410366 2410016 2022-07-30T00:42:58Z 2402:7500:916:306E:6805:A2C5:4D03:BE94 /* Base 36 */ wikitext text/x-wiki A '''quasi-minimal prime''' is a [[w:Prime number|prime number]] for which there is no shorter [[w:Subsequence|subsequence]] of its [[w:Numerical digit|digit]]s in a given [[w:Radix|base]] ''b'' that form a prime > ''b''. For example, 857 is a quasi-minimal prime in [[w:Decimal|decimal]] because there is no prime > 10 among the shorter subsequences of the digits: 8, 5, 7, 85, 87, 57. The subsequence does not have to consist of consecutive digits, so 149 is not a quasi-minimal prime in decimal (because 19 is prime and 19 > 10). But it does have to be in the same order; so, for example, 991 is still a quasi-minimal prime in decimal even though a subset of the digits can form the shorter prime 19 > 10 by changing the order. (using A−Z to represent digit values 10 to 35) For the quasi-minimal primes in bases up to 36, I have only solved (found all quasi-minimal primes and proved that these are all such primes) bases 2~12, 14~15, 18, 20, 22, 24, 30 (bases 11, 22, 30 need primality proving of the probable primes). For the remain bases 13, 16~17, 19, 21, 23, 25~29, 31~36, there are some ''x''{''d''}''y'' (with ''x'', ''y'' strings (may be [[w:Empty string|empty]]) with digits in base ''b'', ''d'' digit in base ''b'') families which are not solved (not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers > base (''b'')). I left as a challenge to readers the task of solving (finding all quasi-minimal primes and proving that these are all such primes) bases 13, 16~17, 19, 21, 23, 25~29, 31~36 (this will be a hard problem, e.g. base 23 has a quasi-minimal prime 9E₈₀₀₈₇₃, and base 36 has quasi-minimal prime P₈₁₉₉₃SZ). Proving the set of the quasi-minimal primes in base ''b'' is ''S'', is equivalent to: * Prove that all elements in ''S'', when read as base ''b'' representation, are primes > ''b''. * Prove that all [[w:Proper subset|proper]] subsequence of all elements in ''S'', when read as base ''b'' representation, which are > ''b'', are composite. * Prove that all primes > ''b'', when written in base ''b'', contain at least one element in ''S'' as subsequence (equivalently, prove that all strings not containing any element in ''S'' as subsequence, when read as base ''b'' representation, which are > ''b'', are composite). e.g. proving the set of the quasi-minimal primes in base ''b'' = 10 is {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}, is equivalent to: * Prove that all of 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027 are primes > 10. * Prove that all proper subsequence of all elements in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} which are > 10 are composite. * Prove that all primes > 10 contain at least one element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence (equivalently, prove that all numbers > 10 not containing any element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence are composite). ==Condensed table== {|class=wikitable |''b''||number of quasi-minimal primes base ''b''||base-''b'' form of largest known quasi-minimal prime base ''b''||length of largest known quasi-minimal prime base ''b''||algebraic ((''a''×''b''^''n''+''c'')/''d'') form of largest known quasi-minimal prime base ''b'' |- |2||1||11||2||3 |- |3||3||111||3||13 |- |4||5||221||3||41 |- |5||22||10₉₃13||96||5⁹⁵+8 |- |6||11||40041||5||5209 |- |7||71||3₁₆1||17||(7¹⁷−5)/2 |- |8||75||4₂₂₀7||221||(4×8²²¹+17)/7 |- |9||151||30₁₁₅₈11||1161||3×9¹¹⁶⁰+10 |- |10||77||50₂₈27||31||5×10³⁰+27 |- |11^*||1068||57₆₂₆₆₈||62669||(57×11⁶²⁶⁶⁸−7)/10 |- |12||106||40₃₉77||42||4×12⁴¹+91 |- |13^*||3195~3197||80₃₂₀₁₇111||32021||8×13³²⁰²⁰+183 |- |14||650||4D₁₉₆₉₈||19699||5×14¹⁹⁶⁹⁸−1 |- |15||1284||7₁₅₅97||157||(15¹⁵⁷+59)/2 |- |16^*||2346~2347||4₇₂₇₈₅DD||72787||(4×16⁷²⁷⁸⁷+2291)/15 |- |17^*||10407~10428||F70₁₈₆₇₆₇1||186770||262×17¹⁸⁶⁷⁶⁸+1 |- |18||549||C0₆₂₆₈C5||6271||12×18⁶²⁷⁰+221 |- |20||3314||G0₆₂₆₉D||6271||16×20⁶²⁷⁰+13 |- |21^*||13375~13396||CF₄₇₉₁₄₇0K||479150||(51×21⁴⁷⁹¹⁴⁹−1243)/4 |- |22^*||8003||BK₂₂₀₀₁5||22003||(251×22²²⁰⁰²−335)/21 |- |24||3409||N00N₈₁₂₉LN||8134||13249×24⁸¹³¹−49 |- |30^*||2619||OT₃₄₂₀₅||34206||25×30³⁴²⁰⁵−1 |- |36^*||35257~35263||P₈₁₉₉₃SZ||81995||(5×36⁸¹⁹⁹⁵+821)/7 |} ^* Data assumes the primality of the [[w:probable prime|probable prime]]s. Except bases ''b'' = 13, 16, 17, 21, all bases in this table are completely solved (if we allow strong probable primes > 10²⁰⁰⁰⁰), also, except bases ''b'' = 11, 13, 16, 17, 21, 22, 30, 36, all bases in this table are completely solved even if we only allow definitely primes (thus, we can complete the classification of the quasi-minimal primes in these bases, i.e. the “quasi-minimal problems” in these bases are now theorems), for the quasi-minimal primes see the data below. Base ''b'' = 13 has 3195 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 13, then they must be of the form 9{5} or A{3}A (we are unable to determine if these two families contain a prime or not, i.e. these two families have no known prime members, nor can these two families be ruled out as only containing composites), and must have at least 82000 digits in base 13, besides, since these two families can contain at most one quasi-minimal prime, there are at most 3197 quasi-minimal primes in base 13. (i.e. the quasi-minimal primes in base 13 are the 3195 known quasi-minimal primes in base 13 (they are given in the data section) plus the smallest prime in the family 9{5} in base 13 (if exists) plus the smallest prime in the family A{3}A in base 13 (if exists)) Base ''b'' = 16 has 2346 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 16, then they must be of the form {3}AF (we are unable to determine if this family contains a prime or not, i.e. this family have no known prime members, nor can this family be ruled out as only containing composites), and must have at least 76000 digits in base 16, besides, since this family can contain at most one quasi-minimal prime, there are at most 2347 quasi-minimal primes in base 16. (i.e. the quasi-minimal primes in base 16 are the 2346 known quasi-minimal primes in base 16 (they are given in the data section) plus the smallest prime in the family {3}AF in base 16 (if exists)) ==Data for quasi-minimal primes== ===Base 2=== 11 ===Base 3=== 12, 21, 111 ===Base 4=== 11, 13, 23, 31, 221 ===Base 5=== 12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013 ===Base 6=== 11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041 ===Base 7=== 14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331 ===Base 8=== 13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447 ===Base 9=== 12, 14, 18, 21, 25, 32, 34, 41, 45, 47, 52, 58, 65, 67, 74, 78, 81, 87, 117, 131, 135, 151, 155, 175, 177, 238, 272, 308, 315, 331, 337, 355, 371, 375, 377, 438, 504, 515, 517, 531, 537, 557, 564, 601, 638, 661, 702, 711, 722, 735, 737, 751, 755, 757, 771, 805, 838, 1011, 1015, 1101, 1701, 2027, 2207, 3017, 3057, 3101, 3501, 3561, 3611, 3688, 3868, 5035, 5051, 5071, 5101, 5501, 5554, 5705, 5707, 7017, 7075, 7105, 7301, 8535, 8544, 8555, 8854, 20777, 22227, 22777, 30161, 33388, 50161, 50611, 53335, 55111, 55535, 55551, 57061, 57775, 70631, 71007, 77207, 100037, 100071, 100761, 105007, 270707, 301111, 305111, 333035, 333385, 333835, 338885, 350007, 500075, 530005, 555611, 631111, 720707, 2770007, 3030335, 7776662, 30300005, 30333335, 38333335, 51116111, 70000361, 300030005, 300033305, 351111111, 1300000007, 5161111111, 8333333335, 300000000035, 311111111161, 544444444444, 2000000000007, 5700000000001, 7270000000007, 88888888833335, 100000000000507, 5111111111111161, 7277777777777777707, 8888888888888888888335, 30000000000000000000051, 1000000000000000000000000057, 56111111111111111111111111111111111111, 7666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666662, 27777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777707, 300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011 ===Base 10=== 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027 ===Base 11=== 12, 16, 18, 21, 27, 29, 34, 38, 3A, 43, 49, 54, 56, 61, 65, 67, 72, 76, 81, 89, 92, 94, 98, 9A, A3, 10A, 115, 117, 133, 139, 153, 155, 171, 193, 197, 199, 1AA, 225, 232, 236, 25A, 263, 315, 319, 331, 335, 351, 353, 362, 373, 379, 391, 395, 407, 414, 452, 458, 478, 47A, 485, 4A5, 4A7, 502, 508, 511, 513, 533, 535, 539, 551, 571, 579, 588, 595, 623, 632, 70A, 711, 715, 731, 733, 737, 755, 759, 775, 791, 797, 7AA, 803, 847, 858, 85A, 874, 885, 887, 913, 919, 931, 937, 957, 959, 975, 995, A07, A1A, A25, A45, A74, A7A, A85, AA1, AA7, 1101, 11A9, 1305, 1451, 1457, 15A7, 175A, 17A5, 17A9, 2023, 2045, 2052, 2083, 20A5, 2333, 2A05, 2A52, 3013, 3026, 3059, 3097, 3206, 3222, 3233, 3307, 3332, 3505, 4025, 4151, 4157, 4175, 4405, 4445, 4487, 450A, 4575, 5017, 5031, 5059, 5075, 5097, 5099, 5105, 515A, 517A, 520A, 5301, 5583, 5705, 577A, 5853, 5873, 5909, 5A17, 5A57, 5A77, 5A8A, 6683, 66A9, 7019, 7073, 7079, 7088, 7093, 7095, 7309, 7451, 7501, 7507, 7578, 757A, 75A7, 7787, 7804, 7844, 7848, 7853, 7877, 78A4, 7A04, 7A57, 7A79, 7A95, 8078, 8245, 8333, 8355, 8366, 8375, 8425, 8553, 8663, 8708, 8777, 878A, 8A05, 9053, 9101, 9107, 9305, 9505, 9703, A052, A119, A151, A175, A515, A517, A575, A577, A5A8, A719, A779, A911, AAA9, 10011, 10075, 10091, 10109, 10411, 10444, 10705, 10709, 10774, 10901, 11104, 11131, 11144, 11191, 1141A, 114A1, 13757, 1411A, 14477, 144A4, 14A04, 14A11, 17045, 17704, 1774A, 17777, 177A4, 17A47, 1A091, 1A109, 1A114, 1A404, 1A411, 1A709, 20005, 20555, 22203, 25228, 25282, 25552, 25822, 28522, 30037, 30701, 30707, 31113, 33777, 35009, 35757, 39997, 40045, 4041A, 40441, 4045A, 404A1, 4111A, 411A1, 42005, 44401, 44474, 444A1, 44555, 44577, 445AA, 44744, 44A01, 47471, 47477, 47701, 5057A, 50903, 5228A, 52A22, 52A55, 52A82, 55007, 550A9, 55205, 55522, 55557, 55593, 55805, 57007, 57573, 57773, 57807, 5822A, 58307, 58505, 58A22, 59773, 59917, 59973, 59977, 59999, 5A015, 5A2A2, 5AA99, 60836, 60863, 68636, 6A609, 6A669, 6A696, 6A906, 6A966, 70048, 70103, 70471, 70583, 70714, 71474, 717A4, 71A09, 74084, 74444, 74448, 74477, 744A8, 74747, 74774, 7488A, 74A48, 75773, 77144, 77401, 77447, 77799, 77A09, 78008, 78783, 7884A, 78888, 788A8, 79939, 79993, 79999, 7A051, 7A444, 7A471, 80005, 80252, 80405, 80522, 80757, 80AA5, 83002, 84045, 85307, 86883, 88863, 8A788, 90073, 90707, 90901, 95003, 97779, 97939, 99111, 99177, 99973, A0111, A0669, A0966, A0999, A0A09, A1404, A4177, A4401, A4717, A5228, A52AA, A5558, A580A, A5822, A58AA, A5A59, A5AA2, A6096, A6966, A6999, A7051, A7778, A7808, A9055, A9091, A9699, A9969, AA52A, AA58A, 100019, 100079, 101113, 101119, 101911, 107003, 140004, 144011, 144404, 1A0019, 1A0141, 1A5001, 1A7005, 1A9001, 222223, 222823, 300107, 300202, 300323, 303203, 307577, 310007, 332003, 370777, 400555, 401A11, 404001, 404111, 405AAA, 41A011, 440A41, 441011, 451777, 455555, 470051, 470444, 474404, 4A0401, 4A4041, 500015, 500053, 500077, 500507, 505577, 522A2A, 525223, 528A2A, 531707, 550777, 553707, 5555A9, 555A99, 557707, 55A559, 5807A7, 580A0A, 580A55, 58A0AA, 590007, 599907, 5A2228, 5A2822, 5A2AAA, 5A552A, 5AA22A, 5AAA22, 60A069, 683006, 6A0096, 6A0A96, 6A9099, 6A9909, 700778, 701074, 701777, 704408, 704417, 704457, 704484, 707041, 707441, 707708, 707744, 707784, 710777, 717044, 717077, 740008, 74484A, 770441, 770744, 770748, 770771, 777017, 777071, 777448, 777484, 777701, 7778A8, 777A19, 777A48, 778883, 78A808, 790003, 7A1009, 7A4408, 7A7708, 80A555, 828283, 828883, 840555, 850505, 868306, 873005, 883202, 900701, 909739, 909979, 909991, 970771, 977701, 979909, 990739, 990777, 990793, 997099, 999709, 999901, A00009, A00599, A01901, A05509, A0A058, A0A955, A10114, A555A2, A55999, A59991, A5A222, A5A22A, A60609, A66069, A66906, A69006, A79005, A87888, A90099, A90996, A96006, A96666, A97177, A97771, AA0A58, AA5A22, AAA522, 1000501, 1011141, 1030007, 1070047, 111114A, 1111A14, 1111A41, 1144441, 14A4444, 1700005, 1700474, 1A44444, 2555505, 2845055, 3030023, 3100003, 3333397, 4000111, 4011111, 41A1111, 4411111, 444441A, 4444771, 4470004, 4505005, 4744417, 4774441, 4777404, 4777417, 4777747, 4A11111, 4A40001, 5000093, 50005A7, 5005777, 5050553, 5055503, 5070777, 5222222, 5222AAA, 52AAAA2, 52AAAAA, 5505053, 5552AAA, 5555599, 5555A58, 5558A0A, 5558A55, 5558AAA, 55A0009, 55AAA52, 580000A, 5822222, 58AAAAA, 5A2222A, 5AA2222, 6000A69, 6000A96, 6A00069, 7000417, 7000741, 7000835, 7000857, 7007177, 7008305, 7014447, 7017444, 7074177, 7077477, 7077741, 7077747, 7100447, 7174404, 717444A, 7400404, 7700717, 7701077, 7701707, 7707778, 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DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD ===Base 15=== 12, 14, 18, 1E, 21, 27, 2B, 2D, 32, 38, 3E, 41, 47, 4B, 4D, 54, 58, 5E, 67, 6B, 6D, 72, 74, 78, 87, 8B, 92, 94, 9E, A1, A7, AD, B2, B8, BE, C1, CB, CD, D2, D4, E1, ED, 111, 11B, 131, 137, 13B, 13D, 157, 15B, 15D, 171, 177, 197, 19D, 1B7, 1BB, 1D1, 1DB, 1DD, 234, 298, 311, 31B, 337, 33D, 344, 351, 357, 35B, 364, 377, 391, 39B, 39D, 3A4, 3BD, 3C4, 3D7, 3DB, 3DD, 452, 51B, 51D, 531, 53B, 551, 55D, 562, 571, 577, 5A2, 5B1, 5B7, 5BB, 5BD, 5C2, 5D1, 5D7, 634, 652, 681, 698, 717, 71B, 731, 737, 757, 75D, 77D, 79B, 79D, 7B1, 7B7, 7BD, 7D7, 7DD, 801, 852, 88D, 8D8, 91D, 93B, 93D, 95B, 95D, 971, 977, 97B, 97D, 988, 991, 9BD, 9C8, 9D1, A98, AAB, B1D, B31, B3B, B44, B51, B57, B7B, B7D, B97, B9B, BB7, BC4, BD1, BD7, BDD, C07, C34, C52, C7E, C98, CC7, CE7, D0E, D1D, D31, D51, D5B, D68, D77, D7B, D91, D97, DA8, DAE, DCE, DD1, EB4, EEB, 107B, 1091, 10B1, 1107, 110D, 1561, 1651, 1691, 1B01, 2052, 2502, 2522, 303B, 307D, 3097, 30BB, 30D1, 3107, 3361, 3701, 3907, 3B01, 3B0B, 3C97, 4434, 4498, 4834, 4898, 49A8, 4E34, 5037, 507D, 5091, 509B, 5107, 5161, 5202, 53C7, 5552, 570B, 590B, 590D, 59C7, 5A5B, 5C97, 5D0D, 5DAB, 6061, 6151, 6191, 6511, 6601, 6911, 707B, 7091, 7097, 70AE, 70BB, 70CE, 70DB, 7561, 760E, 7691, 76CE, 7907, 7961, 7A0E, 7A3B, 7AEE, 7B0B, 7BAB, 7C0E, 7C77, 7CAE, 7D0B, 7D61, 7DAB, 7E5B, 7E6E, 7E7B, 7EBB, 8098, 811D, 8191, 835D, 853D, 8881, 8908, 8951, 8968, 899D, 8D3D, 8D5D, 8D6E, 8DDD, 8E98, 9011, 9037, 9097, 90D7, 9301, 93C7, 95C7, 9611, 9631, 96A8, 9811, 9851, 989D, 990B, 990D, 998D, 99AB, 99C7, 99D8, 9A08, 9A9B, 9AA8, 9ABB, 9B61, 9BC7, 9D0B, 9DAB, 9DC7, 9DD8, A052, A304, A502, A55B, A9BB, AB04, AB64, B09D, B107, B10B, B161, B1AB, B1C7, B30D, B3C7, B50B, B664, B691, B6A4, B707, B761, B90D, B961, BA5B, BABB, BBAB, BBB4, BC37, BC77, C777, C937, C997, D011, D03D, D05D, D09B, D0B1, D0BD, D101, D10B, D30D, D3AB, D507, D50D, D66E, D761, D7DE, D811, D85D, D86E, D89D, D8C8, D8E8, D9AB, D9D8, DA3B, DA9B, DABB, DB01, DB61, DBAB, DC88, DD07, DD0B, DD7E, DD8D, DDE7, DE6E, E252, E33B, E522, E57B, E7AE, E7CE, E898, E997, E9A8, E9BB, EA34, EB5B, EE98, EEC7, 10017, 10B0D, 170AB, 17A0B, 19001, 19601, 1A09B, 1D0C7, 22E52, 2EA52, 30017, 3001D, 300B1, 301C7, 30334, 30631, 307AB, 3300B, 3333B, 36031, 36301, 37A0B, 37BBB, 39997, 3A30B, 3B0C7, 3D001, 3D601, 40034, 40968, 43334, 49668, 49998, 50022, 5009D, 501C7, 50222, 50507, 505C7, 50611, 50C57, 53007, 53997, 55537, 5555B, 5557B, 5599B, 56101, 56691, 56961, 5700D, 5755B, 59001, 59557, 59997, 5999D, 599DB, 59DDD, 5D99B, 5DD3D, 5DD9D, 60931, 63031, 65691, 66951, 69031, 69361, 69561, 70011, 70051, 7005B, 7006E, 7030D, 703AB, 70501, 70701, 707C7, 71601, 71951, 7300D, 7333B, 75001, 7555B, 75911, 76011, 76051, 766EE, 76EEE, 7700B, 77191, 77661, 7776E, 77771, 777BB, 77911, 77BBB, 79001, 7A05B, 7A66E, 7AA6E, 7AAAE, 7ACCE, 7C6EE, 7CCEE, 7CECE, 7CEEE, 7D3BB, 7E7C7, 7EECE, 80034, 80304, 80434, 809DD, 80A34, 84A34, 850DD, 85961, 86661, 88151, 88331, 88511, 88591, 88898, 890DD, 89998, 89D0D, 8D90D, 8E434, 90017, 90051, 900A8, 900DB, 901C7, 90C57, 90D8D, 91007, 91061, 9199B, 95997, 96068, 96561, 99397, 99537, 9999B, 999B7, 999D7, 999DB, 999DD, 99BBB, 99DBB, 99DD7, 99DDD, 9B007, 9B00B, 9B0AB, 9BB11, 9BBBB, 9D007, 9D08D, 9D537, 9D9BB, 9D9DB, 9DD57, 9DDB7, 9DDDB, 9DDDD, A0A34, A0B5B, A0BBB, A0E34, A2E52, A330B, A8434, A8834, A8E34, A909B, AAA34, AAE52, AB0BB, AB334, ABB34, AE034, AE834, AE99B, AEA52, AEE52, B0011, B0071, B0077, B00B1, B0611, B0A64, B500D, B599D, B6101, B7771, B7911, BA064, BAAA4, BAB34, BB061, BB304, BB53D, BB601, BBB91, BBB9D, BBBBD, BDA0B, BDBBB, D0088, D00D7, D0307, D05C7, D070D, D0888, D0B07, D0BC7, D0C08, D0DC7, D0DD8, D1661, D59DD, D5D3D, D5DDD, D6611, D700D, D8D0D, D900B, D9908, D999D, D9BBB, D9D9D, D9DDB, DB007, DB00D, DB1B1, DB53D, DB59D, DB99D, DBBB1, DD0D8, DD33B, DD3B7, DD3BB, DD57D, DD898, DD9DD, DDB37, DDBDB, DDD08, DDD3D, DDD5D, DDD7D, DDD88, DDD9D, DDDB7, DDDC8, DDDD7, DDE98, DE037, DE998, DEB07, E0098, E00C7, E0537, E0557, E077B, E0834, E0968, E3334, E37AB, E39C7, E4034, E5307, E55AB, E705B, E750B, E766E, E76EE, E8304, E8434, E9608, E9C37, EAE52, EBB0B, EC557, EC597, EC957, 1000BD, 1009AB, 10A90B, 1900AB, 190661, 19099B, 190A0B, 1A900B, 222A52, 2AAA52, 31000D, 330331, 333334, 3733AB, 373ABB, 3BBB61, 430004, 490068, 490608, 5000DB, 500D0B, 505557, 505A0B, 50D00B, 50DDDB, 50DDDD, 522222, 5500AB, 5500C7, 550957, 550A0B, 555A9B, 559057, 560011, 590661, 633331, 666331, 666591, 666661, 7050AB, 705A0B, 706101, 70A50B, 7300AB, 761661, 76666E, 777011, 777101, 77750B, 777A5B, 777CEE, 779051, 791501, 7E7797, 7ECCCE, 7EEE97, 800D9D, 808834, 836631, 83D661, 843004, 856611, 884034, 884304, 888E34, 88A434, 88AE34, 8A4034, 8AEE34, 8E8034, 8E8E34, 8EEE34, 9000BB, 9001AB, 900B07, 900D98, 903661, 905661, 906651, 9080DD, 9099A8, 909D9B, 90A668, 90DD9B, 90DDBB, 910001, 9100AB, 91A00B, 930007, 950001, 956661, 9909A8, 995907, 999068, 999507, 999907, 9B0B1B, 9B0BB1, 9BB01B, 9C5597, 9C5957, 9D09DD, 9D0D9D, 9D800D, 9DB307, 9DD09D, A00034, A0033B, A033B4, A2A252, AAAA52, ABBBBB, B00004, B0001B, B0003D, B00A04, B0555B, B07191, B07711, B07777, B0B911, B0BDBB, B77011, B777C7, BB0001, BB0034, BB035D, BB055B, BB0BDB, BB9101, BBB0DB, BBB50D, BBBB01, BBD0BB, C55397, C55557, C55597, D0003B, D00057, D0007D, D000B7, D000C8, D008DD, D00DAB, D0333B, D05537, D099DD, D09DDD, D0DDBB, D555C7, D5C537, D88008, D88088, D888EE, D909DD, D9D0DD, D9DD0D, DB0BBB, DBBB0B, DBBB0D, DC0008, DC5537, DDDDD8, DDDEBB, DDE99B, DE0808, DE0C57, DE300B, DE5537, DE8888, DEE088, DEE307, DEE888, DEEE37, DEEE57, DEEEC8, E0000B, E007BB, E00A52, E03BC7, E07ABB, E09B07, E0A99B, E0C397, E0E76E, E50057, E55007, E55597, E55937, E730AB, E73A0B, E80E34, E88834, E8E034, E90008, E95557, EA099B, EE4304, EE5057, EE5507, EE8E34, EE9307, EEE434, 100001D, 1000A9B, 1000DC7, 22AA252, 3000BC7, 3033301, 3076661, 333B304, 33B3034, 3B33304, 3D66661, 50007AB, 5005957, 5500597, 5550057, 5559007, 5559597, 5595007, 5966661, 5DDDDDB, 6366631, 7010001, 7066651, 7100061, 733BBBB, 766A6AE, 77505AB, 7776501, 777775B, 777AACE, 777ECCE, 777EEAE, 7CCCCCE, 7E30A0B, 7EEEEAE, 8300004, 8363331, 8693331, 880E834, 8833304, 8888034, 8888434, 888A034, 88A3334, 88E8834, 88EE034, 88EE304, 8AA3334, 8D0009D, 8EE8834, 9000361, 9000668, 9003331, 9005557, 9006008, 9008D0D, 9083331, 9090968, 90BBB01, 90D0908, 9500661, 9555597, 9555957, 9660008, 9900968, 9995597, 9996008, 9999557, 9999597, 9999908, 9A66668, A003B34, A003BB4, AA22252, B00B034, B00B35D, B033334, B0B6661, B0BB01B, B100001, B333304, B777777, B99999D, BA60004, BAA0334, BBB001B, BBB6611, BBBBB11, BBBD00B, BD000AB, D0000DB, D009098, D00CCC8, D00D908, D00D99D, D03000B, D0BB0BB, D0D9008, D0D9998, D1000C7, D800008, D8DDEEE, D90080D, DBBBBBB, DD09998, DDD5557, DDDDBBB, DDDDDBD, DDDE8EE, DECC008, DECCCC8, DEE0CC8, DEEC0C8, E000397, E0003BB, E000434, E00076E, E000937, E007A5B, E00909B, E0090B7, E009307, E00B077, E00E434, E00E797, E00E937, E05999B, E09009B, E0900B7, E0E0937, E0E7E97, E0EAA52, E0EEA52, E555057, E5555C7, E7777C7, E77E797, E88EE34, E999998, EA5999B, EB000BB, EB0BBBB, EE00434, EE0E797, EEE076E, EEE706E, EEE8834, EEEE557, EEEE797, 30333331, 30B66661, 33000034, 33030004, 33B33004, 500575AB, 55000007, 5500075B, 55500907, 55555057, 55555907, 55559507, 60003301, 60033001, 60330001, 7000003D, 70106661, 70666611, 77000001, 7777770B, 777777C7, 77777ACE, 77777EAE, 777E30AB, 777E3A0B, 7CCCC66E, 800005DD, 88AA0834, 90000008, 900008DD, 90099668, 90500557, 90555007, 90666668, 90909998, 90990998, 90996668, 9099999D, 90D00098, 90D90998, 95500057, 99099098, 99555057, 99900998, 99966608, 99966668, 99999668, 99999998, 9D009008, 9D090998, A0803334, A2222252, AAA52222, B00005AB, B000B55B, B0BBBB5B, B3330034, BB0BBB1B, BBAA3334, BBB0BB1B, BBB0BB5B, BBDB000B, D000BBBB, D00100C7, D8888888, D900008D, D9000098, DBB000BB, DC0CCCC8, DCC0CCC8, DCCCC008, DD000908, DD09009D, DDDDDDAB, DDDDDEEE, DDDEEE8E, DDDEEEE8, DEE80008, E0777E97, E0E0E397, E0E77797, E0EE0397, E7777797, E9066668, EE00E397, EE077797, EE0E0397, EEE00797, EEE07E97, EEE0AA52, EEE55397, EEE55557, EEEAAA52, EEEEE834, EEEEEA52, 300003331, 300007661, 300330031, 333000004, 333300001, 333B00034, 3700000AB, 3B3300034, 500000057, 555555007, 555555557, 5DDDDDDDD, 600000331, 7500000AB, 75000A00B, 75A00000B, 761000001, 77000E0C7, 777700EC7, 7777730AB, 7777777AE, 77777EE97, 7777E7E97, 777999997, 7A500000B, 7BBBBBB5B, 88888A834, 900000031, 900666608, 909990098, 90D009998, 950000557, 966666008, 990000007, 990555507, 999999997, A000000B4, A0005999B, AAEEEEE34, B000AA334, BBBBB005B, BBBBBBB5B, D09999998, D0D90009D, D800000DD, D90009998, DCCCC0CC8, DE88EEEEE, DEEEEEE88, E000B7777, E000BBBBB, E003ABBBB, EE0000797, EE0EEE397, EE5555557, EE777EE97, EEEEEE537, EEEEEE937, 2222222252, 3000000071, 3330030001, 3333303001, 3333330001, 500000007B, 5555555097, 7000000071, 77000000C7, 8333333331, 8888883334, 8888888834, 888888AA34, 900000009B, 900000009D, 900000DD9D, 9000099998, 9955555507, 9D0000099D, 9D05555557, AB0000005B, B000000DAB, B00000BBDB, BB00BB0B5B, BB0BB00B5B, D000099998, D00090008D, D0D000909D, D0DDDDDDDB, D300000007, D88EEEEEEE, D900999998, DD00900008, DDD6EEEEEE, DDDDDDD6EE, DDDDDDDDDE, DDDEEEEEEE, DEEEEE8008, E000000797, 7777777CCCE, 88888830004, 90000009D9D, 99955555557, 9999999999D, B00000D00AB, BB000BBB05B, BBBB0000B5B, D000009080D, D000090800D, D090800000D, DDDDDDD999B, DDDDDDDDD9B, EEEEEE00397, EEEEEEE0397, 333000000301, 5000000000DD, 73A00000000B, 9000000000B7, 903333333331, ABB00000000B, D000000001C7, DCCCCCCCCCC8, E0EEEEEEE397, 19A000000000B, 3333333333331, 3BBBBBBBBBBBB, 9333333333331, A00000000099B, B00000000050D, EEEEEEEEEE76E, 1000000000999B, 71000000000001, 908D000000000D, BBBBBBBBBB6661, 77777777777777B, BB00000000BBB5B, DEEEEEEEEEEEEEE, 7777777777777E97, B0BBBBBBBBBBBB1B, BB0000000000DB0B, D000000000000998, D908000000000000D, DDDDDDDDDDDDDDDDB, E9666666666666668, 3330000000000000031, D00000000000000908D, E0BBBBBBBBBBBBBBBBB, 2EEEEEEEEEEEEEEEEE52, 77777777777777777ECE, 5000000000000000005AB, 777777777777777777997, 7BBBBBBBBBBBBBBBBBBBB, BB0000000000000000DBB, DD000000000000000909D, D900000000000000000DDD, DD0000000000000000099D, BBBBBBBBBBBBBBBBBBBBBB1, B00000000000000000000005B, B0700000000000000000000001, B70000000000000000000000001, 705000000000000000000000000B, 633000000000000000000000000001, EBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB, 500000000000000000000000000000000017, 77777777777777777777777777777777777777777777777777777777777CCE, 7777777777777777777777777777777777777777777777777777777777777777777777777CE, 96666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666608, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE397, 7777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777797 ===Base 16=== 11, 13, 17, 1D, 1F, 25, 29, 2B, 2F, 35, 3B, 3D, 43, 47, 49, 4F, 53, 59, 61, 65, 67, 6B, 6D, 71, 7F, 83, 89, 8B, 95, 97, 9D, A3, A7, AD, B3, B5, BF, C1, C5, C7, D3, DF, E3, E5, E9, EF, F1, FB, 14B, 15B, 185, 199, 1A5, 1BB, 1C9, 1EB, 223, 22D, 233, 241, 277, 281, 287, 28D, 2A1, 2D7, 2DD, 2E7, 301, 337, 373, 377, 38F, 3A1, 3A9, 41B, 42D, 445, 455, 45D, 481, 4B1, 4BD, 4CD, 4D5, 4E1, 4EB, 50B, 515, 51B, 527, 551, 557, 55D, 577, 581, 58F, 5AB, 5CB, 5CF, 5D1, 5D5, 5DB, 5E7, 623, 709, 727, 737, 745, 74B, 755, 757, 773, 779, 78D, 7BB, 7C3, 7C9, 7CD, 7DB, 7EB, 7ED, 805, 80F, 815, 821, 827, 841, 851, 85D, 85F, 8A5, 8DD, 8E1, 8F5, 923, 98F, 99B, 9A9, 9EB, A21, A6F, A81, A85, A99, A9F, AA9, AAB, ACF, B1B, B2D, B7B, B8D, B99, B9B, BB7, BB9, BCB, BDD, BE1, C0B, CB9, CBB, CEB, D01, D21, D2D, D55, D69, D79, D81, D85, D87, D8D, DAB, DB7, DBD, DC9, DCD, DD5, DDB, DE7, E21, E27, E4B, E7D, E87, EB1, EB7, ED1, EDB, EED, F07, F0D, F4D, FD9, FFD, 1069, 1505, 1609, 1669, 16A9, 19AB, 1A69, 1AB9, 2027, 204D, 2063, 207D, 20C3, 20ED, 2221, 22E1, 2327, 244D, 26C3, 274D, 2E01, 2E0D, 2ECD, 3023, 3079, 3109, 3263, 3341, 36AF, 3941, 3991, 39AF, 3E41, 3E81, 3EE1, 3EE7, 3F79, 4021, 40DB, 440B, 444B, 44A1, 44AB, 44DB, 4541, 45BB, 4A41, 4B0B, 4BBB, 4C4B, 4D41, 4DED, 5045, 50A1, 50ED, 540D, 5441, 555B, 556F, 5585, 560F, 56FF, 5705, 574D, 580D, 582D, 5855, 588D, 5A01, 5AA1, 5B01, 5B4B, 5B87, 5BB1, 5BEB, 5C4D, 5CDD, 5CED, 5DD7, 5DDD, 5E0D, 5E2D, 5EBB, 68FF, 6A69, 6AC9, 6C8F, 6CA9, 6CAF, 6F8F, 6FAF, 7033, 7063, 7075, 7087, 70A5, 70AB, 7303, 7393, 74DD, 754D, 7603, 7633, 7663, 7669, 7705, 772D, 775D, 77D5, 7807, 7877, 7885, 7939, 7969, 7993, 79AB, 7A05, 7A69, 7A9B, 7AA5, 7B77, 7BA9, 7D4D, 7D75, 7D77, 8077, 808D, 80D7, 80E7, 8587, 86CF, 8777, 8785, 8885, 88CF, 88ED, 88FD, 8C6F, 8C8F, 8E8D, 8EE7, 8F2D, 8F8D, 9031, 9041, 90AF, 90B9, 9221, 9319, 9401, 944B, 9881, 9931, 9941, 9991, 99AF, 9A0F, 9A1B, 9A4B, 9AFF, 9BA1, 9BB1, 9CAF, 9E81, 9EA1, 9FAF, A001, A05B, A0C9, A105, A10B, A4CB, A55B, A6C9, A88F, A91B, A9B1, A9BB, AA15, AB01, AB0B, AB19, ABBB, AC09, AF09, B041, B04B, B069, B07D, B087, B0B1, B0ED, B1A9, B201, B40B, B40D, B609, B70D, B7A9, B807, B9A1, BA41, BAA1, BB4B, BBB1, BBDB, BBED, BD19, BD41, BDBB, BDEB, BE07, BEE7, C0D9, C203, C24D, C6A9, C88D, C88F, C8CF, C8ED, C9AF, C9CB, CA09, CA4B, CA69, CAC9, CC0D, CC23, CC4D, CC9B, CD09, CDD9, CE4D, CEDD, CFA9, CFCD, D04B, D099, D405, D415, D44B, D4A5, D4DD, D50D, D70B, D74D, D77B, D7CB, D91B, D991, DA05, DA09, DA15, DA51, DB91, DBEB, DD7D, DDA1, DDED, DE0B, DE41, DE4D, DEA1, E02D, E07B, E0D7, E1CB, E2CD, E401, E801, EABB, EACB, EAEB, EBAB, EC4D, ECDD, ED07, EDD7, EE7B, EE81, EEAB, EEE1, F08F, F0A9, F227, F2ED, F3AF, F485, F58D, F72D, F763, F769, F787, F7A5, F7E7, F82D, F86F, F877, F88D, F8D7, F8E7, F8FF, FCCD, FED7, FF85, FF8F, FFA9, 100AB, 10BA9, 1A0CB, 1BA09, 200E1, 2C603, 2CC03, 30227, 303AF, 30AAF, 32003, 32207, 32CC3, 330AF, 33169, 33221, 33391, 33881, 33AFF, 38807, 38887, 3AFFF, 3F203, 3F887, 3FAFF, 400BB, 4084D, 40A4B, 42001, 44221, 44401, 444D1, 4480D, 4488D, 44CCB, 44D4D, 44E8D, 4804D, 4840D, 4A0CB, 4A54B, 4CACB, 4D0DD, 4D40D, 4D44D, 5004D, 50075, 502CD, 5044D, 50887, 50EE1, 5448D, 548ED, 55A45, 55F45, 5844D, 5A4A5, 5AE41, 5B0CD, 5B44D, 5BBCD, 5D4ED, 5E0E1, 5EB4D, 5EC8D, 5ECCD, 5EE41, 5F06F, 5F7DD, 5F885, 5F8CD, 5FC8D, 5FF75, 6088F, 60AFF, 630AF, 633AF, 660A9, 668CF, 669AF, 66A09, 66A0F, 66FA9, 6886F, 6A00F, 6A0FF, 6A8AF, 6AFFF, 7002D, 7024D, 70B0D, 70B7D, 7200D, 73363, 73999, 7444D, 770B7, 777D7, 77B07, 77D7D, 77DD7, 79003, 79999, 7B00D, 7D05D, 7D7DD, 8007D, 800D1, 8074D, 82CCD, 82E4D, 8448D, 8484D, 8704D, 8724D, 87887, 88001, 8800D, 880CD, 88507, 88555, 8866F, 8872D, 8877D, 888D1, 888D7, 88AA1, 88C2D, 88D57, 88D75, 88D77, 8AFAF, 8C2CD, 8C40D, 8C8CD, 8CCED, 8CE2D, 8CFED, 8E007, 8E20D, 8E24D, 8F6FF, 8FAAF, 900CB, 901AB, 90901, 909A1, 90AB1, 90AE1, 90EE1, 910AB, 93331, 940AB, 963AF, 966AF, 99019, 99109, 99A01, 9AAE1, 9B00B, 9B0AB, 9B441, 9BABB, 9BBBB, 9E441, A00BB, A0405, A044B, A08AF, A0A51, A0B91, A0C4B, A1B09, A54A5, A5B41, A6609, A904B, A94A1, A9C4B, A9E01, A9E41, AA0A1, AA441, AA501, AA8AF, AAEE1, AAF45, AAF8F, ABBA1, ACC69, AE0BB, AE0EB, AEAE1, AEE0B, AEEA1, AEECB, AF045, AF4A5, AFA8F, B00A1, B00D7, B044D, B0777, B0A0B, B0A91, B0BBD, B0BCD, B0C09, B0DA9, B0EAB, B2207, B4001, B6669, B7707, B7D07, B8081, B9021, BA091, BA109, BA4BB, BB001, BB0EB, BB8A1, BBBEB, BBE0B, BBEBB, BC009, BCECD, BD0A9, BE44D, BEB0D, BEBBB, BEEBB, C0263, C02C3, C02ED, C040D, C0CA9, C0CCD, C2663, C2CED, C32C3, C3323, C400D, C40ED, C44CB, C44ED, C480D, C484D, C4CAB, C60AF, C686F, C6A0F, C86FF, C8C2D, CAA0F, CAFAF, CBCED, CC0AF, CC44B, CC82D, CC8FF, CCAF9, CCAFF, CCCFD, CCFAF, CD00D, CD4CB, CD4ED, CDDDD, CF2C3, CFC8F, CFE8D, D0045, D07DD, D09BB, D0D4D, D0DD7, D0EBB, D0EEB, D1009, D1045, D10B9, D1BA9, D54BB, D54ED, D5AE1, D5D07, D5EE1, D70DD, D7707, D7777, D77DD, D7DD7, D9441, D9AE1, D9B0B, DA9A1, DA9E1, DAA41, DAAA1, DBB0B, DBBA1, DC4CB, DD227, DD44D, DDDD7, E0081, E00E1, E010B, E088D, E08CD, E0B0D, E0BBD, E100B, E4D0D, E777B, E77AB, E7CCB, E844D, E848D, E884D, E88A1, EB0BB, EBB4D, EBBEB, EBEEB, EC8CD, ECBCD, ECC8D, ED04D, EE001, EE0EB, EE4A1, EEEBB, F0085, F09AF, F0C23, F0CAF, F2663, F2C03, F3799, F3887, F4A05, F4AA5, F506F, F5845, F5885, F5C2D, F5ECD, F5F45, F66A9, F688F, F6AFF, F7399, F777D, F8545, F8555, F8AAF, F8F87, F9AAF, FA0F9, FA405, FA669, FAFF9, FC263, FCA0F, FCAFF, FCE8D, FCF23, FD777, FDDDD, FDEDD, FEC2D, FEC8D, FF545, FF6AF, FF739, FF775, FF9AF, FFC23, 100055, 100555, 10A9CB, 1A090B, 1A900B, 1CACCB, 1CCACB, 20DEE1, 266003, 3000AF, 300A0F, 300AFF, 308087, 308E07, 3323E1, 333A0F, 339331, 33CA0F, 33CF23, 33CFAF, 33F323, 380087, 3A00AF, 3A0F0F, 3AA0FF, 3AAF0F, 3C33AF, 3C3A0F, 3C3FAF, 3CCAAF, 3F0FAF, 3F32C3, 3FF0AF, 3FFAAF, 4004CB, 400A05, 4048ED, 404DDD, 40AA05, 40D04D, 40DD4D, 40E0DD, 40E48D, 440041, 44008D, 44044D, 4404DD, 44440D, 4448ED, 4484ED, 448E4D, 44E44D, 48888D, 4AA005, 4DD00D, 4DD04D, 4DDD0D, 4E048D, 4E448D, 4E880D, 5000DD, 500201, 50066F, 5008CD, 500C2D, 500D7D, 50C20D, 520C0D, 544EDD, 54AA05, 54AAA5, 54ED4D, 566AAF, 57D00D, 580087, 5A5545, 5C20CD, 5C8CCD, 5CC2CD, 5D000D, 5D070D, 5F666F, 5FAA45, 5FFF45, 60008F, 600A0F, 603AAF, 6060AF, 6066AF, 60A0AF, 63AA0F, 6663AF, 66668F, 666AAF, 668A8F, 66AFF9, 68888F, 693AAF, 7007B7, 70404D, 70770B, 70770D, 707BE7, 70DD0D, 733339, 733699, 74004D, 74040D, 77007B, 770CCB, 777B4D, 777BE7, 777CCB, 77ACCB, 77B74D, 77D0DD, 7A0CCB, 7B744D, 7CACCB, 7DDD99, 80044D, 800807, 80200D, 8044ED, 80C04D, 80CC2D, 80E44D, 8404ED, 84888D, 84E04D, 84E40D, 86686F, 8668AF, 8686AF, 86F66F, 86FFFF, 87000D, 87744D, 880807, 886AFF, 88824D, 88870D, 888787, 88884D, 88886F, 88887D, 88888D, 888C4D, 888FAF, 88AA8F, 88CC8D, 88F6AF, 88F8AF, 88FA8F, 88FF6F, 88FF87, 88FFAF, 8A8FFF, 8C0C2D, 8C802D, 8CCFFF, 8CE00D, 8CE0CD, 8CFCCF, 8E00CD, 8E044D, 8E0CCD, 8EC0CD, 8F68AF, 8F88F7, 8FCFCF, 8FF887, 8FFCCF, 8FFF6F, 9002E1, 9004AB, 9008A1, 900919, 900ABB, 900B21, 90B801, 90CCCB, 9332E1, 944441, 94ACCB, 990001, 9900A1, 9A4441, 9A4AA1, 9AA4A1, 9AAA41, 9AAAAF, 9B66C9, 9BBA0B, 9BC0C9, 9BC669, 9BC6C9, 9C4ACB, A0094B, A00ECB, A09441, A0A08F, A0E0CB, A0ECCB, A0F669, A40A05, A4AAA5, A50E41, A5AA45, A60069, A8FAFF, A9AA41, AA5E41, AAA4A5, AAA545, AC6669, ACCC4B, ACCCC9, AEAA41, AFF405, AFF669, AFFA45, AFFFF9, B00921, B00BEB, B00CC9, B00D91, B08801, B0D077, B70077, B70E77, B77E77, B88877, B88881, B94421, BAE00B, BB00AB, BB0DA1, BB444D, BB44D1, BB8881, BBBBBD, BBBC4D, BBCCCD, BC0CC9, BC66C9, BCC669, BCC6C9, BCCC09, BE000D, BE00BD, BE0B4D, BE0CCD, BEA00B, BECCCD, C0084D, C00A0F, C0608F, C0668F, C0844D, C0A0FF, C0AFF9, C0C3AF, C0C68F, C0CAAF, C0CDED, C0D0ED, C0E80D, C0EC2D, C0EC8D, C0FA0F, C0FAAF, C2CC63, C30CAF, C333AF, C3CAAF, C3CCAF, C4048D, C40D4D, C4404D, C4408D, C4440D, C44DDD, C4ACCB, C4DCCB, C4DD4D, C6068F, C66AAF, C68AAF, C6AA8F, C8044D, C8440D, C8666F, CA00FF, CA0FFF, CAAAAF, CAAFFF, CAFF0F, CBE0CD, CC008F, CC0C8F, CC3CAF, CC4ACB, CC608F, CC66AF, CCBECD, CCC4AB, CCCA0F, CCCC8F, CCCE8D, CE0C8D, CF0F23, CF0FAF, CFAFFF, CFCAAF, CFFAFF, D0005D, D00BA9, D05EDD, D077D7, D10CCB, D22207, D4000B, D4040D, D4044D, D40CCB, D70077, D7D00D, D90009, D900BB, DB00BB, DB4441, DD400D, DDD109, DDD1A9, DDD919, DDD941, DED00D, E00D4D, E00EEB, E0AAE1, E0AE41, E0AEA1, E0B44D, E0BCCD, E0BEBB, E0D0DD, E0E441, E4048D, E4448D, E800CD, E8200D, EA0E41, EAA0E1, EBB00B, ECCCAB, EDDDDD, EEBE0B, F00263, F0056F, F00A45, F02C63, F03F23, F05405, F060AF, F08585, F0A4A5, F0F2C3, F0F323, F2CCC3, F33203, F33C23, F5F66F, F5FF6F, F68CCF, F6AA8F, F888AF, FA0F45, FAA045, FAA545, FAFC69, FC0AAF, FC66AF, FCCCAF, FCFFAF, FF0323, FF056F, FF3203, FF7903, FFA045, FFA4A5, FFAA45, FFC0AF, FFF4A5, FFF575, FFFA45, FFFCAF, 10A009B, 20000D1, 2CCC663, 30A00FF, 30CCCAF, 30FA00F, 30FCCAF, 3333C23, 333C2C3, 33C3AAF, 33FCAAF, 33FFFAF, 3A0A00F, 3AAAA0F, 3AF000F, 3AFAAAF, 3C0CA0F, 3CCC3AF, 3CFF323, 3F33F23, 3FAA00F, 3FF3323, 4004441, 400DDD1, 400E00D, 400ED0D, 404404D, 404448D, 404E4DD, 440EDDD, 4440EDD, 44444ED, 4444E4D, 44DDDDD, 4A000A5, 4CCCCAB, 4D0CCCB, 4E4404D, 4E4444D, 4E4DDDD, 5000021, 5004221, 5006AAF, 500FF6F, 5042201, 508CCCD, 5400005, 5400AA5, 5555405, 5808007, 5AA4005, 5C0008D, 5CCC8CD, 5D4444D, 5EEEEEB, 5F40005, 5F554A5, 5F6AAAF, 60000AF, 60006A9, 600866F, 6008AAF, 600AA8F, 600F6A9, 606608F, 606686F, 608666F, 60AA08F, 60AAA8F, 66000AF, 66666A9, 6666AF9, 6866A8F, 6AAAAAF, 70070D7, 70077DD, 700DDDD, 707077D, 707D007, 70D00DD, 770077D, 770400D, 770740D, 7777775, 77777B7, 77777DD, 7777ACB, 77B88E7, 77DD00D, 77DDDDD, 7D0D00D, 7DD0D07, 7DDD00D, 800002D, 8000CED, 80C0E0D, 80CECCD, 840400D, 844000D, 844E00D, 868688F, 880444D, 884404D, 887D007, 8888801, 8888881, 8888E07, 8888F77, 8888FE7, 88A8AFF, 88AAAFF, 88FAFFF, 8A8AAAF, 8A8AAFF, 8AAA8FF, 8C00ECD, 8C8444D, 8E4400D, 8FCCCCF, 900BBAB, 90CC4AB, 9908AA1, 99E0E01, 9B00801, 9B6CCC9, A000FF9, A006069, A00A8FF, A01CCCB, A05F545, A0BEEEB, A0E4AA1, AA0008F, AA08FFF, AA40AA5, AA8FFFF, AAAA405, AE04AA1, AE44441, AE4AAA1, AECCCCB, AF40005, AFA5A45, AFFFC69, B000BAB, B000EBB, B0D0007, B222227, B6CCCC9, B8880A1, BA000EB, BA0BEEB, BAEEEEB, BB000CD, BB00C0D, BB0B00D, BC6CC69, BC6CCC9, BCCCC69, BCCCCED, C0000A9, 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4444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444DD ===Base 17=== 12, 16, 1C, 1E, 23, 27, 29, 2D, 32, 38, 3A, 3G, 43, 45, 4B, 4F, 54, 5C, 5G, 61, 65, 67, 6B, 78, 7C, 81, 83, 8D, 8F, 94, 9A, 9E, A3, A9, AB, B4, B6, BA, BC, C7, D2, D6, D8, DC, E1, E3, ED, F2, F8, FE, FG, G5, G9, GB, 104, 111, 115, 117, 11B, 137, 139, 13D, 14A, 14G, 155, 159, 15F, 171, 17B, 17D, 188, 191, 197, 19F, 1A4, 1A8, 1B3, 1BB, 1BF, 1DB, 1DD, 1F3, 1FD, 1G8, 1GA, 1GG, 20F, 214, 221, 225, 241, 25A, 25E, 285, 2B8, 2C5, 2CF, 2E5, 2EB, 2F6, 30E, 313, 331, 33B, 346, 34C, 351, 35F, 36E, 375, 37B, 391, 39B, 39D, 3B7, 3B9, 3BF, 3D3, 3D5, 3D9, 3DF, 3E4, 3EC, 3F1, 3F7, 407, 418, 447, 44D, 472, 474, 47E, 47G, 489, 49C, 4A1, 4C1, 4CD, 4D4, 4G1, 502, 506, 508, 50E, 519, 522, 528, 52A, 52E, 533, 53F, 551, 55D, 562, 566, 573, 577, 57F, 582, 593, 599, 59B, 59F, 5A6, 5B5, 5D1, 5D3, 5EA, 5EE, 5F9, 60D, 62F, 634, 649, 689, 692, 6CD, 6EF, 6F4, 6FA, 704, 706, 70G, 71D, 726, 737, 739, 73D, 73F, 753, 755, 764, 766, 76G, 771, 77B, 793, 7AA, 7AE, 7B3, 7BB, 7D7, 7E6, 7F3, 7F9, 7FF, 7G2, 7GE, 7GG, 825, 82B, 849, 852, 85E, 869, 876, 87A, 87G, 88B, 892, 898, 89C, 8C5, 8E7, 8G7, 908, 90G, 913, 91F, 92C, 935, 937, 93B, 951, 953, 957, 95D, 968, 96G, 979, 97B, 98C, 98G, 99D, 9B1, 9B3, 9B9, 9BD, 9BF, 9DB, 9DF, 9F1, 9F5, 9G6, A07, A0D, A1A, A2F, A4D, A72, A7A, A7E, AA1, AA7, ACF, ADA, AG1, AG7, B02, B08, B17, B1D, B28, B2G, B57, B71, B73, B79, B7F, B88, B8E, B8G, B9B, B9F, BB5, BB7, BD7, BDD, BEG, BFF, BGG, C01, C2F, C3E, C56, C6D, C89, C92, C9G, CA5, CBG, CC1, CC5, CF4, CFA, D04, D0A, D15, D3D, D3F, D55, D59, D5B, D71, D75, D7D, D91, D97, D99, D9D, DA4, DAG, DB3, DDB, DF1, DF7, DF9, DFF, E05, E0B, E2B, E52, E58, E69, E92, E9C, EAF, EB8, EC9, ECB, EE5, F04, F15, F1B, F35, F3B, F46, F51, F53, F64, F6A, F73, F79, F95, FAC, FB1, FCA, FD5, FDB, FF1, FF7, FFD, G0D, G0F, G18, G1A, G1G, G2F, G34, G63, G7G, GA7, GC3, GDG, GEF, GFA, GG7, GGD, 1013, 101D, 1033, 1035, 1051, 105B, 105D, 1077, 108A, 109B, 10AG, 10B1, 10B7, 10BD, 10FB, 1149, 1189, 11AF, 11G3, 1303, 130B, 1314, 1341, 1479, 14D9, 1501, 1503, 15A1, 15B8, 1734, 1749, 17AF, 17G3, 1844, 185B, 1875, 1877, 18AG, 18B5, 1903, 1909, 1958, 19BG, 19G3, 1A5D, 1A75, 1A7F, 1ADF, 1AF1, 1B01, 1B09, 1B18, 1B85, 1B89, 1BDG, 1BGD, 1D07, 1D49, 1D9G, 1DF4, 1F09, 1F47, 1F5A, 1F74, 1F7A, 1FA1, 1FAF, 2018, 201G, 202B, 208B, 20G1, 215B, 218G, 21AG, 21B1, 222F, 22AF, 22BG, 22EF, 22F4, 22GF, 251B, 2526, 25F1, 266F, 26FC, 280B, 2A05, 2A58, 2AFC, 2AGF, 2B1B, 2B1F, 2BGE, 2C1G, 2C2B, 2C8B, 2CG1, 2E2F, 2EGF, 2F0C, 2F55, 2FAA, 2FC4, 2FFF, 2GA1, 2GFC, 2GG1, 2GGF, 301B, 301F, 3037, 3053, 3057, 3079, 3095, 30B3, 30BD, 30C4, 31F4, 330D, 3334, 333E, 3349, 3376, 337E, 33CD, 33EF, 3411, 3417, 3499, 3503, 3505, 3509, 353E, 35E5, 35EB, 3604, 36FD, 3701, 3741, 374D, 376F, 3796, 37D4, 37F4, 3956, 3B03, 3B05, 3B0B, 3BBE, 3C04, 3C15, 3C19, 3C4E, 3C59, 3C64, 3CB3, 3CDB, 3CE6, 3D07, 3D14, 3DDE, 3E77, 3E79, 3E7F, 3E99, 3EEE, 3EFB, 3F05, 3F0D, 3FCB, 3FF4, 4009, 4021, 4069, 4098, 40DG, 40GD, 419D, 4201, 4401, 4492, 46AD, 46C9, 46DA, 4719, 476A, 4779, 479D, 47A6, 4906, 4911, 4917, 4919, 491D, 492G, 4982, 4988, 49D7, 49D9, 49GG, 4ADE, 4AE7, 4C49, 4C96, 4CC9, 4D79, 4DAE, 4DEG, 4E7A, 4E96, 4EG7, 4G6D, 4G87, 501B, 5037, 5059, 507D, 50BB, 50BF, 50D7, 50DD, 50F1, 5105, 51A7, 51AD, 521B, 525F, 52FB, 5307, 5356, 53BE, 53DE, 53E9, 5507, 550B, 5587, 5598, 55EF, 560A, 568E, 56AA, 56F3, 5709, 5725, 572B, 575A, 575E, 5769, 57A1, 57B2, 5868, 586E, 58AE, 58B9, 590D, 5918, 5952, 5958, 596D, 5A17, 5A1F, 5ADD, 5ADF, 5AE8, 5B07, 5B21, 5B2F, 5B3E, 5BEF, 5DA7, 5DEB, 5E57, 5E5F, 5E86, 5E97, 5EB9, 5EBF, 5EF5, 5F01, 5F1A, 5F6F, 5FA7, 5FDA, 60AF, 60G3, 64AD, 64DE, 64DG, 663E, 666D, 66AF, 693D, 69CG, 69D3, 69D9, 69G8, 69GC, 6ADE, 6AGD, 6C98, 6D33, 6D4E, 6D93, 6D9F, 6DDD, 6DEE, 6DF3, 6DFD, 6DGE, 6E09, 6G36, 6G4D, 6G6D, 6GD4, 6GDE, 6GFC, 702E, 7057, 705B, 7073, 7079, 7095, 70B5, 70BD, 70D1, 70E2, 70F5, 7107, 7149, 719G, 71BG, 71F4, 724E, 724G, 725F, 72A2, 72BF, 72EE, 72GA, 7314, 733E, 7341, 7363, 73EB, 7419, 742A, 742G, 7442, 74EG, 7501, 750F, 751A, 756D, 757E, 75A1, 75A7, 75BE, 75DA, 75E9, 75F6, 7622, 769F, 76EA, 7734, 773E, 776D, 779G, 77AF, 7905, 790B, 7976, 79B2, 79F6, 79GD, 7A1F, 7A5D, 7AD5, 7ADF, 7AF1, 7AFD, 7B01, 7B09, 7B2F, 7B52, 7B72, 7BE5, 7D01, 7D05, 7D9G, 7DAF, 7DBG, 7E0A, 7E75, 7EA2, 7EA4, 7EB5, 7EBF, 7EE2, 7EF7, 7EG4, 7F0B, 7F14, 7F5A, 7F76, 7FA7, 7G1F, 7G46, 7GA6, 7GD3, 7GDF, 8009, 8058, 80B8, 80E9, 84A7, 850A, 8557, 857B, 85A8, 870E, 8744, 8777, 879B, 87B5, 87B7, 87EE, 8805, 8872, 8887, 8889, 88E9, 8906, 8959, 8966, 89GG, 8A87, 8AE5, 8B0G, 8B59, 8B95, 8B97, 8CB2, 8CB8, 8CE9, 8E56, 8EE9, 9026, 9031, 903D, 907F, 9091, 909B, 90FB, 9101, 910D, 9118, 917G, 9185, 9189, 91B8, 9202, 9288, 92B5, 92FB, 92GG, 93C1, 9505, 950B, 950F, 952B, 956F, 9592, 9596, 9598, 9602, 96D9, 96FD, 971G, 9725, 9752, 97DG, 9855, 9862, 9895, 9899, 98B7, 98BB, 9901, 990B, 9921, 992F, 99G3, 9B0B, 9B2B, 9B8B, 9BB8, 9BBG, 9C19, 9C1B, 9C31, 9C59, 9C95, 9CD5, 9CFB, 9CGC, 9D03, 9D07, 9D7G, 9DG1, 9DGD, 9F0B, 9F76, 9FCB, 9G11, 9G1D, 9G28, 9G3F, 9G7D, 9GCC, 9GD7, 9GF7, 9GFD, 9GG8, A025, A041, A058, A0C5, A0F6, A0GF, A11F, A184, A1F7, A21G, A258, A401, A421, A476, A511, A517, A57D, A5A8, A5E8, A6AD, A6FC, A6GF, A751, A77F, A7F5, A7FD, A7G6, A847, AACD, AC1G, AC41, AC58, AC5E, ACGD, AD0E, AD0G, AD1F, AD51, ADD5, ADE4, ADF5, ADGE, AE56, AE74, AEF6, AEFA, AF77, AF7D, AFA4, AFCC, AFD7, AFDD, AGAF, AGF4, B00G, B037, B055, B05B, B075, B0D5, B0FD, B10F, B198, B25F, B2F1, B2F5, B307, B309, B35E, B3EF, B50D, B589, B7BE, B7BG, B7E7, B875, B952, B958, B97G, B99G, B9G7, B9GD, BB01, BB2F, BB3E, BB89, BB98, BBDE, BD03, BD09, BD5E, BDE5, BDEB, BDG1, BE5F, BF01, BF0D, BG13, BG1F, BG3F, BGD1, BGE2, BGE8, C00B, C034, C05A, C0AF, C0EF, C0GF, C153, C15B, C199, C1B9, C1D1, C1D5, C1F9, C205, C21A, C21G, C252, C258, C2B2, C335, C33D, C35D, C364, C395, C3B3, C3F5, C3FB, C3FD, C414, C41A, C469, C496, C4DA, C4GD, C535, C55B, C5B1, C5BD, C5D9, C5DF, C5E8, C5F3, C5F5, C6E9, C85A, C885, C8B8, C8BE, C8CB, C8E5, C919, C931, C959, C95F, C9D3, CA0F, CA18, CA1G, CAD4, CADE, CAEF, CAGD, CB22, CB33, CB35, CB3F, CB5D, CB82, CB99, CBB1, CBFB, CC49, CCCB, CCDE, CD11, CD1D, CD39, CD4A, CD53, CD93, CDAE, CDD5, CDF3, CDFD, CDG4, CE49, CE5A, CE8B, CF13, CF19, CF5D, CF5F, CFB9, CFBF, CFD9, CFDF, CG14, CG41, CG6F, CGCF, CGF6, CGG1, D01F, D039, D079, D09B, D09F, D0B7, D0BB, D0D1, D0EG, D0GG, D10D, D19G, D1G3, D30B, D347, D3BE, D4E4, D50D, D57E, D5AD, D5FA, D707, D73E, D7E7, D7GF, DA1F, DA57, DAAE, DB01, DB09, DB0D, DB7E, DB9G, DD05, DD7E, DDA5, DDFA, DDG3, DE0G, DE44, DE4A, DE77, DEAE, DEB9, DEBB, DF03, DF05, DG0E, DGDF, E009, E06F, E072, E07G, E089, E0CF, E0E9, E0G7, E47A, E498, E4E7, E50A, E559, E55F, E575, E5B9, E5BF, E5F5, E5F7, E6FC, E722, E724, E72A, E72E, E744, E746, E75B, E76E, E79B, E7A4, E7A6, E7AG, E7B5, E7B7, E7EG, E7G4, E887, E89G, E8E9, E906, E955, E95B, E95F, E988, E99F, E9F9, E9G8, E9GG, EA25, EA7G, EAC5, EAE7, EB7B, EBF5, EBF7, EBFB, EC6F, ECCF, ECEF, EE72, EE76, EE89, EE9G, EF0A, EF44, EF77, EF97, EFA4, EFB5, EFC6, EFFF, EG6F, EG74, EGE7, EGFC, EGGF, F019, F01F, F075, F091, F09B, F0BF, F0FB, F10F, F1A7, F1AD, F1D4, F376, F3CD, F3F4, F40A, F411, F444, F44A, F497, F499, F49D, F4D7, F509, F57A, F5AD, F5F6, F6D3, F6D9, F70D, F741, F747, F76D, F7F6, F7FA, F907, F976, F9CB, FA11, FA7D, FADD, FB09, FC4C, FC5D, FC5F, FC91, FCB9, FD1A, FD41, FD47, FDF4, FF0B, FF56, G021, G07A, G0A1, G0E7, G11F, G17F, G1DF, G1F1, G1F7, G201, G2A1, G306, G311, G36C, G377, G37F, G3CC, G3CE, G3D1, G476, G487, G4DE, G6AF, G6D4, G6F6, G6GF, G713, G724, G731, G742, G74E, G76E, G7A2, G872, G874, GA21, GAC1, GC6F, GCAF, GCD4, GCDA, GCG1, GD73, GD7F, GDAE, GDDF, GDEA, GDFD, GE47, GE7E, GF13, GF33, GF3F, GF4C, GF71, GF7F, GFDD, GG01, GG21, GGAF, GGC1, 1000G, 10053, 100AA, 100B9, 100F1, 100FF, 10301, 10587, 10705, 1075A, 107GF, 10895, 108B9, 10985, 1099G, 10B98, 10B9G, 10D03, 10D0F, 10D7A, 10DG3, 10DG7, 10G1F, 10G3F, 110GF, 1140D, 11D93, 11DG4, 11F0A, 11G4D, 11GD4, 13333, 133FF, 13F44, 14109, 14499, 150A7, 153B1, 1570A, 17005, 17799, 177AG, 17995, 17A7G, 17G47, 18079, 18507, 185A7, 18B07, 18B9G, 19333, 199B5, 1A00A, 1A00G, 1A0F5, 1AAAA, 1AAAG, 1AF05, 1AFFA, 1B07G, 1B10G, 1B807, 1D001, 1D1AA, 1D7G4, 1DG03, 1DG41, 1F001, 1F00F, 1F01A, 1F0A7, 1F199, 1F1F9, 1F414, 1F449, 1F7F5, 1F999, 1FF0A, 1FFAA, 1FFB5, 1G073, 1G14D, 1G1F4, 1G301, 1G477, 1GD01, 1GD47, 1GF07, 1GFF4, 20005, 200A1, 2010A, 20586, 20588, 20A01, 20B11, 20B15, 20BEE, 20C1A, 20CBE, 210B5, 21A1F, 21A51, 21F1A, 21G1F, 21GFF, 222BE, 228B2, 228BE, 22BE2, 22C0B, 22F0A, 252BB, 25505, 25552, 26GAF, 2A001, 2A1FF, 2A55F, 2AEEF, 2AF44, 2B051, 2B20E, 2BB2B, 2BBBG, 2BE22, 2BEE2, 2BEEE, 2BF0B, 2C0BE, 2C18A, 2F101, 2F1FA, 2F44C, 2FCBB, 2G1FF, 2GA6F, 2GF44, 30035, 300B1, 300FB, 30101, 303C5, 30444, 30497, 304D1, 304D7, 30703, 30714, 30734, 30763, 30774, 30CF5, 30CFD, 30D41, 30FC5, 3100B, 31779, 31F5B, 31FB5, 31FFF, 330C5, 330F4, 33357, 33373, 33379, 33555, 33557, 33777, 3379F, 337FD, 33997, 33D44, 33D4E, 33F3D, 33FF5, 34019, 34044, 340D1, 353DD, 35535, 355B3, 355E6, 35BB3, 35DDD, 3636D, 364DD, 3663D, 36DD4, 37003, 3700F, 3717F, 373EE, 37609, 3774E, 37773, 37797, 37977, 3797F, 37EEF, 39007, 390C5, 39777, 39973, 3B355, 3B553, 3BBDB, 3BDB1, 3C03D, 3C0F5, 3C10F, 3C141, 3C444, 3CBE5, 3CD0D, 3CE5B, 3CEBB, 3CEF9, 3D401, 3DEBE, 3E006, 3E066, 3E57E, 3E5E9, 3E666, 3E90F, 3EF6F, 3F33D, 3F3C4, 3F5BB, 3FB33, 3FDDD, 3FF59, 4006D, 400DE, 4011D, 401D9, 40414, 4041G, 404C9, 40966, 40D11, 40D19, 40D1D, 40E49, 41019, 411DA, 41AAG, 4210A, 44049, 4410G, 44144, 441G4, 44441, 444E9, 446E9, 44986, 44E49, 4609G, 460E9, 466DE, 469DD, 46E9G, 4711A, 476D9, 4770D, 47A77, 47D09, 49099, 490D1, 49226, 49622, 49699, 496DD, 49996, 4999G, 499G7, 49G22, 49G77, 4A7DD, 4AA6D, 4ADD7, 4C0E9, 4C999, 4D1DA, 4DADD, 4DD01, 4DD1G, 4DD7A, 4DDA7, 4DDE9, 4DG0G, 4DGAA, 4DGGA, 4DGGE, 4E049, 4E449, 4E49G, 4E4E9, 4E797, 4G7DD, 4GDAA, 4GDD7, 50011, 50079, 50095, 500B1, 500F3, 501A5, 501AF, 50503, 507A5, 50AF7, 50F03, 50F7A, 510A1, 510DA, 511AA, 511DF, 5135B, 515B7, 5180B, 51A0F, 51F0A, 520B1, 53005, 531BD, 53559, 53609, 53B11, 55205, 55357, 553E6, 5555B, 5556E, 55588, 5558A, 555F3, 555FB, 556AF, 556E9, 55759, 5575B, 55805, 55885, 55896, 558B8, 55926, 55BE2, 55E8B, 55F57, 560FF, 5700D, 570A5, 570DA, 575B9, 576AD, 576DA, 579D5, 57A05, 57A52, 57B9D, 57DBD, 58057, 58509, 5855A, 585A7, 587EB, 58857, 588E8, 58A75, 58B0B, 58B87, 58BBE, 58BEB, 58E5B, 591D5, 59201, 59256, 59715, 59807, 5A88A, 5AA88, 5AFAD, 5B001, 5B00B, 5B1F1, 5B31B, 5B7E2, 5B80B, 5BB13, 5BBE8, 5BBFB, 5BE87, 5BE8B, 5BF37, 5BFBD, 5D00F, 5DA05, 5DA5A, 5DAE5, 5DBBD, 5DD95, 5DDAA, 5DFDD, 5E879, 5E8B7, 5E8BB, 5F07A, 5F0AD, 5F37D, 5F70A, 5F7BD, 5FB7B, 5FBBB, 5FBF3, 5FFF3, 6003E, 60098, 603E6, 606GF, 60986, 609C8, 60G6F, 60GCF, 6336D, 633E9, 63CCE, 63E06, 63E66, 6609G, 660E9, 66AD4, 66D4A, 66DG4, 66DGG, 66E98, 66FD9, 66GF6, 69806, 69866, 69C86, 69CC8, 6A66F, 6AAGF, 6AF06, 6AF66, 6AGGF, 6C6G3, 6C6GF, 6CCGF, 6CG03, 6DA0E, 6DAEA, 6DD9G, 6DDE9, 6DEGA, 6DGD3, 6E986, 6EEE9, 6F69D, 6F6DF, 6F96D, 6FD03, 6FD09, 6G003, 6G3F3, 6G3FF, 6G6CF, 6GAAF, 6GCCF, 70031, 70099, 700BF, 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77777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777 ====Additional known quasi-minimal primes (not necessarily the next)==== 570₅₁₃₁₀1, 49₁₁₁₃₃₃, 970₁₆₆₀₄₇1, F70₁₈₆₇₆₇1 ===Base 18=== 11, 15, 1B, 1D, 21, 25, 27, 2B, 2H, 35, 37, 3D, 3H, 41, 47, 4B, 4H, 57, 5B, 5D, 5H, 61, 65, 71, 75, 7B, 7D, 85, 87, 8D, 91, 95, 9B, 9H, A1, AB, AD, AH, B1, BD, C7, CB, CD, CH, D5, D7, DH, E5, EB, EH, F1, F7, FB, FD, G5, H1, H5, H7, HB, 107, 167, 16H, 177, 17H, 1G7, 1HH, 20D, 24D, 26D, 29D, 30B, 36B, 381, 3BB, 405, 445, 44D, 49D, 4A5, 4DD, 4F5, 4GD, 501, 545, 5E1, 607, 62D, 64D, 66B, 66H, 67H, 68B, 697, 6A7, 6BB, 6E7, 6G7, 6GB, 6HH, 767, 76H, 77H, 797, 7HH, 801, 80H, 831, 83B, 86B, 88H, 8BB, 8FH, 8GH, 94D, 96D, 977, 9DD, 9ED, 9GD, A77, AC5, AE7, B07, B0H, B55, B77, B8B, B97, BB5, BB7, BBH, BE7, BFH, BGB, C01, C31, CA5, CG1, D2D, D4D, D81, DBB, DD1, DDB, DGD, E0D, E17, E31, E4D, E67, E6D, EA7, EDD, EE1, EED, EG7, F0H, F45, F8H, FC5, FFH, G0D, G17, G2D, G6B, G6H, GBB, GBH, GD1, GDD, GE1, GE7, GED, GFH, GG7, GGB, GHD, GHH, H0D, H2D, H8H, H9D, HGH, HHD, 100H, 19E7, 1A97, 1EE7, 1G8H, 1GGH, 22ED, 22GD, 2DED, 2E2D, 3001, 3031, 30C1, 30E1, 3331, 33G1, 3CC1, 40ED, 45C5, 46ED, 4CC5, 5331, 5551, 55G1, 5C05, 608H, 60ED, 60FH, 60HD, 666D, 66ED, 699D, 6B67, 6BGH, 6D0D, 6DDD, 6E9D, 6EGD, 6G0H, 6G9D, 6HGD, 700H, 70A7, 7A07, 7FGH, 7G77, 808B, 8881, 88G1, 88GB, 8BHH, 8EG1, 8GC1, 8H6H, 900D, 90E7, 90G7, 9667, 9907, 999D, 99E7, 9A67, 9A97, 9E97, 9EE7, 9G07, 9G67, 9GA7, AA45, AA97, AGA7, B005, B03B, B06B, B0C5, B60B, B63B, BAA5, BAA7, BCC5, BFA5, BG8H, C045, C055, C555, C5C1, C5F5, CC05, CC81, CCC5, D06D, D09D, D0ED, D38B, D3E1, D60D, D6DD, D8GB, DD6D, DE9D, DG01, E001, E097, E0G1, E8C1, EDC1, EE97, EGC1, EGG1, EGGD, FH6H, G007, G00B, G00H, G03B, G067, G097, G0C1, G0G1, G1GH, G33B, G38B, G3G1, G70H, G777, G88B, GA67, GAA7, GG81, GGC1, GGGH, H0FH, H66D, HEGD, HFHH, 1AAA7, 222DD, 30GG1, 3388B, 33E01, 38G8B, 3G3C1, 3GGG1, 4002D, 500C5, 50C55, 50CF5, 53GG1, 558C1, 55CC5, 55CF5, 58GG1, 5C8C1, 5CFF5, 5G881, 5GG31, 6000H, 6003B, 6006D, 600DB, 6033B, 606GD, 60D0B, 66GGD, 6D03B, 6D33B, 6H6DD, 6HD6D, 6HDED, 70G07, 70GGH, 777A7, 7AAG7, 7G0GH, 80G0B, 8888B, 8CCE1, 90067, 90097, 9022D, 99967, 99997, 9A007, 9A0A7, 9AA07, 9AAA7, 9E007, A0045, A0455, A0667, A09G7, A0A07, A0G07, A0G97, A9997, AA0A7, AAG67, B0AF5, B6GGH, B7GGH, B8HHH, BA045, BAF05, BG667, C0F05, C5005, C5581, C88C1, C8CC1, C8CE1, CCF55, D03C1, D060B, D080B, D0CC1, D0G0B, D0G8B, D3G3B, D600B, DDDED, DG331, DG80B, E8G81, E9007, F6GGH, G018H, G0301, G0331, G466D, G6667, G66GD, GD08B, GG18H, GG6GD, GGG4D, H060H, HGGGD, HHH6H, 199AA7, 40006D, 40600D, 46600D, 5055C5, 5505C5, 55CCC1, 588CC1, 58CCC1, 60000D, 60009D, 7077G7, 7707G7, 777G07, 88000B, 9099A7, A000A7, A009A7, A09067, A099A7, A0AAA7, A90AA7, A99AA7, AA0007, AA6667, AAAG07, BFFF05, BFFFF5, C0FFF5, CCECC1, CECCC1, CF0FF5, CFF005, D0008B, D0033B, D0088B, D0333B, D033GB, D03G31, D0633B, DD990D, DGGG31, FHHHHH, G00081, G6GGGD, G8GGG1, GGG001, GGG331, GGGGG1, GGGGGD, H0006H, H00H6H, HH600H, 222222D, 22DDDDD, 333333B, 5CCCCC1, 70007G7, 88CCCC1, 9000007, 9000A07, A000G67, AAAA667, BBBB33B, C000CF5, C000FF5, CCCCCE1, CCCCEC1, D00063B, D00GG31, D63333B, DCCCCC1, DDDDD9D, DGCCCC1, GCCCCC1, GG00031, 4022222D, 6000GGGD, 66666667, 770000G7, AAAAA007, B6666667, BBBBBB3B, CFFFFF55, D00000C1, D0000EC1, 455555555, 5555550C5, 667777777, A00000967, A00009097, A00009967, A45555555, AAAAAAA07, BHHHHHHHH, CCCCCCCC1, CF0000005, CFFFFFF05, D00000G3B, E0CCCCCC1, G00000031, 70000000G7, A000000097, D000003301, 777777700G7, A0000900007, D0000000001, D000000GGG1, 677777777777, 8HHHHHHHHHHH, 2DDDDDDDDDDDD, 55555555555C5, 77AAAAAAAAAA7, D00000000006B, D0000000003GB, AAAAAAAAAAAAAA7, D0000000000000B, 77777777777777G7, CCFFFFFFFFFFFFFF5, BBBBBBBBBBBBBBBBBBB6B, CFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF5, GG0000000000000000000000000000001, HDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD, C00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000F5, 80000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000B, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHFH, 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4EEEEEEEEEB, 4GGGGGGGGG7, 5EEEEEEEEEB, 600000000I1, 60000000A33, 66666666629, 666666DDD0D, 6AAAAAAAGA3, 700GGGGGGG7, 7777777A7A7, 90I29999999, 9GGGGGGGGE7, 9IIAAAAAAA9, AAAAAAAA3G9, AAAAAAAAA39, AAAAAAAAAAD, AAAAAAAAGAD, C6CGGGGGGG3, CCCCCCCCG99, D0000000007, D0000000C07, D7DDDDDDDDD, DC000000007, EBBBBBBBB8B, EEE8BBBBBBB, EEEEEEEB8BB, EEEEEEEEE47, EEEEEEEEE8B, EJ0JJJJJJJJ, EJJJJJJJ0JJ, EJJJJJJJJAJ, F000000EF09, F0A44444441, F6666666689, FFFFAJJJJJJ, G00000G00C3, G1JJJJJJJJJ, G4444444447, GGGGGGGGG09, H0000000H41, H004444IIIB, HHHHHHHHHEH, I00AAAAAAA9, I0AAAAAAAF9, IIBBBBBB8BB, J050000000J, J77777777A7, JAEEEEEEEE7, JJJ77777777, JJJJJ0JJJ5J, JJJJJ777777, JJJJJJAJA77, 333333333GI9, 600000000089, 600000000D0D, 6AAAAAAAAA63, 6DDDDDDD0DDD, 6GGGGGGGCCC3, 7C3333333333, 7CC0GGGGGGG7, 902999999999, 9A4444444447, 9FAAAAAAAAA9, A777777777E7, AAAAAAAAAII9, AGGGGGGGGGA3, B0000000005B, B00000000IEH, B0000000E00H, B000000E000H, CCC7DDDDDDDD, CGGGGGGGGGG7, ECJJJJJJJJJJ, EEEEEEE4E447, FFFFFFAAAAA9, GAGGGGGGGAG3, GCGCCCCCCCC9, H000000006F1, H00000004441, H00044444441, HHHHHA00000H, I000000001G1, IAAAAAAAAA09, J0000000500J, J0000000JCJJ, J0000050000J, JJJ00000005J, JJJJJ500000J, JJJJJEEEEEC7, JJJJJJEECCC7, JJJJJJJ0005J, 555555555555H, 5BBBBBBBBBBBB, 6AAAAAAAAAAA3, A99FFFFFFFFF9, AAA7777777777, AAAAAAAAAAA93, AJ77777777777, CCCCCCCCC2229, D0000000000I1, DDDDDDDDDDD4D, DEEEEEEEEEE07, E00000000000H, EEEEEEEEEEEEB, G9000000000C3, GAGGGGGGGGG63, GGGGGGGGGGG93, HHHHHHHHHHHI3, J00000000050J, J00000CJJJJJJ, JJJ0CJJJJJJJJ, JJJJJJJJJJAJJ, JJJJJJJJJJJA7, JJJJJJJJJJJG1, 333333333I3IA3, 500000000000CJ, A0000000000099, A0000000000999, AAAAAAAAAAAAA9, B00E000000000H, CCCCCCCCCCC029, EIBBBBBBBBBBBB, FFFFAAAAAAAAA9, G00000000000G3, H0000000000001, I0AAAAAAAAAAA9, I9AAAAAAAAAAA9, JJJJJJJJJJEEC7, JJJJJJJJJJJ50J, 3333333333333A3, 6666666666668I9, 777777777777771, 7GGGGGGGGGGGGG7, 84400000000000B, AAAAAAAAAAAF009, B000000000000IH, E999999999999F9, EEEEEEEEBBBBBBB, FFCCCCCCCCCCC29, G0000000000006D, GGC000000000007, HEHHHHHHHHHHHHH, J00000000000001, J0000000000055J, J0000CJJJJJJJJJ, 500000000000E0AJ, 500000000000F00J, 6000000000000001, C6GGGGGGGGGGGGG3, CCCDDDDDDDDDDDDD, CJJJJJJJJJJJJJAJ, E222222222222229, E999999999999999, EE66666666666689, FFFFFFFFFFFFFFA9, HHHHHHHHHHHHHGA3, HHHHHHHHHI666663, IIIIIIAAAAAAAAA9, JJJEEEEEEEEEEEC7, 222222222222228I9, 444444444444444G7, EEEEEEBBBBBBBBBBB, EEJJJJJJJJJJJJJJJ, F000000000000E0F9, GC000000000000007, GGGGGGGGGGGCCCCC9, JJJJJJJJJJJJJ7777, JJJJJJJJJJJJJJJJ1, 10000000000000007D, 733333333333333333, HHHA0000000000000H, I3AAAAAAAAAAAAAAA3, IIIBBBBBBBBBBBBB8B, AAAAE44444444444447, D0000000000000000CJ, EEEEBBBBBBBBBBBBBBB, FCCCCCCCCCCCCCCCC29, GGGGGGGCCCCCCCCCCC9, GGGGGGGGGCCCCCCCCC9, GGGGGGGGGGGGGGGGG63, HHHHHHHI66666666663, J000000000000000ECJ, JJJCJJJJJJJJJJJJJJJ, 22222222222222222289, 5000000000000000FFFJ, 94444444444444444447, 97777777777777777777, A00000000000000000C9, B800000000000000000B, GGGGGGGGGGGGGGGGGGG3, IIIEBBBBBBBBBBBBBBBB, 40800000000000000000B, 710000000000000000007, J000000000000000005CJ, 76DDDDDDDDDDDDDDDDDDDD, EJJJJJJJJJJJJJJJJJJJJJ, HHHI666666666666666663, 4040400000000000000000B, 80000000000000000000A61, GJJJJJJJJJJJJJJJJJJJJJJ, I0IIIIIIIIIIIIIIIIIII29, D0DDDDDDDDDDDDDDDDDDDCC7, EHHHHHHHHHHHHHHHHHHHHHAH, IBBBBBBBBBBBBBBBBBBB8BBB, J5000000000000000000000J, J0000000000000000000CJJJJ, JJJJJJJJJJJJJJJJJJJ0J0J5J, EC000000000000000000000077, H000000000000000000000222B, HA00000000000000000000000H, HI666666666666666666666663, J000000000000000000000005J, JJJJJJJJJJJJJJJJJJJJJJJEC7, GGGGGGGGGGGGGGGGGGGGGGG9999, J00000000000000000000000E0J, J0CJJJJJJJJJJJJJJJJJJJJJJJJ, JEEEEEEEEEEEEEEEEEEEEEEEEEC7, 333333333333333333333333333G9, EEEEEEE4444444444444444444447, G9000000000000000000000000003, J000000000000000000000000CJJJ, AE7777777777777777777777777777, EEEEEEEEEEEEEEEEEEEEEEEEEECCC7, JJJJJJJJJJJJJJJJJJJJJJJJJ0JJ5J, AFFFFFFFFFFFFFFFFFFFFFFFFFFFFF9, JJJJJJJJJJJJJJJJJJJJJJJJJJJ0J5J, CJJJJJJJJJJJJJJJJJJJJJJJJJJJJJC7, IBBBBBBBBBBBBBBBBBBBBBBBBBBBB8BB, 8I00000000000000000000000000000A1, A77777777777777777777777777777A77, EEBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB, G00000000000000000000000000000007, H3HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH, G0000000000000000000000000000000C3, GCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC9, 666666666666666666666666666666666689, CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC29, CCDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD, GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGCCC9, IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII29, J1EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE7, C2222222222222222222222222222222222222229, J00000000000000000000000000000000000000CJ, DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD7D, GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG47, 4480000000000000000000000000000000000000000000B, A777777777777777777777777777777777777777777777777, I00000000000000000000000000000000000000000000004G1, 6DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD0D0D, IBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB8B, D0D0DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD7, 4000000000000000000000000000000000000000000000000000005B, GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGC9, 7777777777777777777777777777777777777777777777777777777777A77, CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCG9, F000000000000000000000000000000000000000000000000000000000EF9, 500000000000000000000000000000000000000000000000000000000000FJ, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH6A3, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHA3, JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJ77, 1000000000000000000000000000000000000000000000000000000000000000007, AAA4444444444444444444444444444444444444444444444444444444444444447, BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB8BB, EEEEE44444444444444444444444444444444444444444444444444444444444447, B000000000000000000000000000000000000000000000000000000000000000000E0H, D00000000000000000000000000000000000000000000000000000000000000000000000J, 80I0000000000000000000000000000000000000000000000000000000000000000000000001, A44444444444444444444444444444444444444444444444444444444444444444444444444444447, D00I00000000000000000000000000000000000000000000000000000000000000000000000000001, EI66666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666669, 8B000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000B, I8000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001, 92222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222229, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHA000H, JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJ5AJ, 800000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000061, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH3H, EEE44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447, IIBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB, 8I00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001, E444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447, DI0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001, G600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000003, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHA0H, 777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777A7, J777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777, JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJCCC7, 4000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000404B, EC0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000007, GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG99, 3AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA3, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEC7, JCJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJ, JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJ05J, 500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000AJ, CDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD, G00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000D ===Base 21=== ====Additional known quasi-minimal primes (not necessarily the next)==== 40₄₇₃₃₃9G, CF₄₇₉₁₄₇0K ===Base 22=== 11, 17, 19, 1F, 1J, 1L, 23, 29, 2F, 2H, 31, 35, 37, 3D, 3H, 41, 49, 4D, 4F, 4J, 4L, 53, 5H, 5L, 65, 67, 6H, 6J, 73, 79, 7D, 7J, 83, 85, 8F, 8H, 8L, 91, 9D, A3, A7, A9, AD, AJ, AL, B9, BF, BL, C5, C7, CD, CH, CJ, D7, DL, E3, E5, E9, F1, F7, FH, FJ, G1, G7, GF, GL, H5, H9, HF, I1, I5, ID, J1, J3, JD, JF, JL, K3, K9, KH, KL, L1, L5, LH, 103, 12D, 145, 155, 15D, 163, 18D, 1A5, 1BD, 1BH, 1C3, 1D3, 1DH, 1EH, 1G3, 1GH, 1I3, 1K5, 1KD, 221, 227, 22J, 22L, 245, 247, 25D, 25J, 271, 277, 287, 28J, 2A5, 2B7, 2BD, 2BJ, 2D5, 2E1, 2E7, 2ED, 2EL, 2K1, 2KJ, 2LL, 30J, 343, 389, 39J, 3B3, 3GJ, 3IJ, 3J9, 3JJ, 3KF, 3LJ, 427, 443, 445, 457, 4A5, 4C3, 4E7, 4G5, 4I7, 4K5, 4K7, 515, 52D, 551, 559, 55D, 55J, 575, 58D, 59F, 5B1, 5C9, 5CF, 5D1, 5D5, 5DD, 5E1, 5ED, 5G5, 5GJ, 5J5, 5JJ, 5K1, 5KJ, 60D, 61D, 62L, 661, 66D, 689, 6C1, 6D1, 6DD, 6G3, 6KF, 701, 721, 751, 76L, 775, 77F, 77H, 787, 7A5, 7AF, 7B1, 7B7, 7CL, 7E1, 7FF, 7FL, 7H7, 7HH, 7K5, 809, 81D, 821, 827, 82D, 847, 869, 871, 881, 889, 899, 8A1, 8BD, 8BJ, 8D1, 8DJ, 8GJ, 8J7, 907, 90H, 92L, 93J, 943, 947, 95F, 977, 997, 9AF, 9B5, 9EF, 9F5, 9H3, 9HL, 9I7, 9J9, 9JJ, 9K5, A25, A45, A51, A61, A6F, AAH, AB5, ABH, ACF, AG5, AGH, AHH, AK1, B15, B2D, B2J, B33, B45, B47, B57, B71, B75, B81, B87, B8J, BB3, BB7, BC3, BDD, BE7, BEJ, BGD, BGH, BH3, BHD, BHJ, BIH, BJ7, BKJ, CA1, CAF, CB3, CC1, CEF, CG3, CKF, D09, D0J, D13, D21, D33, D39, D3F, D4H, D5J, D63, D81, D8D, DAH, DBD, DBH, DBJ, DCF, DD3, DEJ, DFF, DG9, DGH, DHD, DI3, DIF, DJ9, DK1, DK5, E0F, E0H, E27, E2D, E2L, E47, E7H, E87, E8J, EA1, EAH, EB1, EDH, EEJ, EFF, EFL, EH1, EIF, EIL, EJH, EJJ, EKD, ELF, F25, F43, FB5, FD3, FDD, FDF, FEF, FEL, FFD, FG9, G09, G0D, G25, G3J, G5D, G5J, G63, G8D, G99, GC3, GC9, GD9, GEH, GG5, GJ5, GJ9, GJH, H03, H1D, H21, H2J, H2L, H33, H63, H77, H8J, HCL, HDD, HE1, HE7, HGH, HGJ, HH7, HHL, HI3, HIH, HJH, HK1, HKD, HL3, I07, I0J, I43, I47, I7L, I9J, IBH, IEL, IG3, IHH, IIJ, IJ7, IL7, J07, J55, J69, J8J, J99, J9J, JA5, JAH, JE7, JEH, JHH, JI9, JJ5, JJ9, JJH, JK7, K2J, K51, K5D, K75, K81, KA1, KB1, KB7, KBJ, KD1, KEJ, KG5, KIF, KJ5, KKD, KKJ, L0D, L47, L7F, L97, LAF, LD3, LD9, LDD, LEF, LGD, LI7, LJ7, LJJ, LLD, 104H, 10D5, 1205, 12B5, 140H, 1433, 144H, 14AH, 14B3, 16ED, 1AIH, 1B43, 1DD5, 1DDD, 1E6D, 1EGD, 1G05, 1GDD, 1GED, 1GGD, 1HB3, 1HHH, 1IAH, 200L, 2015, 2051, 20A1, 20DJ, 20GD, 20IL, 21B5, 21DD, 220D, 226D, 228D, 22B5, 22G5, 22K5, 22KD, 2555, 2557, 2581, 25C1, 26A1, 26B1, 2725, 2755, 2801, 2861, 288D, 28B1, 28KD, 2AA1, 2B25, 2B51, 2BB1, 2C81, 2D6D, 2DA1, 2DDJ, 2DGD, 2G0J, 2GB5, 2GDD, 2GGJ, 2I0L, 2I6L, 2ICL, 2J05, 2JK5, 2K07, 2K2D, 2K55, 2K6D, 2KB5, 2KI7, 2L2D, 2L8D, 2LK7, 302L, 30G3, 320L, 32IL, 332L, 33G3, 33G9, 36AF, 36EF, 382J, 388J, 39EL, 3AAF, 3BBJ, 3CG9, 3E2J, 3E6F, 3E6L, 3EEF, 3FAF, 3G69, 3GI3, 3GI9, 3IG9, 3LEL, 3LG3, 4025, 404H, 4063, 4075, 40AH, 40B5, 40B7, 40GH, 4225, 4363, 4447, 444H, 447H, 4487, 44B7, 44H7, 4525, 4555, 45B5, 4663, 4777, 47GH, 4807, 4B0H, 4BB5, 4BG3, 4EBH, 4G4H, 4GB3, 4HH3, 4I03, 4I63, 4IGH, 5069, 5077, 50KD, 5255, 52IJ, 5455, 5477, 5507, 5527, 556F, 557F, 5587, 56EF, 56FF, 56GD, 56KD, 5771, 57C1, 57EF, 5807, 580J, 589J, 58E7, 58IJ, 58K7, 5905, 5945, 5957, 5969, 5989, 598J, 5999, 59EJ, 59K7, 5AEF, 5B25, 5B55, 5BA5, 5BI7, 5BIJ, 5BK5, 5BK7, 5C21, 5EAF, 5EE7, 5F09, 5F6F, 5F95, 5FKD, 5G6D, 5G89, 5I09, 5I0F, 5I27, 5I6F, 5I89, 5IB7, 5IBJ, 5IFF, 5J77, 5K07, 5K25, 5K27, 5K6F, 5K87, 5KK7, 6013, 60A1, 60AF, 60EL, 6281, 62B1, 62KD, 63G9, 6403, 6643, 66EL, 66G9, 68KD, 69B3, 6A01, 6A0F, 6AAF, 6AFF, 6B21, 6B8D, 6BBD, 6BK1, 6D03, 6D43, 6D69, 6D6F, 6D93, 6D9F, 6E21, 6E81, 6E8D, 6EBD, 6ECL, 6ELL, 6FD9, 6GBD, 6GG9, 6IEF, 6K21, 6K8D, 6KBD, 6KE1, 6L43, 6LB3, 6LDF, 6LEL, 6LG9, 700F, 7027, 7057, 70EH, 70LF, 710H, 71G5, 7207, 7225, 7255, 727L, 7505, 7507, 755F, 75EF, 75F5, 75I7, 766F, 7681, 76CF, 7771, 7781, 77IL, 7861, 7A0H, 7A4H, 7BB5, 7C6F, 7EEH, 7EHL, 7FG5, 7G45, 7HEL, 7I0L, 7I27, 7IE7, 7IGH, 7K07, 7K61, 7K77, 7KC1, 7L27, 7LCF, 7LK7, 802J, 80B1, 80GD, 80JJ, 80KJ, 820J, 86E1, 880D, 882J, 88E7, 88I7, 892J, 89EJ, 8B61, 8BI7, 8CI9, 8CK1, 8D6D, 8DD9, 8DGD, 8E01, 8E07, 8EE1, 8EI7, 8EK1, 8EKJ, 8I77, 8I87, 8IC9, 8IK7, 8J0J, 8K6D, 8KI7, 8KIJ, 8KK7, 9025, 9055, 9089, 9275, 92EJ, 92G5, 92GJ, 93EL, 9455, 9505, 9557, 9599, 96B3, 970L, 976F, 97G5, 982J, 98E7, 98I9, 98K7, 9905, 990J, 9925, 9995, 999H, 99BH, 99EL, 99G9, 99GJ, 99LJ, 9A55, 9B03, 9B27, 9BGJ, 9BH7, 9BK7, 9CG9, 9E6L, 9EHJ, 9EKJ, 9ELL, 9G39, 9G45, 9G4H, 9GA5, 9GAH, 9GBJ, 9GEJ, 9GHH, 9GHJ, 9GI9, 9GIJ, 9H7H, 9H87, 9HAH, 9HB7, 9I89, 9I9H, 9IB3, 9IEH, 9IKF, 9J7H, 9J87, 9JB7, 9JGH, 9K0F, 9K27, 9KGJ, 9KKF, 9L89, 9L9J, 9LG9, 9LGJ, 9LIJ, 9LKF, A081, A14H, A1IH, A201, A2C1, AA15, AA81, AAIF, ABC1, AC01, AHC1, AIIF, AKEF, B005, B01D, B01H, B051, B05D, B0A5, B0B1, B0BJ, B0DH, B0E1, B0GJ, B0H1, B0JH, B143, B16D, B255, B2B5, B2C1, B2G5, B2K7, B4EH, B4G3, B501, B5A5, B5BJ, B5C1, B5IJ, B5K5, B621, B663, B6I3, B6K1, B777, B7I7, B80D, B88D, B8ED, B8KD, BB01, BB0J, BB21, BB25, BB4H, BB6D, BBD5, BBE1, BBEH, BBH1, BBIJ, BBJH, BCK1, BD61, BDB5, BDGJ, BEE1, BG55, BGB5, BGGJ, BGIJ, BH27, BH7H, BHC1, BHH1, BHK7, BI77, BIGJ, BJ0H, BJBH, BJIJ, BJJJ, BK25, BK27, BK55, BKA5, BKK1, C043, C143, C2B1, C32L, C3G9, C601, C6EL, C6G9, C8B1, CBK1, CC2L, CC89, CE21, CECL, CF2L, CF89, CG69, CIG9, D003, D015, D045, D05D, D06D, D06F, D0DH, D0H1, D1D5, D1GD, D205, D26D, D2DJ, D32J, D38J, D403, D525, D561, D56D, D5AF, D5F5, D5GD, D6C9, D6D9, D6DF, D6E1, D6ED, D6F9, D6FD, D8IJ, D8JJ, D945, D993, D99F, D9C9, D9F3, D9G5, D9KJ, DA15, DAA5, DAE1, DAKF, DB25, DB61, DBC1, DBG3, DC43, DC99, DCE1, DD0F, DD1D, DD2D, DD2J, DD5D, DD9H, DDA5, DDD5, DDDJ, DDED, DDH1, DDHJ, DDI9, DDJ5, DE0D, DED1, DEDF, DEFD, DF03, DF45, DF55, DF5D, DF6D, DG05, DGGJ, DH0H, DH43, DHC1, DHEH, DHH3, DHJJ, DI8J, DIC9, DII9, DIJH, DJG5, DJHJ, DJKJ, DK6D, DKAF, DKDD, DKGD, E081, E0BJ, E0DD, E0DJ, E0E1, E0ED, E0J7, E0K7, E0KJ, E1GD, E1IH, E201, E281, E66F, E6BD, E6ED, E6K1, E6LL, E7EF, E861, E86D, E88D, EB0D, EB4H, EBHH, EBI7, EC6F, ECEL, ED0D, ED1D, ED6D, EDAF, EDDD, EDFD, EE01, EE0D, EE1H, EE21, EECF, EEE1, EEGH, EEH7, EF8D, EGGD, EGHH, EGIH, EGIJ, EH07, EH0L, EH8D, EHBH, EHBJ, EHK7, EHL7, EI4H, EIK7, EIKJ, EJ77, EK07, EK0J, EKC1, EKIJ, EKK1, EKK7, EL6L, ELBD, ELCL, ELDJ, ELK7, F059, F20L, F26L, F28D, F2GD, F2KD, F32L, F3AF, F3G3, F455, F56F, F595, F5KD, F6AF, F6GD, F88D, F8I9, F955, F995, F9B3, F9G3, F9KF, FB03, FB8D, FBI3, FC89, FD99, FDA5, FDI9, FEGD, FF59, FG2D, FGI3, FGKD, FI89, FIB3, FIKF, FK2D, FK45, FKD5, FL6D, FLG3, G003, G04H, G055, G0BH, G0HH, G0HJ, G0I3, G0K5, G0KJ, G26D, G2DJ, G2GJ, G303, G333, G393, G3G9, G3I3, G403, G40H, G433, G4AH, G4GH, G4H3, G4IH, G589, G5G9, G80J, G89J, G94H, G9G3, GAA5, GAIH, GB0H, GB43, GB6D, GBB5, GBG3, GBGJ, GBJJ, GD0H, GD45, GDD5, GDKD, GG2D, GG39, GG8J, GG9H, GGAH, GGDJ, GGEJ, GGGH, GGH3, GGKJ, GH0H, GHG3, GHH3, GHHH, GHJJ, GI03, GI2J, GI93, GIAH, GIEJ, GIG9, GII3, GJJJ, GJKJ, GK05, GKDD, GKGD, GKGJ, GKIJ, H081, H087, H0AH, H0BJ, H0D1, H0HJ, H0LL, H1HH, H20D, H22D, H26D, H2I7, H2K7, H30L, H3KJ, H40H, H447, H4G3, H4H3, H4HH, H66L, H6ED, H6IL, H7C1, H7EL, H80D, H861, H887, H88D, H8B1, H8B7, H8C1, H8I7, HA0H, HB1H, HB27, HBB1, HBBJ, HBD3, HBI7, HCB1, HCC3, HD3J, HD61, HDC1, HDEH, HDHH, HE4H, HEHD, HELJ, HH0D, HH4H, HH61, HH6D, HH81, HH8D, HHC1, HHDH, HHG3, HHH1, HI6L, HIB7, HIBJ, HJJJ, HKI7, HKK7, HL27, HL2D, HLBJ, HLEJ, HLEL, HLK7, I0AF, I2CL, I2GJ, I32L, I33J, I6B3, I6EF, I727, I74H, I82J, I877, I88J, I8C9, I8EJ, I8JJ, IA0F, IAAF, IB03, IB63, IBGJ, IE2J, IEB7, IEBJ, IECF, IEGJ, IEHJ, IEKJ, IF89, IFKF, IG8J, IGBJ, IGGH, IGI9, IHBJ, IHC3, IHJJ, IHKJ, II9H, IIEF, IIHL, III7, IIIH, IJ09, IJBJ, IK0F, IK6F, IKCF, IKEF, IKJJ, IKK7, IL2L, ILLJ, J025, J05J, J0EJ, J0K5, J44H, J487, J50J, J589, J5J7, J757, J7GH, J975, J9B7, J9GH, JB0H, JB77, JBJJ, JC09, JCC9, JEGJ, JG89, JGBJ, JHI7, JI4H, JII7, JJ27, JJ2J, JJ87, JJGJ, JJJ7, JK05, JKB5, JKK5, K015, K08D, K0ED, K0JJ, K0K7, K105, K16D, K201, K225, K255, K2K5, K4B5, K50J, K557, K587, K5K7, K621, K62D, K6BD, K6E1, K761, K777, K7I7, K7KF, K80J, K88J, KBA5, KBB5, KC0F, KD25, KD5F, KDF5, KDJJ, KDKF, KE0D, KE6D, KEAF, KEED, KEK1, KFD5, KGGD, KI27, KIE7, KIGJ, KIJJ, KJ77, KJGJ, KK07, KK61, KK6F, KK87, KKCF, KKK7, L0G3, L0G9, L22D, L26L, L2DJ, L2GJ, L2IL, L2K7, L433, L4I3, L6DF, L887, L8B7, L99J, L9KF, LB27, LB77, LBDJ, LBED, LC43, LC89, LCG9, LD2J, LD6F, LF2L, LF89, LFG3, LFKF, LG43, LG69, LG8J, LGBJ, LIG9, LILJ, LK8D, LKED, LKKF, LL2L, LLIJ, 100G5, 10225, 10DED, 10DGD, 10H6D, 13333, 1DBG5, 1DEED, 1EDED, 1HEED, 200G5, 200GJ, 2010D, 2016D, 20225, 205B5, 20681, 20B55, 20BC1, 20D1D, 20D61, 20DD1, 20DDD, 20GK5, 20IEJ, 20IK7, 20J0J, 20JI7, 20K25, 20KK5, 21025, 21G6D, 22255, 22DDD, 25001, 250B5, 250K7, 25B05, 25IK7, 25KK5, 266CL, 26GGD, 26IIL, 26LKD, 2A0C1, 2B0G5, 2BB55, 2BC61, 2C0CL, 2C60L, 2C6CL, 2CCIL, 2D01D, 2D22D, 2DC61, 2DD01, 2DDB1, 2DDC1, 2DJIJ, 2G6GD, 2GIJJ, 2ILGJ, 2J0JJ, 2JIGJ, 2JJ0J, 2JJEJ, 2K025, 2KDDD, 2L6KD, 2LDIJ, 2LGIJ, 2LIEJ, 303EL, 306EL, 306G9, 30AFF, 30ECF, 30ELL, 30GG9, 3266L, 32CCL, 332EJ, 3333J, 333AF, 33CEL, 33E0L, 33IEF, 360G9, 363EL, 36E0L, 390G9, 399G3, 3AFIF, 3C0EL, 3E0LL, 3EC0L, 3ELLL, 3FG33, 3GG03, 3GG33, 3I26L, 3IAFF, 3L2CL, 3L6G9, 3LGG9, 40007, 40087, 400G3, 403G3, 40477, 40BBH, 40EHH, 40HEH, 43003, 43033, 43303, 43II3, 44BAH, 44BHH, 44EEH, 46033, 460I3, 46333, 470IH, 47407, 47BEH, 48B77, 4A00H, 4AI4H, 4AIIH, 4B055, 4B4BH, 4B4HH, 4B505, 4BBAH, 4BE4H, 4EE4H, 4EGGH, 4EIIH, 4G00H, 4GBAH, 4GHAH, 4GHBH, 4GI33, 4HBAH, 4I0EH, 4I40H, 4IA4H, 4IHB3, 5002J, 5006F, 50087, 500B7, 500F9, 50407, 504B5, 50525, 505B5, 5066F, 50681, 50761, 507KF, 508C1, 508EJ, 508J9, 50927, 50987, 509I9, 509J7, 50A01, 50DIJ, 50F45, 50GGD, 50II9, 50IKF, 50J89, 50K45, 5106D, 52081, 520B5, 520EJ, 520K5, 52E0J, 52II7, 52K05, 54007, 54887, 550B5, 55205, 552K5, 55405, 55577, 55E77, 55F45, 55KI7, 56009, 5600F, 560I9, 5660F, 5666F, 566F9, 56801, 56909, 56F69, 56I69, 572K7, 57407, 576A1, 577E7, 57I77, 57IKF, 57K47, 57KE7, 58061, 580C1, 589B7, 58II9, 59009, 590I9, 595A5, 59887, 5992J, 59EB7, 5A001, 5AIAF, 5B00D, 5B6BD, 5BBB5, 5BBDJ, 5C681, 5C801, 5C861, 5D0AF, 5D88J, 5DAAF, 5DK0F, 5DKEF, 5DKKF, 5E7K7, 5EI2J, 5EKKF, 5F045, 5F0IF, 5F405, 5F5IF, 5F699, 5FB0D, 5FB6D, 5FBBD, 5FFIF, 5FGGD, 5FI69, 5GG69, 5GGG9, 5I2EJ, 5I777, 5II77, 5II99, 5IIK7, 5IKI7, 5J089, 5J8B7, 5JGG9, 5K405, 5K447, 5K577, 5K7E7, 5KB05, 5KD0F, 5KEI7, 5KKB5, 60463, 606EF, 608B1, 608E1, 60EK1, 62GGD, 633EL, 63AIF, 63IAF, 64333, 643I3, 660EF, 666B3, 66B03, 66BI3, 66EEF, 66FAF, 66IB3, 6888D, 68BE1, 68GGD, 69EEL, 6AAB1, 6ABA1, 6B063, 6B0I3, 6B603, 6BEED, 6BI03, 6C0G9, 6C9EL, 6C9G9, 6CC43, 6CCEL, 6CE0L, 6D9I9, 6DCI9, 6DEAF, 6DI99, 6EE6L, 6EEFD, 6F22D, 6FEED, 6FFAF, 6GGKD, 6GII9, 6IAIF, 6IIG9, 6K0K1, 6KK01, 6L8ED, 70055, 702B5, 7044H, 70477, 704IH, 70555, 70CCF, 70EI7, 70G0H, 70GIH, 70IK7, 70ILL, 70K47, 70KKF, 714IH, 72B05, 74005, 74047, 74407, 7440H, 74477, 744IH, 74707, 74AEH, 74B25, 74B55, 74GBH, 74I0H, 752K7, 755K7, 75E77, 75K47, 76AA1, 76I0F, 77047, 77407, 77447, 776K1, 77AC1, 77EI7, 77IK7, 77K27, 77KK1, 77L2L, 7A8C1, 7AAA1, 7AAC1, 7B025, 7BB0H, 7C0IF, 7E4GH, 7EKKF, 7F055, 7F405, 7G0AH, 7G0IH, 7G5B5, 7GB55, 7GGBH, 7GGIH, 7GI4H, 7I0AH, 7I0IF, 7I7K7, 7II0F, 7IIEH, 7IILL, 7ILLF, 7K0KF, 7K6EF, 7K7K1, 7KEE7, 7KEKF, 7KK0F, 7KK71, 7L0IF, 7LI6F, 7LIIL, 7LILF, 80077, 80707, 807I7, 808K7, 80B07, 80E0J, 80IEJ, 80KC1, 82JJJ, 870I7, 87E77, 88007, 8808J, 880IJ, 886KD, 88807, 8886D, 888GD, 88GED, 88GKD, 88I8J, 88IKJ, 88K07, 88K77, 88KGD, 89IKJ, 8B001, 8B0K1, 8B0K7, 8B7K7, 8BCE1, 8BK07, 8CB01, 8CBB1, 8DDKD, 8DEED, 8E777, 8EGDD, 8GGG9, 8I2EJ, 8I8KJ, 8I9B7, 8IIE7, 8IK8J, 8JE2J, 8JJIJ, 8K00D, 8K00J, 8K08J, 8K6K1, 8K8JJ, 8KC01, 9009J, 900B3, 900KF, 905EJ, 905I9, 90959, 90AA5, 90BI3, 90E0L, 90FKF, 90G33, 90GG3, 90GG9, 90GI3, 90ILJ, 90J45, 92225, 92255, 925K7, 930G9, 93FG3, 93GG3, 93GG9, 94205, 944AH, 944GH, 94A4H, 94HBH, 94HEH, 94I4H, 95009, 950EJ, 950I9, 952K7, 95525, 956I9, 95887, 9592J, 959BJ, 95E0J, 95EBJ, 95GG9, 95IEJ, 96EEL, 9744H, 974IH, 97BEH, 97C0F, 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7777777707I7, 777777EEEEK7, 7F0000000045, 7IIIIIIIILIF, 7IKKKKKKKKKF, 800000000E61, 80000000E06D, 877777777777, 900000000E2J, 90000008800J, 9000000900G3, 9000000IIIG9, 9999999999KF, 9GGGGGGGGGG3, A0B000000001, A0BBBBBBBBB1, AA0000000AEF, AA000000A0EF, AAAAAAAAA2A1, ACBBBBBBBBB1, B00000000027, B00000000K0D, B0GGGGGGGG43, B100000000HH, BAAAAAAAAKK5, C0000000E60L, CCCCCCCCCCEL, CCEEEEEEE00L, CEE0EEEEEE0L, D2JJJJJJJJJJ, DAAAAAAAAA0F, DAAAAAAAAAAF, DD5555505555, DDA000000001, DDDDDDDDDDKD, E00000000K21, E0C00000000L, E4HHHHHHHHEH, E770000000I7, EC000000000L, EE77000000I7, EEEE4BEEEEEH, EEEEEEEEE4BH, EEEEEEEEE7I7, EEEEEEEEEECL, EEEEEEEEEII7, EEEEEEEEKEE7, F00000000GBD, F00000006BED, F0000F0098C9, F0FF00000C2L, FFFFF0000B63, FGGGGGGGGG6D, FLLLLLLLLLB3, G00000000A05, G00000008E2J, G0G00000002J, GG6666666669, GGGGGGGGGG43, GGGGGGGGGGI3, GGGGGGGGGKED, GJE00000000J, H00000004B4H, H0000000KJ47, H0H0000000BH, HB00000000HH, HBH00000000H, HBH0000000BH, I000000L8II9, I0IIIIIIIKFF, I0LLLLLLL989, IAFFFFFFFFFF, IEIEEEEEEEE7, IFFFFFFFFFAF, IFFFFFIIIIAF, IIIIIIIIIKFF, J00000005GG9, J0000000G0JJ, J50000000GG9, J90000000045, JBBBBBBBBBB5, JJ00000000BJ, JJ0000000JIJ, JJ000000JJIJ, K00000000BBD, K00000000EE7, K000000080I7, K0000000K5EF, K000000KKK45, K0000EEEE7E7, K0EEEEE7EEE7, KE0000000601, KEEEEE07EEE7, KEEEEE0E7EE7, KEEEEEEEE0I7, KJJJJ000000J, KK0000055525, KKKK00000045, L000000003EL, L000000060B3, L0000888880J, L000088888KJ, L7IIIIIIIIIL, LL0LLLLLLK27, LLL00000000J, LLLLL6000043, LLLLL9999989, LLLLLLLGGGB3, LLLLLLLI8II9, LLLLLLLLKCCF, LLLLLLLLL643, LLLLLLLLL877, LLLLLLLLLK27, 20000000005K7, 2000000000B61, 2200000000025, 3003A0000000F, 40HHHHHHHHHBH, 4333333333333, 46IIIIIIIIII3, 4H0HHHHHHHHHH, 5000000000057, 5000000002057, 5000000002C61, 50000000D00KF, 5860000000001, 5D000000000KF, 5E0000000000J, 5GGGGGGGGGGGD, 5KF000000000D, 60000000000G9, 6D00000000EEF, 70000000000AH, 70000000077I7, 70000000777I7, 70777770000I7, 7770000000II7, 77777777770I7, 777777777II77, 777EEEEEEEEK7, 80000000000KD, 800000000EE6D, 800000008DDDD, 80000000KKKE1, 80008888888EJ, 800880000000J, 8B00000000007, 9EEEEEEEEEEEL, 9FFFFFFFFFKFF, 9LLLLLLLLECCL, A000000002BA1, A00000000EKKF, A0000000EEEEH, A0000CBBBBBB1, AAAAAAAAAAAA1, B0000000000K1, B0000000K000D, B0JBBBBBBBBB5, BAAA555555555, BAAAAAAAAAA05, BE0000000000D, BHHHHHHHHHHHH, C000000000E6L, C00000000CE6L, D0000000000ED, D000000000DD1, D0HHHHHHHHHHH, DD00A00000001, DD55555555505, DDDDDDDDDDDGD, DDEAAAAAAAAAF, ECC0LLLLLLLLL, EE000000000I7, EEEEEEEEEEEB7, EEEEEEEEEEHLL, EEEEEEEEEEKE7, EEEEEEEEEHLLL, EIEEEEEEEE7E7, EIEEEEEEEEEE7, EIIEEEEEEEE77, F000000000GG3, F00000000B6ED, F00000000K66F, F0000000K0F6F, F000000EEEE8D, F0F00000098C9, FF00000000B63, FF000000098C9, H00000000002D, H0000000000K7, H000000000AA1, H000000000EED, H000000000EEH, H00000000HAA1, HBB0H0000000H, HH000000000BH, HHEEEEEEEEBEH, HHHEEEEEEEEEH, HHHHHHHHHEBBD, IBIIIIIIIIII3, IEEE7EEEEEEE7, IIEEEEE7EEEE7, IIIIIIIIII0KF, IIIIIIIIILIB3, J00000000BBBJ, JGE000000000J, K000000000AEF, K000000000B6D, K00000000E7CF, K000000K005EF, K00000KKK0045, K00KKKKKKK045, K0222222222DD, KB0000000000D, KD555555555B5, KGG000000000J, KKKKKKKKK0K45, KKKKKKKKKK045, L00000000EGGJ, L000LLLLLLIB3, L0FLLLLLLLLB3, L0LLLLLLLLIB3, L70777777772L, L777777777727, L7LLLLLLLLLIL, LEBBBBBBBBBBD, LEEEEEEEEEEK7, LL0000000LECL, 100000000000B5, 5000000000016D, 5000000000088J, 5000088888888J, 50K000000000B5, 555555555555EF, 55IIIIIIIIIIIF, 5K0000000000FD, 60000000000ECF, 60ECCCCCCCCCCF, 6ECCCCCCCCCCCF, 7000000000I04H, 770000000700I7, 7777077777772L, 7I777777777777, 80000000000DDD, 80000000008DDD, 8088000000000J, 8GGGGGGGGGGEED, 9000000000808J, 990000000000G3, 99F000000000G3, 9LLLLLLLLLLL7L, A000000000A0A1, A00000000HBAA1, B00000000BBBED, B0GGGGGGGGGGG3, B5555555555505, BA555555555505, CCCCCCCCCCCLG9, D00000000050EF, EEEEEEEEE4EEIH, EEEHEEEEEEEEEH, F000000000KF6F, F000000009F8C9, F05IIIIIIIIIIF, F0F00000000C2L, F5IIIIIIIIIIIF, FF000000000C2L, FF000000009F89, G00000000000AH, GI000G0000000H, H0000000000I27, H000000000II27, HEEEEEEBEEEEEH, IEEEEEEEEEEE4H, IF0000000000B3, IIEEEEEEEEE7E7, IIIIIIII0IIIKF, IIIIIIIIIBIII3, J0000000000GJJ, JJJJJJJE00000J, K00000000006GD, K0000000000CFF, K0000000000JI7, K000000000B055, K00000000K0KB5, K022222222222D, K0KKKKKKK00045, K0KKKKKKKKK545, K6CCCCCCCCCCCF, KEEEEEEEEEEE07, KKKKKKKKKKK0B5, L0LLLLLLLLLK77, L88888888888EJ, LK0000000000FF, LLLLLLLLLK00E7, LLLLLLLLLL9G33, LLLLLLLLLLGI33, LLLLLLLLLLL3G3, LLLLLLLLLLL9EL, LLLLLLLLLLLGI3, 200000000000JJ7, 2A0000000000001, 2CCCCCCCCCCCCCL, 2K00000000000K5, 3000000000009G9, 4HHHHHHHHHHHEEH, 50000000000010D, 500000000000KB5, 50E0000000000I7, 700000077777II7, 700777777777II7, 7700000000007I7, 7700700000000I7, 7707000000000I7, 777777777777727, 7IIIIIIIIIIIIIF, 7LLLIIIIIIIIIIF, 80008888888888J, 80088888888888J, 80D00000000000D, 89000000000080J, A0000000000I4IH, B00000000000HEH, B000000000BBEBD, B0BBBBBBBBBBBB5, BBBBBBBBBBBBBB1, BBBBE000000000D, C00000000000LEL, CEEEEEEEE0EEE0L, D555555555550A5, DHHHHHHHHHHHHHH, EKEEEEEEEEEEEE7, FFFFFFFFFFFFG45, FFFFFFFFFFFIIAF, G000000000000B5, G0GGGGGGGGGGGG3, GG000000000002J, H000000000000B1, I77777777777777, K000000000K0BK5, K0000000KKKKK25, K000EEEEEEEEEE7, K05555555555KB5, KEEEEEEEEEEE7E7, KEKEEEEEEEEEEE7, KK00000000005EF, KK0K000000000B5, L000000000006B3, L0000000000ILB3, L0000000000LIB3, LCLEEEEEEEEEEEL, LLLLLLLLLL00KE7, LLLLLLLLLLILLB3, LLLLLLLLLLLK0E7, LLLLLLLLLLLL4G3, 20000000000000K7, 3AF000000000000F, 4000000000000IEH, 4IIIIIIIIIIIII33, 500000000000IJG9, 509B00000000000J, 59000000000000BJ, 6000000000008KK1, 60I00000000000B3, 6GGGGGGGGGGGGGED, 70000000000000I7, 700000000000EKE7, 7070000000007II7, 70777777777777I7, 7077777777777II7, 77000007000000I7, 80000000000000DD, 80000000000000E1, 888888888888888J, 900000000000088J, 988000000000000J, A000000000000001, A000000000000015, A0000000000002A1, BBBB0000000000ED, BH00000000000001, CLEEEEEEEEEEEEEL, D055555555555555, DDDHHHHHHHHHHHHH, EEEEEEEEEEEEEEEH, EELLLLLLLLLLLLB7, EHHHHHHHHHHHHHHH, GI0G00000000000H, HEEEEEEEEEBEEEEH, HHHHHHHHHHHHHH2D, I0000000000000B3, IEEEEEEEEEEEE7E7, J0000000000000GH, J000000000000JBJ, J00000000000JIJJ, JJ00000000000IJJ, JJJJE0000000000J, K000000000000K25, KFFFFFFFFFFFFCFF, L00000000000000J, LLLLLLLLLLL0LIB3, LLLLLLLLLLLECLLL, LLLLLLLLLLLLILB3, LLLLLLLLLLLLLG33, LLLLLLLLLLLLLKFF, 2KK00000000000005, 44EHHHHHHHHHHHHHH, 55555555555555BB5, 5B0BBBBBBBBBBBBBD, 7000000000000I40H, 707777777777777K7, 77000000000000I77, 8000000000000008D, A0000000000000CB1, A0000000000000EEH, B00000000000000D1, BAA55555555555555, BIIIIIIIIIIIIII63, C00000000000002IL, C0000000000000CEL, CCEEEEEEEEEEEEEEL, CEEEEEEEEEE0EEEEL, CEEEEEEEEEEEE0EEL, D000002222222222D, D5555505555555555, D5555555555505555, DGGGGGGGGGGGGGEED, F0000000000000EBD, F000000000000262D, F000000000000E0BD, F000000000000F6B3, F000000000000K6FF, F000000F6000000B3, GGGGGGGGGGGGGGGGD, HHEEBEEEEEEEEEEEH, HHHHHHHHHHHHH2GGD, HHHHHHHHHHHHHGBBD, JJE0000000000000J, JJJJJJJJJJJJJK00J, K0000000000000B55, K80000000000000I7, L0000000000000IB3, L0000000000009E2J, LLLLLLLLLLEB00007, LLLLLLLLLLLLLBGG3, 2D0000000000000001, 3000000000000000EF, 500000000000000EI7, 700000000000000I4H, 77777777777777720L, 8000000000000KKKK1, 9000000000000000GJ, B00000000000000K6D, BBD00000000000000H, D5K00000000000000F, ELLLLLLLLLLLLLLEB7, F00000000000000989, F0000000000F6000B3, FFFFFFFFFFFFFFFIAF, G4GGGGGGGGGGGGGGG3, GGGGGGGGGGGGGGGEED, HGGGGGGGGGGGGGGGED, HHHHHHHHHHHHHHHGBD, J00000000000000945, JG00000000000000JJ, K00000000000000045, K00000000000000057, KEEEEEEEEEEEEEEEE7, L0000000000000LECL, LLLLLLLLLLLLLLEB07, 20000000000000000J5, 2DDDDDDDDDDDDDDDD0D, 2DDDDDDDDDDDDDDDDDD, 500088888888888888J, 555555555555555K5B5, 7KKKKKKKKKKKKKKKKKF, 9700000000000000045, A0000000000000000IF, FL00000000000000K0F, GGGGGGGGGG3GGGGGGG3, GGGGGGGGGGGGGGGGGB3, GGI000000000000000H, H000000000000000E0D, I0000000000000008I9, IIIIIIIIIIIIIIIIIKF, KKKKKKKKKKKKKKKK545, L0000000000000006EL, LCEEEEEEEEEEEEEEEEL, LLLLLLLLLLLLLLL0KCF, 50000000000000000D9J, 50000000000000000DKF, 6FFFFFFFFFFFFFFFF0B3, 70000000000000000BBH, 7077777777777777772L, 800000000000000000B7, 99LLLLLLLLLLLLLLLLG3, A0000000000000BBBBA1, B000000000000000004H, B0000000000000000D43, B0IIIIIIIIIIIIIIIII3, E0000000000000000021, E00000000000000000CL, F000000000000000EE8D, F5000000000000000045, H0000000000000000JIJ, H000000000000000BBBH, H2000000000000000007, I00000000000000004GH, IIE7EEEEEEEEEEEEEEE7, JJJJJJJJJJJJJJJJJJIJ, K00000000000000000CF, A0000000000000000AEAF, B000000000000000000KD, BIIIIIIIIIIIIIIIIIII3, G0000000000000000002J, H00000000000000000E6D, K0KK000000000000000B5, LLLLLLLLLLLLLLLLLECCL, 4HHHHHHHHHHHHHHHHHHHBH, 55555555555555555555B5, 5K000000000000000000DF, 8D0000000000000000000D, BBBBBBBBBBBBBBBBBBBBB5, D00000000000002222222D, F00000F6000000000000B3, F000F600000000000000B3, GA0000000000000000000H, GGGGGGGGGGGGGGG3GGGGG3, GGGGGGGGGGGGGGGGGGG3G3, K00000000000000000KBK5, L00J0000000000000000C9, 60000000000000000000B03, B0BBBBBBBBBBBBBBBBBBBBD, ECCLLLLLLLLLLLLLLLLLLLL, IIIIIIIIIIIIIIIIIIII9B3, IIIIIIIIIIIIIIIIIIIILB3, J00000000000000000J0JIJ, JJJJJJJJJJJJJJJJJJJE00J, JJJJJJJJJJJJJJJJJJJJK0J, 500000000000000000000095, 7777777777777777777777I7, A00000000000000000004IIH, B1000000000000000000000H, CEEEEEEEEEEEEEEEEEEE0E0L, ECLLLLLLLLLLLLLLLLLLLLLL, F00000000000000000000C2L, LLLLLLLLLLLLLLLLLLLL0IB3, 5E00000000000000000000II7, CEEEEEEEEEEEEEEEEEEEEE0EL, EK60000000000000000000001, F00000000000000000000BE0D, F0F60000000000000000000B3, HH1000000000000000000000H, IEEEEEEEEEEEEEEEEEEEEEEE7, IIIIIIIIIIIIIIIIIIIIIIIB3, KD55555555555555555555555, 40HHHHHHHHHHHHHHHHHHHHHHHH, 5B000000000000000000000007, 6000000000000000000000KKK1, B00000000000000000000000ED, B0000000000000000000000BBD, BAAAAAAAAAAAAAAAAAAAAAAA55, DH000000000000000000000001, L0000000000000000000000ECL, 500000000000000000000000I8J, 700000000000000000000000447, 800000000000000000000000E6D, CCCCCCCCCCCCCCCCCCCCCCCCCG9, H00000000000000000000000J47, J000000000000000000000000C9, JJJJJJJJJJJJJJJJJJJJJJJJKJJ, K0000000000000000000000KKB5, LKFFFFFFFFFFFFFFFFFFFFFFFFF, 5IIIIIIIIIIIIIIIIIIIIIIIIIIF, D555555555555555555555550555, EEAAAAAAAAAAAAAAAAAAAAAAAAAF, HHHHHHHHHHHHHHHHHHHHHHHHEEBH, K66666666666666666666666666F, LLLLLLLLLLLLLLLLLLLLLLLLEEB7, D5555555555555555555555555A55, GGGGGGGGGGGGGGGGGGGGGGGGGGGG3, GIG0000000000000000000000000H, HH00000000000000000000000001H, K0000000000000000000000005KEF, 5BBBBBBBBBBBBBBBBBBBBBBBBBBBBD, HB0000000000000000000000000001, K000000000000000000000000505EF, L7777777777777777777777777772L, 2000000000000000000000000000CB1, C8CCCCCCCCCCCCCCCCCCCCCCCCCCCC9, IKKKKKKKKKKKKKKKKKKKKKKKKKKKKFF, JE0000000000000000000000000000J, K000000000000000000000000000261, A0000000000000000000000000004I4H, HD000000000000000000000000000001, K000000000000000000000000000EC01, K0FFFFFFFFFFFFFFFFFFFFFFFFFFFFCF, D0002222222222222222222222222222D, FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFL2L, I700000000000000000000000000000GH, K00000000000000000000000000000E61, 20000000000000000000000000000000JJ, DD5KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, EBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBD, FBB000000000000000000000000000000D, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHBBD, L0000000000000000000000000000000877, 59B00000000000000000000000000000000J, B00000000000000000000000000000000063, D000000000000000000000000000000A0BB1, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHEBD, KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKK45, 50E0000000000000000000000000000000007, 60000000000000000000000000000000000KK1, BAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAK5, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE7K7, ELLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL0B7, IKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKAF, 3000000000000000000000000000000000003AF, A00000000000000000000000000000000000EKF, BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBD, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE00I7, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHEBH, CEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE0L, D00000000000000000000000000000000000000B1, 400000000000000000000000000000000000000033, D500000000000000000000000000000000000000KF, E6CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCF, 2000000000000000000000000000000000000000BB5, B555555555555555555555555555555555555555555, 33A0000000000000000000000000000000000000000F, 700000000000000000000000000000000000000000K7, 900000000000000000000000000000000000000008EJ, DD00000000000000000000000000000000000000000E1, F000000000000000000000000000000000000006006B3, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHD, 60000000000000000000000000000000000000000000B3, D0000000000000000000000000000000000000000000EEH, D9J00000000000000000000000000000000000000000005, DDE00000000000000000000000000000000000000000001, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHEEEH, 5E0000000000000000000000000000000000000000000007, 88800000000000000000000000000000000000000000000J, 97LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL, D55555555555555555555555555555555555555555555555, D555555555555555555555555555555555555555555555A5, G2222222222222222222222222222222222222222222222D, KFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFCF, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLEB7, 70000000000000000000000000000000000000000000000GH, KE000000000000000000000000000000000000000000000061, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLECL, H0000000000000000000000000000000000000000000000000JJ, D000000000000000000000000000000000000000000000002222D, 97IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIL, B0000000000000000000000000000000000000000000000000000AH, D5555KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, BAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA5, FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF6B3, H000000000000000000000000000000000000000000000000000000ED, 7000000000000000000000000000000000000000000000000000000045, B0000000000000000000000000000000000000000000000000000000I3, C0000000000000000000000000000000000000000000000000000000EL, D500000000000000000000000000000000000000000000000000000005, K00000000000000000000000000000000000000000000000000000J887, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL0KE7, 44HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH, F0000000000000000000000000000000000000000000000000000000L89, J000000000000000000000000000000000000000000000000000000JJIJ, K222222222222222222222222222222222222222222222222222222222DD, KCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCF, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLKCF, D55KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, K000000000000000000000000000000000000000000000000000000000000077, 22222222222222222222222222222222222222222222222222222222222222222D, K000000000000000000000000000000000000000000000000000000000000008IJ, DFA00000000000000000000000000000000000000000000000000000000000000005, CC4IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII3, JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJE0J, 777777777777777777777777777777777777777777777777777777777777777777777777EK7, 6000000000000000000000000000000000000000000000000000000000000000000000000000000043, KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKB5, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHEEH, 4HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHEH, 50000000000000000000000000000000000000000000000000000000000000000000000000000000002C1, K0000000000000000000000000000000000000000000000000000000000000000000000000000000000055EF, H700000000000000000000000000000000000000000000000000000000000000000000000000000000000000000H, 80000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000K1, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE0I7, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHEH, 5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BB5, J000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BIJ, C4IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII3, F0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000066B3, G0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000A5, D5KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHBH, L0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000IKF, 4IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII3, A400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000H, DKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, 4HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH, E0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000071, 7LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLIL, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLK77, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEI7, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLIB3, I7G00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000H, D0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000005EEF, IKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, C0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000G9, 77EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEK7, 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66FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFB3, L0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000B63, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLG3, E60000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000L, IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIAF, 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J00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000IGGJ, 77777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777K7, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLKE7, 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BKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKK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80BN, 80FH, 80LN, 80MN, 840N, 848N, 866B, 86FB, 880B, 880N, 884N, 88CB, 88FB, 88LN, 88MN, 8BBB, 8BLB, 8C6B, 8CCH, 8CFH, 8F0B, 8FHH, 8FLB, 8H0H, 8HCH, 8IGN, 8ILN, 8KKH, 8L8B, 8LBB, 8LFB, 8LIN, 8M8N, 8MLN, 8N0N, 8NGN, 8NLN, 9061, 9091, 90EJ, 90F1, 90GJ, 90K1, 940J, 9501, 95F1, 9CC1, 9E0J, 9E95, 9F61, 9FI5, 9G3J, 9II5, 9K01, 9KK1, 9M01, A007, A05D, A0AD, A33D, A3AD, A3F5, A44J, A727, A9EJ, AA0D, AAAD, AAAJ, ACM7, AD8D, ADKD, AE27, AE9J, AEAJ, AEE7, AIAD, AIDD, AIID, AJ9J, AK5D, AM07, AM27, AM35, AMK5, B08B, B0CB, B0GB, B18B, B1CB, B80B, B8CB, BB21, BB4N, BBCB, BBF1, BBFB, BBK1, BC8B, BCF1, BCLB, BF1B, BF8B, BFB1, BFM1, BGC1, BGF1, BK21, BL8B, BLFB, BM1B, BM3N, BMB1, BMMN, BNF1, C00D, C06B, C077, C0D7, C0H7, C0L7, C0LB, C0M1, C60H, C6LB, C76H, C7E7, CAID, CC01, CCFH, CCKH, CDLB, CGE7, CH07, CHE7, CI8D, CIAD, CK0D, CL8B, CLDB, CLE7, CM01, CM07, CME7, CMM1, D007, D08D, D0C7, D0HH, D0LN, D0M7, D0NN, D207, D2KH, D2M7, D777, D7E7, D80H, D8LD, DA27, DAC7, DAM7, DBFB, DBMB, DC77, DCLB, DDL7, DE77, DF0H, DF2H, DFFH, DFMB, DH27, DH8H, DHC7, DHHH, DILN, DK0H, DK2H, DK8H, DKHH, DLIN, DLL7, DLM7, DLMB, DM07, DMH7, DMMB, DNGN, E07J, E09J, E335, E355, E555, E5A5, E5K5, E79J, E93J, E995, EA35, EE95, EKE5, F00B, F00H, F06H, F08B, F0I5, F11B, F18B, F1L1, F20H, F26H, F2FH, F355, F661, F6K1, F80B, F86B, F8BB, FBGB, FBK1, FBLB, FC0B, FC6H, FCLB, FEK5, FGB1, FH05, FH0H, FH35, FH6H, FHCH, FHF5, FHHH, FI05, FK91, FKK1, FL1B, FLB1, FLBB, FM61, FMBB, FMK1, G00J, G021, G027, G0EJ, G0JJ, G0M1, G0M7, G1CB, G2E7, G40J, G4AJ, G4IJ, G4JJ, G6C1, G701, G94J, G9IJ, GAEJ, GAJJ, GB21, GBM1, GC01, GCF1, GCLB, GE0J, GEAJ, GEE7, GEG7, GEIJ, GEL7, GFM1, GGE7, GGMB, GI0J, GIIJ, GIIN, GJ9J, GM27, GMB1, GNM7, H005, H0K5, H0M5, H207, H2E7, H335, H3I5, H595, H5K5, H60H, H68H, H76H, H80H, H8HH, HAAJ, HE7J, HEC7, HGE7, HGM7, HH35, HI55, HIM5, I00J, I035, I08D, I0CD, I4JJ, IC0D, ICID, II0D, II0J, II35, IIAD, IILD, IIM5, IIMN, IJ9J, ILCD, IM05, IM35, IMNN, INLD, J03J, J0HJ, J0JH, J2CH, J39J, J3ID, J60H, J62H, J8CH, J9IJ, JGAJ, JGJJ, JH9J, JI0J, JIDD, JJ0H, JJCD, JJJD, JJLD, JL3D, JLCD, K0E5, K0I5, K0K1, K0KH, K191, K211, K2F1, K2G1, K591, K5AD, K6F1, K6G1, K9I5, KA0D, KAAD, KAM5, KCCH, KCHH, KD8D, KDDD, KFI5, KG01, KG61, KH0H, KHHH, KI55, KIDD, KK21, KK8H, KKF1, KKK1, KKKD, KM8H, KMHH, KMK5, L027, L0M7, L1C1, L1MB, L211, L727, L8BB, L8BN, L8FB, L8LN, L9C1, L9M1, LB8B, LBC1, LBM1, LCAD, LD77, LDIN, LDL7, LDM7, LF8B, LG21, LIIN, LLLN, LLN7, LM07, LM1B, LM77, LMG1, LN77, LNM1, M00N, M01B, M03N, M055, M077, M08B, M0B1, M0C1, M0GB, M0K1, M0M7, M0N7, M18B, M1BB, M1MB, M26H, M335, M3GN, M3M5, M3MN, M3NN, M501, M53N, M5M1, M5NN, M6BB, M6C1, M6G1, M6KH, M88B, M88N, M8BB, M8NN, MBB1, MC01, MCC1, MCKH, MCM1, ME07, ME35, MEK5, MGGB, MGMB, MH35, MH8H, MHE7, MHM5, MK8H, MKC1, MKG1, MKHH, MKK5, ML8N, MM01, MM8B, MMC1, MMLN, MMM5, MMMN, MMN1, MN0N, MN27, MNGN, MNLN, N007, N027, N077, N0C7, N0DN, N0IN, N1M1, N227, N2M7, N661, N707, N727, N8LD, NA27, NA3D, NC07, ND0D, ND0N, NDLD, NF11, NF61, NGG7, NILN, 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5M0M1, 5MNNN, 5N03D, 5N3LD, 5NA8D, 5NADD, 5NDGN, 5NNGN, 600KH, 61661, 616L1, 61MM1, 66161, 66611, 666B1, 666L1, 666LB, 66BB1, 66BG1, 66BLB, 66G61, 66KF1, 66LBB, 66LG1, 6BBBB, 6BFCB, 6CC11, 6F1M1, 6F66B, 6F6B1, 6F6L1, 6FBCB, 6FLMB, 6FMCB, 6FMM1, 6FMMB, 6GCM1, 6GM61, 6GMC1, 6K1C1, 6K1K1, 6KK11, 6KKL1, 6KL11, 6L1G1, 6LBFB, 6LCC1, 6LFMB, 6MM61, 6MM6B, 70291, 702C1, 702G1, 72CF1, 72EE7, 7433J, 7443J, 77A07, 79901, 799F1, 7AAEJ, 7EE27, 7H9EJ, 7K2KH, 7KK2H, 7KKKH, 7KKMH, 7L2C1, 800NN, 806BB, 808BB, 808LB, 80F8B, 80IIN, 833ID, 860KH, 8886B, 888NN, 8BG4N, 8CH6H, 8CKHH, 8FC0H, 8FFCH, 8HHHH, 8IIIN, 8K0HH, 8LL4N, 8M0NN, 8MNNN, 8NNND, 9000J, 900M1, 9034J, 90IIJ, 94IIJ, 96CM1, 96KF1, 96MM1, 990C1, 990M1, 99591, 99961, 999C1, 99F91, 99FM1, 99KF1, 99M61, 99MK1, 9AAA5, 9FEE5, 9FFA5, 9FFF5, 9II4J, 9K6C1, 9K9C1, 9K9F1, 9KF91, 9M6M1, 9MK61, A02M7, A0A35, A0AM5, A0C77, A0D0D, A0DDD, A0EC7, A0M55, A0MM7, A2ME7, A3335, A33M5, A3555, A3MM5, A550D, A58ID, A5D0D, A5DDD, A74AJ, A7E07, AA0M5, AA3ID, AA3M5, AA83D, AA8ID, AAAM5, AACID, AAICD, AAM05, AC08D, AC0ID, AC0KD, AC8ID, ACA8D, AFA35, AIC8D, AJAJJ, AK0KD, AKI0D, AKKM5, AKM05, AM505, B00FB, B00LB, B0LBB, B1MBB, B1MMB, B2GM1, B3MGN, B88BB, BB3MN, BBB3N, BBB8B, BBBBN, BBBMN, BBCC1, BBGB1, BBGLB, BBMC1, BBMGB, BBNM1, BFB0B, BFBMB, BFGLB, BFGMB, BGGLB, BGMM1, BLMMB, BMC0B, BMGM1, BMMK1, BMNM1, C006H, C00CH, C00M7, C0C11, C0CC1, C0E07, C0F6H, C0GG7, C0K8D, C0KDH, C0KHH, C1CC1, C1CM1, C1MC1, C70F1, C886B, CC06H, CCCCH, CCCF1, CCCM1, CDHH7, CE707, CE7L7, CEE77, CEEG7, CEL07, CFFHH, CGLL7, CHC6H, CHCCH, CHH7H, CHHH7, CHHM7, CI0LD, CK0HH, CKHCH, CL007, CLD07, CLLL7, CM777, CMMM7, D008H, D07L7, D0DE7, D0EL7, D0L77, D0M8H, D0NA7, D22E7, D70A7, D7227, DBLBB, DCL07, DD7A7, DDAE7, DDD8D, DDE07, DDK0D, DDM77, DEE27, DKK8D, DL227, DL707, DLBBB, DLDFB, DLE27, DM2E7, DMM77, DMME7, DN0A7, DNIIN, DNLLN, E000J, E0227, E02E7, E0AAJ, E0GL7, E0I0J, E2E07, E7AAJ, E7E27, EAA55, EAK55, EC777, ECEE7, ECEL7, ECGL7, ECL77, EEC77, EECL7, EEEC7, EEG27, EEGG7, EG0E7, EG0L7, EG207, EGE07, EGG07, EGG27, EGL07, EI9IJ, F02HH, F06LB, F0C6B, F0CCH, F0CHH, F0E55, F0EA5, F0FH5, F0GGB, F0M2H, F0MGB, F0MMB, F1BMB, F3FF5, F3I35, F3II5, F6BCB, FA035, FB1MB, FBBM1, FBBMB, FBM8B, FC0HH, FC88B, FCFHH, FFC0H, FFCFH, FFCHH, FFF0H, FFF6H, FFFH5, FFHI5, FFI55, FG1MB, FGGCB, FGM1B, FHK55, FI5I5, FKEA5, FKKI5, FKL11, FM16B, FM1CB, FM62H, FM6CB, FMBG1, FMM0B, G0001, G00E7, G039J, G06F1, G07C1, G07F1, G0A9J, G0G07, G0GG7, G0I4J, G0I9J, G0LL7, G22M7, G2M07, G339J, G433J, G62M1, G6M61, G6MM1, G903J, G933J, GAA9J, GBCC1, GE007, GGGGB, GGGL7, GJ0IJ, GL007, GL2M7, GLL07, GLLM7, GMGCB, GMM07, GMM1B, H007H, H05I5, H0CCH, H0CM7, H0GG7, H0H27, H0H8H, H0HC7, H0HH5, H0I35, H0MM7, H3555, H35F5, H3F55, H3FF5, H5055, H50F5, H7HEJ, H9995, H9GEJ, HCC0H, HCC6H, HCGG7, HCHM7, HE027, HE0G7, HEE07, HEEG7, HEG27, HF0F5, HF505, HF555, HFF05, HFKF5, HG0G7, HH06H, HH08H, HH0H5, HH5I5, HH7HJ, HH9EJ, HH9I5, HHCHH, HHE07, HHGG7, HHH05, HHH7H, HHH9J, HHKI5, HHM05, HHM07, HHM27, HI0I5, HJ86H, HJC0H, HK055, HK9F5, HKF05, HKFF5, HKFK5, HKII5, HKK95, HM2M7, HME27, HMKM5, HMM05, HMM27, HMM55, HMMM7, I00DD, I00M5, I044J, I0505, I09IJ, I0AAD, I0D0D, I0DDD, I0I05, I0IDD, I0II5, I0IJJ, I0JIJ, I33M5, I4I4J, I5INN, I888N, I8NND, I8NNN, I904J, I94IJ, IA8ID, IAADD, IAI8D, IDDDD, IDINN, II88N, II8NN, IID8D, IIDIN, III8N, IIIDD, IIIID, IIIIN, IIIND, IIN8D, IINDD, IJJ0J, IJJJJ, ILILN, ILLIN, IMM8N, INA0D, INDNN, INGIN, INNDN, INNND, J000H, J002H, J00AJ, J00GJ, J00KH, J02KH, J068H, J080H, J090J, J0A9J, J0AAJ, J0C0H, J0G0J, J0IIJ, J0JGJ, J0JIJ, J0K8H, J2K0H, J2KKH, J6K8H, J86KH, JC00H, JC0KH, JCCCH, JCK0H, JCKCH, JDDLD, JG93J, JIIJJ, JJ0IJ, JJ2KH, JJ9GJ, JJCCH, JJG9J, JJGIJ, JJJ9J, JJJJH, JJK8H, JK08H, JK0CH, JK8KH, JKC0H, JKKKH, K0001, K0091, K020H, K02C1, K03ID, K0611, K06L1, K083D, K08HH, K0961, K09C1, K0CF1, K0F91, K0KM5, K0LG1, K1G21, K20HH, K29K1, K2KHH, K5001, K500D, K58ID, K5D0D, K5L11, K6621, K6C11, K6LC1, K8CKH, K8KCH, K96C1, K99E5, K9F91, K9FA5, K9FE5, K9K91, KA55D, KC011, KCF11, KD02H, KD0MH, KD20H, KDM2H, KEA55, KEAA5, KEK95, KEKK5, KF1G1, KF1K1, KF611, KF6L1, KFEA5, KH8CH, KI005, KIMM5, KK05D, KK0AD, KK0DH, KK2CH, KK2KH, KK33D, KK961, KK9C1, KKA5D, KKD0D, KKE55, KKI0D, KKIID, KKIM5, KKK0H, KKKM5, KLGC1, KMK2H, L188B, L1991, L2007, L22M7, L2EE7, L2MM7, L333D, L3LIN, L7291, L72G1, L88IN, L8C8B, L9991, LBB1B, LBBBB, LBBBN, LD0E7, LDBBN, LE207, LFMCB, LGCC1, LL227, LL3IN, LL48N, LLM27, LLMM7, LMBCB, LME27, LMMBB, LMMM7, LN33D, LN3AD, LNAAD, LNACD, M0007, M0061, M00K5, M0207, M066B, M0BCB, M0EE5, M0G01, M0GM1, M0M8N, M27KH, M2E27, M2M07, M2M27, M5005, M5555, M66CB, M66K1, M6K61, M6MCB, M7007, M7EE7, M8C0B, M8KCH, M8MGN, MBBGB, MBGM1, MCCCH, MCHCH, MEE55, MGBC1, MGMM1, MH227, MH2M7, MHH7H, MKM55, ML3LN, MM0CB, MM16B, MM227, MM661, MM6K1, MME55, MMEE7, MMKE5, MMM07, MMM6B, MMMB1, MMMGB, MMMM7, MNM61, MNN3N, N00CD, N00KD, N03LN, N0A8D, N0AM7, N0D8D, N0KKD, N0L3N, N0LAD, N0NDD, N16L1, N3GIN, N3LAD, N3LIN, N3NNN, N61L1, N96M1, N9M61, NA0CD, NAK0D, NAKKD, NCA8D, NCM77, NDGIN, NDIIN, NDLLN, NF991, NGM07, NIIIN, NINNN, NKKDD, NLNAD, NN0LN, NN191, NN3NN, NN6L1, NN83D, NNAAD, NNDIN, NNGIN, NNL3N, NNLND, NNM61, NNNIN, 166G21, 16G621, 19MMM1, 1BBBMB, 1BBGM1, 1GCCC1, 1GCCM1, 1MMM1B, 200E27, 2E0027, 2HH0HH, 2HHC0H, 2KK0HH, 2M0E27, 2M22E7, 30NNNN, 3333M5, 333AID, 333I35, 33I555, 3F5FF5, 3I3MM5, 3I88GN, 3II8LN, 3IIII5, 400IJJ, 40J00J, 40JJ3J, 40JJJJ, 44403J, 444I0J, 44IJJJ, 44J0JJ, 44JJIJ, 48I8IN, 4I440J, 4II8IN, 4IJ0IJ, 4JIIIJ, 4JJ0JJ, 4JJJ0J, 50033D, 5003AD, 5008ID, 500D8D, 500G01, 500L11, 500LAD, 500MG1, 503LAD, 508ILD, 50DDLD, 50ILAD, 50M001, 516G61, 519MM1, 538NNN, 53NNNN, 55005D, 5508ID, 550D8D, 558ILD, 55F5I5, 56G661, 58333D, 58NNNN, 5999F5, 59AAF5, 5DNNNN, 5F55I5, 5FMMM1, 5G6661, 5K9AA5, 5KK9F5, 5KKK95, 5M0001, 5NDD8D, 5NDINN, 5NN33D, 5NNLAD, 5NNNAD, 5NNNDN, 608K0H, 61CCM1, 61G621, 661G21, 666621, 6666CB, 6666F1, 66K661, 6BCCC1, 6BKKC1, 6F6BBB, 6G6621, 6GCCC1, 6GMMM1, 6K6K61, 6M666B, 70A077, 70L991, 7722E7, 772E27, 7772E7, 777A27, 777L27, 77A777, 77EL27, 7A7077, 7A7777, 7E7227, 7L2E27, 7LEL27, 7LL2E7, 7LLE27, 7LLL27, 800GIN, 80NINN, 80NNNN, 8BBMGN, 8C888B, 8C88LB, 8MM0GN, 900001, 90043J, 959MM1, 96K661, 9999F1, 9999K1, 999AF5, 999FF5, 99EEE5, 99K991, 99MMM1, 9AAFF5, 9EIIIJ, 9F9991, 9F9MM1, 9FEAA5, 9G444J, 9K9991, 9M6661, A000CD, A000KD, A000M5, A0083D, A00I0D, A00M05, A022E7, A07E77, A0FF35, A0K3ID, A0K83D, A4AJJJ, A77777, AA0035, AA0355, AAA035, ADDD0D, ADDDDD, AF0035, AFFF35, AKK8ID, AM0M05, AM7777, B0F0MB, BBBBM1, BBLBMB, BFBBBB, BFM0MB, BFMMMB, BLBBMB, BLMBBB, C00071, C000E7, C007C1, C00G07, C07KKH, C0CC6H, C0CH6H, C0EEE7, C0HHHH, C777L7, C77L77, C7L777, C7LL07, C8088B, CAAK8D, CAKKAD, CC000H, CC0CHH, CD000H, CD0KKH, CE0007, CEE0E7, CELL77, CG0007, CGGL07, CH0CHH, CHCH0H, CHHH6H, CK0C0H, CKAK8D, CKKA8D, CL7707, D002FH, D0D0KD, D0DA77, D0DKKD, D0IIIN, D0K0DD, D0KDKD, D0KKDD, D0KKKH, DC0EE7, DCEEE7, DD0227, DD0D27, DD0DKD, DD0KKD, DD2E27, DDD0D7, DDD0LD, DDD227, DDDA77, DDDBCB, DDDCE7, DDDDFB, DDDMM7, DDEEE7, DDMBCB, DEEC07, DH000H, DHMEE7, DIIIGN, DK0KDD, DKMKKH, DMBBBB, DMEEE7, DMMM27, E00G27, E07727, E0C707, E0CE77, E0E027, E0EEG7, E0EGE7, E0EL27, E0GE27, E0L207, E0LE27, E0LL27, E2E2E7, E7L2E7, E900IJ, E9EEE5, EAAKK5, EC00E7, EC0G07, EC7007, ECEG07, EE0G07, EE0GE7, EE72E7, EE7L27, EECE07, EECEG7, EEEEE5, EEEK55, EEEKA5, EEEL27, EEGLL7, EELE27, EGLLL7, EKK595, EKKA55, EKKAK5, EKKKK5, F000E5, F0AA35, F0F035, F0FFFH, F0HKK5, F0KKE5, F16BB1, F16MM1, F1BBBB, F1MC6B, F666BB, F66BBB, F6GMM1, FB0BBB, FB1BBB, FBBB0B, FBMMG1, FC0FFH, FCFCCH, FEEE55, FEEEA5, FF03F5, FF0FFH, FF3F35, FFEE35, FFF2CH, FFFCCH, FFFFE5, FFI335, FFKFE5, FGLMMB, FK55I5, FKFE55, FLM8CB, FMC66B, FMMC6B, G0AA4J, G0CCC1, G0LE07, G666F1, GG0007, GG00G7, GG0L07, GGLLL7, GGLMM7, GI444J, GJJ33J, GLLE27, GLMMCB, GM0661, GMMM61, H00G07, H05555, H09FF5, H0C0E7, H0CE07, H0CEE7, H0E227, H0H007, H0H5F5, H0H995, H0HHE7, H0HHH7, H55505, H55II5, H5FII5, H99FF5, HEG007, HFFK55, HH0007, HH02M7, HH0C0H, HH7AEJ, HHC0E7, HHE227, HHH0C7, HHH0M7, HHH995, HHHC0H, HHHE27, HHHEAJ, HHHH07, HHHH8H, HHHHE7, HHHHI5, HHHHJH, HHHJ8H, HHHJCH, HHJ00H, HHK095, HHKKM5, HKK0F5, HKK5F5, HKKK55, HKKKK5, HKM555, HMEEE7, I00555, I05555, I0I94J, I333I5, I33555, I444IJ, I55055, I55505, I55555, IAAC8D, ID000D, IDD0LD, II9I4J, III4IJ, III505, IIIC8D, IIJIJJ, IJIIIJ, IM8LLN, IN00AD, INAACD, INCAAD, ININGN, J00CCH, J0IJJJ, J0J09J, J3333D, JIJIIJ, JJ68KH, JJIJIJ, JJJAJJ, JJJHGJ, JJJJAJ, JJJJGJ, JJJJIJ, K0008H, K00161, K001G1, K001L1, K002CH, K002HH, K00521, K00AKD, K00C0H, K00GF1, K00I0D, K00K95, K00M05, K01621, K05021, K0505D, K051G1, K059F5, K05K95, K0C0C1, K0L291, K0M005, K0M505, K1K661, K2CK0H, K33IAD, K3IIID, K5550D, K56121, K59AA5, K612K1, K61CC1, K66661, K6K611, K900C1, K99661, K9AFF5, K9C001, KAKI8D, KC00C1, KDK00D, KF9991, KI0IID, KK000D, KK01L1, KK0661, KK0I8D, KK0L11, KK0M2H, KK5661, KK59F5, KK61C1, KK9995, KK9EE5, KKA3ID, KKA83D, KKAI8D, KKC001, KKC0C1, KKC1C1, KKCCC1, KKD2HH, KKK595, KKK9A5, KKKK95, KKKKKH, KKKMCH, KKM505, KKMEE5, KKMKCH, KM0005, L222E7, L33AAD, L38I8N, LCCC11, LDFBCB, LEL2E7, LELE27, LELL27, LGMMM1, LLE2E7, LM2ME7, M000M5, M006MB, M00E27, M00MM1, M02227, M06M61, M06MM1, M0E227, M0EE27, M0KME5, M0M5GN, M0MM61, M0MMCB, M0MNNN, M0NNM1, M38LLN, M5K505, M77707, M7E227, M7E727, M8CHHH, MBMMCB, MBMMM1, MEEE77, MHH027, MHH505, MHHC6H, MHHH6H, MHHK05, MKK001, MM2ME7, MM7707, MM7E77, MMBMK1, MMM2E7, MMMC0B, MMMK61, N0003D, N0008D, N0030N, N030NN, N0C0AD, N0CKAD, N0DKDD, N0N3GN, N0NN3N, N333AD, N777A7, N77A77, NAACKD, NAAKDD, NACAKD, NACKAD, NC0AKD, NC0KAD, NCA0KD, NCKAKD, NDNNLN, NNNLAD, NNNNLD, 1BBBBBB, 1BBBBG1, 1M6MMMB, 1MMBBBB, 1MMMMK1, 2000227, 2000EE7, 20EEEE7, 2C0FFFH, 2E2EEE7, 2KKKHCH, 2MEE227, 2MEEE27, 333333D, 3333355, 3335555, 333FFF5, 333IIID, 388NNNN, 38INNNN, 3INNNNN, 4000IMN, 4000JJJ, 400IIIN, 444444J, 44JJJJJ, 488888N, 4IIJIIJ, 4JJJ33J, 50002M1, 5001G21, 5006621, 500LGM1, 555083D, 55555I5, 5616G21, 59MMMM1, 5K999A5, 61CCCC1, 66666K1, 6K0000H, 6K0080H, 70000A7, 70077A7, 70700A7, 7070A77, 77700A7, 77770A7, 7777227, 7777E27, 77L2227, 7LE22E7, 888888B, 888888N, 8888BBN, 8888IIN, 888B88B, 888I8IN, 88IINNN, 88NIINN, 88NNIIN, 8INNNNN, 90444IJ, 904I44J, 9666661, 9666FK1, 9666K61, 9966FK1, A00KK0D, A0K000D, AAAAA35, AAKKI8D, BB8888B, BBB0BLB, BBBB1BB, BBBBBB1, BBBBBGB, BBBBBLB, BBBLMBB, C0007KH, C000F11, C00FFFH, C00HH0H, C00K00H, C0C0HHH, C0CCHHH, C0CHH0H, C0CHHCH, C0FFFFH, C0H0H0H, C0KKC0H, CC0HH0H, CCCCC11, CCCCCC1, CCHHHHH, CDKKKKH, CEL7777, CGGG0G7, CGGGGG7, CH00HHH, CHGGGG7, CHHHH0H, CHHHHCH, CHHHHHH, CK0000H, CKDKKKH, D00DDKD, DD0DDD7, DDBBBLB, DDD2EE7, DDDBBLB, DDDDD27, DDDDDBB, DDDDDC7, DDDDDKD, DDDDDMB, DDDDEE7, DDDDKKD, DDDDLDB, DDDFBBB, DDDLFCB, DDDMEE7, DDM2227, DHHEEE7, DK000KD, DK00D0D, DK0D00D, DNN000N, E000CL7, E000EG7, E000GE7, E00C0G7, E00CE07, E00EE27, E0C00G7, E0C0EG7, E0CE007, E0EC0G7, E0EE207, E0G0007, E20EE27, E22EEE7, E2EE227, E2EEE27, E772227, E77LL27, E7L2227, E9IIIIJ, EAKKKA5, EC000G7, ECG00G7, EE00L27, EE0E0G7, EE20EE7, EEE0EG7, EEE22E7, EEEE727, EEEEE27, EEEEG07, EEEEGE7, EEEKKK5, EELLL27, EI0IIIJ, EKKKAA5, ELLLE27, F00FA35, F0333F5, F0F0FE5, F333335, FAAFF35, FCF0FCH, FEEEE35, FF03335, FF0FA35, FF0FE35, FF0FMCH, FFF0A35, FFF0F35, FFFAF35, FFFF5I5, FFFFM2H, FFFI3I5, FFFIII5, FFH5555, FH55555, FL1MMM1, G0000G7, G0000L7, GGGG007, GLE2227, GLLLLE7, GLLLLL7, H000007, H0000C7, H000HCH, H000HM7, H000M27, H000ME7, H00G227, H00HHM7, H02M227, H0C0HHH, H0CH00H, H0E0007, H0FFF35, H0FFFF5, H0H0ME7, H0HFII5, H0HHHCH, H0M0227, H555555, H5F5FF5, HC000G7, HC00H0H, HCCHHCH, HCHHHCH, HCHHHHH, HEEEE27, HFF5FF5, HFF5FI5, HFKKK05, HG00007, HH00E27, HH0G227, HHH2MM7, HHH55F5, HHH9FF5, HHHFFK5, HHHFK55, HHHH7EJ, HHHHCM7, HHHHHAJ, HHHHHF5, HHHHHHJ, HKK5505, I000055, I00A0ID, I0I4IIJ, I0IIIIJ, I88NIIN, III0055, III0I55, III444J, IINNLIN, J000IJJ, K0000DH, K0000KD, K00033D, K000A5D, K000K5D, K00555D, K009995, K00K00D, K00K8ID, K00KI8D, K00KIAD, K01GCC1, K05033D, K0999F5, K2KKKCH, K53333D, K956661, K999991, KCCC1C1, KFFFE55, KFFKKE5, KFKFKE5, KK009A5, KK00C11, KK01GC1, KK99001, KKIII05, KKK09F5, KKKE9E5, KKKEAK5, KKKKI05, KKKKKE5, KMMEEE5, L1BBBG1, LBMMMCB, LBMMMMB, LDEEE07, LEE22E7, LEE2E27, LEEE2E7, LLLLE27, M0000CB, M000C6B, M02EEE7, M0K0005, M0M0005, M2CHHHH, M2HHHHH, M6MMMM1, MC0000B, MCHHHHH, ME7E777, MEE7777, MEEE2E7, MG06661, MHHHCCH, MHHHH27, MHHHHH7, MHM0027, MM6666B, MM77777, MMC000B, MMM7727, MMNM777, N000NLN, N00333D, N003AAD, N0A00DD, N0NN33D, N0NNLLN, N30000N, N777777, NDNNNNN, NN0N0GN, NN0N30N, NNN300N, NNN333D, NNN3LLN, NNNDDDD, NNNNN3N, NNNNNND, 33333F35, 33FFFF35, 3555FFF5, 3FFFFF55, 3NNNNNLN, 40000I0J, 40I0IIIJ, 444440IJ, 4J0000IJ, 500006G1, 5D00DDDD, 5L1MMMM1, 5MMMMMG1, 5NNDDDDD, 5NNNNDDD, 5NNNNN8D, 6000080H, 777777A7, 77777A77, 7944444J, 800000IN, 996666K1, 999999I5, 9999FEA5, A00003ID, AAAAFF35, BBBGMMMB, C0000011, C000007H, C0CC0H0H, C666666B, CCCH0HHH, CCHH0HCH, CE777777, CEEEEE07, CHH0H00H, D00000GN, D000D0LD, D000IIGN, D0DDDDD7, DDD0E2E7, DDDDDDDB, DDDDDME7, DEEEELE7, DEEELEE7, DEELEEE7, DELEE0E7, E00000C7, E00000G7, E0000CG7, E000C0E7, E000G007, E00CG007, E00E0CG7, E0C00007, E0CGGGG7, E0GGGGG7, E20000E7, EAAKAAA5, EAKKAAA5, EE00E727, EE020007, EEEE2027, ELEE2227, F00003F5, F0000A35, F0003335, F0FFFA35, F1999991, F1999MM1, FAAAAF35, FBBBBBG1, FEAAAAA5, FF000A35, FF00FF35, FF0KEEE5, FFAAAF35, FFF555I5, FFFF33I5, FFFFF035, FFFFF3F5, FFFFFKI5, FFFFFMHH, FKFKEEE5, FKKFEEE5, FMMMMMCB, FMMMMMM1, G2000007, GGGGGMM7, GJJJJJ0J, GJJJJJ3J, H0000E27, H0000G27, H000C0G7, H000CEG7, H000CHH7, H000E0E7, H000EE27, H00CHHG7, H00EEE27, H00M0EE7, H05FF5F5, H0E00EE7, H55FF5F5, HCHH0H0H, HE000EE7, HFFIIII5, HGGG2227, HH00CEG7, HH00H0CH, HH0EEEE7, HH0FFFI5, HHEEEEE7, HHH000CH, HHH00EG7, HHHC00G7, HHHFFFF5, HHHHHKK5, HHHK5F55, HK5555F5, I4IIIIIJ, IA0000ID, II0005I5, III000I5, III055I5, III5NNNN, IIIII9IJ, IIIIIII5, IIINNNGN, JAJJJJJJ, JJAJJJJJ, K000005D, K00009A5, K0000M55, K000M555, K008IIID, K00D0K0D, K00III8D, K00K550D, K00LCC11, K0999951, K0D0000H, K0K00595, K0K9AAF5, K0KK0095, K0KK9FF5, K3333IID, KFKFEEE5, KFKKEEE5, KK00000H, KK0000M5, KK099991, KK55583D, KKKEEEA5, KKKKKKI5, LEEEE227, LLLEEE27, LLLL2E07, M000006B, M000M6CB, M0MMMMM1, M222EEE7, M777E777, M7E77777, ME222EE7, ME2EEEE7, MEE222E7, MM0NNNNN, MME77727, MMM6MMM1, N00003GN, N0000ADD, N000N0GN, N000NNND, N033333D, N0NN0NGN, NDDDDKDD, NN000N3N, NN00N03N, NN03000N, NNN003GN, NNNNDNLN, NNNNNADD, 199999MM1, 200FFFFFH, 222MEEEE7, 2FFFFFFCH, 30000000N, 30N00000N, 400000J3J, 500000M01, 5000166G1, 5000666G1, 50DDDDDDD, 8NN33333D, 999999991, C000000FH, C00000K0H, C000H00HH, C77777707, CH00H000H, CHH0000HH, D00000DKD, D00000DLD, D0000200H, D0000KK0D, D000KK00D, D00D0DDLD, D0D00DDLD, D0LEEEEE7, DDBBBBBBB, DDD000KDD, DDDDDDDE7, DK00000DD, DNNNNNNNN, E00000E27, E00007L27, E0000E727, E0E000C07, E20000027, EAAAAKAA5, EAKAAAAK5, EE0000C07, EEE000E27, EEEEEEGL7, F00FFFF35, F0FFFFF35, FF0000035, FF0FFFF35, FF5555FI5, FFFFFFA35, FFFFFFF35, FFFFFFFI5, FKKKKEEE5, FMMMMMMMB, GGGGG2227, GGGGGG207, GJJJJJJJJ, GLMMMMMMB, H000022M7, H000222M7, H000EEEE7, H0EEEEEE7, H0H0000CH, H0IIIIII5, HCH00000H, HE0EEEEE7, HFFFFFI35, HFFFFKKK5, HHHHHHG27, HHHHHHH55, HHHHHHHH7, HHHHHHM55, HHHKK5555, HHIIIII05, HIIIIII05, HKK5555I5, I000000AD, I000000ID, I000A000D, I00A0000D, IIIII0555, IIIIIII9J, K000000AD, K00000595, K000009F5, K0000550D, K099999A5, K0C00000H, K0I00000D, KK0000595, KK0000HCH, KKK000095, KKKFKFFE5, KKKIIIII5, M77777777, MEEEE2227, MMMMMMMM1, N0000000D, N0000003N, N000003NN, N00000N3N, N00000NGN, N00N000GN, N00NNNN8D, N0NNN00GN, N0NNNN3AD, N0NNNNNGN, N999999M1, NN0NNNNGN, NNN000NGN, NNNNNDD8D, NNNNNN0GN, 16MMMMMMMB, 1MMMMMMBCB, 3333333335, 33333333I5, 400000000N, 40IIIIIIJJ, 4IIIIIIIJJ, 4IIIIIIJIJ, 50000000M1, 70F9999991, 777E777727, 9999995MM1, ADD000000D, C00000088B, C000000CF1, C00000F0HH, CH00000H0H, D00KD0000D, D0D0DDDDLD, D0E2EEEEE7, D2EEEEEEE7, DBBBBBBBBB, DD000000KD, DD0000DDLD, DLE0EEEEE7, EEE0000727, EEEAAAAAA5, EEEEEE00G7, EEEEEEE0G7, F000000F35, F00FFKEEE5, F00KFFEEE5, F0M666666B, FCFFFFFFFH, FFFFFFF2HH, FFFKKKEEE5, GGGGGGG227, H00000C06H, H0000HHH6H, H555FFFFF5, H55FFFFF55, H5FFFFFF55, HF5FFFFFF5, HHHH0H0HCH, HHHHH0HHCH, HHHHHH0HCH, HHHHHHHHM5, HHHIIIIII5, IIDNNNNNLN, IIIIIJJIIJ, IIINNNNNLN, IINNNNNNGN, INNNNNNNLN, J0000000IJ, K0000II8ID, K099999995, K0I0000AID, K0K0009FF5, K9999999F5, KK00000095, KKFFFKEEE5, LLLLLLLME7, LLLMEEEEE7, LMEEEEEEE7, M000000005, MHHHHHHHH5, MK00000005, MMMMMMMBCB, NN000000GN, NN0000NNGN, NN99999991, 2HHHHHHHHHH, 38NNNNNNNNN, 3MNNNNNNNNN, 40IIIIIIIIJ, 4AJJJJJJJJJ, 4J000000J0J, 4JJJJJJJJJJ, 506666666G1, 5DDDDDDDDLD, 999999999F5, 99999999EA5, 99999999FE5, A0000000035, C0000000007, C00000000G7, C00000000KH, CEEEEEEEEL7, D0000000FMH, D000DDDDDLD, D0KD000000D, DDDD00000KD, DEEE0EEEEE7, DEL0EEEEEE7, DELEEEEEE07, E0000E20007, E7777777727, EE000000207, EEE20000007, FFFFFFFFMCH, FM66666666B, GGGGGGGG2M7, HFFFFFFFF55, HHHHHHHHH6H, HHHHHHHHHCH, HHHHHK55555, I9IIIIIIIIJ, IIIIIIII44J, IIIIIIIJJIJ, IINNNNNNNNN, JDDDDDDDDDD, JJIIIIIIIIJ, K00000I8IID, LLLLLLLLL27, 9999999EEAA5, AI000000000D, C77700000007, CH0HH000000H, D00D000000LD, DEEEEEEE0EE7, DN000000000N, EAKAAAAAAAA5, EKAAAAAAAAK5, F6666666666B, H000HHHHHH6H, H55FFFFFFFF5, HFFFFFFFFKK5, K00000000I8D, K999999999A5, 3555555555FF5, 5000000000001, 6G66666666661, 99999999999A5, C00000000000H, CFFFFFFFFFFCH, CHHH00000000H, D00000000K0KD, D0000000K00KD, D0D00000000LD, DEEEEEEEEEL07, E000E20000007, E00E200000007, EEE0000000027, GGGGGGGGGGGM7, GGGGGGGGGGM07, H00000000CHHH, H00HC0000000H, HFFFFFFFFFFK5, I0A000000000D, J000000000J9J, K000000000095, K000000000M2H, M0000000000M1, M0EEEEEEEEEE7, MHHHHHHHHHHHH, MMNNNNNNNNNNN, MNNNNNNNNNNNN, N000000DDDDDD, N000DDDDDDDDD, NNDDDDDDDDDDD, 22EEEEEEEEEEE7, 35FFFFFFFFFFF5, 400000000000JJ, 800000000000GN, DDDDDDDDDDD077, DDDDDDDDDDDDD7, E0000000000L27, EAAAAAAAAAAKA5, EEG00000000007, H0000000000C6H, I5500000000005, II0000000000I5, M0666666666661, M6MMMMMMMMMMMB, 4JJ00000000000J, 506666666666661, BGMMMMMMMMMMMCB, CFFFFFFFFFFFFFH, D0000000000KD0D, D0HEEEEEEEEEEE7, F0BBBBBBBBBBBBB, HGGGGGGGGGGGGG7, K0000000000000D, K00000000000MCH, M0M6MMMMMMMMMMB, 5DDDDDDDDDDDDDDD, C00000000000008B, D00000000000000H, DEEEEEEEEEEEE0L7, DEEEEEEEEEEEEEL7, EEE2EEEEEEEEEEE7, GM66666666666661, H5FFFFFFFFFFFFF5, IIIIIIIIIIIIIJJJ, BGMMMMMMMMMMMMMMB, DLEEEEEEEEEEEEEE7, H0000000000000CHH, H000000000C0000HH, I000000000000000D, IIIIIIIIIIIIIIIJJ, INNNNNNNNNNNNNNNN, J000000000000009J, M666666666666666B, N0000000000000LLN, N00DDDDDDDDDDDDDD, 355555555555555555, 60000000000000008H, 6M6666666666666661, C000000000000000F1, N0DDDDDDDDDDDDDDDD, 666666666666666666B, 800000000000000000N, AD000000000000000DD, DEEEEEEEEEEEEEEEEE7, I500000000000000005, 20000000000000000027, 4000000000000000003J, 400000000000000000IJ, 99999999999999999995, DD00DDDDDDDDDDDDDDLD, E2EEEEEEEEEEEEEEEEE7, N00000000000000000LN, 500000000066666666661, EE0000000000000000727, GGGGGGGGGGGGGGGGGGG07, H0000000000000000006H, 40000000000000IIIIIIIJ, AD0000000000000000000D, K0000000000000000000M5, CL777777777777777777777, D000000000000000000000N, D0000000000000000000IIN, HHHHHHHHHHHHHHHHHHHHHK5, NDDDDDDDDDDDDDDDDDDDDDD, 1MMMMMMMMMMMMMMMMMMMMMBB, D00DDDDDDDDDDDDDDDDDDDLD, FFFFFFFFFFFFFFFFFFFFFFFH, 4J0000000000000000000000J, 566666666666666666666666G1, EKKAAAAAAAAAAAAAAAAAAAAAA5, 6666666666666666666666666G1, AJJJJJJJJJJJJJJJJJJJJJJJJJJJ, H00000000000000000000000008H, N0000000000000000000000000GN, DD0000000000000000000000000LD, IIIIIIIIIIIIIIIIIIIIIIIIIIIIJ, G0666666666666666666666666666661, GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG7, K000000000000000000000000000000000H, EAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA5, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLM7, M2EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE7, C000000000000000000000000000000000000000001, MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMCB, E00000000000000000000000000000000000000000000727, 777777777777777777777777777777777777777777777777727, EG000000000000000000000000000000000000000000000000000007, D000000000000000000000000000000000000000000000000000000000LD, 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===Base 36=== 11, 15, 17, 1B, 1H, 1N, 1P, 1V, 1Z, 21, 27, 2B, 2H, 2P, 2T, 2V, 2Z, 31, 35, 3J, 3N, 3T, 3V, 45, 47, 4D, 4J, 4N, 4T, 4Z, 51, 5B, 5D, 5H, 5J, 5V, 67, 6B, 6D, 6H, 6N, 6P, 6Z, 75, 7B, 7H, 7J, 7P, 7T, 7V, 85, 8J, 8N, 8P, 8T, 97, 9D, 9N, 9P, 9T, 9Z, A7, AD, AJ, AN, AT, B1, B5, BD, BN, BP, BZ, C1, C7, CB, CH, CP, CT, CV, CZ, DB, DJ, DN, DV, DZ, E5, EH, EJ, F1, F7, FH, FN, FT, FV, G1, GB, GH, GN, GP, GV, H1, H5, H7, HJ, HT, HV, HZ, I5, IB, ID, IP, IT, IZ, J7, JH, JP, JZ, K7, KD, KJ, KN, KV, L1, L5, LD, LH, LV, M5, MH, MJ, MT, MV, MZ, N1, NB, NP, NT, NV, NZ, OD, OH, OJ, ON, P7, PB, PJ, PT, Q1, Q5, QB, QH, QV, QZ, R5, RB, RJ, RP, S1, S5, SB, SD, SN, SP, SV, T5, T7, TH, TJ, TP, U7, UB, UD, UH, UN, UT, V1, V7, VD, VZ, W1, WB, WJ, WT, WZ, X5, XD, XP, XT, XZ, Y5, Y7, YD, YP, YZ, ZH, ZJ, ZN, ZT, ZV, 12D, 16J, 18D, 1CD, 1CJ, 1DD, 1GT, 1JD, 1JJ, 1LT, 1QJ, 1RD, 1RT, 1ST, 1XJ, 1YJ, 1YT, 22D, 22J, 22N, 23D, 255, 2A5, 2CD, 2ED, 2EN, 2F5, 2GD, 2GJ, 2IJ, 2JN, 2LN, 2MN, 2ND, 2O5, 2QD, 2QJ, 2U5, 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MYMU1, MYRK1, MYUU1, MYXR1, N000D, N0055, N006J, N007N, N00CN, N00IH, N00O7, N00S7, N029J, N02DD, N0307, N044H, N049H, N0505, N05M7, N05Q7, N06LJ, N070N, N0A55, N0CEN, N0CNN, N0CYN, N0DMD, N0DRD, N0F05, N0FF5, N0FMD, N0GED, N0HNN, N0I4H, N0IS7, N0IXJ, N0JLN, N0JNN, N0K4H, N0KIH, N0L6J, N0LI7, N0LLN, N0M0N, N0MQ7, N0NEN, N0O07, N0O77, N0Q0N, N0QEN, N0RRH, N0S07, N0SS7, N0U0J, N0ULJ, N0W77, N0WC5, N0WL7, N0XO7, N26JJ, N29XJ, N2CNN, N2CYN, N2MMD, N2N6J, N2N9J, N2NLJ, N2YYN, N30ED, N32RD, N33GD, N33MD, N3GGD, N3IW7, N3MR7, N3MX7, N3RGD, N3RI7, N3W2D, N3W77, N3X07, N3X77, N400H, N409H, N44YH, N494H, N49YH, N4AYH, N50K5, N5307, N5537, N5577, N57Q7, N5FK5, N5KK5, N5Q77, N5SQ7, N69LJ, N6L9J, N6U0J, N700D, N7077, N70EN, N70FD, N73RD, N7777, N777N, N7DMD, N7E2N, N7FED, N7M3D, N7MWD, N7SR7, N7WMD, N92LJ, N9XXJ, NA505, NA5F5, NAFK5, NAK05, NAKF5, NARYH, NAWW5, NC00N, NC0EN, NC0NN, NCE0N, NCN0N, NCNQN, NCW05, ND3GD, ND7DD, NDAW5, NDDGD, NDDRD, NDE3D, NDWM7, NDWW7, NE0GD, NED3D, NEDDD, NEL2N, NEMMD, NEMRD, NF0F5, NFEMD, NFKF5, NFWC5, NFWW5, NGDED, NGW2D, NH00H, NH2NN, NHH0H, NHMCN, NHMMD, NHNNN, NI00H, NI0EN, NI0MN, NI0NH, NI0NJ, NI0XJ, NI44H, NI7R7, NI7S7, NII77, NIIM7, NIIMN, NIIYN, NIM0N, NIN0H, NIN0J, NINJJ, NIW37, NIWM7, NIX77, NIXW7, NJ0EN, NJ0NN, NJ30D, NJ33D, NJFMD, NJIEN, NJJ2N, NJJIN, NJJJJ, NJJJN, NJL0J, NJLNN, NJN0N, NJSJJ, NK005, NK04H, NK3KH, NK5F5, NKA05, NKFK5, NKK55, NKKF5, NL09J, NL2IN, NLCNN, NLII7, NLIIN, NLJLN, NLL2J, NLLNJ, NLLNN, NLNCN, NLNIN, NLNLJ, NLNLN, NM00N, NM73D, NM7DD, NM7RD, NME0N, NMELN, NMLL7, NMLQ7, NMMRD, NMQEN, NMRD7, NMRR7, NMWWD, NN02J, NN0AH, NN0EN, NN30H, NN525, NN6UJ, NN92J, NNA55, NNASH, NNCLN, NNCMN, NNELN, NNF25, NNHHN, NNJ2J, NNJJJ, NNLNN, NNN55, NNN9J, NNNMN, NNNNJ, NNNSH, NNNXJ, NNQYN, NNS3H, NNU9J, NNW55, NNXJJ, NO077, NO0W7, NO777, NOI77, NOIM7, NOWL7, NQ00N, NQ0YN, NQ77N, NQ7EN, NQLLN, NQLNN, NQQQN, NQQYN, NRARH, NRAYH, NRDI7, NRHYH, NRQL7, NRRHH, NRW2D, NS0L7, NSA3H, NSHHH, NSIIJ, NSIJJ, NSS07, NSSJJ, NSSL7, NSSS7, NSULJ, NU00J, NUL6J, NUSIJ, NW0C5, NW3W7, NW5W5, NWI77, NWIM7, NWIO7, NWW55, NWWM7, NX0I7, NX0O7, NX3R7, NX9XJ, NXIM7, NXRL7, NXSSJ, NXWI7, NXWW7, NXXSJ, NY44H, NY94H, NYYIN, O0057, O005T, O0077, O0091, O009B, O00BT, O00EV, O00OV, O00TT, O00WP, O00WV, O040V, O0507, O056T, O065T, O08AV, O08E7, O0901, O09AB, O09I1, O09K1, O09WV, O0A4V, O0AI1, O0E9V, O0EAB, O0EAV, O0EFP, O0EKB, O0EOB, O0EU1, O0FMP, O0K6T, O0K91, O0MM1, O0O9B, O0OAB, O0OI1, O0OOB, O0OT1, O0OUV, O0OW7, O0OYT, O0T8V, O0TOT, O0TOV, O0TTV, O0TUV, O0TX1, O0U91, O0UE1, O0W87, O0WOV, O0YOT, O100T, O1EET, O4PAV, O4U0V, O5005, O50EP, O50I7, O50K5, O555P, O55FP, O55UP, O5EFP, O5I07, O5K05, O5KK5, O5OYT, O6LGT, O77E7, O7Q77, O80E7, O8AAZ, O8AFZ, O8AOV, O8FFZ, O8FKB, O8FZB, O8IM7, O8ZRZ, O8ZZB, O900V, O90WV, O96OV, O99LB, O9K01, O9OU1, O9R6V, O9U01, O9UI1, OA091, OA0O1, OA4AV, OA4IV, OA68V, OA8AZ, OA9I1, OAA4V, OAAFZ, OAAO1, OAAOV, OAAZZ, OAFAZ, OAFYB, OAI0V, OAI91, OAOAV, OAOE1, OAOU1, OAUE1, OAUI1, OAY01, OAY0B, OAYI1, OAZZZ, OB66T, OBE0T, OBEET, OE0BT, OE0FB, OE4AV, OE6ET, OE8AV, OE9EB, OEA8V, OEAA1, OEAFB, OEBET, OEBI7, OEEO7, OEFFP, OEFLP, OEKA1, OEKAB, OEKU1, OELMP, OEML7, OEO07, OEO0B, OEO4V, OEO6V, OEOAV, OEOO7, OEOOB, OEOVB, OEPU1, OERET, OERRV, OETY1, OEUK1, OEUU1, OEV9B, OEVFP, OEW8V, OEWW7, OF0FP, OF0OB, OFFFP, OFFMP, OFFZP, OFFZZ, OFMMP, OFMWP, OFOQP, OFPFZ, OFPUZ, OFZ0P, OFZFZ, OFZOB, OFZZZ, OG065, OG0YT, OI0EV, OI7Q7, OI9WV, OIIE7, OIQ77, OIQW7, OIR6V, OIYO1, OK8KB, OKAE1, OKAY1, OKE6T, OKEU1, OKFYB, OKIY1, OKYA1, OKYI1, OL08B, OL4EP, OL777, OL8E7, OLBI7, OLEL7, OLFOP, OLGYT, OLIW7, OLLSZ, OLLZZ, OLOUP, OLR6T, OLUMP, OMAU1, OMOU1, OMXM1, OO0AB, OO0I7, OO0T1, OO1ET, OO4AV, OO4MP, OO4QP, OO6KT, OO6LT, OO6OV, OO7O7, OOA0B, OOAIV, OOAOV, OOBGT, OOEMP, OOERT, OOET1, OOEU1, OOEY1, OOGTT, OOI4V, OOIRV, OOKET, OOLBT, OOMMP, OOMQ7, OOMU1, OOMWP, OOO0B, OOO6T, OOO77, OOOE7, OOOKT, OOOOV, OOORT, OOOW7, OOPY1, OOQ77, OOT01, OOT91, OOTAV, OOTOV, OOTUV, OOU91, OOUUV, OOWIV, OOWMP, OOWVP, OOY01, OOY0B, OOYKB, OOYU1, OOYYB, OP0EV, OP44V, OP8FZ, OPA91, OPAAZ, OPAOV, OPEAV, OPI4V, OPO4V, OPOIV, OPOUV, OPOY1, OPUAZ, OPUEV, OPWUV, OPYU1, OQ7W7, OQ8E7, OQLE7, OQQL7, OR8AZ, ORA8Z, ORKLT, ORS8Z, OS8RZ, OSR8Z, OSS8Z, OT09V, OT0K1, OT66T, OTT8V, OTUOV, OU091, OU0E1, OU0MP, OU4LP, OUAAZ, OUASZ, OUL4P, OUP8Z, OUP91, OUPAZ, OUPY1, OUU0P, OUUPZ, OUYM1, OUZ91, OV0FB, OV0LB, OV54P, OV5WP, OV9K5, OVALB, OVB6T, OVKFB, OVLEB, OVLFP, OVQWP, OW0EV, OW7O7, OW7Q7, OW7W7, OW90V, OWEEP, OWIWV, OWLW7, OWOM7, OWOVP, OWQ87, OWU0V, OWW9V, OWWL7, OWWOV, OXAA1, OXU01, OXUI1, OXXI1, OXXX1, OY00T, OY0E1, OY0RT, OY0U1, OY0YB, OY98B, OYE0T, OYFKB, OYKFB, OYLYB, OYYYB, OZ0OB, OZ0WP, OZ0YB, OZ9EB, OZ9K1, OZ9O1, OZALB, OZAYB, OZEAB, OZEOB, OZFEB, OZFFP, OZFFZ, OZFRZ, OZFZP, OZO0B, OZO91, OZOYB, OZPRZ, OZR8Z, OZUUP, OZZYB, P00W5, P02G5, P049V, P04YV, P0A0V, P0AW5, P0CC5, P0CU5, P0E9V, P0EWV, P0F25, P0FC5, P0G25, P0K3Z, P0KOZ, P0KRH, P0O55, P0OEV, P0PG5, P0PO5, P0PW5, P0R0H, P0ROZ, P0U25, P0UIV, P0UVP, P0V0P, P0VKP, P0W9V, P0YIV, P13WD, P22K5, P2D95, P2NC5, P33ZZ, P38DD, P3GGD, P3SAH, P409V, P444V, P44O1, P44Y1, P4AIV, P4AYV, P4E0V, P4OA1, P4OK1, P4OO1, P4U81, P4UEV, P4UR1, P55EP, P5GZ5, P6A41, P803H, P80AZ, P838D, P8AAZ, P8D8D, P8FKZ, P8K0Z, P8KKZ, P8KXH, P8OE1, P8UE1, P8X0H, P8X3H, P8XKH, P909V, P9941, P99EV, P99U1, P9E9V, P9K91, P9OE1, P9OK1, P9W9V, P9X9H, PA00H, PA441, PA5G5, PA8FZ, PA9O1, PAA8Z, PAI4V, PAI9V, PAMKP, PAOEV, PAR8H, PAUM1, PAVKP, PAZEP, PC0W5, PCAG5, PCAU5, PCCA5, PCWN5, PD0QP, PD0SH, PD33D, PD3AH, PD3MD, PDGC5, PDK3H, PDK8H, PDSAH, PDUAP, PE99V, PEA91, PEAEP, PEEA1, PEEE1, PEER1, PEI9V, PEK61, PEQQP, PERE1, PERK1, PEUY1, PEY0V, PF2K5, PF3CD, PF83D, PFC2D, PFCWD, PFDAP, PFEKP, PFMCD, PFMMD, PFO55, PFR2D, PFZEP, PG295, PG2WD, PG8CD, PGCD5, PGDC5, PGF5Z, PGG8D, PGKF5, PGKK5, PGO95, PGW8D, PH2MD, PHD8D, PK0O5, PK441, PK53Z, PK5GZ, PK83H, PK8KZ, PKAA1, PKAU1, PKKO5, PKP3Z, PKRXH, PKSIH, PMAA1, PMAAP, PMAU1, PMF3D, PMFCD, PMMAP, PMMMD, PMMMP, PMMPD, PMRR1, PMUO1, PN2D5, PN2W5, PNA3H, PNAF5, PNFK5, PNHFD, PNK55, PNKF5, PNSKH, PO09V, PO0EV, PO4AV, PO90V, POAY1, POI4V, POOE1, POUEV, POUY1, POWUV, POWWV, POYA1, POZAZ, POZFZ, PP0QP, PP2CN, PP2MD, PP33D, PP595, PP5W5, PP8CD, PPCNN, PPCRD, PPCU5, PPCW5, PPCWD, PPD05, PPDUP, PPECN, PPF8D, PPFC5, PPFW5, PPGC5, PPGF5, PPGFZ, PPKGZ, PPMCD, PPMCN, PPMEN, PPMND, PPNF5, PPNMN, PPNWD, PPP3Z, PPP95, PPPC5, PPPRZ, PPPZD, PPQKP, PPS0Z, PPW2N, PPWND, PPWWN, PPZC5, PPZSZ, PQQQP, PR1GD, PR8AH, PRA0H, PREK1, PRKO1, PRMM1, PRNFD, PRRM1, PRRRH, PRW2N, PS0ZZ, PS80H, PS80Z, PS8AH, PS8ZZ, PSIKH, PSR3Z, PU025, PU0IV, PU49V, PU4YV, PU8AZ, PU9EV, PUE91, PUE9V, PUI9V, PURK1, PUUM1, PUUR1, PUVAP, PUYIV, PUYO1, PUZMP, PV00P, PV2CN, PV2RN, PVAKP, PVFAP, PVFFP, PVFKP, PVK0P, PVKAP, PVNCN, PVNEN, PW005, PW0C5, PWCWD, PWF8D, PWNC5, PWO9V, PWOWV, PWWOV, PWWWN, PWWWV, PX0IH, PXAIH, PXIAH, PXIKH, PYKE1, PYKO1, PYRK1, PYRM1, PZ0G5, PZ1WD, PZ295, PZ2C5, PZ2G5, PZ481, PZ641, PZE61, PZEAP, PZG25, PZK61, PZKOZ, PZOFZ, PZOO1, PZR41, PZRZZ, PZS0Z, PZSUZ, PZUMP, Q00DT, Q00JD, Q00QT, Q00YN, Q020J, Q03RD, Q0CQN, Q0DRD, Q0FRD, Q0JTT, Q0Q3D, Q0QRN, Q0QXN, Q0XEN, Q0XIJ, Q0XLN, Q0XQN, Q0YYN, Q20RD, Q2D0D, Q32WD, Q3D3D, Q4EKP, Q4PQP, Q7777, Q77Q7, Q7ER7, Q7EW7, Q7LR7, Q7LS7, Q7LW7, Q7QQN, Q7QR7, Q7QYN, Q7RE7, Q7SS7, QAAPP, QAOOP, QAOWP, QAUAP, QAUWP, QAWEP, QAWLP, QCEQN, QCNYN, QCQEN, QD0GD, QD0RD, QD20D, QD33D, QD3DD, QDAWP, QDRET, QDWKP, QE7QN, QECLN, QEEL7, QEES7, QELCN, QEQCN, QEQTT, QERE7, QESR7, QF2WD, QFGUJ, QFR2D, QG00J, QG0IJ, QG7Q7, QG7R7, QGDDD, QGGGD, QGGQD, QGJ2J, QGMQ7, QGOO7, QGQQ7, QGRR7, QJ02J, QJ03D, QJ0SJ, QJ33D, QJ3WD, QJD0D, QJI0J, QK44P, QKPKP, QL7W7, QLES7, QLLL7, QLNNN, QLQ77, QLQL7, QLQM7, QLQPN, QLR87, QLSRT, QMLR7, QMQQ7, QN0YN, QN70N, QNCYN, QNLW7, QNNQN, QNNYN, QNQNN, QO777, QO7E7, QO8M7, QOUWP, QP3RD, QPDAP, QPEEP, QPFRD, QPGGD, QPPQP, QPR2D, QPRGD, QQ0RN, QQ0YN, QQ4LP, QQ4QP, QQAEP, QQCYN, QQELT, QQEPN, QQEQP, QQGDD, QQGGD, QQGO7, QQGQD, QQKKP, QQLAP, QQLPN, QQLRT, QQMR7, QQN77, QQNNN, QQNQN, QQPDD, QQPEN, QQPGD, QQPQP, QQPWD, QQQ0N, QQQ3D, QQQ87, QQQKP, QQQL7, QQQM7, QQQPD, QQQQN, QQQWD, QQRL7, QQRN7, QQRYN, QQSST, QQTDT, QQTQT, QQU4P, QQUAP, QQUUP, QQWPD, QQXEN, QQXQN, QQYXN, QR00N, QR0XN, QR2FD, QR32D, QREKT, QRN0N, QRNL7, QRQQD, QRRE7, QRXXN, QSES7, QSS2J, QSSE7, QSSS7, QSSSJ, QTDET, QTQ2D, QTQLT, QUAAP, QUAWP, QUPPP, QUUAP, QW7G7, QWG77, QWGM7, QWLEP, QWOEP, QWOLP, QX77N, QX7EN, QXELN, QXEQN, QXLLN, QXLQN, QXNQN, QXOE7, QXQXN, QXXNN, R0061, R008D, R00IH, R00NN, R00XN, R013D, R02YN, R030D, R04R1, R06E1, R06ET, R07M1, R08AH, R0A8H, R0AYH, R0D07, R0DDT, R0EK1, R0EYT, R0F3D, R0G8D, R0GDD, R0GGD, R0GQ7, R0H0N, R0KE1, R0LQ7, R0N0N, R0RYN, R0T2N, R0TLT, R0TNN, R0TYN, R0X0N, R0YST, R100T, R130D, R13WD, R166T, R1E0T, R1F3D, R1GFD, R1TGD, R200N, R28WD, R2W8D, R2WFD, R32DD, R330D, R332D, R3W8D, R44XV, R4AAV, R4AYV, R4MO1, R4OI1, R4RO1, R4U4V, R4X9V, R6001, R60K1, R66LT, R66YV, R6E01, R6EE1, R6MK1, R6SST, R6X01, R6X9V, R6XE1, R6XYV, R6Y6V, R7OO1, R7XE1, R80G7, R88FD, R8F8D, R8IE7, R8X8H, R9OAV, R9UVV, R9VOV, R9X6V, RA08H, RAA9V, RAS8Z, RAX0H, RD00D, RD8I7, RD8IH, RDA0H, RDDQT, RDQDT, RE061, RE6KT, RE6LT, REEK1, REEKT, RER01, RERK1, RERR1, RF0DD, RFD0D, RG28D, RGGFD, RGGWD, RGRM7, RGWQD, RH00H, RH0EN, RH8HH, RH9HH, RH9XH, RHH9H, RHHIN, RHHNN, RHIWN, RHNEN, RHWIN, RHWWN, RHX8H, RHXEN, RHY9H, RI00H, RI2IN, RIHEN, RIIWN, RIMRN, RK00T, RKEE1, RKELT, RKL0T, RKQLT, RLEI7, RLQM7, RLR07, RLRE7, RLSSZ, RLSYT, RLZE7, RLZI7, RLZLZ, RM0G7, RM0Q7, RMIWN, RMKO1, RMLQ7, RMOX1, RMR07, RMRR1, RN2YN, RN30D, RND3D, RNI2N, RNIMN, RNMEN, RNW2D, RO0TT, RO66T, RO6KT, ROAAV, ROOKT, ROOOT, ROU9V, ROXM1, ROZK1, RQK0T, RQLKT, RQLST, RQMR7, RQQLT, RQQQT, RQQWD, RQRE7, RQTQD, RR44V, RR4YV, RR601, RR66V, RR6WV, RR9HH, RR9UV, RRA9V, RRAYV, RRE01, RREE1, RRGM7, RRGR7, RRI2N, RRIIN, RRIN7, RRIXN, RRKE1, RRL07, RRLM7, RRM07, RRMG7, RRML7, RRMM1, RRMO1, RRMWN, RRNR7, RRO01, RROI1, RROK1, RRR0N, RRRE1, RRRG7, RRRIN, RRRL7, RRRM1, RRRO1, RRROV, RRRWV, RRU6V, RRV9H, RRVUV, RRVYH, RRWWV, RRXRN, RRY4V, RRY9H, RRYHH, RRYIN, RRYRN, RS8FZ, RSSSZ, RSSYT, RSYST, RT3ZZ, 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NKIKKKKH, NKKKKK3H, NN0000YN, NN000LNJ, NN000NUJ, NN00QLQN, NN0I0I0J, NN0N00LN, NN0NNNQN, NN0NQQNN, NN0NYYYN, NNA0000H, NNARRRRH, NNN0000H, NNN0NQQN, NNNARRRH, NNNN040H, NNNN0NIN, NNNNNNIN, NR000007, NRRRRRM7, NSJSSSSJ, NWWWWWW7, O000008B, O00000YB, O0000ALB, O0000KE1, O0000KK5, O0000MO1, O0000OKB, O0000OO7, O0008KYB, O0008YYB, O000AAM1, O000E0E7, O000EEE7, O000EMMP, O000FKYB, O000KYFB, O000MAA1, O000MEEP, O000OKYB, O000OUM1, O000P4AV, O000TKU1, O000UUM1, O000XA01, O000YY8B, O0080007, O00E0EE7, O00EF0EB, O00EMMEP, O00MUUU1, O00OEE07, O00OUUU1, O00UUA9V, O0777777, O07OIII7, O0F0E00B, O0I0III7, O0K00EA1, O0OOOO0T, O0OUUUU1, O0TKAAU1, O0U0U0M1, O40000UV, O400UUAV, O444444V, O55555K5, O90000U1, O99999I1, OA00008V, OA000YFB, OAAA090V, OAAUYUU1, OAMUUUM1, OE0A000B, OEE00EE7, OFFFFUUZ, OK000EA1, OKKK0005, OLZZZZLZ, OMMMMEWP, OO00UU01, OO0WOOO7, OO0WWOO7, OOO0OOOT, OOOOE00T, OOOOO0TT, OOOOOO87, OOOOOO9B, OOOOOOOB, OOOOOY9B, OTEEEEET, OTUUU001, OUUUUUSZ, OUUUUUUP, OUYUUUU1, OV0000LP, OVF0000B, OWWW000V, OZ00EF0B, OZLZZZLZ, OZZAAA8Z, OZZZZ9AB, OZZZZPFZ, P00000IV, P00000RH, P000099V, P0000GF5, P0000O9V, P0000RFZ, P0000XRH, P000AGF5, P000PFQP, P000S0IH, P000WWIV, P000X08H, P000ZQAP, P00999AV, P00AXRRH, P00PRFZZ, P08000XH, P083000H, P0OFFFFZ, P0P5553Z, P0QAAAAP, P0Z0QAAP, P0ZZZOUZ, P0ZZZRFZ, P44000UV, P4AYUUU1, P4UUUUY1, P5AAAAW5, P80000XH, P800A0XH, P9555555, P999999H, P9UUUEU1, PF000Z25, PFDUUUUP, PFF00Z25, PMUUUYU1, PN000AK5, PNFFFFF5, PNKKKKA5, POUUUUE1, PP0ZZZG5, PPPF0FO5, PPPN00A5, PPPN5005, PPPP0AG5, PPPPFPQP, PPPPN005, PPPPPRWN, PPPPR8FD, PSZZZZ0Z, PSZZZZZZ, PU999991, PUA8ZZZZ, PUAAAMUP, PUUU2KK5, PUUU4UO1, PUUUEEU1, PYAUUUO1, PZZZ0025, PZZZZ595, PZZZZGK5, PZZZZZSZ, Q0000DGD, Q0000FXJ, Q0000GDD, Q0000RET, Q0000XSJ, Q0002XXJ, Q000I0XJ, Q00F000D, Q00Q00LN, Q00Q0CLN, Q00Q0ECN, Q00Q0GFD, Q00QE0CN, Q00QQ0GD, Q0GQQQQD, Q0JQQ0QD, Q0Q00DGD, Q0Q00QFD, Q0Q0Q0FD, Q0QQQ0FD, Q0QQQF0D, Q0UI000J, Q2FXXXXJ, Q33DDDRD, Q7000QEN, QAAAAKKP, QJ0QQQQD, QJQ00QQD, QJQQ00QD, QJQQ0QQD, QLEEEEW7, QLQQQQQP, QLSSSSST, QPAPAAAP, QQ0000GD, QQ0000XN, QQ003DDD, QQ0Q00GD, QQ0Q0GFD, QQAAAAKP, QQPPAAAP, QQQLLNCN, QQQPAPPP, QQQQ7WO7, QQQQDUWP, QQQQLQQP, QQQQQEEP, QQQQQGQ7, QQQQQQG7, QSSSSSRT, QXNNNNNN, QXOOWOO7, QXXXJXXJ, QXXXXJXJ, R00008G7, R0000H2N, R0000QL7, R0000R2N, R0000YRN, R000EETT, R000ETET, R000RMK1, R00666ST, R00E006T, R00I0007, R00N00FD, R00R0MK1, R00RRRXH, R00Y0L0T, R0E0066T, R0HXXXXN, R0RR888H, R0TT000T, R0Y0L00T, R0YL000T, R2F0000D, REEE00R1, REKEEEYT, RERREEE7, RGQQQQGD, RHEXXXXN, RIRWWWWN, RIWWWWIN, RN0000EN, RN000II7, RN0YYYYN, RNIIIIIN, RNRRIIR7, RNYYYYYN, RR0000E1, RR000MR1, RR00R001, RR0R00K1, RR0RMREN, RREREEE7, RRHHXYYH, RRIWWWWN, RRNIIII7, RRREERE7, RRRHHHHH, RRRR00M7, RRRRR0R1, RRRRRRE7, RRRRRRI7, RRRRRRUV, RRRRRRYV, RRRRRUUV, RRRRUUUV, RRRRYR6V, RVVWWWWV, RWWIIIIN, RY00SLST, RY0L000T, RYHHHHHH, RYLS00ST, S000003H, S000030H, S00004AH, S00073EZ, S0007L3Z, S000I40H, S0073EEZ, S008KKKH, S00L00RZ, S00OSSSZ, S00S0OSZ, S00SSOSZ, S07OSSSZ, S0KI000H, S0SRSS6T, S0SSRS6T, S0SSS7OZ, S0Y0KSST, S388888H, S7SSK00Z, S7SZZZR7, S800000H, S80003WH, S8K00K0H, S8K00KKH, S8K3000H, S8KK00KH, S8KK0K0H, S8KKK30H, S9JJJJJJ, S9SSSSGJ, SA30000H, SF00000Z, SFZ0ZZZZ, SFZZZZ0Z, SHHHHH8H, SK000IKH, SK88KKKH, SK8K300H, SKI0K0KH, SKKIK0KH, SKKK800H, SKKKKIKH, SLL7UZZZ, SLSG9SSJ, SLSSLS37, SOSSSSSZ, SRSSSS6T, SS0SS0OZ, SS7778EZ, SS7ZZZR7, SS8E000Z, SSE0OLLZ, SSIJJJJJ, SSLLSL37, SSLLSS37, SSS00SOZ, SSS07OSZ, SSS6RSST, SSS7SSKZ, SSSK000Z, SSSK00OZ, SSSLS3E7, SSSLSL37, SSSLTSOZ, SSSS7LOZ, SSSSL307, SSSSLL37, SSSSRL07, SSSSS7KZ, SSSSSIGJ, SSSSSR07, SSSSSSKZ, SSSZ7SR7, SSSZK00Z, SSUZZZZZ, SSZQEEE7, SU0ZZZ0Z, SYSSSSKT, SZKK000Z, SZQEEEE7, SZZKK00Z, SZZSZSR7, SZZSZZRZ, SZZZ7ZR7, SZZZZSRZ, SZZZZZUZ, T00000CD, T000039B, T0000AOV, T0000UVV, T0000XX1, T000380D, T0008UUV, T0008VAV, T0008VUV, T00099AB, T000DD3D, T000E03D, T000K3KB, T000TTST, T000VU0V, T008VUAV, T00909AB, T00TT0ST, T09A000B, T0A0003B, T0DDMMMD, T0E000A1, T0MDMMMD, T0T0T0ST, T0Y0K0U1, T0YK00U1, T0YUUUU1, T900009B, T900K0A1, T9A0000B, TE0U0U01, TEEEELLZ, TEGGGGGD, TEUUU001, TEZEEEEZ, TK0Y00U1, TK900001, TKKKK9AB, TOOOO0TT, TR00TTTT, TRL0000T, TRRRRUUV, TRRRUUAV, TTGGGG8D, TTGGGGGD, TTR0TT0T, TTTOO0TT, TTTTGG8D, TTTTO00T, TTTTS00T, TTTTTO9V, TTTTTVUV, TTTZC88D, TU0Y0001, TUE0UUU1, TUUU0041, TUUUUU41, TUUUYUU1, TUYUUUU1, TXXXXXYN, TYAUUUU1, TYYYYYEN, TZM0000D, U00001LJ, U0000665, U00006A5, U0000E91, U0000MMP, U0000ULP, U0004091, U0005ULP, U00094U1, U0009991, U00099U1, U000A0LP, U000EUM1, U000J991, U000K4O1, U000LFUP, U000O0I1, U000O991, U000U4O1, U000U941, U000ULFP, U000ULUP, U000Y041, U000ZZ4P, U0099091, U0099401, U0099UU1, U009KO91, U009OKU1, U00AA0LP, U00F0525, U00FFFWP, U00J4001, U00J40K1, U00KU4O1, U00MUUUP, U00O00I1, U00O0991, U00U00P1, U00U0MO1, U00U0UP1, U00U5ULP, U00UALUP, U00UK4O1, U00ULFUP, U00UUALP, U00UUYO1, U00VLL6J, U00WMAAP, U00XXX81, U00Y04O1, U00Z4KPP, U049U001, U0A000LP, U0AAK8AZ, U0AK8A0Z, U0I0000J, U0J40KU1, U0OU90K1, U0U00605, U0U0LFUP, U0U0UMO1, U0U0UYO1, U0U994U1, U0U9UU91, U0UE99U1, U0UE9UU1, U0UKU4O1, U0UU9091, U0UU94U1, U0UUU941, U0UY0UO1, U0XXX081, U0Y4UUU1, U0YUU0O1, U0ZZ4PKP, U5UU2225, U60000GJ, U6FFFF05, U8ZZ000Z, U90000E1, U90004U1, U9400001, U999UU01, U9O999U1, UA08ZZZZ, UA0AAZ8Z, UAAAA80Z, UAAAAMMP, UAAK08AZ, UAZAA8AZ, UAZZAA4P, UE000Y01, UEE00UY1, UEUUUUE1, UF000F25, UFFF0LMP, UFFFFFMP, UFFFMMMP, UFFMMLMP, UI00000J, UJIUUUU1, UJSSSSSJ, UK8Z000Z, UK8ZZ00Z, UKZZZ08Z, UL9SSSSJ, ULG0000J, UMMMAALP, UMOUUUU1, UPUMUUU1, UPUUEUE1, UU0006F5, UU009KE1, UU00UY9V, UU0U04EV, UU0U0OEV, UU0U0YVV, UU0UUYO1, UU0UY0O1, UU0UY0VV, UU5U2KK5, UU5UUU25, UU60FFF5, UUA009EV, UUF55525, UUPEUUE1, UUPMUUU1, UUU00E4V, UUU0U0SZ, UUU400EV, UUU904U1, UUUPUEE1, UUUPUMU1, UUUU22K5, UUUU4MM1, UUUULFUP, UUUUMOY1, UUUUMYU1, UUUUU225, UUUUU991, UUUUUUSZ, UUUUWKAP, UUUUYMU1, UUUY009V, UUY0UO01, UVPPPPPP, UVV0VVVV, UVVVVVPV, UXRRRR01, UYU0UUO1, V00000AP, V00000VB, V00000WP, V0000965, V00009FB, V0000E3B, V0000QOP, V0000ULJ, V0000V8B, V0000YOT, V0006LYJ, V000AO4P, V000APPP, V000E44B, V000E94B, V000EXXB, V000VA3B, V000VV8V, ====Additional known quasi-minimal primes (not necessarily the next)==== P₈₁₉₉₃SZ ==Unsolved families== Families for which not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers > base (''b'')). {|class=wikitable |base (''b'')||unsolved family (base-''b'' form)||unsolved family (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||current search limit of length||factorization of numbers in this family |- |13||9{5}||(113×13^''n''−5)/12||88000||[http://factordb.com/index.php?query=%28113*13%5En-5%29%2F12&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |13||A{3}A||(41×13^''n''+1+27)/4||82000||[http://factordb.com/index.php?query=%2841*13%5E%28n%2B1%29%2B27%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |16||{3}AF||(16^''n''+2+619)/5||76000||[http://factordb.com/index.php?query=%2816%5E%28n%2B2%29%2B619%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |} (If these three families contain primes (and they are excepted to contain primes), then the smallest prime in families 9{5} and A{3}A in base ''b'' = 13 will be index 3196 and 3197 quasi-minimal prime in base ''b'' = 13, and the smallest prime in families {3}AF in base ''b'' = 16 will be index 2347 quasi-minimal prime in base ''b'' = 16) === Base 17 === * 15{0}D * 1{7} * 1F{0}7 * 4{7}A * 51{0}D * 70F{0}D * 8{B}9 * 9{5}9 * 95{F} * A{D}F * B{0}B3 * B{0}DB * {B}2BE * {B}2E * {B}E9 * {B}EE * D0G{D} * E9{B} * F1{9} * FD0{D} * G{7}F === Base 21 === * 2{7}9D * 2F{C}A * 4{3}B * 5{0}DJ * {5}FEK * {7}ID * 99{0}99H * {9}0D * {9}D * B0{H}6H * B3{0}EB * B9{0}E5 * B{D}B * B{H}6H * DH{D} * F{9}D * {F}35 * G{0}FK * H{0}7771 * H{D} * {J}6J === Base 36 === * 7{K}Z * B{0}EUV * HM{0}N * N{0}YYN * O{L}Z * S{0}8H ==Primality certificates for the proven primes > 10²⁹⁹== See also: [[w:Primality certificate|Primality certificate]] and [[w:Elliptic curve primality|Elliptic curve primality]] {|class=wikitable |base (''b'')||index of this quasi-minimal prime in base ''b''||quasi-minimal prime (base-''b'' form)||quasi-minimal prime (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||factordb entry of this prime||primality certificate of this prime |- |9||149||76₃₂₉2||(31×9³³⁰−19)/4||[http://factordb.com/index.php?id=1100000002359003642]||[http://factordb.com/cert.php?id=1100000002359003642] |- |9||150||27₆₈₆07||(23×9⁶⁸⁸−511)/8||[http://factordb.com/index.php?id=1100000002495467486]||[http://factordb.com/cert.php?id=1100000002495467486] |- |9||151||30₁₁₅₈11||3×9¹¹⁶⁰+10||[http://factordb.com/index.php?id=1100000002376318423]||[http://factordb.com/cert.php?id=1100000002376318423] |- |11||1065||A₇₁₃58||11⁷¹⁵−58||[http://factordb.com/index.php?id=1100000003576826487]||[http://factordb.com/cert.php?id=1100000003576826487] |- |11||1066||7₇₅₉44||(7×11⁷⁶¹−367)/10||[http://factordb.com/index.php?id=1100000002505568840]||[http://factordb.com/cert.php?id=1100000002505568840] |- |11||1067||557₁₀₁₁||(607×11¹⁰¹¹−7)/10||[http://factordb.com/index.php?id=1100000002361376522]||[http://factordb.com/cert.php?id=1100000002361376522] |- |13||3165||50₂₇₀44||5×13²⁷²+56||[http://factordb.com/index.php?id=1100000002632397005]||[http://factordb.com/cert.php?id=1100000002632397005] |- |13||3166||9₂₇₁095||(3×13²⁷⁴−6103)/4||[http://factordb.com/index.php?id=1100000003590431654]||[http://factordb.com/cert.php?id=1100000003590431654] |- |13||3167||10₂₈₆7771||13²⁹⁰+16654||[http://factordb.com/index.php?id=1100000003590431633]||[http://factordb.com/cert.php?id=1100000003590431633] |- |13||3168||9₃₀₈1||(3×13³⁰⁹−35)/4||[http://factordb.com/index.php?id=1100000000840126705]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=308&c0=-&EN= 13³⁰⁸−1] |- |13||3169||B₃₄₁C4||(11×13³⁴³+61)/12||[http://factordb.com/index.php?id=1100000003590431618]||[http://factordb.com/cert.php?id=1100000003590431618] |- |13||3170||8B₃₄₃||(107×13³⁴³−11)/12||[http://factordb.com/index.php?id=1100000002321018736]||[http://factordb.com/cert.php?id=1100000002321018736] |- |13||3171||710₃₇₁111||92×13³⁷⁴+183||[http://factordb.com/index.php?id=1100000003590431609]||[http://factordb.com/cert.php?id=1100000003590431609] |- |13||3172||75₃₇₅7||(89×13³⁷⁶+19)/12||[http://factordb.com/index.php?id=1100000003590431596]||[http://factordb.com/cert.php?id=1100000003590431596] |- |13||3173||9B0₃₉₁9||128×13³⁹²+9||[http://factordb.com/index.php?id=1100000002632396790]||[http://factordb.com/cert.php?id=1100000002632396790] |- |13||3174||7B0B₃₉₇||(15923×13³⁹⁷−11)/12||[http://factordb.com/index.php?id=1100000003590431574]||[http://factordb.com/cert.php?id=1100000003590431574] |- |13||3175||10₄₁₄93||13⁴¹⁶+120||[http://factordb.com/index.php?id=1100000002523249240]||[http://factordb.com/cert.php?id=1100000002523249240] |- |13||3176||81010₄₁₅1||17746×13⁴¹⁶+1||[http://factordb.com/index.php?id=1100000003590431555]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3177||8110₄₃₅1||1366×13⁴³⁶+1||[http://factordb.com/index.php?id=1100000002373259109]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3178||B7₄₈₆||(139×13⁴⁸⁶−7)/12||[http://factordb.com/index.php?id=1100000002321015892]||[http://factordb.com/cert.php?id=1100000002321015892] |- |13||3179||B₅₆₃C||(11×13⁵⁶⁴+1)/12||[http://factordb.com/index.php?id=1100000000000217927]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=564&c0=-&EN= 13⁵⁶⁴−1] |- |13||3180||1B₅₇₆||(23×13⁵⁷⁶−11)/12||[http://factordb.com/index.php?id=1100000002321021456]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=576&c0=-&EN= 13⁵⁷⁶−1] |- |13||3181||80₆₉₃87||8×13⁶⁹⁵+111||[http://factordb.com/index.php?id=1100000002615636527]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000002615636532], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000002615636532] |- |13||3182||CC5₇₁₃||(2021×13⁷¹³−5)/12||[http://factordb.com/index.php?id=1100000002615627353]||[http://factordb.com/cert.php?id=1100000002615627353] |- |13||3183||B₈₃₄74||(11×13⁸³⁶−719)/12||[http://factordb.com/index.php?id=1100000003590430871]||[http://factordb.com/cert.php?id=1100000003590430871] |- |13||3184||9₉₆₈B||(3×13⁹⁶⁹+5)/4||[http://factordb.com/index.php?id=1100000000258566244]||[http://factordb.com/cert.php?id=1100000000258566244] |- |13||3185||10₁₂₉₅181||13¹²⁹⁸+274||[http://factordb.com/index.php?id=1100000002615445013]||[http://factordb.com/cert.php?id=1100000002615445013] |- |13||3186||9₁₃₆₂5||(3×13¹³⁶³−19)/4||[http://factordb.com/index.php?id=1100000002321017776]||[http://factordb.com/cert.php?id=1100000002321017776] |- |13||3187||7₁₅₀₄1||(7×13¹⁵⁰⁵−79)/12||[http://factordb.com/index.php?id=1100000002320890755]||[http://factordb.com/cert.php?id=1100000002320890755] |- |13||3188||930₁₅₅₁1||120×13¹⁵⁵²+1||[http://factordb.com/index.php?id=1100000000765961452]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3189||720₂₂₉₇2||93×13²²⁹⁸+2||[http://factordb.com/index.php?id=1100000002632396910]||[http://factordb.com/cert.php?id=1100000002632396910] |- |13||3190||1770₂₇₀₃17||267×13²⁷⁰⁵+20||[http://factordb.com/index.php?id=1100000003590430825]||[http://factordb.com/cert.php?id=1100000003590430825] |- |13||3191||390₆₂₆₆1||48×13⁶²⁶⁷+1||[http://factordb.com/index.php?id=1100000000765961441]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3192||B0₆₅₄₀BBA||11×13⁶⁵⁴³+2012||[http://factordb.com/index.php?id=1100000002616382906]||[http://factordb.com/cert.php?id=1100000002616382906] |- |13||3193||C₁₀₆₃₁92||13¹⁰⁶³³−50||[http://factordb.com/index.php?id=1100000003590493750]||[http://factordb.com/cert.php?id=1100000003590493750] |- |14||649||34D₇₀₈||47×14⁷⁰⁸−1||[http://factordb.com/index.php?id=1100000001540144903]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |14||650||4D₁₉₆₉₈||5×14¹⁹⁶⁹⁸−1||[http://factordb.com/index.php?id=1100000000884560233]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |16||2328||880₂₄₆7||136×16²⁴⁷+7||[http://factordb.com/index.php?id=1100000002468140199]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, and this prime factor is < 10²⁹⁹ |- |16||2329||D4₂₆₃D||(199×16²⁶⁴+131)/15||[http://factordb.com/index.php?id=1100000002468170238]||[http://factordb.com/cert.php?id=1100000002468170238] |- |16||2330||E0₂₆₁4DD||14×16²⁶⁴+1245||[http://factordb.com/index.php?id=1100000003588388352]||[http://factordb.com/cert.php?id=1100000003588388352] |- |16||2331||8C0₂₉₀ED||140×16²⁹²+237||[http://factordb.com/index.php?id=1100000003588388307]||[http://factordb.com/cert.php?id=1100000003588388307] |- |16||2332||DA₃₀₅5||(41×16³⁰⁶−17)/3||[http://factordb.com/index.php?id=1100000003588388284]||[http://factordb.com/cert.php?id=1100000003588388284] |- |16||2333||CE80₄₂₂D||3304×16⁴²³+13||[http://factordb.com/index.php?id=1100000003588388257]||[http://factordb.com/cert.php?id=1100000003588388257] |- |16||2334||5F₅₄₄6F||6×16⁵⁴⁶−145||[http://factordb.com/index.php?id=1100000002604723967]||[http://factordb.com/cert.php?id=1100000002604723967] |- |16||2335||88F₅₄₅||137×16⁵⁴⁵−1||[http://factordb.com/index.php?id=1100000000413679658]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |16||2336||BE0₇₉₂BB||190×16⁷⁹⁴+187||[http://factordb.com/index.php?id=1100000003588387938]||[http://factordb.com/cert.php?id=1100000003588387938] |- |16||2337||D9₁₀₅₂||(68×16¹⁰⁵²−3)/5||[http://factordb.com/index.php?id=1100000002321036020]||[http://factordb.com/cert.php?id=1100000002321036020] |- |16||2338||FAF₁₀₆₂45||251×16¹⁰⁶⁴−187||[http://factordb.com/index.php?id=1100000003588387610]||[http://factordb.com/cert.php?id=1100000003588387610] |- |16||2339||F8₁₅₁₇F||(233×16¹⁵¹⁸+97)/15||[http://factordb.com/index.php?id=1100000000633744824]||[http://factordb.com/cert.php?id=1100000000633744824] |- |16||2340||20₁₇₁₃321||2×16¹⁷¹⁶+801||[http://factordb.com/index.php?id=1100000003588386735]||[http://factordb.com/cert.php?id=1100000003588386735] |- |16||2341||300F₁₉₆₀AF||769×16¹⁹⁶²−81||[http://factordb.com/index.php?id=1100000003588368750]||[http://factordb.com/cert.php?id=1100000003588368750] |- |16||2342||90₃₅₄₂91||9×16³⁵⁴⁴+145||[http://factordb.com/index.php?id=1100000000633424191]||[http://factordb.com/cert.php?id=1100000000633424191] |- |16||2343||5BC₃₇₀₀D||(459×16³⁷⁰¹+1)/5||[http://factordb.com/index.php?id=1100000000993764322]||[http://factordb.com/cert.php?id=1100000000993764322] |- |16||2344||D0B₁₇₈₀₄||(3131×16¹⁷⁸⁰⁴−11)/15||[http://factordb.com/index.php?id=1100000003589278511]||[http://factordb.com/cert.php?id=1100000003589278511] |- |18||547||80₂₉₈B||8×18²⁹⁹+11||[http://factordb.com/index.php?id=1100000002355574745]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has sum-of-two-cubes algebraic factorization, 6×18⁹⁹+1 is an algebraic factor of ''N''+1, factordb entry of 6×18⁹⁹+1 is [http://factordb.com/index.php?id=1100000000900149167] |- |18||548||H₇₆₆FH||18⁷⁶⁸−37||[http://factordb.com/index.php?id=1100000003590430490]||[http://factordb.com/cert.php?id=1100000003590430490] |- |18||549||C0₆₂₆₈C5||12×18⁶²⁷⁰+221||[http://factordb.com/index.php?id=1100000003590442437]||[http://factordb.com/cert.php?id=1100000003590442437] |- |20||3301||H₂₄₇A0H||(17×20²⁵⁰−59677)/19||[http://factordb.com/index.php?id=1100000003590502619]||[http://factordb.com/cert.php?id=1100000003590502619] |- |20||3302||7₂₄₉A7||(7×20²⁵¹+1133)/19||[http://factordb.com/index.php?id=1100000003590502602]||[http://factordb.com/cert.php?id=1100000003590502602] |- |20||3303||J7₂₇₀||(368×20²⁷⁰−7)/19||[http://factordb.com/index.php?id=1100000002325395462]||[http://factordb.com/cert.php?id=1100000002325395462] |- |20||3304||J₃₃₀CCC7||20³³⁴−58953||[http://factordb.com/index.php?id=1100000003590502572]||[http://factordb.com/cert.php?id=1100000003590502572] |- |20||3305||40₃₈₇404B||4×20³⁹¹+32091||[http://factordb.com/index.php?id=1100000003590502563]||[http://factordb.com/cert.php?id=1100000003590502563] |- |20||3306||EC0₄₂₉7||292×20⁴³⁰+7||[http://factordb.com/index.php?id=1100000002633348702]||[http://factordb.com/cert.php?id=1100000002633348702] |- |20||3307||G₄₄₇99||(16×20⁴⁴⁹−2809)/19||[http://factordb.com/index.php?id=1100000000840126753]||[http://factordb.com/cert.php?id=1100000000840126753] |- |20||3308||3A₅₂₇3||(67×20⁵²⁸−143)/19||[http://factordb.com/index.php?id=1100000003590502531]||[http://factordb.com/cert.php?id=1100000003590502531] |- |20||3309||E₅₆₆C7||(14×20⁵⁶⁸−907)/19||[http://factordb.com/index.php?id=1100000003590502516]||[http://factordb.com/cert.php?id=1100000003590502516] |- |20||3310||JCJ₆₂₉||393×20⁶²⁹−1||[http://factordb.com/index.php?id=1100000001559454258]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |20||3311||J₆₅₅05J||20⁶⁵⁸−7881||[http://factordb.com/index.php?id=1100000003590502490]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000003591067052], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000003591067052] |- |20||3312||50₁₁₆₃AJ||5×20¹¹⁶⁵+219||[http://factordb.com/index.php?id=1100000003590502412]||[http://factordb.com/cert.php?id=1100000003590502412] |- |20||3313||CD₂₄₄₉||(241×20²⁴⁴⁹−13)/19||[http://factordb.com/index.php?id=1100000002325393915]||[http://factordb.com/cert.php?id=1100000002325393915] |- |20||3314||G0₆₂₆₉D||16×20⁶²⁷⁰+13||[http://factordb.com/index.php?id=1100000003590539457]||[http://factordb.com/cert.php?id=1100000003590539457] |- |22||7984||I7G0₂₅₄H||8882×22²⁵⁵+17||[http://factordb.com/index.php?id=1100000003591372788]||[http://factordb.com/cert.php?id=1100000003591372788] |- |22||7985||D0₂₅₅5EEF||13×22²⁵⁹+60339||[http://factordb.com/index.php?id=1100000003591371932]||[http://factordb.com/cert.php?id=1100000003591371932] |- |22||7986||IK₃₂₂F||(398×22³²³−125)/21||[http://factordb.com/index.php?id=1100000000840384145]||[http://factordb.com/cert.php?id=1100000000840384145] |- |22||7987||C0₃₄₀G9||12×22³⁴²+361||[http://factordb.com/index.php?id=1100000000840384159]||[http://factordb.com/cert.php?id=1100000000840384159] |- |22||7988||77E₃₄₈K7||(485×22³⁵⁰+373)/3||[http://factordb.com/index.php?id=1100000003591369779]||[http://factordb.com/cert.php?id=1100000003591369779] |- |22||7989||J₃₇₉KJ||(19×22³⁸¹+443)/21||[http://factordb.com/index.php?id=1100000003591369027]||[http://factordb.com/cert.php?id=1100000003591369027] |- |22||7990||J₃₈₈EJ||(19×22³⁹⁰−2329)/21||[http://factordb.com/index.php?id=1100000003591367729]||[http://factordb.com/cert.php?id=1100000003591367729] |- |22||7991||DJ₄₀₀||(292×22⁴⁰⁰−19)/21||[http://factordb.com/index.php?id=1100000002325880110]||[http://factordb.com/cert.php?id=1100000002325880110] |- |22||7992||E₄₀₄K7||(2×22⁴⁰⁶+373)/3||[http://factordb.com/index.php?id=1100000003591366298]||[http://factordb.com/cert.php?id=1100000003591366298] |- |22||7993||66F₄₅₃B3||(971×22⁴⁵⁵−705)/7||[http://factordb.com/index.php?id=1100000003591365809]||[http://factordb.com/cert.php?id=1100000003591365809] |- |22||7994||L0₄₅₄B63||21×22⁴⁵⁷+5459||[http://factordb.com/index.php?id=1100000003591365331]||[http://factordb.com/cert.php?id=1100000003591365331] |- |22||7995||L₄₈₃G3||22⁴⁸⁵−129||[http://factordb.com/index.php?id=1100000003591364730]||[http://factordb.com/cert.php?id=1100000003591364730] |- |22||7996||E60₄₉₆L||314×22⁴⁹⁷+21||[http://factordb.com/index.php?id=1100000000632703239]||[http://factordb.com/cert.php?id=1100000000632703239] |- |22||7997||I₆₂₆AF||(6×22⁶²⁸−1259)/7||[http://factordb.com/index.php?id=1100000000632724334]||[http://factordb.com/cert.php?id=1100000000632724334] |- |22||7998||K0₇₆₀EC1||20×22⁷⁶³+7041||[http://factordb.com/index.php?id=1100000000632724415]||[http://factordb.com/cert.php?id=1100000000632724415] |- |22||7999||J0₇₆₇IGGJ||19×22⁷⁷¹+199779||[http://factordb.com/index.php?id=1100000003591362567]||[http://factordb.com/cert.php?id=1100000003591362567] |- |22||8000||7₉₅₉K7||(22⁹⁶¹+857)/3||[http://factordb.com/index.php?id=1100000003591361817]||[http://factordb.com/cert.php?id=1100000003591361817] |- |22||8001||L₂₃₈₅KE7||22²³⁸⁸−653||[http://factordb.com/index.php?id=1100000003591360774]||[http://factordb.com/cert.php?id=1100000003591360774] |- |22||8002||7₃₈₁₅2L||(22³⁸¹⁷−289)/3||[http://factordb.com/index.php?id=1100000003591359839]||[http://factordb.com/cert.php?id=1100000003591359839] |- |24||3400||I0₂₄₁I5||18×24²⁴³+437||[http://factordb.com/index.php?id=1100000002633360037]||[http://factordb.com/cert.php?id=1100000002633360037] |- |24||3401||D0₂₅₉KKD||13×24²⁶²+12013||[http://factordb.com/index.php?id=1100000003593270725]||[http://factordb.com/cert.php?id=1100000003593270725] |- |24||3402||C7₂₉₈||(283×24²⁹⁸−7)/23||[http://factordb.com/index.php?id=1100000002326181235]||[http://factordb.com/cert.php?id=1100000002326181235] |- |24||3403||20₃₁₃7||2×24³¹⁴+7||[http://factordb.com/index.php?id=1100000002355610241]||[http://factordb.com/cert.php?id=1100000002355610241] |- |24||3404||BC0₃₃₁B||276×24³³²+11||[http://factordb.com/index.php?id=1100000002633359842]||[http://factordb.com/cert.php?id=1100000002633359842] |- |24||3405||N₂₆₄₄LLN||24²⁶⁴⁷−1201||[http://factordb.com/index.php?id=1100000003593270089]||[http://factordb.com/cert.php?id=1100000003593270089] |- |24||3406||D₂₆₉₈LD||(13×24²⁷⁰⁰+4403)/23||[http://factordb.com/index.php?id=1100000003593269876]||[http://factordb.com/cert.php?id=1100000003593269876] |- |24||3407||A0₂₉₅₁8ID||10×24²⁹⁵⁴+5053||[http://factordb.com/index.php?id=1100000003593269654]||[http://factordb.com/cert.php?id=1100000003593269654] |- |24||3408||88N₅₉₅₁||201×24⁵⁹⁵¹−1||[http://factordb.com/index.php?id=1100000003593275880]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |24||3409||N00N₈₁₂₉LN||13249×24⁸¹³¹−49||[http://factordb.com/index.php?id=1100000003593391606]||[http://factordb.com/cert.php?id=1100000003593391606] |- |30||2613||AN₂₀₆||(313×30²⁰⁶−23)/29||[http://factordb.com/index.php?id=1100000002327651073]||[http://factordb.com/cert.php?id=1100000002327651073] |- |30||2614||M₂₄₁QB||(22×30²⁴³+3139)/29||[http://factordb.com/index.php?id=1100000003593408295]||[http://factordb.com/cert.php?id=1100000003593408295] |- |30||2615||M0₅₄₇SS7||22×30⁵⁵⁰+26047||[http://factordb.com/index.php?id=1100000003593407988]||[http://factordb.com/cert.php?id=1100000003593407988] |- |30||2616||C0₁₀₂₂1||12×30¹⁰²³+1||[http://factordb.com/index.php?id=1100000000785448736]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |30||2617||5₄₈₈₂J||(5×30⁴⁸⁸³+401)/29||[http://factordb.com/index.php?id=1100000002327649423]||[http://factordb.com/cert.php?id=1100000002327649423] |- |30||2619||OT₃₄₂₀₅||25×30³⁴²⁰⁵−1||[http://factordb.com/index.php?id=1100000000800812865]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |} ==Unproven PRPs== {|class=wikitable |base (''b'')||index of this quasi-minimal prime in base ''b'' (assuming the primality of all PRP in base ''b'')||unproven PRP (base-''b'' form)||unproven PRP (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||factordb entry of this PRP |- |11||1068||57₆₂₆₆₈||(57×11⁶²⁶⁶⁸−7)/10||[http://factordb.com/index.php?id=1100000003573679860] |- |13||3194||C5₂₃₇₅₅C||(149×13²³⁷⁵⁶+79)/12||[http://factordb.com/index.php?id=1100000003590647776] |- |13||3195||80₃₂₀₁₇111||8×13³²⁰²⁰+183||[http://factordb.com/index.php?id=1100000000490878060] |- |16||2345||DB₃₂₂₃₄||(206×16³²²³⁴−11)/15||[http://factordb.com/index.php?id=1100000002383583629] |- |16||2346||4₇₂₇₈₅DD||(4×16⁷²⁷⁸⁷+2291)/15||[http://factordb.com/index.php?id=1100000003615909841] |- |22||8003||BK₂₂₀₀₁5||(251×22²²⁰⁰²−335)/21||[http://factordb.com/index.php?id=1100000003594696838] |- |30||2618||I0₂₄₆₀₈D||18×30²⁴⁶⁰⁹+13||[http://factordb.com/index.php?id=1100000003593967511] |} All these PRPs pass the [[w:Miller–Rabin primality test|Miller–Rabin primality test]] to bases 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59 and 61, and pass the [[w:Lucas pseudoprime#Strong Lucas pseudoprimes|strong Lucas primality test]] with parameters (''P'', ''Q'') defined by Selfridge's Method ''A'', and [[w:Trial division|trial factored]] to 10¹⁶. (Thus, they pass the [[w:Baillie–PSW primality test|Baillie–PSW primality test]]) ==Proof== ===Base 2=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. ===Base 3=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (2,1), (2,2) * Case (1,1): ** Since 12, 21, 111 are primes, we only need to consider the family 1{0}1 (since any digits 1, 2 between them will produce smaller primes) *** All numbers of the form 1{0}1 are divisible by 2, thus cannot be prime. * Case (1,2): ** 12 is prime, and thus the only minimal prime in this family. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,2): ** Since 21, 12 are primes, we only need to consider the family 2{0,2}2 (since any digits 1 between them will produce smaller primes) *** All numbers of the form 2{0,2}2 are divisible by 2, thus cannot be prime. ===Base 4=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (2,1), (2,3), (3,1), (3,3) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 11, 31, 221 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3 between them will produce smaller primes) *** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 31, 13, 23 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2 between them will produce smaller primes) *** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime. ===Base 5=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (2,4), (3,1), (3,2), (3,3), (3,4), (4,1), (4,2), (4,3), (4,4) * Case (1,1): ** Since 12, 21, 111, 131 are primes, we only need to consider the family 1{0,4}1 (since any digits 1, 2, 3 between them will produce smaller primes) *** All numbers of the form 1{0,4}1 are divisible by 2, thus cannot be prime. * Case (1,2): ** 12 is prime, and thus the only minimal prime in this family. * Case (1,3): ** Since 12, 23, 43, 133 are primes, we only need to consider the family 1{0,1}3 (since any digits 2, 3, 4 between them will produce smaller primes) *** Since 111 is prime, we only need to consider the families 1{0}3 and 1{0}1{0}3 (since any digit combo 11 between (1,3) will produce smaller primes) **** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime. **** For the 1{0}1{0}3 family, since 10103 is prime, we only need to consider the families 1{0}13 and 11{0}3 (since any digit combo 010 between (1,3) will produce smaller primes) ***** The smallest prime of the form 1{0}13 is 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013, which can be written as 1(0^93)13 and equal the prime 5^95+8 ([http://factordb.com/index.php?id=1100000000034686071 factordb]) ***** All numbers of the form 11{0}3 are divisible by 3, thus cannot be prime. * Case (1,4): ** Since 12, 34, 104 are primes, we only need to consider the family 1{1,4}4 (since any digits 0, 2, 3 between them will produce smaller primes) *** Since 111, 414 are primes, we only need to consider the families 1{4}4 and 11{4}4 (since any digit combo 11 or 41 between them will produce smaller primes) **** The smallest prime of the form 1{4}4 is 14444. **** All numbers of the form 11{4}4 are divisible by 2, thus cannot be prime. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,2): ** Since 21, 23, 12, 32 are primes, we only need to consider the family 2{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 2{0,2,4}2 are divisible by 2, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,4): ** Since 21, 23, 34 are primes, we only need to consider the family 2{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 2{0,2,4}4 are divisible by 2, thus cannot be prime. * Case (3,1): ** Since 32, 34, 21 are primes, we only need to consider the family 3{0,1,3}1 (since any digits 2, 4 between them will produce smaller primes) *** Since 313, 111, 131, 3101 are primes, we only need to consider the families 3{0,3}1 and 3{0,3}11 (since any digit combo 10, 11, 13 between (3,1) will produce smaller primes) **** For the 3{0,3}1 family, we can separate this family to four families: ***** For the 30{0,3}01 family, we have the prime 30301, and the remain case is the family 30{0}01. ****** All numbers of the form 30{0}01 are divisible by 2, thus cannot be prime. ***** For the 30{0,3}31 family, note that there must be an even number of 3's between (30,31), or the result number will be divisible by 2 and cannot be prime. ****** Since 33331 is prime, any digit combo 33 between (30,31) will produce smaller primes. ******* Thus, the only possible prime is the smallest prime in the family 30{0}31, and this prime is 300031. ***** For the 33{0,3}01 family, note that there must be an even number of 3's between (33,01), or the result number will be divisible by 2 and cannot be prime. ****** Since 33331 is prime, any digit combo 33 between (33,01) will produce smaller primes. ******* Thus, the only possible prime is the smallest prime in the family 33{0}01, and this prime is 33001. ***** For the 33{0,3}31 family, we have the prime 33331, and the remain case is the family 33{0}31. ****** All numbers of the form 33{0}31 are divisible by 2, thus cannot be prime. **** All numbers of the form 3{0,3}11 are divisible by 3, thus cannot be prime. * Case (3,2): ** 32 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 32, 34, 23, 43, 313 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2, 4 between them will produce smaller primes) *** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime. * Case (3,4): ** 34 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 43, 21, 401 are primes, we only need to consider the family 4{1,4}1 (since any digits 0, 2, 3 between them will produce smaller primes) *** Since 414, 111 are primes, we only need to consider the families 4{4}1 and 4{4}11 (since any digit combo 14 or 11 between them will produce smaller primes) **** The smallest prime of the form 4{4}1 is 44441. **** All numbers of the form 4{4}11 are divisible by 2, thus cannot be prime. * Case (4,2): ** Since 43, 12, 32 are primes, we only need to consider the family 4{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 4{0,2,4}2 are divisible by 2, thus cannot be prime. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,4): ** Since 43, 34, 414 are primes, we only need to consider the family 4{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 4{0,2,4}4 are divisible by 2, thus cannot be prime. ===Base 6=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,5), (2,1), (2,5), (3,1), (3,5), (4,1), (4,5), (5,1), (5,5) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 11, 21, 31, 51 are primes, we only need to consider the family 4{0,4}1 (since any digits 1, 2, 3, 5 between them will produce smaller primes) *** Since 4401 and 4441 are primes, we only need to consider the families 4{0}1 and 4{0}41 (since any digits combo 40 and 44 between them will produce smaller primes) **** All numbers of the form 4{0}1 are divisible by 5, thus cannot be prime. **** The smallest prime of the form 4{0}41 is 40041 * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 15, 25, 35, 45 are primes, we only need to consider the family 5{0,5}5 (since any digits 1, 2, 3, 4 between them will produce smaller primes) *** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. ===Base 7=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6) * Case (1,1): ** Since 14, 16, 41, 61, 131 are primes, we only need to consider the family 1{0,1,2,5}1 (since any digits 3, 4, 6 between them will produce smaller primes) *** Since the digit sum of primes must be odd (otherwise the number will be divisible by 2, thus cannot be prime), there is an odd total number of 1 and 5 in the {} **** If there are >=3 number of 1 and 5 in the {}: ***** If there is 111 in the {}, then we have the prime 11111 ***** If there is 115 in the {}, then the prime 115 is a subsequence ***** If there is 151 in the {}, then the prime 115 is a subsequence ***** If there is 155 in the {}, then the prime 155 is a subsequence ***** If there is 511 in the {}, then the current number is 15111, which has digit sum = 12, but digit sum divisible by 3 will cause the number divisible by 3 and cannot be prime, and we cannot add more 1 or 5 to this number (to avoid 11111, 155, 515, 551 as subsequence), thus we must add at least one 2 to this number, but then the number has both 2 and 5, and will have either 25 or 52 as subsequence, thus cannot be minimal prime ***** If there is 515 in the {}, then the prime 515 is a subsequence ***** If there is 551 in the {}, then the prime 551 is a subsequence ***** If there is 555 in the {}, then the prime 551 is a subsequence **** Thus there is only one 1 (and no 5) or only one 5 (and no 1) in the {}, i.e. we only need to consider the families 1{0,2}1{0,2}1 and 1{0,2}5{0,2}1 ***** For the 1{0,2}1{0,2}1 family, since 1211 is prime, we only need to consider the family 1{0}1{0,2}1 ****** Since all numbers of the form 1{0}1{0}1 are divisible by 3 and cannot be prime, we only need to consider the family 1{0}1{0}2{0}1 ******* Since 11201 is prime, we only need to consider the family 1{0}1{0}21 ******** The smallest prime of the form 11{0}21 is 1100021 ******** All numbers of the form 101{0}21 are divisible by 5, thus cannot be prime ******** The smallest prime of the form 1001{0}21 is 100121 ********* Since this prime has no 0 between 1{0}1 and 21, we do not need to consider more families ***** For the 1{0,2}5{0,2}1 family, since 25 and 52 are primes, we only need to consider the family 1{0}5{0}1 ****** Since 1051 is prime, we only need to consider the family 15{0}1 ******* The smallest prime of the form 15{0}1 is 150001 * Case (1,2): ** Since 14, 16, 32, 52 are primes, we only need to consider the family 1{0,1,2}2 (since any digits 3, 4, 5, 6 between them will produce smaller primes) *** Since 1112 and 1222 are primes, there is at most one 1 and at most one 2 in {} **** If there are one 1 and one 2 in {}, then the digit sum is 6, and the number will be divisible by 6 and cannot be prime. **** If there is one 1 but no 2 in {}, then the digit sum is 4, and the number will be divisible by 2 and cannot be prime. **** If there is no 1 but one 2 in {}, then the form is 1{0}2{0}2 ***** Since 1022 and 1202 are primes, we only need to consider the number 122 ****** 122 is not prime. **** If there is no 1 and no 2 in {}, then the digit sum is 3, and the number will be divisible by 3 and cannot be prime. * Case (1,3): ** Since 14, 16, 23, 43, 113, 133 are primes, we only need to consider the family 1{0,5}3 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes) *** Since 155 is prime, we only need to consider the family 1{0}3 and 1{0}5{0}3 **** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime. **** All numbers of the form 1{0}5{0}3 are divisible by 3, thus cannot be prime. * Case (1,4): ** 14 is prime, and thus the only minimal prime in this family. * Case (1,5): ** Since 14, 16, 25, 65, 115, 155 are primes, we only need to consider the family 1{0,3}5 (since any digits 1, 2, 4, 5, 6 between them will produce smaller primes) *** All numbers of the form 1{0,3}5 are divisible by 3, thus cannot be prime. * Case (1,6): ** 16 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 25, 41, 61, 221 are primes, we only need to consider the family 2{0,1}1 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes) *** Since 2111 is prime, we only need to consider the families 2{0}1 and 2{0}1{0}1 **** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. **** All numbers of the form 2{0}1{0}1 are divisible by 2, thus cannot be prime. * Case (2,2): ** Since 23, 25, 32, 52, 212 are primes, we only need to consider the family 2{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,4): ** Since 23, 25, 14 are primes, we only need to consider the family 2{0,2,4,6}4 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}4 are divisible by 2, thus cannot be prime. * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (2,6): ** Since 23, 25, 16, 56 are primes, we only need to consider the family 2{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}6 are divisible by 2, thus cannot be prime. * Case (3,1): ** Since 32, 41, 61 are primes, we only need to consider the family 3{0,1,3,5}1 (since any digits 2, 4, 6 between them will produce smaller primes) *** Since 551 is prime, we only need to consider the family 3{0,1,3}1 and 3{0,1,3}5{0,1,3}1 (since any digits combo 55 between (3,1) will produce smaller primes) **** For the 3{0,1,3}1 family, since 3031 and 131 are primes, we only need to consider the families 3{0,1}1 and 3{3}3{0,1}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes, thus for the digits between (3,1), all 3's must be before all 0's and 1's, and thus we can let the red 3 in 3{3}3{0,1}1 be the rightmost 3 between (3,1), all digits before this 3 must be 3's, and all digits after this 3 must be either 0's or 1's) ***** For the 3{0,1}1 family: ****** If there are >=2 0's and >=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence. ****** If there are no 1's between (3,1), then the form will be 3{0}1 ******* All numbers of the form 3{0}1 are divisible by 2, thus cannot be prime. ****** If there are no 0's between (3,1), then the form will be 3{1}1 ******* The smallest prime of the form 3{1}1 is 31111 ****** If there are exactly 1 0's between (3,1), then there must be <3 1's between (3,1), or 31111 will be a subsequence. ******* If there are 2 1's between (3,1), then the digit sum is 6, thus the number is divisible by 6 and cannot be prime. ******* If there are 1 1's between (3,1), then the number can only be either 3101 or 3011 ******** Neither 3101 nor 3011 is prime. ******* If there are no 1's between (3,1), then the number must be 301 ******** 301 is not prime. ***** For the 3{3}3{0,1}1 family: ****** If there are at least one 3 between (3,3{0,1}1) and at least one 1 between (3{3}3,1), then 33311 will be a subsequence. ****** If there are no 3 between (3,3{0,1}1), then the form will be 33{0,1}1 ******* If there are at least 3 1's between (33,1), then 31111 will be a subsequence. ******* If there are exactly 2 1's between (33,1), then the digit sum is 12, thus the number is divisible by 3 and cannot be prime. ******* If there are exactly 1 1's between (33,1), then the digit sum is 11, thus the number is divisible by 2 and cannot be prime. ******* If there are no 1's between (33,1), then the form will be 33{0}1 ******** The smallest prime of the form 33{0}1 is 33001 ****** If there are no 1 between (3{3}3,1), then the form will be 3{3}3{0}1 ******* If there are at least 2 0's between (3{3}3,1), then 33001 will be a subsequence. ******* If there are exactly 1 0's between (3{3}3,1), then the form is 3{3}301 ******** The smallest prime of the form 3{3}301 is 33333301 ******* If there are no 0's between (3{3}3,1), then the form is 3{3}31 ******** The smallest prime of the form 3{3}31 is 33333333333333331 **** For the 3{0,1,3}5{0,1,3}1 family, since 335 is prime, we only need to consider the family 3{0,1}5{0,1,3}1 ***** Numbers containing 3 between (3{0,1}5,1): ****** The form is 3{0,1}5{0,1,3}3{0,1,3}1 ******* Since 3031 and 131 are primes, we only need to consider the family 35{3}3{0,1,3}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes) ******** Since 533 is prime, we only need to consider the family 353{0,1}1 (since any digits combo 33 between (35,1) will produce smaller primes) ********* Since 5011 is prime, we only need to consider the family 353{1}{0}1 (since any digits combo 01 between (353,1) will produce smaller primes) ********** If there are at least 3 1's between (353,{0}1), then 31111 will be a subsequence. ********** If there are exactly 2 1's between (353,{0}1), then the digit sum is 20, thus the number is divisible by 2 and cannot be prime. ********** If there are exactly 1 1's between (353,{0}1), then the form is 3531{0}1 *********** The smallest prime of the form 3531{0}1 is 3531001, but it is not minimal prime since 31001 is prime. ********** If there are no 1's between (353,{0}1), then the digit sum is 15, thus the number is divisible by 6 and cannot be prime. ***** Numbers not containing 3 between (3{0,1}5,1): ****** The form is 3{0,1}5{0,1}1 ******* If there are >=2 0's and >=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence. ******* If there are no 1's between (3,1), then the form will be 3{0}5{0}1 ******** All numbers of the form 3{0}5{0}1 are divisible by 3, thus cannot be prime. ******* If there are no 0's between (3,1), then the form will be 3{1}5{1}1 ******** If there are >=3 1's between (3,1), then 31111 will be a subsequence. ******** If there are exactly 2 1's between (3,1), then the number can only be 31151, 31511, 35111 ********* None of 31151, 31511, 35111 are primes. ******** If there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime. ******** If there are no 1's between (3,1), then the number is 351 ********* 351 is not prime. ******* If there are exactly 1 0's between (3,1), then the form will be 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1 ******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are >=3 1's between (3,1), then 31111 will be a subsequence. ******** If there are exactly 2 1's between (3,1), then the number can only be 311051, 310151, 310511, 301151, 301511, 305111, 311501, 315101, 315011, 351101, 351011, 350111 ********* Of these numbers, 311051, 301151, 311501, 351101, 350111 are primes. ********** However, 311051, 301151, 311501 have 115 as subsequence, and 350111 has 5011 as subsequence, thus only 351101 is minimal prime. ******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime. ******** If there are no 1's between (3,1), then the number is 3051 for 3{1}0{1}5{1}1 or 3501 for 3{1}5{1}0{1}1 ********* Neither 3051 nor 3501 is prime. * Case (3,2): ** 32 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 32, 23, 43, 313 are primes, we only need to consider the family 3{0,3,5,6}3 (since any digits 1, 2, 4 between them will produce smaller primes) *** If there are >=2 5's in {}, then 553 will be a subsequence. *** If there are no 5's in {}, then the family will be 3{0,3,6}3 **** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime. *** If there are exactly 1 5's in {}, then the family will be 3{0,3,6}5{0,3,6}3 **** Since 335, 65, 3503, 533, 56 are primes, we only need to consider the family 3{0}53 (since any digit 3, 6 between (3,5{0,3,6}3) and any digit 0, 3, 6 between (3{0,3,6}5,3) will produce smaller primes) ***** The smallest prime of the form 3{0}53 is 300053 * Case (3,4): ** Since 32, 14, 304, 344, 364 are primes, we only need to consider the family 3{3,5}4 (since any digits 0, 1, 2, 4, 6 between them will produce smaller primes) *** Since 3334 and 335 are primes, we only need to consider the family 3{5}4 and 3{5}34 (since any digits combo 33, 35 between them will produce smaller primes) **** The smallest prime of the form 3{5}4 is 35555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555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with 9234 5's, which can be written as 3(5^9234)4 and equal the prime (23*7^9235-11)/6 ([http://factordb.com/index.php?id=1100000002766595757 factordb]) ([http://factordb.com/cert.php?id=1100000002766595757 primality certificate]) (not minimal prime, since 35555 and 5554 are primes) **** The smallest prime of the form 3{5}34 is 355555555555555555555555555555555555555555555555555555555555555534 (not minimal prime, since 35555, 553, and 5554 are primes) * Case (3,5): ** Since 32, 25, 65, 335 are primes, we only need to consider the family 3{0,1,4,5}5 (since any digits 2, 3, 6 between them will produce smaller primes) *** If there are at least one 1's and at least one 5's in {}, then either 155 or 515 will be a subsequence. *** If there are at least one 1's and at least one 4's in {}, then either 14 or 41 will be a subsequence. *** If there are at least two 1's in {}, then 115 will be a subsequence. *** If there are exactly one 1's and no 4's or 5's in {}, then the family will be 3{0}1{0}5 **** All numbers of the form 3{0}1{0}5 are divisible by 3, thus cannot be prime. *** If there is no 1's in {}, then the family will be 3{0,4,5}5 **** If there are at least to 4's in {}, then 344 and 445 will be subsequences. **** If there is no 4's in {}, then the family will be 3{0,5}5 ***** Since 3055 and 3505 are primes, we only need to consider the families 3{0}5 and 3{5}5 ****** All numbers of the form 3{0}5 are divisible by 2, thus cannot be prime. ****** The smallest prime of the form 3{5}5 is 35555 **** If there is exactly one 4's in {}, then the family will be 3{0,5}4{0,5}5 ***** Since 304, 3545 are primes, we only need to consider the families 34{0,5}5 (since any digits 0 or 5 between (3,4{0,5}5) will produce small primes) ****** All numbers of the form 34{0,5}5 are divisible by 5, thus cannot be prime. * Case (3,6): ** Since 32, 16, 56, 346 are primes, we only need to consider the family 3{0,3,6}6 (since any digits 1, 2, 4, 5 between them will produce smaller primes) *** All numbers of the form 3{0,3,6}6 are divisible by 3, thus cannot be prime. * Case (4,1): ** 41 is prime, and thus the only minimal prime in this family. * Case (4,2): ** Since 41, 43, 32, 52 are primes, we only need to consider the family 4{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 4{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,4): ** Since 41, 43, 14 are primes, we only need to consider the family 4{0,2,4,5,6}4 (since any digits 1, 3 between them will produce smaller primes) *** If there is no 5's in {}, then the family will be 4{0,2,4,6}4 **** All numbers of the form 4{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there is at least one 5's in {}, then there cannot be 2 in {} (since if so, then either 25 or 52 will be a subsequence) and there cannot be 6 in {} (since if so, then either 65 or 56 will be a subsequence), thus the family is 4{0,4,5}5{0,4,5}4 **** Since 445, 4504, 544 are primes, we only need to consider the family 4{0,5}5{5}4 (since any digit 4 between (4,5{0,4,5}4) and any digit 0, 4 between (4{0,4,5}5,4) will produce smaller primes) ***** If there are at least two 0's between (4,5{0,4,5}4), then 40054 will be a subsequence. ***** If there is no 0's between (4,5{0,4,5}4), then the family will be 4{5}5{5}4, which is equivalent to 4{5}4 ****** The smallest prime of the form 4{5}4 is 45555555555555554 (not minimal prime, since 4555 and 5554 are primes) ***** If there is exactly one 0's between (4,5{0,4,5}4), then the family will be 4{5}0{5}5{5}4 ****** Since 4504 is prime, we only need to consider the family 40{5}5{5}4 (since any digit 5 between (4,0{5}5{5}4) will produce small primes), which is equivalent to 40{5}4 ******* The smallest prime of the form 40{5}4 is 405555555555555554 (not minimal prime, since 4555 and 5554 are primes) * Case (4,5): ** Since 41, 43, 25, 65, 445 are primes, we only need to consider the family 4{0,5}5 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes) *** If there are at least two 5's in {}, then 4555 will be a subsequence. *** If there is exactly one 5's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime. *** If there is no 5's in {}, then the family will be 4{0}5 **** All numbers of the form 4{0}5 are divisible by 3, thus cannot be prime. * Case (4,6): ** Since 41, 43, 16, 56 are primes, we only need to consider the family 4{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 4{0,2,4,6}6 are divisible by 2, thus cannot be prime. * Case (5,1): ** Since 52, 56, 41, 61, 551 are primes, we only need to consider the family 5{0,1,3}1 (since any digits 2, 4, 5, 6 between them will produce smaller primes) *** If there are at least two 3's in {}, then 533 will be a subsequence. *** If there is no 3's in {}, then the family will be 5{0,1}1 **** Since 5011 is prime, we only need to consider the family 5{1}{0}1 ***** Since 11111 is prime, we only need to consider the families 5{0}1, 51{0}1, 511{0}1, 5111{0}1 (since any digits combo 1111 between (5,1) will produce small primes) ****** All numbers of the form 5{0}1 are divisible by 6, thus cannot be prime. ****** The smallest prime of the form 51{0}1 is 5100000001 ****** All numbers of the form 511{0}1 are divisible by 2, thus cannot be prime. ****** All numbers of the form 5111{0}1 are divisible by 3, thus cannot be prime. *** If there is exactly one 3's in {}, then the family will be 5{0,1}3{0,1}1 **** If there is at least one 1's between (5,3{0,1}1), then 131 will be a subsequence. ***** Thus we only need to consider the family 5{0}3{0,1}1 ****** If there are no 1's between (5{0}3,1), then the digit sum is 12, and the number will be divisible by 3 and cannot be prime. ****** If there are exactly one 1's between (5{0}3,1), then the digit sum is 13, and the number will be divisible by 2 and cannot be prime. ****** If there are exactly three 1's between (5{0}3,1), then the digit sum is 15, and the number will be divisible by 6 and cannot be prime. ****** If there are at least four 1's between (5{0}3,1), then 11111 will be a subsequence. ****** If there are exactly two 1's between (5{0}3,1), then the family will be 5{0}3{0}1{0}1{0}1 ******* Since 5011 is prime, we only need to consider the family 5311{0}1 (since any digit 0 between (5,1{0}1) will produce small primes, this includes the leftmost three {} in 5{0}3{0}1{0}1{0}1, and thus only the rightmost {} can contain 0) ******** The smallest prime of the form 5311{0}1 is 531101 * Case (5,2): ** 52 is prime, and thus the only minimal prime in this family. * Case (5,3): ** Since 52, 56, 23, 43, 533, 553 are primes, we only need to consider the family 5{0,1}3 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes) *** If there are at least two 1's in {}, then 113 will be a subsequence. *** If there is exactly one 1's in {}, then the digit sum is 12, and the number will be divisible by 3 and cannot be prime. *** If there is no 1's in {}, then the digit sum is 11, and the number will be divisible by 2 and cannot be prime. * Case (5,4): ** Since 52, 56, 14, 544 are primes, we only need to consider the family 5{0,3,5}4 (since any digits 1, 2, 4, 6 between them will produce smaller primes) *** If there are no 5's in {}, then the family will be 5{0,3}4 **** All numbers of the form 5{0,3}4 are divisible by 3, thus cannot be prime. *** If there are at least one 5's and at least one 3's in {}, then either 535 or 553 will be a subsequence. *** If there are exactly one 5's and no 3's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime. *** If there are at least two 5's in {}, then 5554 will be a subsequence. * Case (5,5): ** Since 52, 56, 25, 65, 515, 535 are primes, we only need to consider the family 5{0,4,5}5 (since any digits 1, 2, 3, 6 between them will produce smaller primes) *** If there are no 4's in {}, then the family will be 5{0,5}5 **** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. *** If there are no 5's in {}, then the family will be 5{0,4}5 **** All numbers of the form 5{0,4}5 are divisible by 2, thus cannot be prime. *** If there are at least one 4's and at least one 5's in {}, then either 5455 or 5545 will be a subsequence. * Case (5,6): ** 56 is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,2): ** Since 61, 65, 32, 52 are primes, we only need to consider the family 6{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 6{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (6,3): ** Since 61, 65, 23, 43 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5 between them will produce smaller primes) *** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (6,4): ** Since 61, 65, 14 are primes, we only need to consider the family 6{0,2,3,4,6}4 (since any digits 1, 5 between them will produce smaller primes) *** If there is no 3's in {}, then the family will be 6{0,2,4,6}4 **** All numbers of the form 6{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there are exactly two 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4 **** All numbers of the form 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there are at least three 3's in {}, then 3334 will be a subsequence. *** If there is exactly one 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}4 **** If there is 0 between (6,3{0,2,4,6}4), then 6034 will be a subsequence. **** If there is 2 between (6,3{0,2,4,6}4), then 23 will be a subsequence. **** If there is 4 between (6,3{0,2,4,6}4), then 43 will be a subsequence. **** If there is 6 between (6,3{0,2,4,6}4), then 6634 will be a subsequence. **** If there is 0 between (6{0,2,4,6}3,4), then 304 will be a subsequence. **** If there is 2 between (6{0,2,4,6}3,4), then 32 will be a subsequence. **** If there is 4 between (6{0,2,4,6}3,4), then 344 will be a subsequence. **** If there is 6 between (6{0,2,4,6}3,4), then 364 will be a subsequence. **** Thus the number can only be 634 ***** 634 is not prime. * Case (6,5): ** 65 is prime, and thus the only minimal prime in this family. * Case (6,6): ** Since 61, 65, 16, 56 are primes, we only need to consider the family 6{0,2,3,4,6}6 (since any digits 1, 5 between them will produce smaller primes) *** If there is no 3's in {}, then the family will be 6{0,2,4,6}6 **** All numbers of the form 6{0,2,4,6}6 are divisible by 2, thus cannot be prime. *** If there is no 2's and no 4's in {}, then the family will be 6{0,3,6}6 **** All numbers of the form 6{0,3,6}6 are divisible by 3, thus cannot be prime. *** If there is at least one 3's and at least one 2's in {}, then either 32 or 23 will be a subsequence. *** If there is at least one 3's and at least one 4's in {}, then either 346 or 43 will be a subsequence. ===Base 8=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (1,5), (1,7), (2,1), (2,3), (2,5), (2,7), (3,1), (3,3), (3,5), (3,7), (4,1), (4,3), (4,5), (4,7), (5,1), (5,3), (5,5), (5,7), (6,1), (6,3), (6,5), (6,7), (7,1), (7,3), (7,5), (7,7) * Case (1,1): ** Since 13, 15, 21, 51, 111, 141, 161 are primes, we only need to consider the family 1{0,7}1 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes) *** Since 107, 177, 701 are primes, we only need to consider the number 171 and the family 1{0}1 (since any digits combo 07, 70, 77 between them will produce smaller primes) **** 171 is not prime. **** All numbers of the form 1{0}1 factored as 10^n+1 = (2^n+1) * (4^n-2^n+1) (n≥1) (and since if n≥1, 2^n+1 ≥ 2^1+1 = 3 > 1, 4^n-2^n+1 ≥ 4^1-2^1+1 = 3 > 1, this factorization is nontrivial), thus cannot be prime. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (1,7): ** Since 13, 15, 27, 37, 57, 107, 117, 147, 177 are primes, we only need to consider the family 1{6}7 (since any digits 0, 1, 2, 3, 4, 5, 7 between them will produce smaller primes) *** The smallest prime of the form 1{6}7 is 16667 (not minimal prime, since 667 is prime) * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,5): ** Since 21, 23, 27, 15, 35, 45, 65, 75, 225, 255 are primes, we only need to consider the family 2{0}5 (since any digits 1, 2, 3, 4, 5, 6, 7 between them will produce smaller primes) *** All numbers of the form 2{0}5 are divisible by 7, thus cannot be prime. * Case (2,7): ** 27 is prime, and thus the only minimal prime in this family. * Case (3,1): ** Since 35, 37, 21, 51, 301, 361 are primes, we only need to consider the family 3{1,3,4}1 (since any digits 0, 2, 5, 6, 7 between them will produce smaller primes) *** Since 13, 343, 111, 131, 141, 431, 3331, 3411 are primes, we only need to consider the families 3{3}11, 33{1,4}1, 3{3,4}4{4}1 (since any digits combo 11, 13, 14, 33, 41, 43 between them will produce smaller primes) **** All numbers of the form 3{3}11 are divisible by 3, thus cannot be prime. **** For the 33{1,4}1 family, since 111 and 141 are primes, we only need to consider the families 33{4}1 and 33{4}11 (since any digits combo 11, 14 between them will produce smaller primes) ***** The smallest prime of the form 33{4}1 is 3344441 ***** All numbers of the form 33{4}11 are divisible by 301, thus cannot be prime. **** For the 3{3,4}4{4}1 family, since 3331 and 3344441 are primes, we only need to consider the families 3{4}1, 3{4}31, 3{4}341, 3{4}3441, 3{4}34441 (since any digits combo 33 or 34444 between (3,1) will produce smaller primes) ***** All numbers of the form 3{4}1 are divisible by 31, thus cannot be prime. ***** Since 4443 is prime, we only need to consider the numbers 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 (since any digit combo 444 between (3,3{4}1) will produce smaller primes) ****** None of 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 are primes. * Case (3,3): ** Since 35, 37, 13, 23, 53, 73, 343 are primes, we only need to consider the family 3{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes) *** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 21, 51, 401, 431, 471 are primes, we only need to consider the family 4{1,4,6}1 (since any digits 0, 2, 3, 5, 7 between them will produce smaller primes) *** Since 111, 141, 161, 661, 4611 are primes, we only need to consider the families 4{4}11, 4{4,6}4{1,4,6}1, 4{4}6{4}1 (since any digits combo 11, 14, 16, 61, 66 between them will produce smaller primes) **** The smallest prime of the form 4{4}11 is 44444444444444411 (not minimal prime, since 444444441 is prime) **** For the 4{4,6}4{1,4,6}1 family, we can separate this family to 4{4,6}41, 4{4,6}411, 4{4,6}461 ***** For the 4{4,6}41 family, since 661 and 6441 are primes, we only need to consider the families 4{4}41 and 4{4}641 (since any digits combo 64 or 66 between (4,41) will produce smaller primes) ****** The smallest prime of the form 4{4}41 is 444444441 ****** The smallest prime of the form 4{4}641 is 444641 ***** For the 4{4,6}411 family, since 661 and 6441 are primes, we only need to consider the families 4{4}411 and 4{4}6411 (since any digits combo 64 or 66 between (4,411) will produce smaller primes) ****** The smallest prime of the form 4{4}411 is 444444441 ****** The smallest prime of the form 4{4}6411 is 4444444444444446411 (not minimal prime, since 444444441 and 444641 are primes) ***** For the 4{4,6}461 family, since 661 is prime, we only need to consider the family 4{4}461 ****** The smallest prime of the form 4{4}461 is 4444444461 (not minimal prime, since 444444441 is prime) **** For the 4{4}6{4}1 family, since 6441 is prime, we only need to consider the families 4{4}61 and 4{4}641 (since any digits combo 44 between (4{4}6,1) will produce smaller primes) ***** The smallest prime of the form 4{4}61 is 4444444461 (not minimal prime, since 444444441 is prime) ***** The smallest prime of the form 4{4}641 is 444641 * Case (4,3): ** Since 45, 13, 23, 53, 73, 433, 463 are primes, we only need to consider the family 4{0,4}3 (since any digits 1, 2, 3, 5, 6, 7 between them will produce smaller primes) *** Since 4043 and 4443 are primes, we only need to consider the families 4{0}3 and 44{0}3 (since any digits combo 04, 44 between them will produce smaller primes) **** All numbers of the form 4{0}3 are divisible by 7, thus cannot be prime. **** All numbers of the form 44{0}3 are divisible by 3, thus cannot be prime. * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (4,7): ** Since 45, 27, 37, 57, 407, 417, 467 are primes, we only need to consider the family 4{4,7}7 (since any digits 0, 1, 2, 3, 5, 6 between them will produce smaller primes) *** Since 747 is prime, we only need to consider the families 4{4}7, 4{4}77, 4{7}7, 44{7}7 (since any digits combo 74 between (4,7) will produce smaller primes) **** The smallest prime of the form 4{4}7 is 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447, with 220 4's, which can be written as (4^220)7 and equal the prime (4*8^221+17)/7 ([http://factordb.com/index.php?id=1100000000416605822 factordb]) **** The smallest prime of the form 4{4}77 is 4444477 **** The smallest prime of the form 4{7}7 is 47777 **** The smallest prime of the form 44{7}7 is 4477777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777, with 851 7's, which can be written as 44(7^851) and equal the prime 37*8^851-1 ([http://factordb.com/index.php?id=1100000000413677646 factordb]) (not minimal prime, since 47777 is prime) * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,3): ** 53 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 53, 57, 15, 35, 45, 65, 75 are primes, we only need to consider the family 5{0,2,5}5 (since any digits 1, 3, 4, 6, 7 between them will produce smaller primes) *** Since 225, 255, 5205 are primes, we only need to consider the families 5{0,5}5 and 5{0,5}25 (since any digits combo 20, 22, 25 between them will produce smaller primes) **** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. **** For the 5{0,5}25 family, since 500025 and 505525 are primes, we only need to consider the number 500525 the families 5{5}25, 5{5}025, 5{5}0025, 5{5}0525, 5{5}00525, 5{5}05025 (since any digits combo 000, 055 between (5,25) will produce smaller primes) ***** 500525 is not prime. ***** The smallest prime of the form 5{5}25 is 555555555555525 ***** The smallest prime of the form 5{5}025 is 55555025 ***** The smallest prime of the form 5{5}0025 is 5555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555550025, with 184 5's, which can be written as (5^183)0025 and equal the prime (5*8^187-20333)/7 ([http://factordb.com/index.php?id=1100000002350205912 factordb]) (not minimal prime, since 55555025 and 555555555555525 are primes) ***** The smallest prime of the form 5{5}0525 is 5550525 ***** The smallest prime of the form 5{5}00525 is 5500525 ***** The smallest prime of the form 5{5}05025 is 5555555555555555555555505025 (not minimal prime, since 5550525, 55555025, and 555555555555525 are primes) * Case (5,7): ** 57 is prime, and thus the only minimal prime in this family. * Case (6,1): ** Since 65, 21, 51, 631, 661 are primes, we only need to consider the family 6{0,1,4,7}1 (since any digits 2, 3, 5, 6 between them will produce smaller primes) *** Numbers containing 4: (note that the number cannot contain two or more 4's, or 6441 will be a subsequence) **** The form is 6{0,1,7}4{0,1,7}1 ***** Since 141, 401, 471 are primes, we only need to consider the family 6{0,7}4{1}1 ****** Since 111 is prime, we only need to consider the families 6{0,7}41 and 6{0,7}411 ******* For the 6{0,7}41 family, since 60741 is prime, we only need to consider the family 6{7}{0}41 ******** Since 6777 is prime, we only need to consider the families 6{0}41, 67{0}41, 677{0}41 ********* All numbers of the form 6{0}41 are divisible by 3, thus cannot be prime. ********* All numbers of the form 67{0}41 are divisible by 13, thus cannot be prime. ********* All numbers of the form 677{0}41 are divisible by 3, thus cannot be prime. ******* For the 6{0,7}411 family, since 60411 is prime, we only need to consider the family 6{7}411 ******** The smallest prime of the form 6{7}411 is 67777411 (not minimal prime, since 6777 is prime) *** Numbers not containing 4: **** The form is 6{0,1,7}1 ***** Since 111 is prime, we only need to consider the families 6{0,7}1 and 6{0,7}1{0,7}1 ****** All numbers of the form 6{0,7}1 are divisible by 7, thus cannot be prime. ****** For the 6{0,7}1{0,7}1 family, since 711 and 6101 are primes, we only need to consider the family 6{0}1{7}1 ******* Since 60171 is prime, we only need to consider the families 6{0}11 and 61{7}1 ******** All numbers of the form 6{0}11 are divisible by 3, thus cannot be prime. ******** The smallest prime of the form 61{7}1 is 617771 (not minimal prime, since 6777 is prime) * Case (6,3): ** Since 65, 13, 23, 53, 73, 643 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes) *** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (6,5): ** 65 is prime, and thus the only minimal prime in this family. * Case (6,7): ** Since 65, 27, 37, 57, 667 are primes, we only need to consider the family 6{0,1,4,7}7 (since any digits 2, 3, 5, 6 between them will produce smaller primes) *** Since 107, 117, 147, 177, 407, 417, 717, 747, 6007, 6477, 6707, 6777 are primes, there cannot be digits combo 00, 10, 11, 14, 17, 40, 41, 47, 70, 71, 74, 77 between them **** If there is 1 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}17 ***** All numbers of the form 6{0}17 are divisible by 3, thus cannot be prime. **** If there is 7 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}77 ***** All numbers of the form 6{0}77 are divisible by 3, thus cannot be prime. **** If there is neither 1 nor 7 between them, then the form is 6{0,4}7 ***** Since 6007, 407 at primes, we only need to consider the families 6{4}7 and 60{4}7 (since any digits combo 00, 40 between them will produce smaller primes) ****** All numbers of the form 6{4}7 are divisible by 3, 5, or 15, thus cannot be prime. ****** All numbers of the form 60{4}7 are divisible by 21, thus cannot be prime. * Case (7,1): ** Since 73, 75, 21, 51, 701, 711 are primes, we only need to consider the family 7{4,6,7}1 (since any digits 0, 1, 2, 3, 5 between them will produce smaller primes) *** Since 747, 767, 471, 661, 7461, 7641 are primes, we only need to consider the families 7{4,7}4{4}1, 7{7}61, 7{7}7{4,6,7}1 (since any digits combo 46, 47, 64, 66, 67 between them will produce smaller primes) **** For the 7{4,7}4{4}1 family, since 747, 471 are primes, we only need to consider the family 7{7}{4}1 (since any digits combo 47 between (7,4{4}1) will produce smaller primes) ***** The smallest prime of the form 7{7}1 is 7777777777771 ***** The smallest prime of the form 7{7}41 is 777777777777777777777777777777777777777777777777777777777777777777777777777777741, with 79 7's, which can be written as (7^79)41 and equal the prime 8^81-31 ([http://factordb.com/index.php?id=1100000000294462449 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777441, with 84 7's, which can be written as (7^84)441 and equal the prime 8^87-223 ([http://factordb.com/index.php?id=1100000000294462776 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}4441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777774441, with 233 7's, which can be written as (7^233)4441 and equal the prime 8^237-1759 ([http://factordb.com/index.php?id=1100000002352073382 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}44441 is 7777777777777777777777777777777777777777777777777777777744441, with 56 7's, which can be written as (7^56)44441 and equal the prime 8^61-14047 ([http://factordb.com/index.php?id=1100000002350250002 factordb]) (not minimal prime, since 7777777777771 is prime) ***** All numbers of the form 7{7}444441 are divisible by 7, thus cannot be prime. ***** The smallest prime of the form 7{7}4444441 is 77774444441 ****** Since this prime has just 4 7's, we only need to consider the families with <=3 7's ******* The smallest prime of the form 7{4}1 is 744444441 ******* All numbers of the form 77{4}1 are divisible by 5, thus cannot be prime. ******* The smallest prime of the form 777{4}1 is 777444444444441 (not minimal prime, since 444444441 and 744444441 are primes) * Case (7,3): ** 73 is prime, and thus the only minimal prime in this family. * Case (7,5): ** 75 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 73, 75, 27, 37, 57, 717, 747, 767 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. ===Base 10=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (1,7), (1,9), (2,1), (2,3), (2,7), (2,9), (3,1), (3,3), (3,7), (3,9), (4,1), (4,3), (4,7), (4,9), (5,1), (5,3), (5,7), (5,9), (6,1), (6,3), (6,7), (6,9), (7,1), (7,3), (7,7), (7,9), (8,1), (8,3), (8,7), (8,9), (9,1), (9,3), (9,7), (9,9) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (1,7): ** 17 is prime, and thus the only minimal prime in this family. * Case (1,9): ** 19 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 29, 11, 31, 41, 61, 71, 251, 281 are primes, we only need to consider the family 2{0,2}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 2221 and 20201 are primes, we only need to consider the families 2{0}1, 2{0}21, 22{0}1 (since any digits combo 22 or 020 between them will produce smaller primes) **** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. **** The smallest prime of the form 2{0}21 is 20021 **** The smallest prime of the form 22{0}1 is 22000001 * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,7): ** Since 23, 29, 17, 37, 47, 67, 97, 227, 257, 277 are primes, we only need to consider the family 2{0,8}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 9 between them will produce smaller primes) *** Since 887 and 2087 are primes, we only need to consider the families 2{0}7 and 28{0}7 (since any digit combo 08 or 88 between them will produce smaller primes) **** All numbers of the form 2{0}7 are divisible by 3, thus cannot be prime. **** All numbers of the form 28{0}7 are divisible by 7, thus cannot be prime. * Case (2,9): ** 29 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 31, 37, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 3{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 3{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (3,9): ** Since 31, 37, 19, 29, 59, 79, 89, 349 are primes, we only need to consider the family 3{0,3,6,9}9 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 3{0,3,6,9}9 are divisible by 3, thus cannot be prime. * Case (4,1): ** 41 is prime, and thus the only minimal prime in this family. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,7): ** 47 is prime, and thus the only minimal prime in this family. * Case (4,9): ** Since 41, 43, 47, 19, 29, 59, 79, 89, 409, 449, 499 are primes, we only need to consider the family 4{6}9 (since any digits 0, 1, 2, 3, 4, 5, 7, 8, 9 between them will produce smaller primes) *** All numbers of the form 4{6}9 are divisible by 7, thus cannot be prime. * Case (5,1): ** Since 53, 59, 11, 31, 41, 61, 71, 521 are primes, we only need to consider the family 5{0,5,8}1 (since any digits 1, 2, 3, 4, 6, 7, 9 between them will produce smaller primes) *** Since 881 is prime, we only need to consider the families 5{0,5}1 and 5{0,5}8{0,5}1 (since any digit combo 88 between them will produce smaller primes) **** For the 5{0,5}1 family, since 5051 and 5501 are primes, we only need to consider the families 5{0}1 and 5{5}1 (since any digit combo 05 or 50 between them will produce smaller primes) ***** All numbers of the form 5{0}1 are divisible by 3, thus cannot be prime. ***** The smallest prime of the form 5{5}1 is 555555555551 **** For the 5{0,5}8{0,5}1 family, since 5081, 5581, 5801, 5851 are primes, we only need to consider the number 581 ***** 581 is not prime. * Case (5,3): ** 53 is prime, and thus the only minimal prime in this family. * Case (5,7): ** Since 53, 59, 17, 37, 47, 67, 97, 557, 577, 587 are primes, we only need to consider the family 5{0,2}7 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 227 and 50207 are primes, we only need to consider the families 5{0}7, 5{0}27, 52{0}7 (since any digits combo 22 or 020 between them will produce smaller primes) **** All numbers of the form 5{0}7 are divisible by 3, thus cannot be prime. **** The smallest prime of the form 5{0}27 is 5000000000000000000000000000027 **** The smallest prime of the form 52{0}7 is 5200007 * Case (5,9): ** 59 is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,3): ** Since 61, 67, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 6{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 6{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (6,7): ** 67 is prime, and thus the only minimal prime in this family. * Case (6,9): ** Since 61, 67, 19, 29, 59, 79, 89 are primes, we only need to consider the family 6{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes) *** Since 449 is prime, we only need to consider the families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes) **** All numbers of the form 6{0,3,6,9}9 are divisible by 3, thus cannot be prime. **** For the 6{0,3,6,9}4{0,3,6,9}9 family, since 409, 43, 6469, 499 are primes, we only need to consider the family 6{0,3,6,9}49 ***** Since 349, 6949 are primes, we only need to consider the family 6{0,6}49 ****** Since 60649 is prime, we only need to consider the family 6{6}{0}49 (since any digits combo 06 between {6,49} will produce smaller primes) ******* The smallest prime of the form 6{6}49 is 666649 ******** Since this prime has just 4 6's, we only need to consider the families with <=3 6's ********* The smallest prime of the form 6{0}49 is 60000049 ********* The smallest prime of the form 66{0}49 is 66000049 ********* The smallest prime of the form 666{0}49 is 66600049 * Case (7,1): ** 71 is prime, and thus the only minimal prime in this family. * Case (7,3): ** 73 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 71, 73, 79, 17, 37, 47, 67, 97, 727, 757, 787 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9 between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. * Case (7,9): ** 79 is prime, and thus the only minimal prime in this family. * Case (8,1): ** Since 83, 89, 11, 31, 41, 61, 71, 821, 881 are primes, we only need to consider the family 8{0,5}1 (since any digits 1, 2, 3, 4, 6, 7, 8, 9 between them will produce smaller primes) *** Since 8501 is prime, we only need to consider the family 8{0}{5}1 (since any digits combo 50 between them will produce smaller primes) **** Since 80051 is prime, we only need to consider the families 8{0}1, 8{5}1, 80{5}1 (since any digits combo 005 between them will produce smaller primes) ***** All numbers of the form 8{0}1 are divisible by 3, thus cannot be prime. ***** The smallest prime of the form 8{5}1 is 8555555555555555555551 (not minimal prime, since 555555555551 is prime) ***** The smallest prime of the form 80{5}1 is 80555551 * Case (8,3): ** 83 is prime, and thus the only minimal prime in this family. * Case (8,7): ** Since 83, 89, 17, 37, 47, 67, 97, 827, 857, 877, 887 are primes, we only need to consider the family 8{0}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** All numbers of the form 8{0}7 are divisible by 3, thus cannot be prime. * Case (8,9): ** 89 is prime, and thus the only minimal prime in this family. * Case (9,1): ** Since 97, 11, 31, 41, 61, 71, 991 are primes, we only need to consider the family 9{0,2,5,8}1 (since any digits 1, 3, 4, 6, 7, 9 between them will produce smaller primes) *** Since 251, 281, 521, 821, 881, 9001, 9221, 9551, 9851 are primes, we only need to consider the families 9{2,5,8}0{2,5,8}1, 9{0}2{0}1, 9{0}5{0,8}1, 9{0,5}8{0}1 (since any digits combo 00, 22, 25, 28, 52, 55, 82, 85, 88 between them will produce smaller primes) **** For the 9{2,5,8}0{2,5,8}1 family, since any digits combo 22, 25, 28, 52, 55, 82, 85, 88 between (9,1) will produce smaller primes, we only need to consider the numbers 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801 ***** 95801 is the only prime among 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. **** For the 9{0}2{0}1 family, since 9001 is prime, we only need to consider the numbers 921, 9201, 9021 ***** None of 921, 9201, 9021 are primes. **** For the 9{0}5{0,8}1 family, since 9001 and 881 are primes, we only need to consider the numbers 951, 9051, 9501, 9581, 90581, 95081, 95801 ***** 95801 is the only prime among 951, 9051, 9501, 9581, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. **** For the 9{0,5}8{0}1 family, since 9001 and 5581 are primes, we only need to consider the numbers 981, 9081, 9581, 9801, 90581, 95081, 95801 ***** 95801 is the only prime among 981, 9081, 9581, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. * Case (9,3): ** Since 97, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 9{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 9{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (9,7): ** 97 is prime, and thus the only minimal prime in this family. * Case (9,9): ** Since 97, 19, 29, 59, 79, 89 are primes, we only need to consider the family 9{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes) *** Since 449 is prime, we only need to consider the families 9{0,3,6,9}9 and 9{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes) **** All numbers of the form 9{0,3,6,9}9 are divisible by 3, thus cannot be prime. **** For the 9{0,3,6,9}4{0,3,6,9}9 family, since 9049, 349, 9649, 9949 are primes, we only need to consider the family 94{0,3,6,9}9 ***** Since 409, 43, 499 are primes, we only need to consider the family 94{6}9 (since any digits 0, 3, 9 between (94,9) will produce smaller primes) ****** The smallest prime of the form 94{6}9 is 946669 ===Base 12=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,5), (1,7), (1,B), (2,1), (2,5), (2,7), (2,B), (3,1), (3,5), (3,7), (3,B), (4,1), (4,5), (4,7), (4,B), (5,1), (5,5), (5,7), (5,B), (6,1), (6,5), (6,7), (6,B), (7,1), (7,5), (7,7), (7,B), (8,1), (8,5), (8,7), (8,B), (9,1), (9,5), (9,7), (9,B), (A,1), (A,5), (A,7), (A,B), (B,1), (B,5), (B,7), (B,B) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (1,7): ** 17 is prime, and thus the only minimal prime in this family. * Case (1,B): ** 1B is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 25, 27, 11, 31, 51, 61, 81, 91, 221, 241, 2A1, 2B1 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B between them will produce smaller primes) *** The smallest prime of the form 2{0}1 is 2001 * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (2,7): ** 27 is prime, and thus the only minimal prime in this family. * Case (2,B): ** Since 25, 27, 1B, 3B, 4B, 5B, 6B, 8B, AB, 2BB are primes, we only need to consider the family 2{0,2,9}B (since any digits 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes) *** Since 90B, 200B, 202B, 222B, 229B, 292B, 299B are primes, we only need to consider the numbers 20B, 22B, 29B, 209B, 220B (since any digits combo 00, 02, 22, 29, 90, 92, 99 between them will produce smaller primes) **** None of 20B, 22B, 29B, 209B, 220B are primes. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (3,B): ** 3B is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 4B, 11, 31, 51, 61, 81, 91, 401, 421, 471 are primes, we only need to consider the family 4{4,A}1 (since any digit 0, 1, 2, 3, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since A41 and 4441 are primes, we only need to consider the families 4{A}1 and 44{A}1 (since any digit combo 44, A4 between them will produce smaller primes) **** All numbers of the form 4{A}1 are divisible by 5, thus cannot be prime. **** The smallest prime of the form 44{A}1 is 44AAA1 * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (4,7): ** Since 45, 4B, 17, 27, 37, 57, 67, 87, A7, B7, 447, 497 are primes, we only need to consider the family 4{0,7}7 (since any digit 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes) *** Since 4707 and 4777 are primes, we only need to consider the families 4{0}7 and 4{0}77 (since any digit combo 70, 77 between them will produce smaller primes) **** All numbers of the form 4{0}7 are divisible by B, thus cannot be prime. **** The smallest prime of the form 4{0}77 is 400000000000000000000000000000000000000077 * Case (4,B): ** 4B is prime, and thus the only minimal prime in this family. * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 57, 5B, 15, 25, 35, 45, 75, 85, 95, B5, 565 are primes, we only need to consider the family 5{0,5,A}5 (since any digits 1, 2, 3, 4, 6, 7, 8, 9, B between them will produce smaller primes) *** All numbers of the form 5{0,5,A}5 are divisible by 5, thus cannot be prime. * Case (5,7): ** 57 is prime, and thus the only minimal prime in this family. * Case (5,B): ** 5B is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,5): ** Since 61, 67, 6B, 15, 25, 35, 45, 75, 85, 95, B5, 655, 665 are primes, we only need to consider the family 6{0,A}5 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since 6A05 and 6AA5 are primes, we only need to consider the families 6{0}5 and 6{0}A5 (since any digit combo A0, AA between them will produce smaller primes) **** All numbers of the form 6{0}5 are divisible by B, thus cannot be prime. **** The smallest prime of the form 6{0}A5 is 600A5 * Case (6,7): ** 67 is prime, and thus the only minimal prime in this family. * Case (6,B): ** 6B is prime, and thus the only minimal prime in this family. * Case (7,1): ** Since 75, 11, 31, 51, 61, 81, 91, 701, 721, 771, 7A1 are primes, we only need to consider the family 7{4,B}1 (since any digits 0, 1, 2, 3, 5, 6, 7, 8, 9, A between them will produce smaller primes) *** Since 7BB, 7441 and 7B41 are primes, we only need to consider the numbers 741, 7B1, 74B1 **** None of 741, 7B1, 74B1 are primes. * Case (7,5): ** 75 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 75, 17, 27, 37, 57, 67, 87, A7, B7, 747, 797 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. * Case (7,B): ** Since 75, 1B, 3B, 4B, 5B, 6B, 8B, AB, 70B, 77B, 7BB are primes, we only need to consider the family 7{2,9}B (since any digits 0, 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes) *** Since 222B, 729B is prime, we only need to consider the families 7{9}B, 7{9}2B, 7{9}22B (since any digits combo 222, 29 between them will produce smaller primes) **** The smallest prime of the form 7{9}B is 7999B **** The smallest prime of the form 7{9}2B is 79992B (not minimal prime, since 992B and 7999B are primes) **** The smallest prime of the form 7{9}22B is 79922B (not minimal prime, since 992B is prime) * Case (8,1): ** 81 is prime, and thus the only minimal prime in this family. * Case (8,5): ** 85 is prime, and thus the only minimal prime in this family. * Case (8,7): ** 87 is prime, and thus the only minimal prime in this family. * Case (8,B): ** 8B is prime, and thus the only minimal prime in this family. * Case (9,1): ** 91 is prime, and thus the only minimal prime in this family. * Case (9,5): ** 95 is prime, and thus the only minimal prime in this family. * Case (9,7): ** Since 91, 95, 17, 27, 37, 57, 67, 87, A7, B7, 907 are primes, we only need to consider the family 9{4,7,9}7 (since any digit 0, 1, 2, 3, 5, 6, 8, A, B between them will produce smaller primes) *** Since 447, 497, 747, 797, 9777, 9947, 9997 are primes, we only need to consider the numbers 947, 977, 997, 9477, 9977 (since any digits combo 44, 49, 74, 77, 79, 94, 99 between them will produce smaller primes) **** None of 947, 977, 997, 9477, 9977 are primes. * Case (9,B): ** Since 91, 95, 1B, 3B, 4B, 5B, 6B, 8B, AB, 90B, 9BB are primes, we only need to consider the family 9{2,7,9}B (since any digit 0, 1, 3, 4, 5, 6, 8, A, B between them will produce smaller primes) *** Since 27, 77B, 929B, 992B, 997B are primes, we only need to consider the families 9{2,7}2{2}B, 97{2,9}B, 9{7,9}9{9}B (since any digits combo 27, 29, 77, 92, 97 between them will produce smaller primes) **** For the 9{2,7}2{2}B family, since 27 and 77B are primes, we only need to consider the families 9{2}2{2}B and 97{2}2{2}B (since any digits combo 27, 77 between (9,2{2}B) will produce smaller primes) ***** The smallest prime of the form 9{2}2{2}B is 9222B (not minimal prime, since 222B is prime) ***** The smallest prime of the form 97{2}2{2}B is 9722222222222B (not minimal prime, since 222B is prime) **** For the 97{2,9}B family, since 729B and 929B are primes, we only need to consider the family 97{9}{2}B (since any digits combo 29 between (97,B) will produce smaller primes) ***** Since 222B is prime, we only need to consider the families 97{9}B, 97{9}2B, 97{9}22B (since any digit combo 222 between (97,B) will produce smaller primes) ****** All numbers of the form 97{9}B are divisible by 11, thus cannot be prime. ****** The smallest prime of the form 97{9}2B is 979999992B (not minimal prime, since 9999B is prime) ****** All numbers of the form 97{9}22B are divisible by 11, thus cannot be prime. **** For the 9{7,9}9{9}B family, since 77B and 9999B are primes, we only need to consider the numbers 99B, 999B, 979B, 9799B, 9979B ***** None of 99B, 999B, 979B, 9799B, 9979B are primes. * Case (A,1): ** Since A7, AB, 11, 31, 51, 61, 81, 91, A41 are primes, we only need to consider the family A{0,2,A}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since 221, 2A1, A0A1, A201 are primes, we only need to consider the families A{A}{0}1 and A{A}{0}21 (since any digits combo 0A, 20, 22, 2A between them will produce smaller primes) **** For the A{A}{0}1 family: ***** All numbers of the form A{0}1 are divisible by B, thus cannot be prime. ***** The smallest prime of the form AA{0}1 is AA000001 ***** The smallest prime of the form AAA{0}1 is AAA0001 ***** The smallest prime of the form AAAA{0}1 is AAAA1 ****** Since this prime has no 0's, we do not need to consider the families {A}1, {A}01, {A}001, etc. **** All numbers of the form A{A}{0}21 are divisible by 5, thus cannot be prime. * Case (A,5): ** Since A7, AB, 15, 25, 35, 45, 75, 85, 95, B5 are primes, we only need to consider the family A{0,5,6,A}5 (since any digits 1, 2, 3, 4, 7, 8, 9, B between them will produce smaller primes) *** Since 565, 655, 665, A605, A6A5, AA65 are primes, we only need to consider the families A{0,5,A}5 and A{0}65 (since any digits combo 56, 60, 65, 66, 6A, A6 between them will produce smaller primes) **** All numbers of the form A{0,5,A}5 are divisible by 5, thus cannot be prime. **** The smallest prime of the form A{0}65 is A00065 * Case (A,7): ** A7 is prime, and thus the only minimal prime in this family. * Case (A,B): ** AB is prime, and thus the only minimal prime in this family. * Case (B,1): ** Since B5, B7, 11, 31, 51, 61, 81, 91, B21 are primes, we only need to consider the family B{0,4,A,B}1 (since any digits 1, 2, 3, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 4B, AB, 401, A41, B001, B0B1, BB01, BB41 are primes, we only need to consider the families B{A}0{4,A}1, B{0,4}4{4,A}1, B{0,4,A,B}A{0,A}1, B{B}B{A,B}1 (since any digits combo 00, 0B, 40, 4B, A4, AB, B0, B4 between them will produce smaller primes) **** For the B{A}0{4,A}1 family, since A41 is prime, we only need consider the families B0{4}{A}1 and B{A}0{A}1 ***** For the B0{4}{A}1 family, since B04A1 is prime, we only need to consider the families B0{4}1 and B0{A}1 ****** The smallest prime of the form B0{4}1 is B04441 (not minimal prime, since 4441 is prime) ****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime) ***** For the B{A}0{A}1 family, since A0A1 is prime, we only need to consider the families B{A}01 and B0{A}1 ****** The smallest prime of the form B{A}01 is BAA01 ****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime) **** For the B{0,4}4{4,A}1 family, since 4441 is prime, we only need to consider the families B{0}4{4,A}1 and B{0,4}4{A}1 ***** For the B{0}4{4,A}1 family, since B001 is prime, we only need to consider the families B4{4,A}1 and B04{4,A}1 ****** For the B4{4,A}1 family, since A41 is prime, we only need to consider the family B4{4}{A}1 ******* Since 4441 and BAAA1 are primes, we only need to consider the numbers B41, B441, B4A1, B44A1, B4AA1, B44AA1 ******** None of B41, B441, B4A1, B44A1, B4AA1, B44AA1 are primes. ****** For the B04{4,A}1 family, since B04A1 is prime, we only need to consider the family B04{4}1 ******* The smallest prime of the form B04{4}1 is B04441 (not minimal prime, since 4441 is prime) ***** For the B{0,4}4{A}1 family, since 401, 4441, B001 are primes, we only need to consider the families B4{A}1, B04{A}1, B44{A}1, B044{A}1 (since any digits combo 00, 40, 44 between (B,4{A}1) will produce smaller primes) ****** The smallest prime of the form B4{A}1 is B4AAA1 (not minimal prime, since BAAA1 is prime) ****** The smallest prime of the form B04{A}1 is B04A1 ****** The smallest prime of the form B44{A}1 is B44AAAAAAA1 (not minimal prime, since BAAA1 is prime) ****** The smallest prime of the form B044{A}1 is B044A1 (not minimal prime, since B04A1 is prime) **** For the B{0,4,A,B}A{0,A}1 family, since all numbers in this family with 0 between (B,1) are in the B{A}0{4,A}1 family, and all numbers in this family with 4 between (B,1) are in the B{0,4}4{4,A}1 family, we only need to consider the family B{A,B}A{A}1 ***** Since BAAA1 is prime, we only need to consider the families B{A,B}A1 and B{A,B}AA1 ****** For the B{A,B}A1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}A1 and B{B}AA1 ******* All numbers of the form B{B}A1 are divisible by B, thus cannot be prime. ******* The smallest prime of the form B{B}AA1 is BBBAA1 ****** For the B{A,B}AA1 family, since BAAA1 is prime, we only need to consider the families B{B}AA1 ******* The smallest prime of the form B{B}AA1 is BBBAA1 **** For the B{B}B{A,B}1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}B1, B{B}BA1, B{B}BAA1 (since any digits combo AB or AAA between (B{B}B,1) will produce smaller primes) ***** The smallest prime of the form B{B}B1 is BBBB1 ***** All numbers of the form B{B}BA1 are divisible by B, thus cannot be prime. ***** The smallest prime of the form B{B}BAA1 is BBBAA1 * Case (B,5): ** B5 is prime, and thus the only minimal prime in this family. * Case (B,7): ** B7 is prime, and thus the only minimal prime in this family. * Case (B,B): ** Since B5, B7, 1B, 3B, 4B, 5B, 6B, 8B, AB, B2B are primes, we only need to consider the family B{0,9,B}B (since any digits 1, 2, 3, 4, 5, 6, 7, 8, A between them will produce smaller primes) *** Since 90B and 9BB are primes, we only need to consider the families B{0,B}{9}B **** Since 9999B is prime, we only need to consider the families B{0,B}B, B{0,B}9B, B{0,B}99B, B{0,B}999B ***** All numbers of the form B{0,B}B are divisible by B, thus cannot be prime. ***** For the B{0,B}9B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}9B and B{B}9B (since any digits combo 0B, B0 between (B,9B) will produce smaller primes) ******* The smallest prime of the form B{0}9B is B0000000000000000000000000009B ******* All numbers of the from B{B}9B is either divisible by 11 (if totally number of B's is even) or factored as 10^(2*n)-21 = (10^n-5) * (10^n+5) (if totally number of B's is odd number 2*n-1 (n≥1)) (and since if n≥1, 10^n-5 ≥ 10^1-5 = 7 > 1, 10^n+5 ≥ 10^1+5 = 15 > 1, this factorization is nontrivial), thus cannot be prime. ***** For the B{0,B}99B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}99B and B{B}99B (since any digits combo 0B, B0 between (B,99B) will produce smaller primes) ******* The smallest prime of the form B{0}99B is B00099B ******* The smallest prime of the form B{B}99B is BBBBBB99B ***** For the B{0,B}999B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}999B and B{B}999B (since any digits combo 0B, B0 between (B,999B) will produce smaller primes) ******* The smallest prime of the form B{0}999B is B0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000999B, with 1765 0's, which can be written as B(0^1765)999B and equal the prime 11*12^1769+16967 ([http://factordb.com/index.php?id=1100000002378273165 factordb]) ([http://factordb.com/cert.php?id=1100000002378273165 primality certificate]) (not minimal prime, since B00099B and B0000000000000000000000000009B are primes) ******* The smallest prime of the form B{B}999B is BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB999B, with 245 B's, which can be written as (B^244)999B and equal the prime 12^248-3769 ([http://factordb.com/index.php?id=1100000002378270237 factordb]) (not minimal prime, since BBBBBB99B is prime) == Examples of families which can be ruled out as contain no primes > ''b'' == It is not known if this problem is solvable: Problem: Given strings ''x'', ''y'', ''z'', and a base ''b'', does there exist a prime number whose base-''b'' expansion is of the form ''x''{''y''}''z''? It will be necessary for our algorithm to determine if families of the form ''x''{''y''}''z'' contain a prime > ''b'' or not. We use two different heuristic strategies to show that such families contain no primes > ''b''. In the first strategy, we mimic the well-known technique of “covering congruences”, by finding some finite set ''S'' of primes ''p'' such that every number in a given family is divisible by some element of ''S''. In the second strategy, we attempt to find an algebraic factorization, such as difference-of-squares factorization, difference-of-cubes factorization, and Aurifeuillian factorization for numbers of the form ''x''⁴+4''y''⁴. Examples of first strategy: (we can show that the corresponding numbers are > all elements in ''S'', if ''n'' makes corresponding numbers > ''b'' (i.e. ''n''≥1 for 5{1} in base 9 and 2{5} in base 11 and {4}D in base 16 and {8}F in base 16, ''n''≥0 for other examples), thus these factorizations are nontrivial) * In base 10, all numbers of the form 4{6}9 are divisible by 7 * In base 6, all numbers of the form 4{0}1 are divisible by 5 * In base 15, all numbers of the form 9{6}8 are divisible by 11 * In base 9, all numbers of the form 5{1} are divisible by some element of {2, 5} * In base 11, all numbers of the form 2{5} are divisible by some element of {2, 3} * In base 14, all numbers of the form B{0}1 are divisible by some element of {3, 5} * In base 8, all numbers of the form 6{4}7 are divisible by some element of {3, 5, 13} * In base 13, all numbers of the form 3{0}95 are divisible by some element of {5, 7, 17} * In base 16, all numbers of the form {4}D are divisible by some element of {3, 7, 13} * In base 16, all numbers of the form {8}F are divisible by some element of {3, 7, 13} Examples of second strategy: (we can show that both factors are > 1, if ''n'' makes corresponding numbers > ''b'' (i.e. ''n''≥2 for {1} in base 9, ''n''≥0 for 1{0}1 in base 8 and B{4}1 in base 16, ''n''≥1 for other examples), thus these factorizations are nontrivial) * In base 9, all numbers of the form {1} factored as difference of squares * In base 8, all numbers of the form 1{0}1 factored as sum of cubes * In base 9, all numbers of the form 3{8} factored as difference of squares * In base 16, all numbers of the form 8{F} factored as difference of squares * In base 16, all numbers of the form {F}7 factored as difference of squares * In base 9, all numbers of the form 3{1} factored as difference of squares * In base 16, all numbers of the form {4}1 factored as difference of squares * In base 16, all numbers of the form 1{5} factored as difference of squares * In base 16, all numbers of the from {C}D factored as ''x''⁴+4''y''⁴ * In base 16, all numbers of the form B{4}1 factored as difference of squares Examples of combine of the two strategies: (we can show that for the part of the first strategy, the corresponding numbers are > all elements in S, and for the part of the second strategy, both factors are > 1, if n makes corresponding numbers > b, thus these factorizations are nontrivial) * In base 14, numbers of the form 8{D} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 12, numbers of the form {B}9B are divisible by 13 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 14, numbers of the form {D}5 are divisible by 5 if ''n'' is even and factored as difference of squares if ''n'' is odd * In base 17, numbers of the form 1{9} are divisible by 2 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 19, numbers of the form 1{6} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even == Bases 2≤''b''≤1024 such that these families can be ruled out as contain no primes > ''b'' == (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) === 1{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-powers factorization === 1{0}2 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 1{0}3 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} === 1{0}4 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ === 1{0}5 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 0 mod 5: Finite covering set {5} === 1{0}6 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 7: Finite covering set {7} === 1{0}7 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 7: Finite covering set {7} === 1{0}z === (none) === 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === * ''b'' == 1 mod 3: Finite covering set {3} === 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) === (none) === 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === * ''b'' == 1 mod 3: Finite covering set {3} === {1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z) === * ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841) === {1} === * ''b'' = ''m''^''r'' with ''r''>1: Difference-of-''r''th-powers factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 4 (length 2), 8 (length 3), 16 (length 2), 27 (length 3), 36 (length 2), 100 (length 2), 128 (length 7), 196 (length 2), 256 (length 2), 400 (length 2), 512 (length 3), 576 (length 2), 676 (length 2)) === {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) === * ''b'' == 0 mod 2: Finite covering set {2} === 1{2} === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49) === 1{3} === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' such that ''b'' and 3(''b''+2) are both squares: Difference-of-squares factorization (such bases are 25, 361) * ''b'' == 1 mod 2 such that 3(''b''+2) is square: Combine of finite covering set {2} (when length is even) and difference-of-squares factorization (when length is odd) (such bases are 25, 73, 145, 241, 361, 505, 673, 865) === 1{4} === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' such that ''b'' and 4(''b''+3) are both squares: Difference-of-squares factorization === 1{z} === (none) === 2{0}1 === * ''b'' == 1 mod 3: Finite covering set {3} === 2{0}3 === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 5: Finite covering set {5} === 2{1} (not quasi-minimal prime if there is smaller prime of the form {1}) === * ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841) === {2}1 === * ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49) === 2{z} === * ''b'' == 1 mod 2: Finite covering set {2} === 3{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} === 3{0}2 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} === 3{0}4 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 7: Finite covering set {7} === {3}1 === * ''b'' such that ''b'' and 3(2''b''+1) are both squares: Difference-of-squares factorization (such bases are 121) === 3{z} === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === 4{0}1 === * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ === 4{0}3 === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 7: Finite covering set {7} === {4}1 === * ''b'' such that ''b'' and 4(3''b''+1) are both squares: Difference-of-squares factorization (such bases are 16, 225) === 4{z} === * ''b'' == 1 mod 2: Finite covering set {2} === 5{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 5{z} === * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 34 mod 35: Finite covering set {5, 7} * ''b'' = 6''m''² with ''m'' == 2 or 3 mod 5: Combine of finite covering set {5} (when length is odd) and difference-of-squares factorization (when length is even) (such bases are 24, 54, 294, 384, 864, 1014) === 6{0}1 === * ''b'' == 1 mod 7: Finite covering set {7} * ''b'' == 34 mod 35: Finite covering set {5, 7} === 6{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 7{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} === 7{z} === * ''b'' == 1 mod 7: Finite covering set {7} * ''b'' == 20 mod 21: Finite covering set {3, 7} * ''b'' == 83, 307 mod 455: Finite covering set {5, 7, 13} (such bases are 83, 307, 538, 762, 993) * ''b'' = ''m''³: Difference-of-cubes factorization === 8{0}1 === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 20 mod 21: Finite covering set {3, 7} * ''b'' == 47, 83 mod 195: Finite covering set {3, 5, 13} (such bases are 47, 83, 242, 278, 437, 473, 632, 668, 827, 863, 1022) * ''b'' = 467: Finite covering set {3, 5, 7, 19, 37} * ''b'' = 722: Finite covering set {3, 5, 13, 73, 109} * ''b'' = ''m''³: Sum-of-cubes factorization * ''b'' = 128: Cannot have primes since 7''n''+3 cannot be power of 2 === 8{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === 9{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} === 9{z} === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 32 mod 33: Finite covering set {3, 11} === A{0}1 === * ''b'' == 1 mod 11: Finite covering set {11} * ''b'' == 32 mod 33: Finite covering set {3, 11} === A{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} === B{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} === B{z} === * ''b'' == 1 mod 11: Finite covering set {11} * ''b'' == 142 mod 143: Finite covering set {11, 13} * ''b'' = 307: Finite covering set {5, 11, 29} * ''b'' = 901: Finite covering set {7, 11, 13, 19} === C{0}1 === * ''b'' == 1 mod 13: Finite covering set {13} * ''b'' == 142 mod 143: Finite covering set {11, 13} * ''b'' = 296, 901: Finite covering set {7, 11, 13, 19} * ''b'' = 562, 828, 900: Finite covering set {7, 13, 19} * ''b'' = 563: Finite covering set {5, 7, 13, 19, 29} * ''b'' = 597: Finite covering set {5, 13, 29} === {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) === (none) === {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) === * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-power factorization === #{z} (for even bases b, # = b/2−1) === (none) === y{z} === (none) === {y}z === (none) === z{0}1 === (none) === {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) === * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-power factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 128 (length 7), 216 (length 3), 343 (length 3), 729 (length 3)) * ''b'' = 4''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ (base 4 still have primes, since for the corresponding length this factorization is trivial, but it only have this prime, at length 2) === {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) === (none) === {z}1 === (none) === {z}t === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 0 mod 7: Finite covering set {7} === {z}u === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 34 mod 35: Finite covering set {5, 7} === {z}v === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 5: Finite covering set {5} === {z}w === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === {z}x === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} === {z}y === * ''b'' == 0 mod 2: Finite covering set {2} == Large known (probable) primes (length ≥10000) in these families (for bases 2≤''b''≤1024) == Format: base (length) (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) === 1{0}1 === (none) === 1{0}2 === (none) === 1{0}3 === (none) === 1{0}4 === 53 (13403) 113 (10647) === 1{0}z === 113 (20089) 123 (64371) === 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === (none) === 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) === 208 (26682) 607 (11032) 828 (19659) === 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === 201 (31276) 222 (52727) 227 (36323) 327 (135983) 425 (11231) 710 (24112) 717 (37508) 719 (13420) === {1} === 152 (270217) 184 (16703) 200 (17807) 311 (36497) 326 (26713) 331 (25033) 371 (15527) 485 (99523) 629 (32233) 649 (43987) 670 (18617) 684 (22573) 691 (62903) 693 (41189) 731 (15427) 752 (32833) 872 (10093) 932 (20431) === {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) === (none) === 1{z} === 107 (21911) 170 (166429) 278 (43909) 303 (40175) 383 (20957) 515 (58467) 522 (62289) 578 (129469) 590 (15527) 647 (21577) 662 (16591) 698 (127559) 704 (62035) 845 (39407) 938 (40423) 969 (24097) 989 (26869) === 2{0}1 === 101 (192276) 206 (46206) 218 (333926) 236 (161230) 257 (12184) 305 (16808) 467 (126776) 578 (44166) 626 (174204) 695 (94626) 752 (26164) 788 (72918) 869 (49150) 887 (27772) 899 (15732) 932 (13644) === 2{z} === 432 (16003) === 3{0}1 === (none) === 3{z} === 72 (1119850) 212 (34414) 218 (23050) 270 (89662) 303 (198358) 312 (51566) 422 (21738) 480 (93610) 513 (38032) 527 (46074) 566 (23874) 650 (498102) 686 (16584) 758 (15574) 783 (12508) 800 (33838) 921 (98668) 947 (10056) === 4{0}1 === 107 (32587) 227 (13347) 257 (160423) 355 (10990) 410 (144079) 440 (56087) 452 (14155) 482 (30691) 542 (15983) 579 (67776) 608 (20707) 635 (11723) 650 (96223) 679 (69450) 737 (269303) 740 (58043) 789 (149140) 797 (468703) 920 (103687) 934 (101404) 962 (84235) === 4{z} === 14 (19699) 68 (13575) 254 (15451) 800 (20509) === 5{0}1 === 326 (400786) 350 (20392) 554 (10630) 662 (13390) 926 (40036) === 5{z} === 258 (212135) 272 (148427) 299 (64898) 307 (26263) 354 (25566) 433 (283919) 635 (36163) 678 (40859) 692 (45447) 719 (20552) 768 (70214) 857 (23083) 867 (61411) 972 (36703) === 6{0}1 === 108 (16318) 129 (16797) 409 (369833) 522 (52604) 587 (24120) 643 (164916) 762 (11152) 789 (27297) 986 (21634) === 6{z} === 68 (25396) 332 (15222) 338 (42868) 362 (146342) 488 (33164) 566 (164828) 980 (50878) 986 (12506) 1016 (23336) === 7{0}1 === 398 (17473) 1004 (54849) === 7{z} === 97 (192336) 170 (15423) 194 (38361) 202 (155772) 282 (21413) 283 (164769) 332 (13205) 412 (29792) 560 (19905) 639 (10668) 655 (53009) 811 (31784) 814 (17366) 866 (108591) 908 (61797) 962 (31841) 992 (10605) 997 (15815) === 8{0}1 === 23 (119216) 53 (227184) 158 (123476) 254 (67716) 320 (52004) 410 (279992) 425 (94662) 513 (19076) 518 (11768) 596 (148446) 641 (87702) 684 (23387) 695 (39626) 785 (900326) 788 (11408) 893 (86772) 908 (243440) 920 (107822) 962 (47222) 998 (81240) 1013 (43872) === 8{z} === 138 (35686) 412 (12154) 788 (11326) 990 (23032) === 9{0}1 === 248 (39511) 592 (96870) === 9{z} === 431 (43574) 446 (152028) 458 (126262) 599 (11776) 846 (12781) === A{0}1 === 173 (264235) 198 (47665) 311 (314807) 341 (106009) 449 (18507) 492 (42843) 605 (12395) 708 (17563) 710 (31039) 743 (285479) 744 (137056) 786 (68169) 800 (15105) 802 (149320) 879 (25004) 929 (13065) 977 (125873) 986 (48279) 1004 (10645) === A{z} === 368 (10867) 488 (10231) 534 (80328) 662 (13307) 978 (14066) === B{0}1 === 710 (15272) 740 (33520) 878 (227482) === B{z} === 153 (21660) 186 (112718) 439 (18752) 593 (16064) 602 (36518) 707 (10573) 717 (67707) === C{0}1 === 68 (656922) 219 (29231) 230 (94751) 312 (21163) 334 (83334) 353 (20262) 359 (61295) 457 (10024) 481 (45941) 501 (20140) 593 (42779) 600 (11242) 604 (17371) 641 (26422) 700 (91953) 887 (13961) 919 (45359) 923 (64365) 992 (10300) === {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) === (none) === {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) === (none) === #{z} (for even bases b, # = b/2−1) === (none) === y{z} === 38 (136212) 83 (21496) 113 (286644) 188 (13508) 401 (103670) 417 (21003) 458 (46900) 494 (21580) 518 (129372) 527 (65822) 602 (17644) 608 (36228) 638 (74528) 663 (47557) 723 (24536) 758 (50564) 833 (12220) 904 (13430) 938 (50008) 950 (16248) === z{0}1 === 202 (46774) 251 (102979) 272 (16681) 297 (14314) 298 (60671) 326 (64757) 347 (69661) 363 (142877) 452 (71941) 543 (10042) 564 (38065) 634 (84823) 788 (13541) 869 (12289) 890 (37377) 953 (60995) 1004 (29685) === {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) === 53 (21942) 124 (16426) 175 (31626) 188 (22036) 316 (48538) 365 (25578) 373 (24006) 434 (10090) 530 (11086) 545 (12346) 560 (15072) 596 (12762) 701 (12576) 706 (10656) 821 (13536) 833 (17116) 966 (14820) 983 (11272) === {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) === (none) === {z}1 === (none) === {z}y === 317 (13896) == Bases 2≤''b''≤1024 which have these families as unsolved families == Unsolved families are families which are neither primes (>''b'') found nor can be ruled out as contain no primes > ''b'' (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) 1{0}1: 38, 50, 62, 68, 86, 92, 98, 104, 122, 144, 168, 182, 186, 200, 202, 212, 214, 218, 244, 246, 252, 258, 286, 294, 298, 302, 304, 308, 322, 324, 338, 344, 354, 356, 362, 368, 380, 390, 394, 398, 402, 404, 410, 416, 422, 424, 446, 450, 454, 458, 468, 480, 482, 484, 500, 514, 518, 524, 528, 530, 534, 538, 552, 558, 564, 572, 574, 578, 580, 590, 602, 604, 608, 620, 622, 626, 632, 638, 648, 650, 662, 666, 668, 670, 678, 684, 692, 694, 698, 706, 712, 720, 722, 724, 734, 744, 746, 752, 754, 762, 766, 770, 792, 794, 802, 806, 812, 814, 818, 836, 840, 842, 844, 848, 854, 868, 870, 872, 878, 888, 896, 902, 904, 908, 922, 924, 926, 932, 938, 942, 944, 948, 954, 958, 964, 968, 974, 978, 980, 988, 994, 998, 1002, 1006, 1014, 1016 (length limit: ≥8388608) 1{0}2: 167, 257, 323, 353, 383, 527, 557, 563, 623, 635, 647, 677, 713, 719, 803, 815, 947, 971, 1013 (length limit: 2000) 1{0}3: 646, 718, 998 (length limit: 2000) 1{0}4: 139, 227, 263, 315, 335, 365, 485, 515, 647, 653, 683, 773, 789, 797, 815, 857, 875, 893, 939, 995, 1007 (length limit: 2000) 1{0}5 1{0}6 1{0}7 1{0}8 1{0}9 1{0}A 1{0}B 1{0}C 1{0}D 1{0}E 1{0}F 1{0}G 1{0}z: 173, 179, 257, 277, 302, 333, 362, 392, 422, 452, 467, 488, 512, 527, 545, 570, 575, 614, 622, 650, 677, 680, 704, 707, 734, 740, 827, 830, 851, 872, 886, 887, 902, 904, 908, 929, 932, 942, 947, 949, 962, 973, 1022 (length limit: 2000) 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 198, 213, 318, 327, 353, 375, 513, 591, 647, 732, 734, 738, 759, 948, 951, 957, 1013, 1014 (length limit: 2000) 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}): 575 (length limit: 247000) 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 813, 863, 962, 1017 (length limit: ≥100000) {1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z): 137, 161, 167, 217, 229, 232, 253, 261, 317, 325, 337, 347, 355, 375, 403, 411, 421, 427, 457, 479, 483, 505, 507, 537, 547, 577, 597, 599, 601, 613, 627, 631, 632, 641, 643, 649, 657, 679, 688, 697, 707, 711, 729, 733, 737, 742, 762, 773, 787, 793, 797, 817, 819, 841, 843, 853, 859, 861, 874, 877, 895, 899, 907, 913, 916, 917, 927, 957, 959, 997, 1003, 1009, 1015, 1017 (length limit: 2000) {1}: 185, 269, 281, 380, 384, 385, 394, 452, 465, 511, 574, 601, 631, 632, 636, 711, 713, 759, 771, 795, 861, 866, 881, 938, 948, 951, 956, 963, 1005, 1015 (length limit: ≥100000) 11{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}): 31, 61, 91, 93, 143, 247, 253, 293, 313, 329, 371, 383, 391, 393, 403, 415, 435, 443, 451, 491, 493, 513, 523, 527, 537, 541, 553, 565, 581, 587, 601, 613, 615, 623, 627, 635, 663, 729, 735, 757, 763, 775, 783, 823, 843, 865, 873, 877, 883, 897, 931, 941, 943, 955, 983, 1013, 1015, 1021, 1023 (length limit: 2000) {1}z 1{2}: 265, 355, 379, 391, 481, 649, 661, 709, 745, 811, 877, 977 (length limit: 2000) 1{3}: 107, 133, 179, 281, 305, 365, 473, 485, 487, 491, 535, 541, 601, 617, 665, 737, 775, 787, 802, 827, 905, 911, 928, 953, 955, 995 1{4}: 83, 143, 185, 239, 269, 293, 299, 305, 319, 325, 373, 383, 395, 431, 471, 503, 551, 577, 581, 593, 605, 617, 631, 659, 743, 761, 773, 781, 803, 821, 857, 869, 897, 911, 917, 923, 935, 983, 1019 (length limit: 2000) 1{z}: 581, 992, 1019 (length limit: ≥100000) 2{0}1: 365, 383, 461, 512, 542, 647, 773, 801, 836, 878, 908, 914, 917, 947, 1004 (length limit: ≥100000) 2{0}3: 79, 149, 179, 254, 359, 394, 424, 434, 449, 488, 499, 532, 554, 578, 664, 683, 694, 749, 794, 839, 908, 944, 982 (length limit: 2000) 2{1} (not quasi-minimal prime if there is smaller prime of the form {1}): 109, 117, 137, 147, 157, 175, 177, 201, 227, 235, 256, 269, 271, 297, 310, 331, 335, 397, 417, 427, 430, 437, 442, 451, 465, 467, 481, 502, 517, 547, 557, 567, 572, 577, 591, 597, 607, 627, 649, 654, 655, 667, 679, 687, 691, 697, 715, 727, 739, 759, 766, 782, 787, 796, 797, 808, 817, 821, 829, 841, 852, 877, 881, 899, 903, 907, 937, 947, 955, 1007, 1011, 1021 (length limit: 2000) {2}1: 106, 238, 262, 295, 364, 382, 391, 397, 421, 458, 463, 478, 517, 523, 556, 601, 647, 687, 754, 790, 793, 832, 872, 898, 962, 1002, 1021 (length limit: 2000) 2{z}: 588, 972 (length limit: ≥100000) 3{0}1: 718, 912 (length limit: ≥100000) 3{0}2: 223, 283, 359, 489, 515, 529, 579, 619, 669, 879, 915, 997 (length limit: 2000) 3{0}4: 167, 391, 447, 487, 529, 653, 657, 797, 853, 913, 937 (length limit: 2000) {3}1: 79, 101, 189, 215, 217, 235, 243, 253, 255, 265, 313, 338, 341, 378, 379, 401, 402, 413, 489, 498, 499, 508, 525, 535, 589, 591, 599, 611, 621, 635, 667, 668, 681, 691, 711, 717, 719, 721, 737, 785, 804, 805, 813, 831, 835, 837, 849, 873, 911, 915, 929, 933, 941, 948, 959, 999, 1013, 1019 (length limit: 2000) 3{z}: 275, 438, 647, 653, 812, 927, 968 (length limit: ≥100000) 4{0}1: 32, 53, 155, 174, 204, 212, 230, 332, 334, 335, 395, 467, 512, 593, 767, 803, 848, 875, 1024 (length limit: ≥100000) 4{0}3: 83, 88, 97, 167, 188, 268, 289, 293, 412, 419, 425, 433, 503, 517, 529, 548, 613, 620, 622, 650, 668, 692, 706, 727, 763, 818, 902, 913, 937, 947, 958 (length limit: 2000) {4}1: 46, 77, 103, 107, 119, 152, 198, 203, 211, 217, 229, 257, 263, 291, 296, 305, 332, 371, 374, 407, 413, 416, 440, 445, 446, 464, 467, 500, 542, 545, 548, 557, 566, 586, 587, 605, 611, 614, 632, 638, 641, 653, 659, 698, 701, 731, 733, 736, 755, 786, 812, 820, 821, 827, 830, 887, 896, 899, 901, 922, 923, 935, 941, 953, 977, 983, 991, 1004 (length limit: 2000) 4{z}: 338, 998 (length limit: ≥100000) 5{0}1: 308, 512, 824 (length limit: ≥100000) 5{z}: 234, 412, 549, 553, 573, 619, 750, 878, 894, 954 (length limit: ≥100000) 6{0}1: 212, 509, 579, 625, 774, 794, 993, 999 (length limit: ≥100000) 6{z}: 308, 392, 398, 518, 548, 638, 662, 878 (length limit: ≥100000) 7{0}1: (none) 7{z}: 321, 328, 374, 432, 665, 697, 710, 721, 727, 728, 752, 800, 815, 836, 867, 957, 958, 972 (length limit: ≥100000) 8{0}1: 86, 140, 182, 263, 353, 368, 389, 395, 422, 426, 428, 434, 443, 488, 497, 558, 572, 575, 593, 606, 698, 710, 746, 758, 770, 773, 824, 828, 866, 911, 930, 953, 957, 983, 993, 1014 (length limit: ≥100000) 8{z}: 378, 438, 536, 566, 570, 592, 636, 688, 718, 830, 852, 926, 1010 (length limit: ≥100000) 9{0}1: 724, 884 (length limit: ≥100000) 9{z}: 80, 233, 530, 551, 611, 899, 912, 980 (length limit: ≥100000) A{0}1: 185, 338, 417, 432, 614, 668, 773, 863, 935, 1000 (length limit: ≥100000) A{z}: 214, 422, 444, 452, 458, 542, 638, 668, 804, 872, 950, 962 (length limit: ≥100000) B{0}1: 560, 770, 968 (length limit: ≥100000) B{z}: 263, 615, 912, 978 (length limit: ≥100000) C{0}1: 163, 207, 354, 362, 368, 480, 620, 692, 697, 736, 753, 792, 978, 998, 1019, 1022 (length limit: ≥100000) C{z} D{0}1 D{z} E{0}1 E{z} F{0}1 F{z} G{0}1 {&#35;}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3): 808, 829, 859, 1006 (length limit: 2000) {&#35;}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2): 31, 37, 55, 63, 67, 77, 83, 89, 91, 93, 97, 99, 107, 109, 117, 123, 127, 133, 135, 137, 143, 147, 149, 151, 155, 161, 177, 179, 183, 189, 193, 197, 207, 211, 213, 215, 217, 223, 225, 227, 233, 235, 241, 247, 249, 255, 257, 263, 265, 269, 273, 277, 281, 283, 285, 287, 291, 293, 297, 303, 307, 311, 319, 327, 347, 351, 355, 357, 359, 361, 367, 369, 377, 381, 383, 385, 387, 389, 393, 397, 401, 407, 411, 413, 417, 421, 423, 437, 439, 443, 447, 457, 465, 467, 469, 473, 475, 481, 483, 489, 493, 495, 497, 509, 511, 515, 533, 541, 547, 549, 555, 563, 591, 593, 597, 601, 603, 611, 615, 619, 621, 625, 627, 629, 633, 635, 637, 645, 647, 651, 653, 655, 659, 663, 667, 671, 673, 675, 679, 683, 687, 691, 693, 697, 707, 709, 717, 731, 733, 735, 737, 741, 743, 749, 753, 755, 757, 759, 765, 767, 771, 773, 775, 777, 783, 785, 787, 793, 797, 801, 807, 809, 813, 817, 823, 825, 849, 851, 853, 865, 867, 873, 877, 887, 889, 893, 897, 899, 903, 907, 911, 915, 923, 927, 933, 937, 939, 941, 943, 945, 947, 953, 957, 961, 967, 975, 977, 983, 987, 993, 999, 1003, 1005, 1009, 1017 (length limit: ≥262143) &#35;{z} (for even bases ''b'', # = ''b''/2−1): 108, 278, 296, 338, 386, 494, 626, 920 (length limit: 2000) ${&#35;} (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) x{z} y{z}: 128, 233, 268, 383, 478, 488, 533, 554, 665, 698, 779, 863, 878, 932, 941, 1010 (length limit: ≥200000) z{0}1: 123, 342, 362, 422, 438, 479, 487, 512, 542, 602, 757, 767, 817, 830, 872, 893, 932, 992, 997, 1005, 1007 (length limit: ≥100000) {y}z: 143, 173, 176, 213, 235, 248, 253, 279, 327, 343, 353, 358, 373, 383, 401, 413, 416, 427, 439, 448, 453, 463, 481, 513, 522, 527, 535, 547, 559, 565, 583, 591, 598, 603, 621, 623, 653, 659, 663, 679, 691, 698, 711, 743, 745, 757, 768, 785, 793, 796, 801, 808, 811, 821, 835, 845, 847, 853, 856, 883, 898, 903, 927, 955, 961, 971, 973, 993, 1005, 1013, 1019, 1021 (length limit: 2000) {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family): 97, 103, 113, 186, 187, 220, 304, 306, 309, 335, 414, 416, 428, 433, 445, 459, 486, 498, 539, 550, 557, 587, 592, 597, 598, 617, 624, 637, 659, 665, 671, 677, 696, 717, 726, 730, 740, 754, 766, 790, 851, 873, 890, 914, 923, 929, 943, 944, 965, 984, 985, 996, 1004, 1005 (length limit: ≥17326) zy{z} (not quasi-minimal prime if there is smaller prime of the form y{z}) {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y): 215, 353, 517, 743, 852, 899, 913 (length limit: 2000) {z}01 (not quasi-minimal prime if there is smaller prime of the form {z}1) {z}1: 93, 113, 152, 158, 188, 217, 218, 226, 227, 228, 233, 240, 275, 278, 293, 312, 338, 350, 353, 383, 404, 438, 464, 471, 500, 533, 576, 614, 641, 653, 704, 723, 728, 730, 758, 779, 788, 791, 830, 878, 881, 899, 908, 918, 929, 944, 953, 965, 968, 978, 983, 986, 1013 (length limit: 2000) {z}k {z}l {z}m {z}n {z}o {z}p {z}q {z}r {z}s {z}t {z}u {z}v {z}w: 207, 221, 293, 375, 387, 533, 633, 647, 653, 687, 701, 747, 761, 785, 863, 897, 905, 965, 1017 (length limit: 2000) {z}x: (none) {z}y: 305, 353, 397, 485, 487, 535, 539, 597, 641, 679, 731, 739, 755 (length limit: 2000) == List of lengths for quasi-minimal primes in some simple families == [https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list of lengths for quasi-minimal primes in some simple families for bases 2≤''b''≤1024] NB: this family is not interpretable in this base (e.g. family 7{0}1 and 7{z} in bases <=7, family {z}x in bases <=3) (including the case which this family has either leading zeros (leading zeros do not count) or ending zeros (numbers ending in zero cannot be prime > base) in this base) RC: this family can be proven to only contain composite numbers (only count numbers > base) unknown: this family has no primes or PRPs found, nor can this family be proven to only contain composite numbers (only count numbers > base) Background color: red for title (bases or families), green for length > 10000, orange for 2500 < length ≤ 10000, white for length ≤ 2500, cyan for "RC", pink for "NB", yellow for "unknown". Search limit for lengths: ≥8388608 for 1{0}1, ≥200000 for y{z}, ≥100000 for ''d''{0}1 (''d'' = one of digits in {2, 3, 4, 5, 6, 7, 8, 9, A, B, C}) and ''d''{z} (''d'' = one of digits in {1, 2, 3, 4, 5, 6, 7, 8, 9, A, B}) and z{0}1 and {1}, ≥5000 for 1{0}2, {z}y, 1{0}z, {z}1, {y}z, ≥2500 for other families. == References == * [https://mersenneforum.org/showthread.php?t=24972 mersenneforum thread of this problem] * [https://docs.google.com/document/d/e/2PACX-1vQct6Hx-IkJd5-iIuDuOKkKdw2teGmmHW-P75MPaxqBXB37u0odFBml5rx0PoLa0odTyuW67N_vn96J/pub Minimal elements for the base ''b'' representations of the primes which are > ''b'' for bases ''b''≤16] * [https://primes.utm.edu/glossary/xpage/MinimalPrime.html article “minimal prime” in The Prime Glossary] * [https://en.wikipedia.org/wiki/Minimal_prime_(recreational_mathematics article “minimal prime” in Wikipedia] * [https://www.primepuzzles.net/puzzles/puzz_178.htm the puzzle of minimal primes (when the restriction of prime>base is not required) in The Prime Puzzles & Problems Connection] * [https://www.primepuzzles.net/problems/prob_083.htm the problem of minimal primes in The Prime Puzzles & Problems Connection] * [https://github.com/xayahrainie4793/non-single-digit-primes my data for these M(Lb) sets for 2 ≤ b ≤ 16] * [http://www.cs.uwaterloo.ca/~shallit/Papers/minimal5.pdf Shallit’s proof of base 10 minimal primes, when the restriction of prime>base is not required] * [https://scholar.colorado.edu/downloads/hh63sw661 proofs of minimal primes in bases b≤10, when the restriction of prime>base is not required] * [https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://cs.uwaterloo.ca/~cbright/talks/minimal-slides.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://doi.org/10.1080/10586458.2015.1064048 the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://github.com/curtisbright/mepn-data data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 30, when the restriction of prime>base is not required, search limits of lengths: 1000000 for b=17, 707000 for b=19, 506000 for b=21, 292000 for b=25, 486000 for b=26, 543000 for b=28, 233000 for b=29] * [https://github.com/RaymondDevillers/primes data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 50, when the restriction of prime>base is not required, search limits of lengths: 10000 for all b] * [http://www.bitman.name/math/article/730 article for minimal primes, when the restriction of prime>base is not required] * [http://www.bitman.name/math/table/497 data for minimal primes in bases 2 ≤ b ≤ 16, when the restriction of prime>base is not required] * [http://www.prothsearch.com/sierp.html the Sierpinski problem] * [http://www.prothsearch.com/rieselprob.html the Riesel problem] * [https://oeis.org/A076336/a076336c.html the dual Sierpinski problem] * [http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm generalized Sierpinski conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &gt; b] * [http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm generalized Riesel conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &gt; b] * [http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm list for the status of the generalized Sierpinski conjectures and the generalized Riesel conjectures in bases b≤1030] * [https://www.utm.edu/staff/caldwell/preprints/2to100.pdf article for generalized Sierpinski conjectures in bases b≤100] * [http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf article for the mixed (original+dual) Sierpinski problem] * [http://www.fermatquotient.com/PrimSerien/GenRepu.txt generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤160, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://jeppesn.dk/generalized-fermat.html generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://www.noprimeleftbehind.net/crus/GFN-primes.htm generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1030, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt list of generalized half Fermat primes (primes of the form (b2^n+1)/2) sorted by n, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://harvey563.tripod.com/wills.txt primes of the form (b−1)*bn−1 for bases b≤2049, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Williams_prime_MM_least the smallest primes of the form (b−1)*bn−1 for bases b≤2049, these primes (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Williams_prime_MP_least the smallest primes of the form (b−1)*bn+1 for bases b≤1024, these primes (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n the smallest primes of the form k*bn−1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b>k] * [https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n the smallest primes of the form k*bn+1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b>k] * [https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list for the smallest primes in given simple family in bases b≤1024] * [https://www.rose-hulman.edu/~rickert/Compositeseq/ a problem related to this project] * [http://www.worldofnumbers.com/Appending%201s%20to%20n.txt a problem related to this project] * [https://stdkmd.net/nrr/prime/primecount.txt near- and quasi- repdigit (probable) primes sorted by count] * [https://stdkmd.net/nrr/prime/primedifficulty.txt near- and quasi- repdigit (probable) primes sorted by difficulty] * [http://www.prothsearch.com/fermat.html factoring status of Fermat numbers] * [http://www.rieselprime.de/dl/CRUS_pack.zip srsieve, sr1sieve, sr2sieve, pfgw, and llr softwares] * [https://www.bc-team.org/app.php/dlext/?cat=3 srsieve, sr1sieve, sr2sieve, sr5sieve software] * [https://sourceforge.net/projects/openpfgw/ pfgw software] * [http://jpenne.free.fr/index2.html llr software] * [http://www.ellipsa.eu/public/primo/primo.html PRIMO software] * [https://primes.utm.edu/prove/index.html website for primality proving] * [https://primes.utm.edu/curios/page.php?number_id=22380 the largest base 10 minimal prime in Prime Curios!] * [https://oeis.org/A071062 OEIS sequence for base 10 minimal primes, when the restriction of prime>base is not required] * [https://oeis.org/A326609 OEIS sequence for the largest base b minimal prime, when the restriction of prime>base is not required] * [https://primes.utm.edu/primes/lists/all.txt top proven primes] * [http://www.primenumbers.net/prptop/prptop.php top PRPs] * [http://factordb.com online factor database, including many primes which are minimal primes in a small base] lul2hxo86sitd4qq2zeb3f8prjs4kxr 2410367 2410366 2022-07-30T00:49:38Z 2402:7500:916:306E:6805:A2C5:4D03:BE94 /* Base 36 */ wikitext text/x-wiki A '''quasi-minimal prime''' is a [[w:Prime number|prime number]] for which there is no shorter [[w:Subsequence|subsequence]] of its [[w:Numerical digit|digit]]s in a given [[w:Radix|base]] ''b'' that form a prime > ''b''. For example, 857 is a quasi-minimal prime in [[w:Decimal|decimal]] because there is no prime > 10 among the shorter subsequences of the digits: 8, 5, 7, 85, 87, 57. The subsequence does not have to consist of consecutive digits, so 149 is not a quasi-minimal prime in decimal (because 19 is prime and 19 > 10). But it does have to be in the same order; so, for example, 991 is still a quasi-minimal prime in decimal even though a subset of the digits can form the shorter prime 19 > 10 by changing the order. (using A−Z to represent digit values 10 to 35) For the quasi-minimal primes in bases up to 36, I have only solved (found all quasi-minimal primes and proved that these are all such primes) bases 2~12, 14~15, 18, 20, 22, 24, 30 (bases 11, 22, 30 need primality proving of the probable primes). For the remain bases 13, 16~17, 19, 21, 23, 25~29, 31~36, there are some ''x''{''d''}''y'' (with ''x'', ''y'' strings (may be [[w:Empty string|empty]]) with digits in base ''b'', ''d'' digit in base ''b'') families which are not solved (not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers > base (''b'')). I left as a challenge to readers the task of solving (finding all quasi-minimal primes and proving that these are all such primes) bases 13, 16~17, 19, 21, 23, 25~29, 31~36 (this will be a hard problem, e.g. base 23 has a quasi-minimal prime 9E₈₀₀₈₇₃, and base 36 has quasi-minimal prime P₈₁₉₉₃SZ). Proving the set of the quasi-minimal primes in base ''b'' is ''S'', is equivalent to: * Prove that all elements in ''S'', when read as base ''b'' representation, are primes > ''b''. * Prove that all [[w:Proper subset|proper]] subsequence of all elements in ''S'', when read as base ''b'' representation, which are > ''b'', are composite. * Prove that all primes > ''b'', when written in base ''b'', contain at least one element in ''S'' as subsequence (equivalently, prove that all strings not containing any element in ''S'' as subsequence, when read as base ''b'' representation, which are > ''b'', are composite). e.g. proving the set of the quasi-minimal primes in base ''b'' = 10 is {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}, is equivalent to: * Prove that all of 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027 are primes > 10. * Prove that all proper subsequence of all elements in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} which are > 10 are composite. * Prove that all primes > 10 contain at least one element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence (equivalently, prove that all numbers > 10 not containing any element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence are composite). ==Condensed table== {|class=wikitable |''b''||number of quasi-minimal primes base ''b''||base-''b'' form of largest known quasi-minimal prime base ''b''||length of largest known quasi-minimal prime base ''b''||algebraic ((''a''×''b''^''n''+''c'')/''d'') form of largest known quasi-minimal prime base ''b'' |- |2||1||11||2||3 |- |3||3||111||3||13 |- |4||5||221||3||41 |- |5||22||10₉₃13||96||5⁹⁵+8 |- |6||11||40041||5||5209 |- |7||71||3₁₆1||17||(7¹⁷−5)/2 |- |8||75||4₂₂₀7||221||(4×8²²¹+17)/7 |- |9||151||30₁₁₅₈11||1161||3×9¹¹⁶⁰+10 |- |10||77||50₂₈27||31||5×10³⁰+27 |- |11^*||1068||57₆₂₆₆₈||62669||(57×11⁶²⁶⁶⁸−7)/10 |- |12||106||40₃₉77||42||4×12⁴¹+91 |- |13^*||3195~3197||80₃₂₀₁₇111||32021||8×13³²⁰²⁰+183 |- |14||650||4D₁₉₆₉₈||19699||5×14¹⁹⁶⁹⁸−1 |- |15||1284||7₁₅₅97||157||(15¹⁵⁷+59)/2 |- |16^*||2346~2347||4₇₂₇₈₅DD||72787||(4×16⁷²⁷⁸⁷+2291)/15 |- |17^*||10407~10428||F70₁₈₆₇₆₇1||186770||262×17¹⁸⁶⁷⁶⁸+1 |- |18||549||C0₆₂₆₈C5||6271||12×18⁶²⁷⁰+221 |- |20||3314||G0₆₂₆₉D||6271||16×20⁶²⁷⁰+13 |- |21^*||13375~13396||CF₄₇₉₁₄₇0K||479150||(51×21⁴⁷⁹¹⁴⁹−1243)/4 |- |22^*||8003||BK₂₂₀₀₁5||22003||(251×22²²⁰⁰²−335)/21 |- |24||3409||N00N₈₁₂₉LN||8134||13249×24⁸¹³¹−49 |- |30^*||2619||OT₃₄₂₀₅||34206||25×30³⁴²⁰⁵−1 |- |36^*||35257~35263||P₈₁₉₉₃SZ||81995||(5×36⁸¹⁹⁹⁵+821)/7 |} ^* Data assumes the primality of the [[w:probable prime|probable prime]]s. Except bases ''b'' = 13, 16, 17, 21, all bases in this table are completely solved (if we allow strong probable primes > 10²⁰⁰⁰⁰), also, except bases ''b'' = 11, 13, 16, 17, 21, 22, 30, 36, all bases in this table are completely solved even if we only allow definitely primes (thus, we can complete the classification of the quasi-minimal primes in these bases, i.e. the “quasi-minimal problems” in these bases are now theorems), for the quasi-minimal primes see the data below. Base ''b'' = 13 has 3195 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 13, then they must be of the form 9{5} or A{3}A (we are unable to determine if these two families contain a prime or not, i.e. these two families have no known prime members, nor can these two families be ruled out as only containing composites), and must have at least 82000 digits in base 13, besides, since these two families can contain at most one quasi-minimal prime, there are at most 3197 quasi-minimal primes in base 13. (i.e. the quasi-minimal primes in base 13 are the 3195 known quasi-minimal primes in base 13 (they are given in the data section) plus the smallest prime in the family 9{5} in base 13 (if exists) plus the smallest prime in the family A{3}A in base 13 (if exists)) Base ''b'' = 16 has 2346 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 16, then they must be of the form {3}AF (we are unable to determine if this family contains a prime or not, i.e. this family have no known prime members, nor can this family be ruled out as only containing composites), and must have at least 76000 digits in base 16, besides, since this family can contain at most one quasi-minimal prime, there are at most 2347 quasi-minimal primes in base 16. (i.e. the quasi-minimal primes in base 16 are the 2346 known quasi-minimal primes in base 16 (they are given in the data section) plus the smallest prime in the family {3}AF in base 16 (if exists)) ==Data for quasi-minimal primes== ===Base 2=== 11 ===Base 3=== 12, 21, 111 ===Base 4=== 11, 13, 23, 31, 221 ===Base 5=== 12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013 ===Base 6=== 11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041 ===Base 7=== 14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331 ===Base 8=== 13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447 ===Base 9=== 12, 14, 18, 21, 25, 32, 34, 41, 45, 47, 52, 58, 65, 67, 74, 78, 81, 87, 117, 131, 135, 151, 155, 175, 177, 238, 272, 308, 315, 331, 337, 355, 371, 375, 377, 438, 504, 515, 517, 531, 537, 557, 564, 601, 638, 661, 702, 711, 722, 735, 737, 751, 755, 757, 771, 805, 838, 1011, 1015, 1101, 1701, 2027, 2207, 3017, 3057, 3101, 3501, 3561, 3611, 3688, 3868, 5035, 5051, 5071, 5101, 5501, 5554, 5705, 5707, 7017, 7075, 7105, 7301, 8535, 8544, 8555, 8854, 20777, 22227, 22777, 30161, 33388, 50161, 50611, 53335, 55111, 55535, 55551, 57061, 57775, 70631, 71007, 77207, 100037, 100071, 100761, 105007, 270707, 301111, 305111, 333035, 333385, 333835, 338885, 350007, 500075, 530005, 555611, 631111, 720707, 2770007, 3030335, 7776662, 30300005, 30333335, 38333335, 51116111, 70000361, 300030005, 300033305, 351111111, 1300000007, 5161111111, 8333333335, 300000000035, 311111111161, 544444444444, 2000000000007, 5700000000001, 7270000000007, 88888888833335, 100000000000507, 5111111111111161, 7277777777777777707, 8888888888888888888335, 30000000000000000000051, 1000000000000000000000000057, 56111111111111111111111111111111111111, 7666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666662, 27777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777707, 300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011 ===Base 10=== 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027 ===Base 11=== 12, 16, 18, 21, 27, 29, 34, 38, 3A, 43, 49, 54, 56, 61, 65, 67, 72, 76, 81, 89, 92, 94, 98, 9A, A3, 10A, 115, 117, 133, 139, 153, 155, 171, 193, 197, 199, 1AA, 225, 232, 236, 25A, 263, 315, 319, 331, 335, 351, 353, 362, 373, 379, 391, 395, 407, 414, 452, 458, 478, 47A, 485, 4A5, 4A7, 502, 508, 511, 513, 533, 535, 539, 551, 571, 579, 588, 595, 623, 632, 70A, 711, 715, 731, 733, 737, 755, 759, 775, 791, 797, 7AA, 803, 847, 858, 85A, 874, 885, 887, 913, 919, 931, 937, 957, 959, 975, 995, A07, A1A, A25, A45, A74, A7A, A85, AA1, AA7, 1101, 11A9, 1305, 1451, 1457, 15A7, 175A, 17A5, 17A9, 2023, 2045, 2052, 2083, 20A5, 2333, 2A05, 2A52, 3013, 3026, 3059, 3097, 3206, 3222, 3233, 3307, 3332, 3505, 4025, 4151, 4157, 4175, 4405, 4445, 4487, 450A, 4575, 5017, 5031, 5059, 5075, 5097, 5099, 5105, 515A, 517A, 520A, 5301, 5583, 5705, 577A, 5853, 5873, 5909, 5A17, 5A57, 5A77, 5A8A, 6683, 66A9, 7019, 7073, 7079, 7088, 7093, 7095, 7309, 7451, 7501, 7507, 7578, 757A, 75A7, 7787, 7804, 7844, 7848, 7853, 7877, 78A4, 7A04, 7A57, 7A79, 7A95, 8078, 8245, 8333, 8355, 8366, 8375, 8425, 8553, 8663, 8708, 8777, 878A, 8A05, 9053, 9101, 9107, 9305, 9505, 9703, A052, A119, A151, A175, A515, A517, A575, A577, A5A8, A719, A779, A911, AAA9, 10011, 10075, 10091, 10109, 10411, 10444, 10705, 10709, 10774, 10901, 11104, 11131, 11144, 11191, 1141A, 114A1, 13757, 1411A, 14477, 144A4, 14A04, 14A11, 17045, 17704, 1774A, 17777, 177A4, 17A47, 1A091, 1A109, 1A114, 1A404, 1A411, 1A709, 20005, 20555, 22203, 25228, 25282, 25552, 25822, 28522, 30037, 30701, 30707, 31113, 33777, 35009, 35757, 39997, 40045, 4041A, 40441, 4045A, 404A1, 4111A, 411A1, 42005, 44401, 44474, 444A1, 44555, 44577, 445AA, 44744, 44A01, 47471, 47477, 47701, 5057A, 50903, 5228A, 52A22, 52A55, 52A82, 55007, 550A9, 55205, 55522, 55557, 55593, 55805, 57007, 57573, 57773, 57807, 5822A, 58307, 58505, 58A22, 59773, 59917, 59973, 59977, 59999, 5A015, 5A2A2, 5AA99, 60836, 60863, 68636, 6A609, 6A669, 6A696, 6A906, 6A966, 70048, 70103, 70471, 70583, 70714, 71474, 717A4, 71A09, 74084, 74444, 74448, 74477, 744A8, 74747, 74774, 7488A, 74A48, 75773, 77144, 77401, 77447, 77799, 77A09, 78008, 78783, 7884A, 78888, 788A8, 79939, 79993, 79999, 7A051, 7A444, 7A471, 80005, 80252, 80405, 80522, 80757, 80AA5, 83002, 84045, 85307, 86883, 88863, 8A788, 90073, 90707, 90901, 95003, 97779, 97939, 99111, 99177, 99973, A0111, A0669, A0966, A0999, A0A09, A1404, A4177, A4401, A4717, A5228, A52AA, A5558, A580A, A5822, A58AA, A5A59, A5AA2, A6096, A6966, A6999, A7051, A7778, A7808, A9055, A9091, A9699, A9969, AA52A, AA58A, 100019, 100079, 101113, 101119, 101911, 107003, 140004, 144011, 144404, 1A0019, 1A0141, 1A5001, 1A7005, 1A9001, 222223, 222823, 300107, 300202, 300323, 303203, 307577, 310007, 332003, 370777, 400555, 401A11, 404001, 404111, 405AAA, 41A011, 440A41, 441011, 451777, 455555, 470051, 470444, 474404, 4A0401, 4A4041, 500015, 500053, 500077, 500507, 505577, 522A2A, 525223, 528A2A, 531707, 550777, 553707, 5555A9, 555A99, 557707, 55A559, 5807A7, 580A0A, 580A55, 58A0AA, 590007, 599907, 5A2228, 5A2822, 5A2AAA, 5A552A, 5AA22A, 5AAA22, 60A069, 683006, 6A0096, 6A0A96, 6A9099, 6A9909, 700778, 701074, 701777, 704408, 704417, 704457, 704484, 707041, 707441, 707708, 707744, 707784, 710777, 717044, 717077, 740008, 74484A, 770441, 770744, 770748, 770771, 777017, 777071, 777448, 777484, 777701, 7778A8, 777A19, 777A48, 778883, 78A808, 790003, 7A1009, 7A4408, 7A7708, 80A555, 828283, 828883, 840555, 850505, 868306, 873005, 883202, 900701, 909739, 909979, 909991, 970771, 977701, 979909, 990739, 990777, 990793, 997099, 999709, 999901, A00009, A00599, A01901, A05509, A0A058, A0A955, A10114, A555A2, A55999, A59991, A5A222, A5A22A, A60609, A66069, A66906, A69006, A79005, A87888, A90099, A90996, A96006, A96666, A97177, A97771, AA0A58, AA5A22, AAA522, 1000501, 1011141, 1030007, 1070047, 111114A, 1111A14, 1111A41, 1144441, 14A4444, 1700005, 1700474, 1A44444, 2555505, 2845055, 3030023, 3100003, 3333397, 4000111, 4011111, 41A1111, 4411111, 444441A, 4444771, 4470004, 4505005, 4744417, 4774441, 4777404, 4777417, 4777747, 4A11111, 4A40001, 5000093, 50005A7, 5005777, 5050553, 5055503, 5070777, 5222222, 5222AAA, 52AAAA2, 52AAAAA, 5505053, 5552AAA, 5555599, 5555A58, 5558A0A, 5558A55, 5558AAA, 55A0009, 55AAA52, 580000A, 5822222, 58AAAAA, 5A2222A, 5AA2222, 6000A69, 6000A96, 6A00069, 7000417, 7000741, 7000835, 7000857, 7007177, 7008305, 7014447, 7017444, 7074177, 7077477, 7077741, 7077747, 7100447, 7174404, 717444A, 7400404, 7700717, 7701077, 7701707, 7707778, 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DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD ===Base 15=== 12, 14, 18, 1E, 21, 27, 2B, 2D, 32, 38, 3E, 41, 47, 4B, 4D, 54, 58, 5E, 67, 6B, 6D, 72, 74, 78, 87, 8B, 92, 94, 9E, A1, A7, AD, B2, B8, BE, C1, CB, CD, D2, D4, E1, ED, 111, 11B, 131, 137, 13B, 13D, 157, 15B, 15D, 171, 177, 197, 19D, 1B7, 1BB, 1D1, 1DB, 1DD, 234, 298, 311, 31B, 337, 33D, 344, 351, 357, 35B, 364, 377, 391, 39B, 39D, 3A4, 3BD, 3C4, 3D7, 3DB, 3DD, 452, 51B, 51D, 531, 53B, 551, 55D, 562, 571, 577, 5A2, 5B1, 5B7, 5BB, 5BD, 5C2, 5D1, 5D7, 634, 652, 681, 698, 717, 71B, 731, 737, 757, 75D, 77D, 79B, 79D, 7B1, 7B7, 7BD, 7D7, 7DD, 801, 852, 88D, 8D8, 91D, 93B, 93D, 95B, 95D, 971, 977, 97B, 97D, 988, 991, 9BD, 9C8, 9D1, A98, AAB, B1D, B31, B3B, B44, B51, B57, B7B, B7D, B97, B9B, BB7, BC4, BD1, BD7, BDD, C07, C34, C52, C7E, C98, CC7, CE7, D0E, D1D, D31, D51, D5B, D68, D77, D7B, D91, D97, DA8, DAE, DCE, DD1, EB4, EEB, 107B, 1091, 10B1, 1107, 110D, 1561, 1651, 1691, 1B01, 2052, 2502, 2522, 303B, 307D, 3097, 30BB, 30D1, 3107, 3361, 3701, 3907, 3B01, 3B0B, 3C97, 4434, 4498, 4834, 4898, 49A8, 4E34, 5037, 507D, 5091, 509B, 5107, 5161, 5202, 53C7, 5552, 570B, 590B, 590D, 59C7, 5A5B, 5C97, 5D0D, 5DAB, 6061, 6151, 6191, 6511, 6601, 6911, 707B, 7091, 7097, 70AE, 70BB, 70CE, 70DB, 7561, 760E, 7691, 76CE, 7907, 7961, 7A0E, 7A3B, 7AEE, 7B0B, 7BAB, 7C0E, 7C77, 7CAE, 7D0B, 7D61, 7DAB, 7E5B, 7E6E, 7E7B, 7EBB, 8098, 811D, 8191, 835D, 853D, 8881, 8908, 8951, 8968, 899D, 8D3D, 8D5D, 8D6E, 8DDD, 8E98, 9011, 9037, 9097, 90D7, 9301, 93C7, 95C7, 9611, 9631, 96A8, 9811, 9851, 989D, 990B, 990D, 998D, 99AB, 99C7, 99D8, 9A08, 9A9B, 9AA8, 9ABB, 9B61, 9BC7, 9D0B, 9DAB, 9DC7, 9DD8, A052, A304, A502, A55B, A9BB, AB04, AB64, B09D, B107, B10B, B161, B1AB, B1C7, B30D, B3C7, B50B, B664, B691, B6A4, B707, B761, B90D, B961, BA5B, BABB, BBAB, BBB4, BC37, BC77, C777, C937, C997, D011, D03D, D05D, D09B, D0B1, D0BD, D101, D10B, D30D, D3AB, D507, D50D, D66E, D761, D7DE, D811, D85D, D86E, D89D, D8C8, D8E8, D9AB, D9D8, DA3B, DA9B, DABB, DB01, DB61, DBAB, DC88, DD07, DD0B, DD7E, DD8D, DDE7, DE6E, E252, E33B, E522, E57B, E7AE, E7CE, E898, E997, E9A8, E9BB, EA34, EB5B, EE98, EEC7, 10017, 10B0D, 170AB, 17A0B, 19001, 19601, 1A09B, 1D0C7, 22E52, 2EA52, 30017, 3001D, 300B1, 301C7, 30334, 30631, 307AB, 3300B, 3333B, 36031, 36301, 37A0B, 37BBB, 39997, 3A30B, 3B0C7, 3D001, 3D601, 40034, 40968, 43334, 49668, 49998, 50022, 5009D, 501C7, 50222, 50507, 505C7, 50611, 50C57, 53007, 53997, 55537, 5555B, 5557B, 5599B, 56101, 56691, 56961, 5700D, 5755B, 59001, 59557, 59997, 5999D, 599DB, 59DDD, 5D99B, 5DD3D, 5DD9D, 60931, 63031, 65691, 66951, 69031, 69361, 69561, 70011, 70051, 7005B, 7006E, 7030D, 703AB, 70501, 70701, 707C7, 71601, 71951, 7300D, 7333B, 75001, 7555B, 75911, 76011, 76051, 766EE, 76EEE, 7700B, 77191, 77661, 7776E, 77771, 777BB, 77911, 77BBB, 79001, 7A05B, 7A66E, 7AA6E, 7AAAE, 7ACCE, 7C6EE, 7CCEE, 7CECE, 7CEEE, 7D3BB, 7E7C7, 7EECE, 80034, 80304, 80434, 809DD, 80A34, 84A34, 850DD, 85961, 86661, 88151, 88331, 88511, 88591, 88898, 890DD, 89998, 89D0D, 8D90D, 8E434, 90017, 90051, 900A8, 900DB, 901C7, 90C57, 90D8D, 91007, 91061, 9199B, 95997, 96068, 96561, 99397, 99537, 9999B, 999B7, 999D7, 999DB, 999DD, 99BBB, 99DBB, 99DD7, 99DDD, 9B007, 9B00B, 9B0AB, 9BB11, 9BBBB, 9D007, 9D08D, 9D537, 9D9BB, 9D9DB, 9DD57, 9DDB7, 9DDDB, 9DDDD, A0A34, A0B5B, A0BBB, A0E34, A2E52, A330B, A8434, A8834, A8E34, A909B, AAA34, AAE52, AB0BB, AB334, ABB34, AE034, AE834, AE99B, AEA52, AEE52, B0011, B0071, B0077, B00B1, B0611, B0A64, B500D, B599D, B6101, B7771, B7911, BA064, BAAA4, BAB34, BB061, BB304, BB53D, BB601, BBB91, BBB9D, BBBBD, BDA0B, BDBBB, D0088, D00D7, D0307, D05C7, D070D, D0888, D0B07, D0BC7, D0C08, D0DC7, D0DD8, D1661, D59DD, D5D3D, D5DDD, D6611, D700D, D8D0D, D900B, D9908, D999D, D9BBB, D9D9D, D9DDB, DB007, DB00D, DB1B1, DB53D, DB59D, DB99D, DBBB1, DD0D8, DD33B, DD3B7, DD3BB, DD57D, DD898, DD9DD, DDB37, DDBDB, DDD08, DDD3D, DDD5D, DDD7D, DDD88, DDD9D, DDDB7, DDDC8, DDDD7, DDE98, DE037, DE998, DEB07, E0098, E00C7, E0537, E0557, E077B, E0834, E0968, E3334, E37AB, E39C7, E4034, E5307, E55AB, E705B, E750B, E766E, E76EE, E8304, E8434, E9608, E9C37, EAE52, EBB0B, EC557, EC597, EC957, 1000BD, 1009AB, 10A90B, 1900AB, 190661, 19099B, 190A0B, 1A900B, 222A52, 2AAA52, 31000D, 330331, 333334, 3733AB, 373ABB, 3BBB61, 430004, 490068, 490608, 5000DB, 500D0B, 505557, 505A0B, 50D00B, 50DDDB, 50DDDD, 522222, 5500AB, 5500C7, 550957, 550A0B, 555A9B, 559057, 560011, 590661, 633331, 666331, 666591, 666661, 7050AB, 705A0B, 706101, 70A50B, 7300AB, 761661, 76666E, 777011, 777101, 77750B, 777A5B, 777CEE, 779051, 791501, 7E7797, 7ECCCE, 7EEE97, 800D9D, 808834, 836631, 83D661, 843004, 856611, 884034, 884304, 888E34, 88A434, 88AE34, 8A4034, 8AEE34, 8E8034, 8E8E34, 8EEE34, 9000BB, 9001AB, 900B07, 900D98, 903661, 905661, 906651, 9080DD, 9099A8, 909D9B, 90A668, 90DD9B, 90DDBB, 910001, 9100AB, 91A00B, 930007, 950001, 956661, 9909A8, 995907, 999068, 999507, 999907, 9B0B1B, 9B0BB1, 9BB01B, 9C5597, 9C5957, 9D09DD, 9D0D9D, 9D800D, 9DB307, 9DD09D, A00034, A0033B, A033B4, A2A252, AAAA52, ABBBBB, B00004, B0001B, B0003D, B00A04, B0555B, B07191, B07711, B07777, B0B911, B0BDBB, B77011, B777C7, BB0001, BB0034, BB035D, BB055B, BB0BDB, BB9101, BBB0DB, BBB50D, BBBB01, BBD0BB, C55397, C55557, C55597, D0003B, D00057, D0007D, D000B7, D000C8, D008DD, D00DAB, D0333B, D05537, D099DD, D09DDD, D0DDBB, D555C7, D5C537, D88008, D88088, D888EE, D909DD, D9D0DD, D9DD0D, DB0BBB, DBBB0B, DBBB0D, DC0008, DC5537, DDDDD8, DDDEBB, DDE99B, DE0808, DE0C57, DE300B, DE5537, DE8888, DEE088, DEE307, DEE888, DEEE37, DEEE57, DEEEC8, E0000B, E007BB, E00A52, E03BC7, E07ABB, E09B07, E0A99B, E0C397, E0E76E, E50057, E55007, E55597, E55937, E730AB, E73A0B, E80E34, E88834, E8E034, E90008, E95557, EA099B, EE4304, EE5057, EE5507, EE8E34, EE9307, EEE434, 100001D, 1000A9B, 1000DC7, 22AA252, 3000BC7, 3033301, 3076661, 333B304, 33B3034, 3B33304, 3D66661, 50007AB, 5005957, 5500597, 5550057, 5559007, 5559597, 5595007, 5966661, 5DDDDDB, 6366631, 7010001, 7066651, 7100061, 733BBBB, 766A6AE, 77505AB, 7776501, 777775B, 777AACE, 777ECCE, 777EEAE, 7CCCCCE, 7E30A0B, 7EEEEAE, 8300004, 8363331, 8693331, 880E834, 8833304, 8888034, 8888434, 888A034, 88A3334, 88E8834, 88EE034, 88EE304, 8AA3334, 8D0009D, 8EE8834, 9000361, 9000668, 9003331, 9005557, 9006008, 9008D0D, 9083331, 9090968, 90BBB01, 90D0908, 9500661, 9555597, 9555957, 9660008, 9900968, 9995597, 9996008, 9999557, 9999597, 9999908, 9A66668, A003B34, A003BB4, AA22252, B00B034, B00B35D, B033334, B0B6661, B0BB01B, B100001, B333304, B777777, B99999D, BA60004, BAA0334, BBB001B, BBB6611, BBBBB11, BBBD00B, BD000AB, D0000DB, D009098, D00CCC8, D00D908, D00D99D, D03000B, D0BB0BB, D0D9008, D0D9998, D1000C7, D800008, D8DDEEE, D90080D, DBBBBBB, DD09998, DDD5557, DDDDBBB, DDDDDBD, DDDE8EE, DECC008, DECCCC8, DEE0CC8, DEEC0C8, E000397, E0003BB, E000434, E00076E, E000937, E007A5B, E00909B, E0090B7, E009307, E00B077, E00E434, E00E797, E00E937, E05999B, E09009B, E0900B7, E0E0937, E0E7E97, E0EAA52, E0EEA52, E555057, E5555C7, E7777C7, E77E797, E88EE34, E999998, EA5999B, EB000BB, EB0BBBB, EE00434, EE0E797, EEE076E, EEE706E, EEE8834, EEEE557, EEEE797, 30333331, 30B66661, 33000034, 33030004, 33B33004, 500575AB, 55000007, 5500075B, 55500907, 55555057, 55555907, 55559507, 60003301, 60033001, 60330001, 7000003D, 70106661, 70666611, 77000001, 7777770B, 777777C7, 77777ACE, 77777EAE, 777E30AB, 777E3A0B, 7CCCC66E, 800005DD, 88AA0834, 90000008, 900008DD, 90099668, 90500557, 90555007, 90666668, 90909998, 90990998, 90996668, 9099999D, 90D00098, 90D90998, 95500057, 99099098, 99555057, 99900998, 99966608, 99966668, 99999668, 99999998, 9D009008, 9D090998, A0803334, A2222252, AAA52222, B00005AB, B000B55B, B0BBBB5B, B3330034, BB0BBB1B, BBAA3334, BBB0BB1B, BBB0BB5B, BBDB000B, D000BBBB, D00100C7, D8888888, D900008D, D9000098, DBB000BB, DC0CCCC8, DCC0CCC8, DCCCC008, DD000908, DD09009D, DDDDDDAB, DDDDDEEE, DDDEEE8E, DDDEEEE8, DEE80008, E0777E97, E0E0E397, E0E77797, E0EE0397, E7777797, E9066668, EE00E397, EE077797, EE0E0397, EEE00797, EEE07E97, EEE0AA52, EEE55397, EEE55557, EEEAAA52, EEEEE834, EEEEEA52, 300003331, 300007661, 300330031, 333000004, 333300001, 333B00034, 3700000AB, 3B3300034, 500000057, 555555007, 555555557, 5DDDDDDDD, 600000331, 7500000AB, 75000A00B, 75A00000B, 761000001, 77000E0C7, 777700EC7, 7777730AB, 7777777AE, 77777EE97, 7777E7E97, 777999997, 7A500000B, 7BBBBBB5B, 88888A834, 900000031, 900666608, 909990098, 90D009998, 950000557, 966666008, 990000007, 990555507, 999999997, A000000B4, A0005999B, AAEEEEE34, B000AA334, BBBBB005B, BBBBBBB5B, D09999998, D0D90009D, D800000DD, D90009998, DCCCC0CC8, DE88EEEEE, DEEEEEE88, E000B7777, E000BBBBB, E003ABBBB, EE0000797, EE0EEE397, EE5555557, EE777EE97, EEEEEE537, EEEEEE937, 2222222252, 3000000071, 3330030001, 3333303001, 3333330001, 500000007B, 5555555097, 7000000071, 77000000C7, 8333333331, 8888883334, 8888888834, 888888AA34, 900000009B, 900000009D, 900000DD9D, 9000099998, 9955555507, 9D0000099D, 9D05555557, AB0000005B, B000000DAB, B00000BBDB, BB00BB0B5B, BB0BB00B5B, D000099998, D00090008D, D0D000909D, D0DDDDDDDB, D300000007, D88EEEEEEE, D900999998, DD00900008, DDD6EEEEEE, DDDDDDD6EE, DDDDDDDDDE, DDDEEEEEEE, DEEEEE8008, E000000797, 7777777CCCE, 88888830004, 90000009D9D, 99955555557, 9999999999D, B00000D00AB, BB000BBB05B, BBBB0000B5B, D000009080D, D000090800D, D090800000D, DDDDDDD999B, DDDDDDDDD9B, EEEEEE00397, EEEEEEE0397, 333000000301, 5000000000DD, 73A00000000B, 9000000000B7, 903333333331, ABB00000000B, D000000001C7, DCCCCCCCCCC8, E0EEEEEEE397, 19A000000000B, 3333333333331, 3BBBBBBBBBBBB, 9333333333331, A00000000099B, B00000000050D, EEEEEEEEEE76E, 1000000000999B, 71000000000001, 908D000000000D, BBBBBBBBBB6661, 77777777777777B, BB00000000BBB5B, DEEEEEEEEEEEEEE, 7777777777777E97, B0BBBBBBBBBBBB1B, BB0000000000DB0B, D000000000000998, D908000000000000D, DDDDDDDDDDDDDDDDB, E9666666666666668, 3330000000000000031, D00000000000000908D, E0BBBBBBBBBBBBBBBBB, 2EEEEEEEEEEEEEEEEE52, 77777777777777777ECE, 5000000000000000005AB, 777777777777777777997, 7BBBBBBBBBBBBBBBBBBBB, BB0000000000000000DBB, DD000000000000000909D, D900000000000000000DDD, DD0000000000000000099D, BBBBBBBBBBBBBBBBBBBBBB1, B00000000000000000000005B, B0700000000000000000000001, B70000000000000000000000001, 705000000000000000000000000B, 633000000000000000000000000001, EBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB, 500000000000000000000000000000000017, 77777777777777777777777777777777777777777777777777777777777CCE, 7777777777777777777777777777777777777777777777777777777777777777777777777CE, 96666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666608, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE397, 7777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777797 ===Base 16=== 11, 13, 17, 1D, 1F, 25, 29, 2B, 2F, 35, 3B, 3D, 43, 47, 49, 4F, 53, 59, 61, 65, 67, 6B, 6D, 71, 7F, 83, 89, 8B, 95, 97, 9D, A3, A7, AD, B3, B5, BF, C1, C5, C7, D3, DF, E3, E5, E9, EF, F1, FB, 14B, 15B, 185, 199, 1A5, 1BB, 1C9, 1EB, 223, 22D, 233, 241, 277, 281, 287, 28D, 2A1, 2D7, 2DD, 2E7, 301, 337, 373, 377, 38F, 3A1, 3A9, 41B, 42D, 445, 455, 45D, 481, 4B1, 4BD, 4CD, 4D5, 4E1, 4EB, 50B, 515, 51B, 527, 551, 557, 55D, 577, 581, 58F, 5AB, 5CB, 5CF, 5D1, 5D5, 5DB, 5E7, 623, 709, 727, 737, 745, 74B, 755, 757, 773, 779, 78D, 7BB, 7C3, 7C9, 7CD, 7DB, 7EB, 7ED, 805, 80F, 815, 821, 827, 841, 851, 85D, 85F, 8A5, 8DD, 8E1, 8F5, 923, 98F, 99B, 9A9, 9EB, A21, A6F, A81, A85, A99, A9F, AA9, AAB, ACF, B1B, B2D, B7B, B8D, B99, B9B, BB7, BB9, BCB, BDD, BE1, C0B, CB9, CBB, CEB, D01, D21, D2D, D55, D69, D79, D81, D85, D87, D8D, DAB, DB7, DBD, DC9, DCD, DD5, DDB, DE7, E21, E27, E4B, E7D, E87, EB1, EB7, ED1, EDB, EED, F07, F0D, F4D, FD9, FFD, 1069, 1505, 1609, 1669, 16A9, 19AB, 1A69, 1AB9, 2027, 204D, 2063, 207D, 20C3, 20ED, 2221, 22E1, 2327, 244D, 26C3, 274D, 2E01, 2E0D, 2ECD, 3023, 3079, 3109, 3263, 3341, 36AF, 3941, 3991, 39AF, 3E41, 3E81, 3EE1, 3EE7, 3F79, 4021, 40DB, 440B, 444B, 44A1, 44AB, 44DB, 4541, 45BB, 4A41, 4B0B, 4BBB, 4C4B, 4D41, 4DED, 5045, 50A1, 50ED, 540D, 5441, 555B, 556F, 5585, 560F, 56FF, 5705, 574D, 580D, 582D, 5855, 588D, 5A01, 5AA1, 5B01, 5B4B, 5B87, 5BB1, 5BEB, 5C4D, 5CDD, 5CED, 5DD7, 5DDD, 5E0D, 5E2D, 5EBB, 68FF, 6A69, 6AC9, 6C8F, 6CA9, 6CAF, 6F8F, 6FAF, 7033, 7063, 7075, 7087, 70A5, 70AB, 7303, 7393, 74DD, 754D, 7603, 7633, 7663, 7669, 7705, 772D, 775D, 77D5, 7807, 7877, 7885, 7939, 7969, 7993, 79AB, 7A05, 7A69, 7A9B, 7AA5, 7B77, 7BA9, 7D4D, 7D75, 7D77, 8077, 808D, 80D7, 80E7, 8587, 86CF, 8777, 8785, 8885, 88CF, 88ED, 88FD, 8C6F, 8C8F, 8E8D, 8EE7, 8F2D, 8F8D, 9031, 9041, 90AF, 90B9, 9221, 9319, 9401, 944B, 9881, 9931, 9941, 9991, 99AF, 9A0F, 9A1B, 9A4B, 9AFF, 9BA1, 9BB1, 9CAF, 9E81, 9EA1, 9FAF, A001, A05B, A0C9, A105, A10B, A4CB, A55B, A6C9, A88F, A91B, A9B1, A9BB, AA15, AB01, AB0B, AB19, ABBB, AC09, AF09, B041, B04B, B069, B07D, B087, B0B1, B0ED, B1A9, B201, B40B, B40D, B609, B70D, B7A9, B807, B9A1, BA41, BAA1, BB4B, BBB1, BBDB, BBED, BD19, BD41, BDBB, BDEB, BE07, BEE7, C0D9, C203, C24D, C6A9, C88D, C88F, C8CF, C8ED, C9AF, C9CB, CA09, CA4B, CA69, CAC9, CC0D, CC23, CC4D, CC9B, CD09, CDD9, CE4D, CEDD, CFA9, CFCD, D04B, D099, D405, D415, D44B, D4A5, D4DD, D50D, D70B, D74D, D77B, D7CB, D91B, D991, DA05, DA09, DA15, DA51, DB91, DBEB, DD7D, DDA1, DDED, DE0B, DE41, DE4D, DEA1, E02D, E07B, E0D7, E1CB, E2CD, E401, E801, EABB, EACB, EAEB, EBAB, EC4D, ECDD, ED07, EDD7, EE7B, EE81, EEAB, EEE1, F08F, F0A9, F227, F2ED, F3AF, F485, F58D, F72D, F763, F769, F787, F7A5, F7E7, F82D, F86F, F877, F88D, F8D7, F8E7, F8FF, FCCD, FED7, FF85, FF8F, FFA9, 100AB, 10BA9, 1A0CB, 1BA09, 200E1, 2C603, 2CC03, 30227, 303AF, 30AAF, 32003, 32207, 32CC3, 330AF, 33169, 33221, 33391, 33881, 33AFF, 38807, 38887, 3AFFF, 3F203, 3F887, 3FAFF, 400BB, 4084D, 40A4B, 42001, 44221, 44401, 444D1, 4480D, 4488D, 44CCB, 44D4D, 44E8D, 4804D, 4840D, 4A0CB, 4A54B, 4CACB, 4D0DD, 4D40D, 4D44D, 5004D, 50075, 502CD, 5044D, 50887, 50EE1, 5448D, 548ED, 55A45, 55F45, 5844D, 5A4A5, 5AE41, 5B0CD, 5B44D, 5BBCD, 5D4ED, 5E0E1, 5EB4D, 5EC8D, 5ECCD, 5EE41, 5F06F, 5F7DD, 5F885, 5F8CD, 5FC8D, 5FF75, 6088F, 60AFF, 630AF, 633AF, 660A9, 668CF, 669AF, 66A09, 66A0F, 66FA9, 6886F, 6A00F, 6A0FF, 6A8AF, 6AFFF, 7002D, 7024D, 70B0D, 70B7D, 7200D, 73363, 73999, 7444D, 770B7, 777D7, 77B07, 77D7D, 77DD7, 79003, 79999, 7B00D, 7D05D, 7D7DD, 8007D, 800D1, 8074D, 82CCD, 82E4D, 8448D, 8484D, 8704D, 8724D, 87887, 88001, 8800D, 880CD, 88507, 88555, 8866F, 8872D, 8877D, 888D1, 888D7, 88AA1, 88C2D, 88D57, 88D75, 88D77, 8AFAF, 8C2CD, 8C40D, 8C8CD, 8CCED, 8CE2D, 8CFED, 8E007, 8E20D, 8E24D, 8F6FF, 8FAAF, 900CB, 901AB, 90901, 909A1, 90AB1, 90AE1, 90EE1, 910AB, 93331, 940AB, 963AF, 966AF, 99019, 99109, 99A01, 9AAE1, 9B00B, 9B0AB, 9B441, 9BABB, 9BBBB, 9E441, A00BB, A0405, A044B, A08AF, A0A51, A0B91, A0C4B, A1B09, A54A5, A5B41, A6609, A904B, A94A1, A9C4B, A9E01, A9E41, AA0A1, AA441, AA501, AA8AF, AAEE1, AAF45, AAF8F, ABBA1, ACC69, AE0BB, AE0EB, AEAE1, AEE0B, AEEA1, AEECB, AF045, AF4A5, AFA8F, B00A1, B00D7, B044D, B0777, B0A0B, B0A91, B0BBD, B0BCD, B0C09, B0DA9, B0EAB, B2207, B4001, B6669, B7707, B7D07, B8081, B9021, BA091, BA109, BA4BB, BB001, BB0EB, BB8A1, BBBEB, BBE0B, BBEBB, BC009, BCECD, BD0A9, BE44D, BEB0D, BEBBB, BEEBB, C0263, C02C3, C02ED, C040D, C0CA9, C0CCD, C2663, C2CED, C32C3, C3323, C400D, C40ED, C44CB, C44ED, C480D, C484D, C4CAB, C60AF, C686F, C6A0F, C86FF, C8C2D, CAA0F, CAFAF, CBCED, CC0AF, CC44B, CC82D, CC8FF, CCAF9, CCAFF, CCCFD, CCFAF, CD00D, CD4CB, CD4ED, CDDDD, CF2C3, CFC8F, CFE8D, D0045, D07DD, D09BB, D0D4D, D0DD7, D0EBB, D0EEB, D1009, D1045, D10B9, D1BA9, D54BB, D54ED, D5AE1, D5D07, D5EE1, D70DD, D7707, D7777, D77DD, D7DD7, D9441, D9AE1, D9B0B, DA9A1, DA9E1, DAA41, DAAA1, DBB0B, DBBA1, DC4CB, DD227, DD44D, DDDD7, E0081, E00E1, E010B, E088D, E08CD, E0B0D, E0BBD, E100B, E4D0D, E777B, E77AB, E7CCB, E844D, E848D, E884D, E88A1, EB0BB, EBB4D, EBBEB, EBEEB, EC8CD, ECBCD, ECC8D, ED04D, EE001, EE0EB, EE4A1, EEEBB, F0085, F09AF, F0C23, F0CAF, F2663, F2C03, F3799, F3887, F4A05, F4AA5, F506F, F5845, F5885, F5C2D, F5ECD, F5F45, F66A9, F688F, F6AFF, F7399, F777D, F8545, F8555, F8AAF, F8F87, F9AAF, FA0F9, FA405, FA669, FAFF9, FC263, FCA0F, FCAFF, FCE8D, FCF23, FD777, FDDDD, FDEDD, FEC2D, FEC8D, FF545, FF6AF, FF739, FF775, FF9AF, FFC23, 100055, 100555, 10A9CB, 1A090B, 1A900B, 1CACCB, 1CCACB, 20DEE1, 266003, 3000AF, 300A0F, 300AFF, 308087, 308E07, 3323E1, 333A0F, 339331, 33CA0F, 33CF23, 33CFAF, 33F323, 380087, 3A00AF, 3A0F0F, 3AA0FF, 3AAF0F, 3C33AF, 3C3A0F, 3C3FAF, 3CCAAF, 3F0FAF, 3F32C3, 3FF0AF, 3FFAAF, 4004CB, 400A05, 4048ED, 404DDD, 40AA05, 40D04D, 40DD4D, 40E0DD, 40E48D, 440041, 44008D, 44044D, 4404DD, 44440D, 4448ED, 4484ED, 448E4D, 44E44D, 48888D, 4AA005, 4DD00D, 4DD04D, 4DDD0D, 4E048D, 4E448D, 4E880D, 5000DD, 500201, 50066F, 5008CD, 500C2D, 500D7D, 50C20D, 520C0D, 544EDD, 54AA05, 54AAA5, 54ED4D, 566AAF, 57D00D, 580087, 5A5545, 5C20CD, 5C8CCD, 5CC2CD, 5D000D, 5D070D, 5F666F, 5FAA45, 5FFF45, 60008F, 600A0F, 603AAF, 6060AF, 6066AF, 60A0AF, 63AA0F, 6663AF, 66668F, 666AAF, 668A8F, 66AFF9, 68888F, 693AAF, 7007B7, 70404D, 70770B, 70770D, 707BE7, 70DD0D, 733339, 733699, 74004D, 74040D, 77007B, 770CCB, 777B4D, 777BE7, 777CCB, 77ACCB, 77B74D, 77D0DD, 7A0CCB, 7B744D, 7CACCB, 7DDD99, 80044D, 800807, 80200D, 8044ED, 80C04D, 80CC2D, 80E44D, 8404ED, 84888D, 84E04D, 84E40D, 86686F, 8668AF, 8686AF, 86F66F, 86FFFF, 87000D, 87744D, 880807, 886AFF, 88824D, 88870D, 888787, 88884D, 88886F, 88887D, 88888D, 888C4D, 888FAF, 88AA8F, 88CC8D, 88F6AF, 88F8AF, 88FA8F, 88FF6F, 88FF87, 88FFAF, 8A8FFF, 8C0C2D, 8C802D, 8CCFFF, 8CE00D, 8CE0CD, 8CFCCF, 8E00CD, 8E044D, 8E0CCD, 8EC0CD, 8F68AF, 8F88F7, 8FCFCF, 8FF887, 8FFCCF, 8FFF6F, 9002E1, 9004AB, 9008A1, 900919, 900ABB, 900B21, 90B801, 90CCCB, 9332E1, 944441, 94ACCB, 990001, 9900A1, 9A4441, 9A4AA1, 9AA4A1, 9AAA41, 9AAAAF, 9B66C9, 9BBA0B, 9BC0C9, 9BC669, 9BC6C9, 9C4ACB, A0094B, A00ECB, A09441, A0A08F, A0E0CB, A0ECCB, A0F669, A40A05, A4AAA5, A50E41, A5AA45, A60069, A8FAFF, A9AA41, AA5E41, AAA4A5, AAA545, AC6669, ACCC4B, ACCCC9, AEAA41, AFF405, AFF669, AFFA45, AFFFF9, B00921, B00BEB, B00CC9, B00D91, B08801, B0D077, B70077, B70E77, B77E77, B88877, B88881, B94421, BAE00B, BB00AB, BB0DA1, BB444D, BB44D1, BB8881, BBBBBD, BBBC4D, BBCCCD, BC0CC9, BC66C9, BCC669, BCC6C9, BCCC09, BE000D, BE00BD, BE0B4D, BE0CCD, BEA00B, BECCCD, C0084D, C00A0F, C0608F, C0668F, C0844D, C0A0FF, C0AFF9, C0C3AF, C0C68F, C0CAAF, C0CDED, C0D0ED, C0E80D, C0EC2D, C0EC8D, C0FA0F, C0FAAF, C2CC63, C30CAF, C333AF, C3CAAF, C3CCAF, C4048D, C40D4D, C4404D, C4408D, C4440D, C44DDD, C4ACCB, C4DCCB, C4DD4D, C6068F, C66AAF, C68AAF, C6AA8F, C8044D, C8440D, C8666F, CA00FF, CA0FFF, CAAAAF, CAAFFF, CAFF0F, CBE0CD, CC008F, CC0C8F, CC3CAF, CC4ACB, CC608F, CC66AF, CCBECD, CCC4AB, CCCA0F, CCCC8F, CCCE8D, CE0C8D, CF0F23, CF0FAF, CFAFFF, CFCAAF, CFFAFF, D0005D, D00BA9, D05EDD, D077D7, D10CCB, D22207, D4000B, D4040D, D4044D, D40CCB, D70077, D7D00D, D90009, D900BB, DB00BB, DB4441, DD400D, DDD109, DDD1A9, DDD919, DDD941, DED00D, E00D4D, E00EEB, E0AAE1, E0AE41, E0AEA1, E0B44D, E0BCCD, E0BEBB, E0D0DD, E0E441, E4048D, E4448D, E800CD, E8200D, EA0E41, EAA0E1, EBB00B, ECCCAB, EDDDDD, EEBE0B, F00263, F0056F, F00A45, F02C63, F03F23, F05405, F060AF, F08585, F0A4A5, F0F2C3, F0F323, F2CCC3, F33203, F33C23, F5F66F, F5FF6F, F68CCF, F6AA8F, F888AF, FA0F45, FAA045, FAA545, FAFC69, FC0AAF, FC66AF, FCCCAF, FCFFAF, FF0323, FF056F, FF3203, FF7903, FFA045, FFA4A5, FFAA45, FFC0AF, FFF4A5, FFF575, FFFA45, FFFCAF, 10A009B, 20000D1, 2CCC663, 30A00FF, 30CCCAF, 30FA00F, 30FCCAF, 3333C23, 333C2C3, 33C3AAF, 33FCAAF, 33FFFAF, 3A0A00F, 3AAAA0F, 3AF000F, 3AFAAAF, 3C0CA0F, 3CCC3AF, 3CFF323, 3F33F23, 3FAA00F, 3FF3323, 4004441, 400DDD1, 400E00D, 400ED0D, 404404D, 404448D, 404E4DD, 440EDDD, 4440EDD, 44444ED, 4444E4D, 44DDDDD, 4A000A5, 4CCCCAB, 4D0CCCB, 4E4404D, 4E4444D, 4E4DDDD, 5000021, 5004221, 5006AAF, 500FF6F, 5042201, 508CCCD, 5400005, 5400AA5, 5555405, 5808007, 5AA4005, 5C0008D, 5CCC8CD, 5D4444D, 5EEEEEB, 5F40005, 5F554A5, 5F6AAAF, 60000AF, 60006A9, 600866F, 6008AAF, 600AA8F, 600F6A9, 606608F, 606686F, 608666F, 60AA08F, 60AAA8F, 66000AF, 66666A9, 6666AF9, 6866A8F, 6AAAAAF, 70070D7, 70077DD, 700DDDD, 707077D, 707D007, 70D00DD, 770077D, 770400D, 770740D, 7777775, 77777B7, 77777DD, 7777ACB, 77B88E7, 77DD00D, 77DDDDD, 7D0D00D, 7DD0D07, 7DDD00D, 800002D, 8000CED, 80C0E0D, 80CECCD, 840400D, 844000D, 844E00D, 868688F, 880444D, 884404D, 887D007, 8888801, 8888881, 8888E07, 8888F77, 8888FE7, 88A8AFF, 88AAAFF, 88FAFFF, 8A8AAAF, 8A8AAFF, 8AAA8FF, 8C00ECD, 8C8444D, 8E4400D, 8FCCCCF, 900BBAB, 90CC4AB, 9908AA1, 99E0E01, 9B00801, 9B6CCC9, A000FF9, A006069, A00A8FF, A01CCCB, A05F545, A0BEEEB, A0E4AA1, AA0008F, AA08FFF, AA40AA5, AA8FFFF, AAAA405, AE04AA1, AE44441, AE4AAA1, AECCCCB, AF40005, AFA5A45, AFFFC69, B000BAB, B000EBB, B0D0007, B222227, B6CCCC9, B8880A1, BA000EB, BA0BEEB, BAEEEEB, BB000CD, BB00C0D, BB0B00D, BC6CC69, BC6CCC9, BCCCC69, BCCCCED, C0000A9, 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4444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444DD ===Base 17=== 12, 16, 1C, 1E, 23, 27, 29, 2D, 32, 38, 3A, 3G, 43, 45, 4B, 4F, 54, 5C, 5G, 61, 65, 67, 6B, 78, 7C, 81, 83, 8D, 8F, 94, 9A, 9E, A3, A9, AB, B4, B6, BA, BC, C7, D2, D6, D8, DC, E1, E3, ED, F2, F8, FE, FG, G5, G9, GB, 104, 111, 115, 117, 11B, 137, 139, 13D, 14A, 14G, 155, 159, 15F, 171, 17B, 17D, 188, 191, 197, 19F, 1A4, 1A8, 1B3, 1BB, 1BF, 1DB, 1DD, 1F3, 1FD, 1G8, 1GA, 1GG, 20F, 214, 221, 225, 241, 25A, 25E, 285, 2B8, 2C5, 2CF, 2E5, 2EB, 2F6, 30E, 313, 331, 33B, 346, 34C, 351, 35F, 36E, 375, 37B, 391, 39B, 39D, 3B7, 3B9, 3BF, 3D3, 3D5, 3D9, 3DF, 3E4, 3EC, 3F1, 3F7, 407, 418, 447, 44D, 472, 474, 47E, 47G, 489, 49C, 4A1, 4C1, 4CD, 4D4, 4G1, 502, 506, 508, 50E, 519, 522, 528, 52A, 52E, 533, 53F, 551, 55D, 562, 566, 573, 577, 57F, 582, 593, 599, 59B, 59F, 5A6, 5B5, 5D1, 5D3, 5EA, 5EE, 5F9, 60D, 62F, 634, 649, 689, 692, 6CD, 6EF, 6F4, 6FA, 704, 706, 70G, 71D, 726, 737, 739, 73D, 73F, 753, 755, 764, 766, 76G, 771, 77B, 793, 7AA, 7AE, 7B3, 7BB, 7D7, 7E6, 7F3, 7F9, 7FF, 7G2, 7GE, 7GG, 825, 82B, 849, 852, 85E, 869, 876, 87A, 87G, 88B, 892, 898, 89C, 8C5, 8E7, 8G7, 908, 90G, 913, 91F, 92C, 935, 937, 93B, 951, 953, 957, 95D, 968, 96G, 979, 97B, 98C, 98G, 99D, 9B1, 9B3, 9B9, 9BD, 9BF, 9DB, 9DF, 9F1, 9F5, 9G6, A07, A0D, A1A, A2F, A4D, A72, A7A, A7E, AA1, AA7, ACF, ADA, AG1, AG7, B02, B08, B17, B1D, B28, B2G, B57, B71, B73, B79, B7F, B88, B8E, B8G, B9B, B9F, BB5, BB7, BD7, BDD, BEG, BFF, BGG, C01, C2F, C3E, C56, C6D, C89, C92, C9G, CA5, CBG, CC1, CC5, CF4, CFA, D04, D0A, D15, D3D, D3F, D55, D59, D5B, D71, D75, D7D, D91, D97, D99, D9D, DA4, DAG, DB3, DDB, DF1, DF7, DF9, DFF, E05, E0B, E2B, E52, E58, E69, E92, E9C, EAF, EB8, EC9, ECB, EE5, F04, F15, F1B, F35, F3B, F46, F51, F53, F64, F6A, F73, F79, F95, FAC, FB1, FCA, FD5, FDB, FF1, FF7, FFD, G0D, G0F, G18, G1A, G1G, G2F, G34, G63, G7G, GA7, GC3, GDG, GEF, GFA, GG7, GGD, 1013, 101D, 1033, 1035, 1051, 105B, 105D, 1077, 108A, 109B, 10AG, 10B1, 10B7, 10BD, 10FB, 1149, 1189, 11AF, 11G3, 1303, 130B, 1314, 1341, 1479, 14D9, 1501, 1503, 15A1, 15B8, 1734, 1749, 17AF, 17G3, 1844, 185B, 1875, 1877, 18AG, 18B5, 1903, 1909, 1958, 19BG, 19G3, 1A5D, 1A75, 1A7F, 1ADF, 1AF1, 1B01, 1B09, 1B18, 1B85, 1B89, 1BDG, 1BGD, 1D07, 1D49, 1D9G, 1DF4, 1F09, 1F47, 1F5A, 1F74, 1F7A, 1FA1, 1FAF, 2018, 201G, 202B, 208B, 20G1, 215B, 218G, 21AG, 21B1, 222F, 22AF, 22BG, 22EF, 22F4, 22GF, 251B, 2526, 25F1, 266F, 26FC, 280B, 2A05, 2A58, 2AFC, 2AGF, 2B1B, 2B1F, 2BGE, 2C1G, 2C2B, 2C8B, 2CG1, 2E2F, 2EGF, 2F0C, 2F55, 2FAA, 2FC4, 2FFF, 2GA1, 2GFC, 2GG1, 2GGF, 301B, 301F, 3037, 3053, 3057, 3079, 3095, 30B3, 30BD, 30C4, 31F4, 330D, 3334, 333E, 3349, 3376, 337E, 33CD, 33EF, 3411, 3417, 3499, 3503, 3505, 3509, 353E, 35E5, 35EB, 3604, 36FD, 3701, 3741, 374D, 376F, 3796, 37D4, 37F4, 3956, 3B03, 3B05, 3B0B, 3BBE, 3C04, 3C15, 3C19, 3C4E, 3C59, 3C64, 3CB3, 3CDB, 3CE6, 3D07, 3D14, 3DDE, 3E77, 3E79, 3E7F, 3E99, 3EEE, 3EFB, 3F05, 3F0D, 3FCB, 3FF4, 4009, 4021, 4069, 4098, 40DG, 40GD, 419D, 4201, 4401, 4492, 46AD, 46C9, 46DA, 4719, 476A, 4779, 479D, 47A6, 4906, 4911, 4917, 4919, 491D, 492G, 4982, 4988, 49D7, 49D9, 49GG, 4ADE, 4AE7, 4C49, 4C96, 4CC9, 4D79, 4DAE, 4DEG, 4E7A, 4E96, 4EG7, 4G6D, 4G87, 501B, 5037, 5059, 507D, 50BB, 50BF, 50D7, 50DD, 50F1, 5105, 51A7, 51AD, 521B, 525F, 52FB, 5307, 5356, 53BE, 53DE, 53E9, 5507, 550B, 5587, 5598, 55EF, 560A, 568E, 56AA, 56F3, 5709, 5725, 572B, 575A, 575E, 5769, 57A1, 57B2, 5868, 586E, 58AE, 58B9, 590D, 5918, 5952, 5958, 596D, 5A17, 5A1F, 5ADD, 5ADF, 5AE8, 5B07, 5B21, 5B2F, 5B3E, 5BEF, 5DA7, 5DEB, 5E57, 5E5F, 5E86, 5E97, 5EB9, 5EBF, 5EF5, 5F01, 5F1A, 5F6F, 5FA7, 5FDA, 60AF, 60G3, 64AD, 64DE, 64DG, 663E, 666D, 66AF, 693D, 69CG, 69D3, 69D9, 69G8, 69GC, 6ADE, 6AGD, 6C98, 6D33, 6D4E, 6D93, 6D9F, 6DDD, 6DEE, 6DF3, 6DFD, 6DGE, 6E09, 6G36, 6G4D, 6G6D, 6GD4, 6GDE, 6GFC, 702E, 7057, 705B, 7073, 7079, 7095, 70B5, 70BD, 70D1, 70E2, 70F5, 7107, 7149, 719G, 71BG, 71F4, 724E, 724G, 725F, 72A2, 72BF, 72EE, 72GA, 7314, 733E, 7341, 7363, 73EB, 7419, 742A, 742G, 7442, 74EG, 7501, 750F, 751A, 756D, 757E, 75A1, 75A7, 75BE, 75DA, 75E9, 75F6, 7622, 769F, 76EA, 7734, 773E, 776D, 779G, 77AF, 7905, 790B, 7976, 79B2, 79F6, 79GD, 7A1F, 7A5D, 7AD5, 7ADF, 7AF1, 7AFD, 7B01, 7B09, 7B2F, 7B52, 7B72, 7BE5, 7D01, 7D05, 7D9G, 7DAF, 7DBG, 7E0A, 7E75, 7EA2, 7EA4, 7EB5, 7EBF, 7EE2, 7EF7, 7EG4, 7F0B, 7F14, 7F5A, 7F76, 7FA7, 7G1F, 7G46, 7GA6, 7GD3, 7GDF, 8009, 8058, 80B8, 80E9, 84A7, 850A, 8557, 857B, 85A8, 870E, 8744, 8777, 879B, 87B5, 87B7, 87EE, 8805, 8872, 8887, 8889, 88E9, 8906, 8959, 8966, 89GG, 8A87, 8AE5, 8B0G, 8B59, 8B95, 8B97, 8CB2, 8CB8, 8CE9, 8E56, 8EE9, 9026, 9031, 903D, 907F, 9091, 909B, 90FB, 9101, 910D, 9118, 917G, 9185, 9189, 91B8, 9202, 9288, 92B5, 92FB, 92GG, 93C1, 9505, 950B, 950F, 952B, 956F, 9592, 9596, 9598, 9602, 96D9, 96FD, 971G, 9725, 9752, 97DG, 9855, 9862, 9895, 9899, 98B7, 98BB, 9901, 990B, 9921, 992F, 99G3, 9B0B, 9B2B, 9B8B, 9BB8, 9BBG, 9C19, 9C1B, 9C31, 9C59, 9C95, 9CD5, 9CFB, 9CGC, 9D03, 9D07, 9D7G, 9DG1, 9DGD, 9F0B, 9F76, 9FCB, 9G11, 9G1D, 9G28, 9G3F, 9G7D, 9GCC, 9GD7, 9GF7, 9GFD, 9GG8, A025, A041, A058, A0C5, A0F6, A0GF, A11F, A184, A1F7, A21G, A258, A401, A421, A476, A511, A517, A57D, A5A8, A5E8, A6AD, A6FC, A6GF, A751, A77F, A7F5, A7FD, A7G6, A847, AACD, AC1G, AC41, AC58, AC5E, ACGD, AD0E, AD0G, AD1F, AD51, ADD5, ADE4, ADF5, ADGE, AE56, AE74, AEF6, AEFA, AF77, AF7D, AFA4, AFCC, AFD7, AFDD, AGAF, AGF4, B00G, B037, B055, B05B, B075, B0D5, B0FD, B10F, B198, B25F, B2F1, B2F5, B307, B309, B35E, B3EF, B50D, B589, B7BE, B7BG, B7E7, B875, B952, B958, B97G, B99G, B9G7, B9GD, BB01, BB2F, BB3E, BB89, BB98, BBDE, BD03, BD09, BD5E, BDE5, BDEB, BDG1, BE5F, BF01, BF0D, BG13, BG1F, BG3F, BGD1, BGE2, BGE8, C00B, C034, C05A, C0AF, C0EF, C0GF, C153, C15B, C199, C1B9, C1D1, C1D5, C1F9, C205, C21A, C21G, C252, C258, C2B2, C335, C33D, C35D, C364, C395, C3B3, C3F5, C3FB, C3FD, C414, C41A, C469, C496, C4DA, C4GD, C535, C55B, C5B1, C5BD, C5D9, C5DF, C5E8, C5F3, C5F5, C6E9, C85A, C885, C8B8, C8BE, C8CB, C8E5, C919, C931, C959, C95F, C9D3, CA0F, CA18, CA1G, CAD4, CADE, CAEF, CAGD, CB22, CB33, CB35, CB3F, CB5D, CB82, CB99, CBB1, CBFB, CC49, CCCB, CCDE, CD11, CD1D, CD39, CD4A, CD53, CD93, CDAE, CDD5, CDF3, CDFD, CDG4, CE49, CE5A, CE8B, CF13, CF19, CF5D, CF5F, CFB9, CFBF, CFD9, CFDF, CG14, CG41, CG6F, CGCF, CGF6, CGG1, D01F, D039, D079, D09B, D09F, D0B7, D0BB, D0D1, D0EG, D0GG, D10D, D19G, D1G3, D30B, D347, D3BE, D4E4, D50D, D57E, D5AD, D5FA, D707, D73E, D7E7, D7GF, DA1F, DA57, DAAE, DB01, DB09, DB0D, DB7E, DB9G, DD05, DD7E, DDA5, DDFA, DDG3, DE0G, DE44, DE4A, DE77, DEAE, DEB9, DEBB, DF03, DF05, DG0E, DGDF, E009, E06F, E072, E07G, E089, E0CF, E0E9, E0G7, E47A, E498, E4E7, E50A, E559, E55F, E575, E5B9, E5BF, E5F5, E5F7, E6FC, E722, E724, E72A, E72E, E744, E746, E75B, E76E, E79B, E7A4, E7A6, E7AG, E7B5, E7B7, E7EG, E7G4, E887, E89G, E8E9, E906, E955, E95B, E95F, E988, E99F, E9F9, E9G8, E9GG, EA25, EA7G, EAC5, EAE7, EB7B, EBF5, EBF7, EBFB, EC6F, ECCF, ECEF, EE72, EE76, EE89, EE9G, EF0A, EF44, EF77, EF97, EFA4, EFB5, EFC6, EFFF, EG6F, EG74, EGE7, EGFC, EGGF, F019, F01F, F075, F091, F09B, F0BF, F0FB, F10F, F1A7, F1AD, F1D4, F376, F3CD, F3F4, F40A, F411, F444, F44A, F497, F499, F49D, F4D7, F509, F57A, F5AD, F5F6, F6D3, F6D9, F70D, F741, F747, F76D, F7F6, F7FA, F907, F976, F9CB, FA11, FA7D, FADD, FB09, FC4C, FC5D, FC5F, FC91, FCB9, FD1A, FD41, FD47, FDF4, FF0B, FF56, G021, G07A, G0A1, G0E7, G11F, G17F, G1DF, G1F1, G1F7, G201, G2A1, G306, G311, G36C, G377, G37F, G3CC, G3CE, G3D1, G476, G487, G4DE, G6AF, G6D4, G6F6, G6GF, G713, G724, G731, G742, G74E, G76E, G7A2, G872, G874, GA21, GAC1, GC6F, GCAF, GCD4, GCDA, GCG1, GD73, GD7F, GDAE, GDDF, GDEA, GDFD, GE47, GE7E, GF13, GF33, GF3F, GF4C, GF71, GF7F, GFDD, GG01, GG21, GGAF, GGC1, 1000G, 10053, 100AA, 100B9, 100F1, 100FF, 10301, 10587, 10705, 1075A, 107GF, 10895, 108B9, 10985, 1099G, 10B98, 10B9G, 10D03, 10D0F, 10D7A, 10DG3, 10DG7, 10G1F, 10G3F, 110GF, 1140D, 11D93, 11DG4, 11F0A, 11G4D, 11GD4, 13333, 133FF, 13F44, 14109, 14499, 150A7, 153B1, 1570A, 17005, 17799, 177AG, 17995, 17A7G, 17G47, 18079, 18507, 185A7, 18B07, 18B9G, 19333, 199B5, 1A00A, 1A00G, 1A0F5, 1AAAA, 1AAAG, 1AF05, 1AFFA, 1B07G, 1B10G, 1B807, 1D001, 1D1AA, 1D7G4, 1DG03, 1DG41, 1F001, 1F00F, 1F01A, 1F0A7, 1F199, 1F1F9, 1F414, 1F449, 1F7F5, 1F999, 1FF0A, 1FFAA, 1FFB5, 1G073, 1G14D, 1G1F4, 1G301, 1G477, 1GD01, 1GD47, 1GF07, 1GFF4, 20005, 200A1, 2010A, 20586, 20588, 20A01, 20B11, 20B15, 20BEE, 20C1A, 20CBE, 210B5, 21A1F, 21A51, 21F1A, 21G1F, 21GFF, 222BE, 228B2, 228BE, 22BE2, 22C0B, 22F0A, 252BB, 25505, 25552, 26GAF, 2A001, 2A1FF, 2A55F, 2AEEF, 2AF44, 2B051, 2B20E, 2BB2B, 2BBBG, 2BE22, 2BEE2, 2BEEE, 2BF0B, 2C0BE, 2C18A, 2F101, 2F1FA, 2F44C, 2FCBB, 2G1FF, 2GA6F, 2GF44, 30035, 300B1, 300FB, 30101, 303C5, 30444, 30497, 304D1, 304D7, 30703, 30714, 30734, 30763, 30774, 30CF5, 30CFD, 30D41, 30FC5, 3100B, 31779, 31F5B, 31FB5, 31FFF, 330C5, 330F4, 33357, 33373, 33379, 33555, 33557, 33777, 3379F, 337FD, 33997, 33D44, 33D4E, 33F3D, 33FF5, 34019, 34044, 340D1, 353DD, 35535, 355B3, 355E6, 35BB3, 35DDD, 3636D, 364DD, 3663D, 36DD4, 37003, 3700F, 3717F, 373EE, 37609, 3774E, 37773, 37797, 37977, 3797F, 37EEF, 39007, 390C5, 39777, 39973, 3B355, 3B553, 3BBDB, 3BDB1, 3C03D, 3C0F5, 3C10F, 3C141, 3C444, 3CBE5, 3CD0D, 3CE5B, 3CEBB, 3CEF9, 3D401, 3DEBE, 3E006, 3E066, 3E57E, 3E5E9, 3E666, 3E90F, 3EF6F, 3F33D, 3F3C4, 3F5BB, 3FB33, 3FDDD, 3FF59, 4006D, 400DE, 4011D, 401D9, 40414, 4041G, 404C9, 40966, 40D11, 40D19, 40D1D, 40E49, 41019, 411DA, 41AAG, 4210A, 44049, 4410G, 44144, 441G4, 44441, 444E9, 446E9, 44986, 44E49, 4609G, 460E9, 466DE, 469DD, 46E9G, 4711A, 476D9, 4770D, 47A77, 47D09, 49099, 490D1, 49226, 49622, 49699, 496DD, 49996, 4999G, 499G7, 49G22, 49G77, 4A7DD, 4AA6D, 4ADD7, 4C0E9, 4C999, 4D1DA, 4DADD, 4DD01, 4DD1G, 4DD7A, 4DDA7, 4DDE9, 4DG0G, 4DGAA, 4DGGA, 4DGGE, 4E049, 4E449, 4E49G, 4E4E9, 4E797, 4G7DD, 4GDAA, 4GDD7, 50011, 50079, 50095, 500B1, 500F3, 501A5, 501AF, 50503, 507A5, 50AF7, 50F03, 50F7A, 510A1, 510DA, 511AA, 511DF, 5135B, 515B7, 5180B, 51A0F, 51F0A, 520B1, 53005, 531BD, 53559, 53609, 53B11, 55205, 55357, 553E6, 5555B, 5556E, 55588, 5558A, 555F3, 555FB, 556AF, 556E9, 55759, 5575B, 55805, 55885, 55896, 558B8, 55926, 55BE2, 55E8B, 55F57, 560FF, 5700D, 570A5, 570DA, 575B9, 576AD, 576DA, 579D5, 57A05, 57A52, 57B9D, 57DBD, 58057, 58509, 5855A, 585A7, 587EB, 58857, 588E8, 58A75, 58B0B, 58B87, 58BBE, 58BEB, 58E5B, 591D5, 59201, 59256, 59715, 59807, 5A88A, 5AA88, 5AFAD, 5B001, 5B00B, 5B1F1, 5B31B, 5B7E2, 5B80B, 5BB13, 5BBE8, 5BBFB, 5BE87, 5BE8B, 5BF37, 5BFBD, 5D00F, 5DA05, 5DA5A, 5DAE5, 5DBBD, 5DD95, 5DDAA, 5DFDD, 5E879, 5E8B7, 5E8BB, 5F07A, 5F0AD, 5F37D, 5F70A, 5F7BD, 5FB7B, 5FBBB, 5FBF3, 5FFF3, 6003E, 60098, 603E6, 606GF, 60986, 609C8, 60G6F, 60GCF, 6336D, 633E9, 63CCE, 63E06, 63E66, 6609G, 660E9, 66AD4, 66D4A, 66DG4, 66DGG, 66E98, 66FD9, 66GF6, 69806, 69866, 69C86, 69CC8, 6A66F, 6AAGF, 6AF06, 6AF66, 6AGGF, 6C6G3, 6C6GF, 6CCGF, 6CG03, 6DA0E, 6DAEA, 6DD9G, 6DDE9, 6DEGA, 6DGD3, 6E986, 6EEE9, 6F69D, 6F6DF, 6F96D, 6FD03, 6FD09, 6G003, 6G3F3, 6G3FF, 6G6CF, 6GAAF, 6GCCF, 70031, 70099, 700BF, 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77777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777 ====Additional known quasi-minimal primes (not necessarily the next)==== 570₅₁₃₁₀1, 49₁₁₁₃₃₃, 970₁₆₆₀₄₇1, F70₁₈₆₇₆₇1 ===Base 18=== 11, 15, 1B, 1D, 21, 25, 27, 2B, 2H, 35, 37, 3D, 3H, 41, 47, 4B, 4H, 57, 5B, 5D, 5H, 61, 65, 71, 75, 7B, 7D, 85, 87, 8D, 91, 95, 9B, 9H, A1, AB, AD, AH, B1, BD, C7, CB, CD, CH, D5, D7, DH, E5, EB, EH, F1, F7, FB, FD, G5, H1, H5, H7, HB, 107, 167, 16H, 177, 17H, 1G7, 1HH, 20D, 24D, 26D, 29D, 30B, 36B, 381, 3BB, 405, 445, 44D, 49D, 4A5, 4DD, 4F5, 4GD, 501, 545, 5E1, 607, 62D, 64D, 66B, 66H, 67H, 68B, 697, 6A7, 6BB, 6E7, 6G7, 6GB, 6HH, 767, 76H, 77H, 797, 7HH, 801, 80H, 831, 83B, 86B, 88H, 8BB, 8FH, 8GH, 94D, 96D, 977, 9DD, 9ED, 9GD, A77, AC5, AE7, B07, B0H, B55, B77, B8B, B97, BB5, BB7, BBH, BE7, BFH, BGB, C01, C31, CA5, CG1, D2D, D4D, D81, DBB, DD1, DDB, DGD, E0D, E17, E31, E4D, E67, E6D, EA7, EDD, EE1, EED, EG7, F0H, F45, F8H, FC5, FFH, G0D, G17, G2D, G6B, G6H, GBB, GBH, GD1, GDD, GE1, GE7, GED, GFH, GG7, GGB, GHD, GHH, H0D, H2D, H8H, H9D, HGH, HHD, 100H, 19E7, 1A97, 1EE7, 1G8H, 1GGH, 22ED, 22GD, 2DED, 2E2D, 3001, 3031, 30C1, 30E1, 3331, 33G1, 3CC1, 40ED, 45C5, 46ED, 4CC5, 5331, 5551, 55G1, 5C05, 608H, 60ED, 60FH, 60HD, 666D, 66ED, 699D, 6B67, 6BGH, 6D0D, 6DDD, 6E9D, 6EGD, 6G0H, 6G9D, 6HGD, 700H, 70A7, 7A07, 7FGH, 7G77, 808B, 8881, 88G1, 88GB, 8BHH, 8EG1, 8GC1, 8H6H, 900D, 90E7, 90G7, 9667, 9907, 999D, 99E7, 9A67, 9A97, 9E97, 9EE7, 9G07, 9G67, 9GA7, AA45, AA97, AGA7, B005, B03B, B06B, B0C5, B60B, B63B, BAA5, BAA7, BCC5, BFA5, BG8H, C045, C055, C555, C5C1, C5F5, CC05, CC81, CCC5, D06D, D09D, D0ED, D38B, D3E1, D60D, D6DD, D8GB, DD6D, DE9D, DG01, E001, E097, E0G1, E8C1, EDC1, EE97, EGC1, EGG1, EGGD, FH6H, G007, G00B, G00H, G03B, G067, G097, G0C1, G0G1, G1GH, G33B, G38B, G3G1, G70H, G777, G88B, GA67, GAA7, GG81, GGC1, GGGH, H0FH, H66D, HEGD, HFHH, 1AAA7, 222DD, 30GG1, 3388B, 33E01, 38G8B, 3G3C1, 3GGG1, 4002D, 500C5, 50C55, 50CF5, 53GG1, 558C1, 55CC5, 55CF5, 58GG1, 5C8C1, 5CFF5, 5G881, 5GG31, 6000H, 6003B, 6006D, 600DB, 6033B, 606GD, 60D0B, 66GGD, 6D03B, 6D33B, 6H6DD, 6HD6D, 6HDED, 70G07, 70GGH, 777A7, 7AAG7, 7G0GH, 80G0B, 8888B, 8CCE1, 90067, 90097, 9022D, 99967, 99997, 9A007, 9A0A7, 9AA07, 9AAA7, 9E007, A0045, A0455, A0667, A09G7, A0A07, A0G07, A0G97, A9997, AA0A7, AAG67, B0AF5, B6GGH, B7GGH, B8HHH, BA045, BAF05, BG667, C0F05, C5005, C5581, C88C1, C8CC1, C8CE1, CCF55, D03C1, D060B, D080B, D0CC1, D0G0B, D0G8B, D3G3B, D600B, DDDED, DG331, DG80B, E8G81, E9007, F6GGH, G018H, G0301, G0331, G466D, G6667, G66GD, GD08B, GG18H, GG6GD, GGG4D, H060H, HGGGD, HHH6H, 199AA7, 40006D, 40600D, 46600D, 5055C5, 5505C5, 55CCC1, 588CC1, 58CCC1, 60000D, 60009D, 7077G7, 7707G7, 777G07, 88000B, 9099A7, A000A7, A009A7, A09067, A099A7, A0AAA7, A90AA7, A99AA7, AA0007, AA6667, AAAG07, BFFF05, BFFFF5, C0FFF5, CCECC1, CECCC1, CF0FF5, CFF005, D0008B, D0033B, D0088B, D0333B, D033GB, D03G31, D0633B, DD990D, DGGG31, FHHHHH, G00081, G6GGGD, G8GGG1, GGG001, GGG331, GGGGG1, GGGGGD, H0006H, H00H6H, HH600H, 222222D, 22DDDDD, 333333B, 5CCCCC1, 70007G7, 88CCCC1, 9000007, 9000A07, A000G67, AAAA667, BBBB33B, C000CF5, C000FF5, CCCCCE1, CCCCEC1, D00063B, D00GG31, D63333B, DCCCCC1, DDDDD9D, DGCCCC1, GCCCCC1, GG00031, 4022222D, 6000GGGD, 66666667, 770000G7, AAAAA007, B6666667, BBBBBB3B, CFFFFF55, D00000C1, D0000EC1, 455555555, 5555550C5, 667777777, A00000967, A00009097, A00009967, A45555555, AAAAAAA07, BHHHHHHHH, CCCCCCCC1, CF0000005, CFFFFFF05, D00000G3B, E0CCCCCC1, G00000031, 70000000G7, A000000097, D000003301, 777777700G7, A0000900007, D0000000001, D000000GGG1, 677777777777, 8HHHHHHHHHHH, 2DDDDDDDDDDDD, 55555555555C5, 77AAAAAAAAAA7, D00000000006B, D0000000003GB, AAAAAAAAAAAAAA7, D0000000000000B, 77777777777777G7, CCFFFFFFFFFFFFFF5, BBBBBBBBBBBBBBBBBBB6B, CFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF5, GG0000000000000000000000000000001, HDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD, C00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000F5, 80000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000B, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHFH, 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4EEEEEEEEEB, 4GGGGGGGGG7, 5EEEEEEEEEB, 600000000I1, 60000000A33, 66666666629, 666666DDD0D, 6AAAAAAAGA3, 700GGGGGGG7, 7777777A7A7, 90I29999999, 9GGGGGGGGE7, 9IIAAAAAAA9, AAAAAAAA3G9, AAAAAAAAA39, AAAAAAAAAAD, AAAAAAAAGAD, C6CGGGGGGG3, CCCCCCCCG99, D0000000007, D0000000C07, D7DDDDDDDDD, DC000000007, EBBBBBBBB8B, EEE8BBBBBBB, EEEEEEEB8BB, EEEEEEEEE47, EEEEEEEEE8B, EJ0JJJJJJJJ, EJJJJJJJ0JJ, EJJJJJJJJAJ, F000000EF09, F0A44444441, F6666666689, FFFFAJJJJJJ, G00000G00C3, G1JJJJJJJJJ, G4444444447, GGGGGGGGG09, H0000000H41, H004444IIIB, HHHHHHHHHEH, I00AAAAAAA9, I0AAAAAAAF9, IIBBBBBB8BB, J050000000J, J77777777A7, JAEEEEEEEE7, JJJ77777777, JJJJJ0JJJ5J, JJJJJ777777, JJJJJJAJA77, 333333333GI9, 600000000089, 600000000D0D, 6AAAAAAAAA63, 6DDDDDDD0DDD, 6GGGGGGGCCC3, 7C3333333333, 7CC0GGGGGGG7, 902999999999, 9A4444444447, 9FAAAAAAAAA9, A777777777E7, AAAAAAAAAII9, AGGGGGGGGGA3, B0000000005B, B00000000IEH, B0000000E00H, B000000E000H, CCC7DDDDDDDD, CGGGGGGGGGG7, ECJJJJJJJJJJ, EEEEEEE4E447, FFFFFFAAAAA9, GAGGGGGGGAG3, GCGCCCCCCCC9, H000000006F1, H00000004441, H00044444441, HHHHHA00000H, I000000001G1, IAAAAAAAAA09, J0000000500J, J0000000JCJJ, J0000050000J, JJJ00000005J, JJJJJ500000J, JJJJJEEEEEC7, JJJJJJEECCC7, JJJJJJJ0005J, 555555555555H, 5BBBBBBBBBBBB, 6AAAAAAAAAAA3, A99FFFFFFFFF9, AAA7777777777, AAAAAAAAAAA93, AJ77777777777, CCCCCCCCC2229, D0000000000I1, DDDDDDDDDDD4D, DEEEEEEEEEE07, E00000000000H, EEEEEEEEEEEEB, G9000000000C3, GAGGGGGGGGG63, GGGGGGGGGGG93, HHHHHHHHHHHI3, J00000000050J, J00000CJJJJJJ, JJJ0CJJJJJJJJ, JJJJJJJJJJAJJ, JJJJJJJJJJJA7, JJJJJJJJJJJG1, 333333333I3IA3, 500000000000CJ, A0000000000099, A0000000000999, AAAAAAAAAAAAA9, B00E000000000H, CCCCCCCCCCC029, EIBBBBBBBBBBBB, FFFFAAAAAAAAA9, G00000000000G3, H0000000000001, I0AAAAAAAAAAA9, I9AAAAAAAAAAA9, JJJJJJJJJJEEC7, JJJJJJJJJJJ50J, 3333333333333A3, 6666666666668I9, 777777777777771, 7GGGGGGGGGGGGG7, 84400000000000B, AAAAAAAAAAAF009, B000000000000IH, E999999999999F9, EEEEEEEEBBBBBBB, FFCCCCCCCCCCC29, G0000000000006D, GGC000000000007, HEHHHHHHHHHHHHH, J00000000000001, J0000000000055J, J0000CJJJJJJJJJ, 500000000000E0AJ, 500000000000F00J, 6000000000000001, C6GGGGGGGGGGGGG3, CCCDDDDDDDDDDDDD, CJJJJJJJJJJJJJAJ, E222222222222229, E999999999999999, EE66666666666689, FFFFFFFFFFFFFFA9, HHHHHHHHHHHHHGA3, HHHHHHHHHI666663, IIIIIIAAAAAAAAA9, JJJEEEEEEEEEEEC7, 222222222222228I9, 444444444444444G7, EEEEEEBBBBBBBBBBB, EEJJJJJJJJJJJJJJJ, F000000000000E0F9, GC000000000000007, GGGGGGGGGGGCCCCC9, JJJJJJJJJJJJJ7777, JJJJJJJJJJJJJJJJ1, 10000000000000007D, 733333333333333333, HHHA0000000000000H, I3AAAAAAAAAAAAAAA3, IIIBBBBBBBBBBBBB8B, AAAAE44444444444447, D0000000000000000CJ, EEEEBBBBBBBBBBBBBBB, FCCCCCCCCCCCCCCCC29, GGGGGGGCCCCCCCCCCC9, GGGGGGGGGCCCCCCCCC9, GGGGGGGGGGGGGGGGG63, HHHHHHHI66666666663, J000000000000000ECJ, JJJCJJJJJJJJJJJJJJJ, 22222222222222222289, 5000000000000000FFFJ, 94444444444444444447, 97777777777777777777, A00000000000000000C9, B800000000000000000B, GGGGGGGGGGGGGGGGGGG3, IIIEBBBBBBBBBBBBBBBB, 40800000000000000000B, 710000000000000000007, J000000000000000005CJ, 76DDDDDDDDDDDDDDDDDDDD, EJJJJJJJJJJJJJJJJJJJJJ, HHHI666666666666666663, 4040400000000000000000B, 80000000000000000000A61, GJJJJJJJJJJJJJJJJJJJJJJ, I0IIIIIIIIIIIIIIIIIII29, D0DDDDDDDDDDDDDDDDDDDCC7, EHHHHHHHHHHHHHHHHHHHHHAH, IBBBBBBBBBBBBBBBBBBB8BBB, J5000000000000000000000J, J0000000000000000000CJJJJ, JJJJJJJJJJJJJJJJJJJ0J0J5J, EC000000000000000000000077, H000000000000000000000222B, HA00000000000000000000000H, HI666666666666666666666663, J000000000000000000000005J, JJJJJJJJJJJJJJJJJJJJJJJEC7, GGGGGGGGGGGGGGGGGGGGGGG9999, J00000000000000000000000E0J, J0CJJJJJJJJJJJJJJJJJJJJJJJJ, JEEEEEEEEEEEEEEEEEEEEEEEEEC7, 333333333333333333333333333G9, EEEEEEE4444444444444444444447, G9000000000000000000000000003, J000000000000000000000000CJJJ, AE7777777777777777777777777777, EEEEEEEEEEEEEEEEEEEEEEEEEECCC7, JJJJJJJJJJJJJJJJJJJJJJJJJ0JJ5J, AFFFFFFFFFFFFFFFFFFFFFFFFFFFFF9, JJJJJJJJJJJJJJJJJJJJJJJJJJJ0J5J, CJJJJJJJJJJJJJJJJJJJJJJJJJJJJJC7, IBBBBBBBBBBBBBBBBBBBBBBBBBBBB8BB, 8I00000000000000000000000000000A1, A77777777777777777777777777777A77, EEBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB, G00000000000000000000000000000007, H3HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH, G0000000000000000000000000000000C3, GCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC9, 666666666666666666666666666666666689, CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC29, CCDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD, GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGCCC9, IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII29, J1EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE7, C2222222222222222222222222222222222222229, J00000000000000000000000000000000000000CJ, DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD7D, GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG47, 4480000000000000000000000000000000000000000000B, A777777777777777777777777777777777777777777777777, I00000000000000000000000000000000000000000000004G1, 6DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD0D0D, IBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB8B, D0D0DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD7, 4000000000000000000000000000000000000000000000000000005B, GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGC9, 7777777777777777777777777777777777777777777777777777777777A77, CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCG9, F000000000000000000000000000000000000000000000000000000000EF9, 500000000000000000000000000000000000000000000000000000000000FJ, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH6A3, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHA3, JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJ77, 1000000000000000000000000000000000000000000000000000000000000000007, AAA4444444444444444444444444444444444444444444444444444444444444447, BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB8BB, EEEEE44444444444444444444444444444444444444444444444444444444444447, B000000000000000000000000000000000000000000000000000000000000000000E0H, D00000000000000000000000000000000000000000000000000000000000000000000000J, 80I0000000000000000000000000000000000000000000000000000000000000000000000001, A44444444444444444444444444444444444444444444444444444444444444444444444444444447, D00I00000000000000000000000000000000000000000000000000000000000000000000000000001, EI66666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666669, 8B000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000B, I8000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001, 92222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222229, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHA000H, JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJ5AJ, 800000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000061, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH3H, EEE44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447, IIBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB, 8I00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001, E444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447, DI0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001, G600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000003, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHA0H, 777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777A7, J777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777, JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJCCC7, 4000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000404B, EC0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000007, GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG99, 3AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA3, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEC7, JCJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJ, JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJ05J, 500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000AJ, CDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD, G00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000D ===Base 21=== ====Additional known quasi-minimal primes (not necessarily the next)==== 40₄₇₃₃₃9G, CF₄₇₉₁₄₇0K ===Base 22=== 11, 17, 19, 1F, 1J, 1L, 23, 29, 2F, 2H, 31, 35, 37, 3D, 3H, 41, 49, 4D, 4F, 4J, 4L, 53, 5H, 5L, 65, 67, 6H, 6J, 73, 79, 7D, 7J, 83, 85, 8F, 8H, 8L, 91, 9D, A3, A7, A9, AD, AJ, AL, B9, BF, BL, C5, C7, CD, CH, CJ, D7, DL, E3, E5, E9, F1, F7, FH, FJ, G1, G7, GF, GL, H5, H9, HF, I1, I5, ID, J1, J3, JD, JF, JL, K3, K9, KH, KL, L1, L5, LH, 103, 12D, 145, 155, 15D, 163, 18D, 1A5, 1BD, 1BH, 1C3, 1D3, 1DH, 1EH, 1G3, 1GH, 1I3, 1K5, 1KD, 221, 227, 22J, 22L, 245, 247, 25D, 25J, 271, 277, 287, 28J, 2A5, 2B7, 2BD, 2BJ, 2D5, 2E1, 2E7, 2ED, 2EL, 2K1, 2KJ, 2LL, 30J, 343, 389, 39J, 3B3, 3GJ, 3IJ, 3J9, 3JJ, 3KF, 3LJ, 427, 443, 445, 457, 4A5, 4C3, 4E7, 4G5, 4I7, 4K5, 4K7, 515, 52D, 551, 559, 55D, 55J, 575, 58D, 59F, 5B1, 5C9, 5CF, 5D1, 5D5, 5DD, 5E1, 5ED, 5G5, 5GJ, 5J5, 5JJ, 5K1, 5KJ, 60D, 61D, 62L, 661, 66D, 689, 6C1, 6D1, 6DD, 6G3, 6KF, 701, 721, 751, 76L, 775, 77F, 77H, 787, 7A5, 7AF, 7B1, 7B7, 7CL, 7E1, 7FF, 7FL, 7H7, 7HH, 7K5, 809, 81D, 821, 827, 82D, 847, 869, 871, 881, 889, 899, 8A1, 8BD, 8BJ, 8D1, 8DJ, 8GJ, 8J7, 907, 90H, 92L, 93J, 943, 947, 95F, 977, 997, 9AF, 9B5, 9EF, 9F5, 9H3, 9HL, 9I7, 9J9, 9JJ, 9K5, A25, A45, A51, A61, A6F, AAH, AB5, ABH, ACF, AG5, AGH, AHH, AK1, B15, B2D, B2J, B33, B45, B47, B57, B71, B75, B81, B87, B8J, BB3, BB7, BC3, BDD, BE7, BEJ, BGD, BGH, BH3, BHD, BHJ, BIH, BJ7, BKJ, CA1, CAF, CB3, CC1, CEF, CG3, CKF, D09, D0J, D13, D21, D33, D39, D3F, D4H, D5J, D63, D81, D8D, DAH, DBD, DBH, DBJ, DCF, DD3, DEJ, DFF, DG9, DGH, DHD, DI3, DIF, DJ9, DK1, DK5, E0F, E0H, E27, E2D, E2L, E47, E7H, E87, E8J, EA1, EAH, EB1, EDH, EEJ, EFF, EFL, EH1, EIF, EIL, EJH, EJJ, EKD, ELF, F25, F43, FB5, FD3, FDD, FDF, FEF, FEL, FFD, FG9, G09, G0D, G25, G3J, G5D, G5J, G63, G8D, G99, GC3, GC9, GD9, GEH, GG5, GJ5, GJ9, GJH, H03, H1D, H21, H2J, H2L, H33, H63, H77, H8J, HCL, HDD, HE1, HE7, HGH, HGJ, HH7, HHL, HI3, HIH, HJH, HK1, HKD, HL3, I07, I0J, I43, I47, I7L, I9J, IBH, IEL, IG3, IHH, IIJ, IJ7, IL7, J07, J55, J69, J8J, J99, J9J, JA5, JAH, JE7, JEH, JHH, JI9, JJ5, JJ9, JJH, JK7, K2J, K51, K5D, K75, K81, KA1, KB1, KB7, KBJ, KD1, KEJ, KG5, KIF, KJ5, KKD, KKJ, L0D, L47, L7F, L97, LAF, LD3, LD9, LDD, LEF, LGD, LI7, LJ7, LJJ, LLD, 104H, 10D5, 1205, 12B5, 140H, 1433, 144H, 14AH, 14B3, 16ED, 1AIH, 1B43, 1DD5, 1DDD, 1E6D, 1EGD, 1G05, 1GDD, 1GED, 1GGD, 1HB3, 1HHH, 1IAH, 200L, 2015, 2051, 20A1, 20DJ, 20GD, 20IL, 21B5, 21DD, 220D, 226D, 228D, 22B5, 22G5, 22K5, 22KD, 2555, 2557, 2581, 25C1, 26A1, 26B1, 2725, 2755, 2801, 2861, 288D, 28B1, 28KD, 2AA1, 2B25, 2B51, 2BB1, 2C81, 2D6D, 2DA1, 2DDJ, 2DGD, 2G0J, 2GB5, 2GDD, 2GGJ, 2I0L, 2I6L, 2ICL, 2J05, 2JK5, 2K07, 2K2D, 2K55, 2K6D, 2KB5, 2KI7, 2L2D, 2L8D, 2LK7, 302L, 30G3, 320L, 32IL, 332L, 33G3, 33G9, 36AF, 36EF, 382J, 388J, 39EL, 3AAF, 3BBJ, 3CG9, 3E2J, 3E6F, 3E6L, 3EEF, 3FAF, 3G69, 3GI3, 3GI9, 3IG9, 3LEL, 3LG3, 4025, 404H, 4063, 4075, 40AH, 40B5, 40B7, 40GH, 4225, 4363, 4447, 444H, 447H, 4487, 44B7, 44H7, 4525, 4555, 45B5, 4663, 4777, 47GH, 4807, 4B0H, 4BB5, 4BG3, 4EBH, 4G4H, 4GB3, 4HH3, 4I03, 4I63, 4IGH, 5069, 5077, 50KD, 5255, 52IJ, 5455, 5477, 5507, 5527, 556F, 557F, 5587, 56EF, 56FF, 56GD, 56KD, 5771, 57C1, 57EF, 5807, 580J, 589J, 58E7, 58IJ, 58K7, 5905, 5945, 5957, 5969, 5989, 598J, 5999, 59EJ, 59K7, 5AEF, 5B25, 5B55, 5BA5, 5BI7, 5BIJ, 5BK5, 5BK7, 5C21, 5EAF, 5EE7, 5F09, 5F6F, 5F95, 5FKD, 5G6D, 5G89, 5I09, 5I0F, 5I27, 5I6F, 5I89, 5IB7, 5IBJ, 5IFF, 5J77, 5K07, 5K25, 5K27, 5K6F, 5K87, 5KK7, 6013, 60A1, 60AF, 60EL, 6281, 62B1, 62KD, 63G9, 6403, 6643, 66EL, 66G9, 68KD, 69B3, 6A01, 6A0F, 6AAF, 6AFF, 6B21, 6B8D, 6BBD, 6BK1, 6D03, 6D43, 6D69, 6D6F, 6D93, 6D9F, 6E21, 6E81, 6E8D, 6EBD, 6ECL, 6ELL, 6FD9, 6GBD, 6GG9, 6IEF, 6K21, 6K8D, 6KBD, 6KE1, 6L43, 6LB3, 6LDF, 6LEL, 6LG9, 700F, 7027, 7057, 70EH, 70LF, 710H, 71G5, 7207, 7225, 7255, 727L, 7505, 7507, 755F, 75EF, 75F5, 75I7, 766F, 7681, 76CF, 7771, 7781, 77IL, 7861, 7A0H, 7A4H, 7BB5, 7C6F, 7EEH, 7EHL, 7FG5, 7G45, 7HEL, 7I0L, 7I27, 7IE7, 7IGH, 7K07, 7K61, 7K77, 7KC1, 7L27, 7LCF, 7LK7, 802J, 80B1, 80GD, 80JJ, 80KJ, 820J, 86E1, 880D, 882J, 88E7, 88I7, 892J, 89EJ, 8B61, 8BI7, 8CI9, 8CK1, 8D6D, 8DD9, 8DGD, 8E01, 8E07, 8EE1, 8EI7, 8EK1, 8EKJ, 8I77, 8I87, 8IC9, 8IK7, 8J0J, 8K6D, 8KI7, 8KIJ, 8KK7, 9025, 9055, 9089, 9275, 92EJ, 92G5, 92GJ, 93EL, 9455, 9505, 9557, 9599, 96B3, 970L, 976F, 97G5, 982J, 98E7, 98I9, 98K7, 9905, 990J, 9925, 9995, 999H, 99BH, 99EL, 99G9, 99GJ, 99LJ, 9A55, 9B03, 9B27, 9BGJ, 9BH7, 9BK7, 9CG9, 9E6L, 9EHJ, 9EKJ, 9ELL, 9G39, 9G45, 9G4H, 9GA5, 9GAH, 9GBJ, 9GEJ, 9GHH, 9GHJ, 9GI9, 9GIJ, 9H7H, 9H87, 9HAH, 9HB7, 9I89, 9I9H, 9IB3, 9IEH, 9IKF, 9J7H, 9J87, 9JB7, 9JGH, 9K0F, 9K27, 9KGJ, 9KKF, 9L89, 9L9J, 9LG9, 9LGJ, 9LIJ, 9LKF, A081, A14H, A1IH, A201, A2C1, AA15, AA81, AAIF, ABC1, AC01, AHC1, AIIF, AKEF, B005, B01D, B01H, B051, B05D, B0A5, B0B1, B0BJ, B0DH, B0E1, B0GJ, B0H1, B0JH, B143, B16D, B255, B2B5, B2C1, B2G5, B2K7, B4EH, B4G3, B501, B5A5, B5BJ, B5C1, B5IJ, B5K5, B621, B663, B6I3, B6K1, B777, B7I7, B80D, B88D, B8ED, B8KD, BB01, BB0J, BB21, BB25, BB4H, BB6D, BBD5, BBE1, BBEH, BBH1, BBIJ, BBJH, BCK1, BD61, BDB5, BDGJ, BEE1, BG55, BGB5, BGGJ, BGIJ, BH27, BH7H, BHC1, BHH1, BHK7, BI77, BIGJ, BJ0H, BJBH, BJIJ, BJJJ, BK25, BK27, BK55, BKA5, BKK1, C043, C143, C2B1, C32L, C3G9, C601, C6EL, C6G9, C8B1, CBK1, CC2L, CC89, CE21, CECL, CF2L, CF89, CG69, CIG9, D003, D015, D045, D05D, D06D, D06F, D0DH, D0H1, D1D5, D1GD, D205, D26D, D2DJ, D32J, D38J, D403, D525, D561, D56D, D5AF, D5F5, D5GD, D6C9, D6D9, D6DF, D6E1, D6ED, D6F9, D6FD, D8IJ, D8JJ, D945, D993, D99F, D9C9, D9F3, D9G5, D9KJ, DA15, DAA5, DAE1, DAKF, DB25, DB61, DBC1, DBG3, DC43, DC99, DCE1, DD0F, DD1D, DD2D, DD2J, DD5D, DD9H, DDA5, DDD5, DDDJ, DDED, DDH1, DDHJ, DDI9, DDJ5, DE0D, DED1, DEDF, DEFD, DF03, DF45, DF55, DF5D, DF6D, DG05, DGGJ, DH0H, DH43, DHC1, DHEH, DHH3, DHJJ, DI8J, DIC9, DII9, DIJH, DJG5, DJHJ, DJKJ, DK6D, DKAF, DKDD, DKGD, E081, E0BJ, E0DD, E0DJ, E0E1, E0ED, E0J7, E0K7, E0KJ, E1GD, E1IH, E201, E281, E66F, E6BD, E6ED, E6K1, E6LL, E7EF, E861, E86D, E88D, EB0D, EB4H, EBHH, EBI7, EC6F, ECEL, ED0D, ED1D, ED6D, EDAF, EDDD, EDFD, EE01, EE0D, EE1H, EE21, EECF, EEE1, EEGH, EEH7, EF8D, EGGD, EGHH, EGIH, EGIJ, EH07, EH0L, EH8D, EHBH, EHBJ, EHK7, EHL7, EI4H, EIK7, EIKJ, EJ77, EK07, EK0J, EKC1, EKIJ, EKK1, EKK7, EL6L, ELBD, ELCL, ELDJ, ELK7, F059, F20L, F26L, F28D, F2GD, F2KD, F32L, F3AF, F3G3, F455, F56F, F595, F5KD, F6AF, F6GD, F88D, F8I9, F955, F995, F9B3, F9G3, F9KF, FB03, FB8D, FBI3, FC89, FD99, FDA5, FDI9, FEGD, FF59, FG2D, FGI3, FGKD, FI89, FIB3, FIKF, FK2D, FK45, FKD5, FL6D, FLG3, G003, G04H, G055, G0BH, G0HH, G0HJ, G0I3, G0K5, G0KJ, G26D, G2DJ, G2GJ, G303, G333, G393, G3G9, G3I3, G403, G40H, G433, G4AH, G4GH, G4H3, G4IH, G589, G5G9, G80J, G89J, G94H, G9G3, GAA5, GAIH, GB0H, GB43, GB6D, GBB5, GBG3, GBGJ, GBJJ, GD0H, GD45, GDD5, GDKD, GG2D, GG39, GG8J, GG9H, GGAH, GGDJ, GGEJ, GGGH, GGH3, GGKJ, GH0H, GHG3, GHH3, GHHH, GHJJ, GI03, GI2J, GI93, GIAH, GIEJ, GIG9, GII3, GJJJ, GJKJ, GK05, GKDD, GKGD, GKGJ, GKIJ, H081, H087, H0AH, H0BJ, H0D1, H0HJ, H0LL, H1HH, H20D, H22D, H26D, H2I7, H2K7, H30L, H3KJ, H40H, H447, H4G3, H4H3, H4HH, H66L, H6ED, H6IL, H7C1, H7EL, H80D, H861, H887, H88D, H8B1, H8B7, H8C1, H8I7, HA0H, HB1H, HB27, HBB1, HBBJ, HBD3, HBI7, HCB1, HCC3, HD3J, HD61, HDC1, HDEH, HDHH, HE4H, HEHD, HELJ, HH0D, HH4H, HH61, HH6D, HH81, HH8D, HHC1, HHDH, HHG3, HHH1, HI6L, HIB7, HIBJ, HJJJ, HKI7, HKK7, HL27, HL2D, HLBJ, HLEJ, HLEL, HLK7, I0AF, I2CL, I2GJ, I32L, I33J, I6B3, I6EF, I727, I74H, I82J, I877, I88J, I8C9, I8EJ, I8JJ, IA0F, IAAF, IB03, IB63, IBGJ, IE2J, IEB7, IEBJ, IECF, IEGJ, IEHJ, IEKJ, IF89, IFKF, IG8J, IGBJ, IGGH, IGI9, IHBJ, IHC3, IHJJ, IHKJ, II9H, IIEF, IIHL, III7, IIIH, IJ09, IJBJ, IK0F, IK6F, IKCF, IKEF, IKJJ, IKK7, IL2L, ILLJ, J025, J05J, J0EJ, J0K5, J44H, J487, J50J, J589, J5J7, J757, J7GH, J975, J9B7, J9GH, JB0H, JB77, JBJJ, JC09, JCC9, JEGJ, JG89, JGBJ, JHI7, JI4H, JII7, JJ27, JJ2J, JJ87, JJGJ, JJJ7, JK05, JKB5, JKK5, K015, K08D, K0ED, K0JJ, K0K7, K105, K16D, K201, K225, K255, K2K5, K4B5, K50J, K557, K587, K5K7, K621, K62D, K6BD, K6E1, K761, K777, K7I7, K7KF, K80J, K88J, KBA5, KBB5, KC0F, KD25, KD5F, KDF5, KDJJ, KDKF, KE0D, KE6D, KEAF, KEED, KEK1, KFD5, KGGD, KI27, KIE7, KIGJ, KIJJ, KJ77, KJGJ, KK07, KK61, KK6F, KK87, KKCF, KKK7, L0G3, L0G9, L22D, L26L, L2DJ, L2GJ, L2IL, L2K7, L433, L4I3, L6DF, L887, L8B7, L99J, L9KF, LB27, LB77, LBDJ, LBED, LC43, LC89, LCG9, LD2J, LD6F, LF2L, LF89, LFG3, LFKF, LG43, LG69, LG8J, LGBJ, LIG9, LILJ, LK8D, LKED, LKKF, LL2L, LLIJ, 100G5, 10225, 10DED, 10DGD, 10H6D, 13333, 1DBG5, 1DEED, 1EDED, 1HEED, 200G5, 200GJ, 2010D, 2016D, 20225, 205B5, 20681, 20B55, 20BC1, 20D1D, 20D61, 20DD1, 20DDD, 20GK5, 20IEJ, 20IK7, 20J0J, 20JI7, 20K25, 20KK5, 21025, 21G6D, 22255, 22DDD, 25001, 250B5, 250K7, 25B05, 25IK7, 25KK5, 266CL, 26GGD, 26IIL, 26LKD, 2A0C1, 2B0G5, 2BB55, 2BC61, 2C0CL, 2C60L, 2C6CL, 2CCIL, 2D01D, 2D22D, 2DC61, 2DD01, 2DDB1, 2DDC1, 2DJIJ, 2G6GD, 2GIJJ, 2ILGJ, 2J0JJ, 2JIGJ, 2JJ0J, 2JJEJ, 2K025, 2KDDD, 2L6KD, 2LDIJ, 2LGIJ, 2LIEJ, 303EL, 306EL, 306G9, 30AFF, 30ECF, 30ELL, 30GG9, 3266L, 32CCL, 332EJ, 3333J, 333AF, 33CEL, 33E0L, 33IEF, 360G9, 363EL, 36E0L, 390G9, 399G3, 3AFIF, 3C0EL, 3E0LL, 3EC0L, 3ELLL, 3FG33, 3GG03, 3GG33, 3I26L, 3IAFF, 3L2CL, 3L6G9, 3LGG9, 40007, 40087, 400G3, 403G3, 40477, 40BBH, 40EHH, 40HEH, 43003, 43033, 43303, 43II3, 44BAH, 44BHH, 44EEH, 46033, 460I3, 46333, 470IH, 47407, 47BEH, 48B77, 4A00H, 4AI4H, 4AIIH, 4B055, 4B4BH, 4B4HH, 4B505, 4BBAH, 4BE4H, 4EE4H, 4EGGH, 4EIIH, 4G00H, 4GBAH, 4GHAH, 4GHBH, 4GI33, 4HBAH, 4I0EH, 4I40H, 4IA4H, 4IHB3, 5002J, 5006F, 50087, 500B7, 500F9, 50407, 504B5, 50525, 505B5, 5066F, 50681, 50761, 507KF, 508C1, 508EJ, 508J9, 50927, 50987, 509I9, 509J7, 50A01, 50DIJ, 50F45, 50GGD, 50II9, 50IKF, 50J89, 50K45, 5106D, 52081, 520B5, 520EJ, 520K5, 52E0J, 52II7, 52K05, 54007, 54887, 550B5, 55205, 552K5, 55405, 55577, 55E77, 55F45, 55KI7, 56009, 5600F, 560I9, 5660F, 5666F, 566F9, 56801, 56909, 56F69, 56I69, 572K7, 57407, 576A1, 577E7, 57I77, 57IKF, 57K47, 57KE7, 58061, 580C1, 589B7, 58II9, 59009, 590I9, 595A5, 59887, 5992J, 59EB7, 5A001, 5AIAF, 5B00D, 5B6BD, 5BBB5, 5BBDJ, 5C681, 5C801, 5C861, 5D0AF, 5D88J, 5DAAF, 5DK0F, 5DKEF, 5DKKF, 5E7K7, 5EI2J, 5EKKF, 5F045, 5F0IF, 5F405, 5F5IF, 5F699, 5FB0D, 5FB6D, 5FBBD, 5FFIF, 5FGGD, 5FI69, 5GG69, 5GGG9, 5I2EJ, 5I777, 5II77, 5II99, 5IIK7, 5IKI7, 5J089, 5J8B7, 5JGG9, 5K405, 5K447, 5K577, 5K7E7, 5KB05, 5KD0F, 5KEI7, 5KKB5, 60463, 606EF, 608B1, 608E1, 60EK1, 62GGD, 633EL, 63AIF, 63IAF, 64333, 643I3, 660EF, 666B3, 66B03, 66BI3, 66EEF, 66FAF, 66IB3, 6888D, 68BE1, 68GGD, 69EEL, 6AAB1, 6ABA1, 6B063, 6B0I3, 6B603, 6BEED, 6BI03, 6C0G9, 6C9EL, 6C9G9, 6CC43, 6CCEL, 6CE0L, 6D9I9, 6DCI9, 6DEAF, 6DI99, 6EE6L, 6EEFD, 6F22D, 6FEED, 6FFAF, 6GGKD, 6GII9, 6IAIF, 6IIG9, 6K0K1, 6KK01, 6L8ED, 70055, 702B5, 7044H, 70477, 704IH, 70555, 70CCF, 70EI7, 70G0H, 70GIH, 70IK7, 70ILL, 70K47, 70KKF, 714IH, 72B05, 74005, 74047, 74407, 7440H, 74477, 744IH, 74707, 74AEH, 74B25, 74B55, 74GBH, 74I0H, 752K7, 755K7, 75E77, 75K47, 76AA1, 76I0F, 77047, 77407, 77447, 776K1, 77AC1, 77EI7, 77IK7, 77K27, 77KK1, 77L2L, 7A8C1, 7AAA1, 7AAC1, 7B025, 7BB0H, 7C0IF, 7E4GH, 7EKKF, 7F055, 7F405, 7G0AH, 7G0IH, 7G5B5, 7GB55, 7GGBH, 7GGIH, 7GI4H, 7I0AH, 7I0IF, 7I7K7, 7II0F, 7IIEH, 7IILL, 7ILLF, 7K0KF, 7K6EF, 7K7K1, 7KEE7, 7KEKF, 7KK0F, 7KK71, 7L0IF, 7LI6F, 7LIIL, 7LILF, 80077, 80707, 807I7, 808K7, 80B07, 80E0J, 80IEJ, 80KC1, 82JJJ, 870I7, 87E77, 88007, 8808J, 880IJ, 886KD, 88807, 8886D, 888GD, 88GED, 88GKD, 88I8J, 88IKJ, 88K07, 88K77, 88KGD, 89IKJ, 8B001, 8B0K1, 8B0K7, 8B7K7, 8BCE1, 8BK07, 8CB01, 8CBB1, 8DDKD, 8DEED, 8E777, 8EGDD, 8GGG9, 8I2EJ, 8I8KJ, 8I9B7, 8IIE7, 8IK8J, 8JE2J, 8JJIJ, 8K00D, 8K00J, 8K08J, 8K6K1, 8K8JJ, 8KC01, 9009J, 900B3, 900KF, 905EJ, 905I9, 90959, 90AA5, 90BI3, 90E0L, 90FKF, 90G33, 90GG3, 90GG9, 90GI3, 90ILJ, 90J45, 92225, 92255, 925K7, 930G9, 93FG3, 93GG3, 93GG9, 94205, 944AH, 944GH, 94A4H, 94HBH, 94HEH, 94I4H, 95009, 950EJ, 950I9, 952K7, 95525, 956I9, 95887, 9592J, 959BJ, 95E0J, 95EBJ, 95GG9, 95IEJ, 96EEL, 9744H, 974IH, 97BEH, 97C0F, 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7777777707I7, 777777EEEEK7, 7F0000000045, 7IIIIIIIILIF, 7IKKKKKKKKKF, 800000000E61, 80000000E06D, 877777777777, 900000000E2J, 90000008800J, 9000000900G3, 9000000IIIG9, 9999999999KF, 9GGGGGGGGGG3, A0B000000001, A0BBBBBBBBB1, AA0000000AEF, AA000000A0EF, AAAAAAAAA2A1, ACBBBBBBBBB1, B00000000027, B00000000K0D, B0GGGGGGGG43, B100000000HH, BAAAAAAAAKK5, C0000000E60L, CCCCCCCCCCEL, CCEEEEEEE00L, CEE0EEEEEE0L, D2JJJJJJJJJJ, DAAAAAAAAA0F, DAAAAAAAAAAF, DD5555505555, DDA000000001, DDDDDDDDDDKD, E00000000K21, E0C00000000L, E4HHHHHHHHEH, E770000000I7, EC000000000L, EE77000000I7, EEEE4BEEEEEH, EEEEEEEEE4BH, EEEEEEEEE7I7, EEEEEEEEEECL, EEEEEEEEEII7, EEEEEEEEKEE7, F00000000GBD, F00000006BED, F0000F0098C9, F0FF00000C2L, FFFFF0000B63, FGGGGGGGGG6D, FLLLLLLLLLB3, G00000000A05, G00000008E2J, G0G00000002J, GG6666666669, GGGGGGGGGG43, GGGGGGGGGGI3, GGGGGGGGGKED, GJE00000000J, H00000004B4H, H0000000KJ47, H0H0000000BH, HB00000000HH, HBH00000000H, HBH0000000BH, I000000L8II9, I0IIIIIIIKFF, I0LLLLLLL989, IAFFFFFFFFFF, IEIEEEEEEEE7, IFFFFFFFFFAF, IFFFFFIIIIAF, IIIIIIIIIKFF, J00000005GG9, J0000000G0JJ, J50000000GG9, J90000000045, JBBBBBBBBBB5, JJ00000000BJ, JJ0000000JIJ, JJ000000JJIJ, K00000000BBD, K00000000EE7, K000000080I7, K0000000K5EF, K000000KKK45, K0000EEEE7E7, K0EEEEE7EEE7, KE0000000601, KEEEEE07EEE7, KEEEEE0E7EE7, KEEEEEEEE0I7, KJJJJ000000J, KK0000055525, KKKK00000045, L000000003EL, L000000060B3, L0000888880J, L000088888KJ, L7IIIIIIIIIL, LL0LLLLLLK27, LLL00000000J, LLLLL6000043, LLLLL9999989, LLLLLLLGGGB3, LLLLLLLI8II9, LLLLLLLLKCCF, LLLLLLLLL643, LLLLLLLLL877, LLLLLLLLLK27, 20000000005K7, 2000000000B61, 2200000000025, 3003A0000000F, 40HHHHHHHHHBH, 4333333333333, 46IIIIIIIIII3, 4H0HHHHHHHHHH, 5000000000057, 5000000002057, 5000000002C61, 50000000D00KF, 5860000000001, 5D000000000KF, 5E0000000000J, 5GGGGGGGGGGGD, 5KF000000000D, 60000000000G9, 6D00000000EEF, 70000000000AH, 70000000077I7, 70000000777I7, 70777770000I7, 7770000000II7, 77777777770I7, 777777777II77, 777EEEEEEEEK7, 80000000000KD, 800000000EE6D, 800000008DDDD, 80000000KKKE1, 80008888888EJ, 800880000000J, 8B00000000007, 9EEEEEEEEEEEL, 9FFFFFFFFFKFF, 9LLLLLLLLECCL, A000000002BA1, A00000000EKKF, A0000000EEEEH, A0000CBBBBBB1, AAAAAAAAAAAA1, B0000000000K1, B0000000K000D, B0JBBBBBBBBB5, BAAA555555555, BAAAAAAAAAA05, BE0000000000D, BHHHHHHHHHHHH, C000000000E6L, C00000000CE6L, D0000000000ED, D000000000DD1, D0HHHHHHHHHHH, DD00A00000001, DD55555555505, DDDDDDDDDDDGD, DDEAAAAAAAAAF, ECC0LLLLLLLLL, EE000000000I7, EEEEEEEEEEEB7, EEEEEEEEEEHLL, EEEEEEEEEEKE7, EEEEEEEEEHLLL, EIEEEEEEEE7E7, EIEEEEEEEEEE7, EIIEEEEEEEE77, F000000000GG3, F00000000B6ED, F00000000K66F, F0000000K0F6F, F000000EEEE8D, F0F00000098C9, FF00000000B63, FF000000098C9, H00000000002D, H0000000000K7, H000000000AA1, H000000000EED, H000000000EEH, H00000000HAA1, HBB0H0000000H, HH000000000BH, HHEEEEEEEEBEH, HHHEEEEEEEEEH, HHHHHHHHHEBBD, IBIIIIIIIIII3, IEEE7EEEEEEE7, IIEEEEE7EEEE7, IIIIIIIIII0KF, IIIIIIIIILIB3, J00000000BBBJ, JGE000000000J, K000000000AEF, K000000000B6D, K00000000E7CF, K000000K005EF, K00000KKK0045, K00KKKKKKK045, K0222222222DD, KB0000000000D, KD555555555B5, KGG000000000J, KKKKKKKKK0K45, KKKKKKKKKK045, L00000000EGGJ, L000LLLLLLIB3, L0FLLLLLLLLB3, L0LLLLLLLLIB3, L70777777772L, L777777777727, L7LLLLLLLLLIL, LEBBBBBBBBBBD, LEEEEEEEEEEK7, LL0000000LECL, 100000000000B5, 5000000000016D, 5000000000088J, 5000088888888J, 50K000000000B5, 555555555555EF, 55IIIIIIIIIIIF, 5K0000000000FD, 60000000000ECF, 60ECCCCCCCCCCF, 6ECCCCCCCCCCCF, 7000000000I04H, 770000000700I7, 7777077777772L, 7I777777777777, 80000000000DDD, 80000000008DDD, 8088000000000J, 8GGGGGGGGGGEED, 9000000000808J, 990000000000G3, 99F000000000G3, 9LLLLLLLLLLL7L, A000000000A0A1, A00000000HBAA1, B00000000BBBED, B0GGGGGGGGGGG3, B5555555555505, BA555555555505, CCCCCCCCCCCLG9, D00000000050EF, EEEEEEEEE4EEIH, EEEHEEEEEEEEEH, F000000000KF6F, F000000009F8C9, F05IIIIIIIIIIF, F0F00000000C2L, F5IIIIIIIIIIIF, FF000000000C2L, FF000000009F89, G00000000000AH, GI000G0000000H, H0000000000I27, H000000000II27, HEEEEEEBEEEEEH, IEEEEEEEEEEE4H, IF0000000000B3, IIEEEEEEEEE7E7, IIIIIIII0IIIKF, IIIIIIIIIBIII3, J0000000000GJJ, JJJJJJJE00000J, K00000000006GD, K0000000000CFF, K0000000000JI7, K000000000B055, K00000000K0KB5, K022222222222D, K0KKKKKKK00045, K0KKKKKKKKK545, K6CCCCCCCCCCCF, KEEEEEEEEEEE07, KKKKKKKKKKK0B5, L0LLLLLLLLLK77, L88888888888EJ, LK0000000000FF, LLLLLLLLLK00E7, LLLLLLLLLL9G33, LLLLLLLLLLGI33, LLLLLLLLLLL3G3, LLLLLLLLLLL9EL, LLLLLLLLLLLGI3, 200000000000JJ7, 2A0000000000001, 2CCCCCCCCCCCCCL, 2K00000000000K5, 3000000000009G9, 4HHHHHHHHHHHEEH, 50000000000010D, 500000000000KB5, 50E0000000000I7, 700000077777II7, 700777777777II7, 7700000000007I7, 7700700000000I7, 7707000000000I7, 777777777777727, 7IIIIIIIIIIIIIF, 7LLLIIIIIIIIIIF, 80008888888888J, 80088888888888J, 80D00000000000D, 89000000000080J, A0000000000I4IH, B00000000000HEH, B000000000BBEBD, B0BBBBBBBBBBBB5, BBBBBBBBBBBBBB1, BBBBE000000000D, C00000000000LEL, CEEEEEEEE0EEE0L, D555555555550A5, DHHHHHHHHHHHHHH, EKEEEEEEEEEEEE7, FFFFFFFFFFFFG45, FFFFFFFFFFFIIAF, G000000000000B5, G0GGGGGGGGGGGG3, GG000000000002J, H000000000000B1, I77777777777777, K000000000K0BK5, K0000000KKKKK25, K000EEEEEEEEEE7, K05555555555KB5, KEEEEEEEEEEE7E7, KEKEEEEEEEEEEE7, KK00000000005EF, KK0K000000000B5, L000000000006B3, L0000000000ILB3, L0000000000LIB3, LCLEEEEEEEEEEEL, LLLLLLLLLL00KE7, LLLLLLLLLLILLB3, LLLLLLLLLLLK0E7, LLLLLLLLLLLL4G3, 20000000000000K7, 3AF000000000000F, 4000000000000IEH, 4IIIIIIIIIIIII33, 500000000000IJG9, 509B00000000000J, 59000000000000BJ, 6000000000008KK1, 60I00000000000B3, 6GGGGGGGGGGGGGED, 70000000000000I7, 700000000000EKE7, 7070000000007II7, 70777777777777I7, 7077777777777II7, 77000007000000I7, 80000000000000DD, 80000000000000E1, 888888888888888J, 900000000000088J, 988000000000000J, A000000000000001, A000000000000015, A0000000000002A1, BBBB0000000000ED, BH00000000000001, CLEEEEEEEEEEEEEL, D055555555555555, DDDHHHHHHHHHHHHH, EEEEEEEEEEEEEEEH, EELLLLLLLLLLLLB7, EHHHHHHHHHHHHHHH, GI0G00000000000H, HEEEEEEEEEBEEEEH, HHHHHHHHHHHHHH2D, I0000000000000B3, IEEEEEEEEEEEE7E7, J0000000000000GH, J000000000000JBJ, J00000000000JIJJ, JJ00000000000IJJ, JJJJE0000000000J, K000000000000K25, KFFFFFFFFFFFFCFF, L00000000000000J, LLLLLLLLLLL0LIB3, LLLLLLLLLLLECLLL, LLLLLLLLLLLLILB3, LLLLLLLLLLLLLG33, LLLLLLLLLLLLLKFF, 2KK00000000000005, 44EHHHHHHHHHHHHHH, 55555555555555BB5, 5B0BBBBBBBBBBBBBD, 7000000000000I40H, 707777777777777K7, 77000000000000I77, 8000000000000008D, A0000000000000CB1, A0000000000000EEH, B00000000000000D1, BAA55555555555555, BIIIIIIIIIIIIII63, C00000000000002IL, C0000000000000CEL, CCEEEEEEEEEEEEEEL, CEEEEEEEEEE0EEEEL, CEEEEEEEEEEEE0EEL, D000002222222222D, D5555505555555555, D5555555555505555, DGGGGGGGGGGGGGEED, F0000000000000EBD, F000000000000262D, F000000000000E0BD, F000000000000F6B3, F000000000000K6FF, F000000F6000000B3, GGGGGGGGGGGGGGGGD, HHEEBEEEEEEEEEEEH, HHHHHHHHHHHHH2GGD, HHHHHHHHHHHHHGBBD, JJE0000000000000J, JJJJJJJJJJJJJK00J, K0000000000000B55, K80000000000000I7, L0000000000000IB3, L0000000000009E2J, LLLLLLLLLLEB00007, LLLLLLLLLLLLLBGG3, 2D0000000000000001, 3000000000000000EF, 500000000000000EI7, 700000000000000I4H, 77777777777777720L, 8000000000000KKKK1, 9000000000000000GJ, B00000000000000K6D, BBD00000000000000H, D5K00000000000000F, ELLLLLLLLLLLLLLEB7, F00000000000000989, F0000000000F6000B3, FFFFFFFFFFFFFFFIAF, G4GGGGGGGGGGGGGGG3, GGGGGGGGGGGGGGGEED, HGGGGGGGGGGGGGGGED, HHHHHHHHHHHHHHHGBD, J00000000000000945, JG00000000000000JJ, K00000000000000045, K00000000000000057, KEEEEEEEEEEEEEEEE7, L0000000000000LECL, LLLLLLLLLLLLLLEB07, 20000000000000000J5, 2DDDDDDDDDDDDDDDD0D, 2DDDDDDDDDDDDDDDDDD, 500088888888888888J, 555555555555555K5B5, 7KKKKKKKKKKKKKKKKKF, 9700000000000000045, A0000000000000000IF, FL00000000000000K0F, GGGGGGGGGG3GGGGGGG3, GGGGGGGGGGGGGGGGGB3, GGI000000000000000H, H000000000000000E0D, I0000000000000008I9, IIIIIIIIIIIIIIIIIKF, KKKKKKKKKKKKKKKK545, L0000000000000006EL, LCEEEEEEEEEEEEEEEEL, LLLLLLLLLLLLLLL0KCF, 50000000000000000D9J, 50000000000000000DKF, 6FFFFFFFFFFFFFFFF0B3, 70000000000000000BBH, 7077777777777777772L, 800000000000000000B7, 99LLLLLLLLLLLLLLLLG3, A0000000000000BBBBA1, B000000000000000004H, B0000000000000000D43, B0IIIIIIIIIIIIIIIII3, E0000000000000000021, E00000000000000000CL, F000000000000000EE8D, F5000000000000000045, H0000000000000000JIJ, H000000000000000BBBH, H2000000000000000007, I00000000000000004GH, IIE7EEEEEEEEEEEEEEE7, JJJJJJJJJJJJJJJJJJIJ, K00000000000000000CF, A0000000000000000AEAF, B000000000000000000KD, BIIIIIIIIIIIIIIIIIII3, G0000000000000000002J, H00000000000000000E6D, K0KK000000000000000B5, LLLLLLLLLLLLLLLLLECCL, 4HHHHHHHHHHHHHHHHHHHBH, 55555555555555555555B5, 5K000000000000000000DF, 8D0000000000000000000D, BBBBBBBBBBBBBBBBBBBBB5, D00000000000002222222D, F00000F6000000000000B3, F000F600000000000000B3, GA0000000000000000000H, GGGGGGGGGGGGGGG3GGGGG3, GGGGGGGGGGGGGGGGGGG3G3, K00000000000000000KBK5, L00J0000000000000000C9, 60000000000000000000B03, B0BBBBBBBBBBBBBBBBBBBBD, ECCLLLLLLLLLLLLLLLLLLLL, IIIIIIIIIIIIIIIIIIII9B3, IIIIIIIIIIIIIIIIIIIILB3, J00000000000000000J0JIJ, JJJJJJJJJJJJJJJJJJJE00J, JJJJJJJJJJJJJJJJJJJJK0J, 500000000000000000000095, 7777777777777777777777I7, A00000000000000000004IIH, B1000000000000000000000H, CEEEEEEEEEEEEEEEEEEE0E0L, ECLLLLLLLLLLLLLLLLLLLLLL, F00000000000000000000C2L, LLLLLLLLLLLLLLLLLLLL0IB3, 5E00000000000000000000II7, CEEEEEEEEEEEEEEEEEEEEE0EL, EK60000000000000000000001, F00000000000000000000BE0D, F0F60000000000000000000B3, HH1000000000000000000000H, IEEEEEEEEEEEEEEEEEEEEEEE7, IIIIIIIIIIIIIIIIIIIIIIIB3, KD55555555555555555555555, 40HHHHHHHHHHHHHHHHHHHHHHHH, 5B000000000000000000000007, 6000000000000000000000KKK1, B00000000000000000000000ED, B0000000000000000000000BBD, BAAAAAAAAAAAAAAAAAAAAAAA55, DH000000000000000000000001, L0000000000000000000000ECL, 500000000000000000000000I8J, 700000000000000000000000447, 800000000000000000000000E6D, CCCCCCCCCCCCCCCCCCCCCCCCCG9, H00000000000000000000000J47, J000000000000000000000000C9, JJJJJJJJJJJJJJJJJJJJJJJJKJJ, K0000000000000000000000KKB5, LKFFFFFFFFFFFFFFFFFFFFFFFFF, 5IIIIIIIIIIIIIIIIIIIIIIIIIIF, D555555555555555555555550555, EEAAAAAAAAAAAAAAAAAAAAAAAAAF, HHHHHHHHHHHHHHHHHHHHHHHHEEBH, K66666666666666666666666666F, LLLLLLLLLLLLLLLLLLLLLLLLEEB7, D5555555555555555555555555A55, GGGGGGGGGGGGGGGGGGGGGGGGGGGG3, GIG0000000000000000000000000H, HH00000000000000000000000001H, K0000000000000000000000005KEF, 5BBBBBBBBBBBBBBBBBBBBBBBBBBBBD, HB0000000000000000000000000001, K000000000000000000000000505EF, L7777777777777777777777777772L, 2000000000000000000000000000CB1, C8CCCCCCCCCCCCCCCCCCCCCCCCCCCC9, IKKKKKKKKKKKKKKKKKKKKKKKKKKKKFF, JE0000000000000000000000000000J, K000000000000000000000000000261, A0000000000000000000000000004I4H, HD000000000000000000000000000001, K000000000000000000000000000EC01, K0FFFFFFFFFFFFFFFFFFFFFFFFFFFFCF, D0002222222222222222222222222222D, FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFL2L, I700000000000000000000000000000GH, K00000000000000000000000000000E61, 20000000000000000000000000000000JJ, DD5KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, EBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBD, FBB000000000000000000000000000000D, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHBBD, L0000000000000000000000000000000877, 59B00000000000000000000000000000000J, B00000000000000000000000000000000063, D000000000000000000000000000000A0BB1, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHEBD, KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKK45, 50E0000000000000000000000000000000007, 60000000000000000000000000000000000KK1, BAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAK5, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE7K7, ELLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL0B7, IKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKAF, 3000000000000000000000000000000000003AF, A00000000000000000000000000000000000EKF, BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBD, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE00I7, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHEBH, CEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE0L, D00000000000000000000000000000000000000B1, 400000000000000000000000000000000000000033, D500000000000000000000000000000000000000KF, E6CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCF, 2000000000000000000000000000000000000000BB5, B555555555555555555555555555555555555555555, 33A0000000000000000000000000000000000000000F, 700000000000000000000000000000000000000000K7, 900000000000000000000000000000000000000008EJ, DD00000000000000000000000000000000000000000E1, F000000000000000000000000000000000000006006B3, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHD, 60000000000000000000000000000000000000000000B3, D0000000000000000000000000000000000000000000EEH, D9J00000000000000000000000000000000000000000005, DDE00000000000000000000000000000000000000000001, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHEEEH, 5E0000000000000000000000000000000000000000000007, 88800000000000000000000000000000000000000000000J, 97LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL, D55555555555555555555555555555555555555555555555, D555555555555555555555555555555555555555555555A5, G2222222222222222222222222222222222222222222222D, KFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFCF, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLEB7, 70000000000000000000000000000000000000000000000GH, KE000000000000000000000000000000000000000000000061, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLECL, H0000000000000000000000000000000000000000000000000JJ, D000000000000000000000000000000000000000000000002222D, 97IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIL, B0000000000000000000000000000000000000000000000000000AH, D5555KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, BAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA5, FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF6B3, H000000000000000000000000000000000000000000000000000000ED, 7000000000000000000000000000000000000000000000000000000045, B0000000000000000000000000000000000000000000000000000000I3, C0000000000000000000000000000000000000000000000000000000EL, D500000000000000000000000000000000000000000000000000000005, K00000000000000000000000000000000000000000000000000000J887, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL0KE7, 44HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH, F0000000000000000000000000000000000000000000000000000000L89, J000000000000000000000000000000000000000000000000000000JJIJ, K222222222222222222222222222222222222222222222222222222222DD, KCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCF, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLKCF, D55KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, K000000000000000000000000000000000000000000000000000000000000077, 22222222222222222222222222222222222222222222222222222222222222222D, K000000000000000000000000000000000000000000000000000000000000008IJ, DFA00000000000000000000000000000000000000000000000000000000000000005, CC4IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII3, JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJE0J, 777777777777777777777777777777777777777777777777777777777777777777777777EK7, 6000000000000000000000000000000000000000000000000000000000000000000000000000000043, KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKB5, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHEEH, 4HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHEH, 50000000000000000000000000000000000000000000000000000000000000000000000000000000002C1, K0000000000000000000000000000000000000000000000000000000000000000000000000000000000055EF, H700000000000000000000000000000000000000000000000000000000000000000000000000000000000000000H, 80000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000K1, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE0I7, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHEH, 5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BB5, J000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BIJ, C4IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII3, F0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000066B3, G0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000A5, D5KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHBH, L0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000IKF, 4IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII3, A400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000H, DKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, 4HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH, E0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000071, 7LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLIL, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLK77, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEI7, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLIB3, I7G00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000H, D0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000005EEF, IKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, C0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000G9, 77EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEK7, 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66FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFB3, L0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000B63, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLG3, E60000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000L, IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIAF, 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J00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000IGGJ, 77777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777K7, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLKE7, 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BKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKK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80BN, 80FH, 80LN, 80MN, 840N, 848N, 866B, 86FB, 880B, 880N, 884N, 88CB, 88FB, 88LN, 88MN, 8BBB, 8BLB, 8C6B, 8CCH, 8CFH, 8F0B, 8FHH, 8FLB, 8H0H, 8HCH, 8IGN, 8ILN, 8KKH, 8L8B, 8LBB, 8LFB, 8LIN, 8M8N, 8MLN, 8N0N, 8NGN, 8NLN, 9061, 9091, 90EJ, 90F1, 90GJ, 90K1, 940J, 9501, 95F1, 9CC1, 9E0J, 9E95, 9F61, 9FI5, 9G3J, 9II5, 9K01, 9KK1, 9M01, A007, A05D, A0AD, A33D, A3AD, A3F5, A44J, A727, A9EJ, AA0D, AAAD, AAAJ, ACM7, AD8D, ADKD, AE27, AE9J, AEAJ, AEE7, AIAD, AIDD, AIID, AJ9J, AK5D, AM07, AM27, AM35, AMK5, B08B, B0CB, B0GB, B18B, B1CB, B80B, B8CB, BB21, BB4N, BBCB, BBF1, BBFB, BBK1, BC8B, BCF1, BCLB, BF1B, BF8B, BFB1, BFM1, BGC1, BGF1, BK21, BL8B, BLFB, BM1B, BM3N, BMB1, BMMN, BNF1, C00D, C06B, C077, C0D7, C0H7, C0L7, C0LB, C0M1, C60H, C6LB, C76H, C7E7, CAID, CC01, CCFH, CCKH, CDLB, CGE7, CH07, CHE7, CI8D, CIAD, CK0D, CL8B, CLDB, CLE7, CM01, CM07, CME7, CMM1, D007, D08D, D0C7, D0HH, D0LN, D0M7, D0NN, D207, D2KH, D2M7, D777, D7E7, D80H, D8LD, DA27, DAC7, DAM7, DBFB, DBMB, DC77, DCLB, DDL7, DE77, DF0H, DF2H, DFFH, DFMB, DH27, DH8H, DHC7, DHHH, DILN, DK0H, DK2H, DK8H, DKHH, DLIN, DLL7, DLM7, DLMB, DM07, DMH7, DMMB, DNGN, E07J, E09J, E335, E355, E555, E5A5, E5K5, E79J, E93J, E995, EA35, EE95, EKE5, F00B, F00H, F06H, F08B, F0I5, F11B, F18B, F1L1, F20H, F26H, F2FH, F355, F661, F6K1, F80B, F86B, F8BB, FBGB, FBK1, FBLB, FC0B, FC6H, FCLB, FEK5, FGB1, FH05, FH0H, FH35, FH6H, FHCH, FHF5, FHHH, FI05, FK91, FKK1, FL1B, FLB1, FLBB, FM61, FMBB, FMK1, G00J, G021, G027, G0EJ, G0JJ, G0M1, G0M7, G1CB, G2E7, G40J, G4AJ, G4IJ, G4JJ, G6C1, G701, G94J, G9IJ, GAEJ, GAJJ, GB21, GBM1, GC01, GCF1, GCLB, GE0J, GEAJ, GEE7, GEG7, GEIJ, GEL7, GFM1, GGE7, GGMB, GI0J, GIIJ, GIIN, GJ9J, GM27, GMB1, GNM7, H005, H0K5, H0M5, H207, H2E7, H335, H3I5, H595, H5K5, H60H, H68H, H76H, H80H, H8HH, HAAJ, HE7J, HEC7, HGE7, HGM7, HH35, HI55, HIM5, I00J, I035, I08D, I0CD, I4JJ, IC0D, ICID, II0D, II0J, II35, IIAD, IILD, IIM5, IIMN, IJ9J, ILCD, IM05, IM35, IMNN, INLD, J03J, J0HJ, J0JH, J2CH, J39J, J3ID, J60H, J62H, J8CH, J9IJ, JGAJ, JGJJ, JH9J, JI0J, JIDD, JJ0H, JJCD, JJJD, JJLD, JL3D, JLCD, K0E5, K0I5, K0K1, K0KH, K191, K211, K2F1, K2G1, K591, K5AD, K6F1, K6G1, K9I5, KA0D, KAAD, KAM5, KCCH, KCHH, KD8D, KDDD, KFI5, KG01, KG61, KH0H, KHHH, KI55, KIDD, KK21, KK8H, KKF1, KKK1, KKKD, KM8H, KMHH, KMK5, L027, L0M7, L1C1, L1MB, L211, L727, L8BB, L8BN, L8FB, L8LN, L9C1, L9M1, LB8B, LBC1, LBM1, LCAD, LD77, LDIN, LDL7, LDM7, LF8B, LG21, LIIN, LLLN, LLN7, LM07, LM1B, LM77, LMG1, LN77, LNM1, M00N, M01B, M03N, M055, M077, M08B, M0B1, M0C1, M0GB, M0K1, M0M7, M0N7, M18B, M1BB, M1MB, M26H, M335, M3GN, M3M5, M3MN, M3NN, M501, M53N, M5M1, M5NN, M6BB, M6C1, M6G1, M6KH, M88B, M88N, M8BB, M8NN, MBB1, MC01, MCC1, MCKH, MCM1, ME07, ME35, MEK5, MGGB, MGMB, MH35, MH8H, MHE7, MHM5, MK8H, MKC1, MKG1, MKHH, MKK5, ML8N, MM01, MM8B, MMC1, MMLN, MMM5, MMMN, MMN1, MN0N, MN27, MNGN, MNLN, N007, N027, N077, N0C7, N0DN, N0IN, N1M1, N227, N2M7, N661, N707, N727, N8LD, NA27, NA3D, NC07, ND0D, ND0N, NDLD, NF11, NF61, NGG7, NILN, 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5M0M1, 5MNNN, 5N03D, 5N3LD, 5NA8D, 5NADD, 5NDGN, 5NNGN, 600KH, 61661, 616L1, 61MM1, 66161, 66611, 666B1, 666L1, 666LB, 66BB1, 66BG1, 66BLB, 66G61, 66KF1, 66LBB, 66LG1, 6BBBB, 6BFCB, 6CC11, 6F1M1, 6F66B, 6F6B1, 6F6L1, 6FBCB, 6FLMB, 6FMCB, 6FMM1, 6FMMB, 6GCM1, 6GM61, 6GMC1, 6K1C1, 6K1K1, 6KK11, 6KKL1, 6KL11, 6L1G1, 6LBFB, 6LCC1, 6LFMB, 6MM61, 6MM6B, 70291, 702C1, 702G1, 72CF1, 72EE7, 7433J, 7443J, 77A07, 79901, 799F1, 7AAEJ, 7EE27, 7H9EJ, 7K2KH, 7KK2H, 7KKKH, 7KKMH, 7L2C1, 800NN, 806BB, 808BB, 808LB, 80F8B, 80IIN, 833ID, 860KH, 8886B, 888NN, 8BG4N, 8CH6H, 8CKHH, 8FC0H, 8FFCH, 8HHHH, 8IIIN, 8K0HH, 8LL4N, 8M0NN, 8MNNN, 8NNND, 9000J, 900M1, 9034J, 90IIJ, 94IIJ, 96CM1, 96KF1, 96MM1, 990C1, 990M1, 99591, 99961, 999C1, 99F91, 99FM1, 99KF1, 99M61, 99MK1, 9AAA5, 9FEE5, 9FFA5, 9FFF5, 9II4J, 9K6C1, 9K9C1, 9K9F1, 9KF91, 9M6M1, 9MK61, A02M7, A0A35, A0AM5, A0C77, A0D0D, A0DDD, A0EC7, A0M55, A0MM7, A2ME7, A3335, A33M5, A3555, A3MM5, A550D, A58ID, A5D0D, A5DDD, A74AJ, A7E07, AA0M5, AA3ID, AA3M5, AA83D, AA8ID, AAAM5, AACID, AAICD, AAM05, AC08D, AC0ID, AC0KD, AC8ID, ACA8D, AFA35, AIC8D, AJAJJ, AK0KD, AKI0D, AKKM5, AKM05, AM505, B00FB, B00LB, B0LBB, B1MBB, B1MMB, B2GM1, B3MGN, B88BB, BB3MN, BBB3N, BBB8B, BBBBN, BBBMN, BBCC1, BBGB1, BBGLB, BBMC1, BBMGB, BBNM1, BFB0B, BFBMB, BFGLB, BFGMB, BGGLB, BGMM1, BLMMB, BMC0B, BMGM1, BMMK1, BMNM1, C006H, C00CH, C00M7, C0C11, C0CC1, C0E07, C0F6H, C0GG7, C0K8D, C0KDH, C0KHH, C1CC1, C1CM1, C1MC1, C70F1, C886B, CC06H, CCCCH, CCCF1, CCCM1, CDHH7, CE707, CE7L7, CEE77, CEEG7, CEL07, CFFHH, CGLL7, CHC6H, CHCCH, CHH7H, CHHH7, CHHM7, CI0LD, CK0HH, CKHCH, CL007, CLD07, CLLL7, CM777, CMMM7, D008H, D07L7, D0DE7, D0EL7, D0L77, D0M8H, D0NA7, D22E7, D70A7, D7227, DBLBB, DCL07, DD7A7, DDAE7, DDD8D, DDE07, DDK0D, DDM77, DEE27, DKK8D, DL227, DL707, DLBBB, DLDFB, DLE27, DM2E7, DMM77, DMME7, DN0A7, DNIIN, DNLLN, E000J, E0227, E02E7, E0AAJ, E0GL7, E0I0J, E2E07, E7AAJ, E7E27, EAA55, EAK55, EC777, ECEE7, ECEL7, ECGL7, ECL77, EEC77, EECL7, EEEC7, EEG27, EEGG7, EG0E7, EG0L7, EG207, EGE07, EGG07, EGG27, EGL07, EI9IJ, F02HH, F06LB, F0C6B, F0CCH, F0CHH, F0E55, F0EA5, F0FH5, F0GGB, F0M2H, F0MGB, F0MMB, F1BMB, F3FF5, F3I35, F3II5, F6BCB, FA035, FB1MB, FBBM1, FBBMB, FBM8B, FC0HH, FC88B, FCFHH, FFC0H, FFCFH, FFCHH, FFF0H, FFF6H, FFFH5, FFHI5, FFI55, FG1MB, FGGCB, FGM1B, FHK55, FI5I5, FKEA5, FKKI5, FKL11, FM16B, FM1CB, FM62H, FM6CB, FMBG1, FMM0B, G0001, G00E7, G039J, G06F1, G07C1, G07F1, G0A9J, G0G07, G0GG7, G0I4J, G0I9J, G0LL7, G22M7, G2M07, G339J, G433J, G62M1, G6M61, G6MM1, G903J, G933J, GAA9J, GBCC1, GE007, GGGGB, GGGL7, GJ0IJ, GL007, GL2M7, GLL07, GLLM7, GMGCB, GMM07, GMM1B, H007H, H05I5, H0CCH, H0CM7, H0GG7, H0H27, H0H8H, H0HC7, H0HH5, H0I35, H0MM7, H3555, H35F5, H3F55, H3FF5, H5055, H50F5, H7HEJ, H9995, H9GEJ, HCC0H, HCC6H, HCGG7, HCHM7, HE027, HE0G7, HEE07, HEEG7, HEG27, HF0F5, HF505, HF555, HFF05, HFKF5, HG0G7, HH06H, HH08H, HH0H5, HH5I5, HH7HJ, HH9EJ, HH9I5, HHCHH, HHE07, HHGG7, HHH05, HHH7H, HHH9J, HHKI5, HHM05, HHM07, HHM27, HI0I5, HJ86H, HJC0H, HK055, HK9F5, HKF05, HKFF5, HKFK5, HKII5, HKK95, HM2M7, HME27, HMKM5, HMM05, HMM27, HMM55, HMMM7, I00DD, I00M5, I044J, I0505, I09IJ, I0AAD, I0D0D, I0DDD, I0I05, I0IDD, I0II5, I0IJJ, I0JIJ, I33M5, I4I4J, I5INN, I888N, I8NND, I8NNN, I904J, I94IJ, IA8ID, IAADD, IAI8D, IDDDD, IDINN, II88N, II8NN, IID8D, IIDIN, III8N, IIIDD, IIIID, IIIIN, IIIND, IIN8D, IINDD, IJJ0J, IJJJJ, ILILN, ILLIN, IMM8N, INA0D, INDNN, INGIN, INNDN, INNND, J000H, J002H, J00AJ, J00GJ, J00KH, J02KH, J068H, J080H, J090J, J0A9J, J0AAJ, J0C0H, J0G0J, J0IIJ, J0JGJ, J0JIJ, J0K8H, J2K0H, J2KKH, J6K8H, J86KH, JC00H, JC0KH, JCCCH, JCK0H, JCKCH, JDDLD, JG93J, JIIJJ, JJ0IJ, JJ2KH, JJ9GJ, JJCCH, JJG9J, JJGIJ, JJJ9J, JJJJH, JJK8H, JK08H, JK0CH, JK8KH, JKC0H, JKKKH, K0001, K0091, K020H, K02C1, K03ID, K0611, K06L1, K083D, K08HH, K0961, K09C1, K0CF1, K0F91, K0KM5, K0LG1, K1G21, K20HH, K29K1, K2KHH, K5001, K500D, K58ID, K5D0D, K5L11, K6621, K6C11, K6LC1, K8CKH, K8KCH, K96C1, K99E5, K9F91, K9FA5, K9FE5, K9K91, KA55D, KC011, KCF11, KD02H, KD0MH, KD20H, KDM2H, KEA55, KEAA5, KEK95, KEKK5, KF1G1, KF1K1, KF611, KF6L1, KFEA5, KH8CH, KI005, KIMM5, KK05D, KK0AD, KK0DH, KK2CH, KK2KH, KK33D, KK961, KK9C1, KKA5D, KKD0D, KKE55, KKI0D, KKIID, KKIM5, KKK0H, KKKM5, KLGC1, KMK2H, L188B, L1991, L2007, L22M7, L2EE7, L2MM7, L333D, L3LIN, L7291, L72G1, L88IN, L8C8B, L9991, LBB1B, LBBBB, LBBBN, LD0E7, LDBBN, LE207, LFMCB, LGCC1, LL227, LL3IN, LL48N, LLM27, LLMM7, LMBCB, LME27, LMMBB, LMMM7, LN33D, LN3AD, LNAAD, LNACD, M0007, M0061, M00K5, M0207, M066B, M0BCB, M0EE5, M0G01, M0GM1, M0M8N, M27KH, M2E27, M2M07, M2M27, M5005, M5555, M66CB, M66K1, M6K61, M6MCB, M7007, M7EE7, M8C0B, M8KCH, M8MGN, MBBGB, MBGM1, MCCCH, MCHCH, MEE55, MGBC1, MGMM1, MH227, MH2M7, MHH7H, MKM55, ML3LN, MM0CB, MM16B, MM227, MM661, MM6K1, MME55, MMEE7, MMKE5, MMM07, MMM6B, MMMB1, MMMGB, MMMM7, MNM61, MNN3N, N00CD, N00KD, N03LN, N0A8D, N0AM7, N0D8D, N0KKD, N0L3N, N0LAD, N0NDD, N16L1, N3GIN, N3LAD, N3LIN, N3NNN, N61L1, N96M1, N9M61, NA0CD, NAK0D, NAKKD, NCA8D, NCM77, NDGIN, NDIIN, NDLLN, NF991, NGM07, NIIIN, NINNN, NKKDD, NLNAD, NN0LN, NN191, NN3NN, NN6L1, NN83D, NNAAD, NNDIN, NNGIN, NNL3N, NNLND, NNM61, NNNIN, 166G21, 16G621, 19MMM1, 1BBBMB, 1BBGM1, 1GCCC1, 1GCCM1, 1MMM1B, 200E27, 2E0027, 2HH0HH, 2HHC0H, 2KK0HH, 2M0E27, 2M22E7, 30NNNN, 3333M5, 333AID, 333I35, 33I555, 3F5FF5, 3I3MM5, 3I88GN, 3II8LN, 3IIII5, 400IJJ, 40J00J, 40JJ3J, 40JJJJ, 44403J, 444I0J, 44IJJJ, 44J0JJ, 44JJIJ, 48I8IN, 4I440J, 4II8IN, 4IJ0IJ, 4JIIIJ, 4JJ0JJ, 4JJJ0J, 50033D, 5003AD, 5008ID, 500D8D, 500G01, 500L11, 500LAD, 500MG1, 503LAD, 508ILD, 50DDLD, 50ILAD, 50M001, 516G61, 519MM1, 538NNN, 53NNNN, 55005D, 5508ID, 550D8D, 558ILD, 55F5I5, 56G661, 58333D, 58NNNN, 5999F5, 59AAF5, 5DNNNN, 5F55I5, 5FMMM1, 5G6661, 5K9AA5, 5KK9F5, 5KKK95, 5M0001, 5NDD8D, 5NDINN, 5NN33D, 5NNLAD, 5NNNAD, 5NNNDN, 608K0H, 61CCM1, 61G621, 661G21, 666621, 6666CB, 6666F1, 66K661, 6BCCC1, 6BKKC1, 6F6BBB, 6G6621, 6GCCC1, 6GMMM1, 6K6K61, 6M666B, 70A077, 70L991, 7722E7, 772E27, 7772E7, 777A27, 777L27, 77A777, 77EL27, 7A7077, 7A7777, 7E7227, 7L2E27, 7LEL27, 7LL2E7, 7LLE27, 7LLL27, 800GIN, 80NINN, 80NNNN, 8BBMGN, 8C888B, 8C88LB, 8MM0GN, 900001, 90043J, 959MM1, 96K661, 9999F1, 9999K1, 999AF5, 999FF5, 99EEE5, 99K991, 99MMM1, 9AAFF5, 9EIIIJ, 9F9991, 9F9MM1, 9FEAA5, 9G444J, 9K9991, 9M6661, A000CD, A000KD, A000M5, A0083D, A00I0D, A00M05, A022E7, A07E77, A0FF35, A0K3ID, A0K83D, A4AJJJ, A77777, AA0035, AA0355, AAA035, ADDD0D, ADDDDD, AF0035, AFFF35, AKK8ID, AM0M05, AM7777, B0F0MB, BBBBM1, BBLBMB, BFBBBB, BFM0MB, BFMMMB, BLBBMB, BLMBBB, C00071, C000E7, C007C1, C00G07, C07KKH, C0CC6H, C0CH6H, C0EEE7, C0HHHH, C777L7, C77L77, C7L777, C7LL07, C8088B, CAAK8D, CAKKAD, CC000H, CC0CHH, CD000H, CD0KKH, CE0007, CEE0E7, CELL77, CG0007, CGGL07, CH0CHH, CHCH0H, CHHH6H, CK0C0H, CKAK8D, CKKA8D, CL7707, D002FH, D0D0KD, D0DA77, D0DKKD, D0IIIN, D0K0DD, D0KDKD, D0KKDD, D0KKKH, DC0EE7, DCEEE7, DD0227, DD0D27, DD0DKD, DD0KKD, DD2E27, DDD0D7, DDD0LD, DDD227, DDDA77, DDDBCB, DDDCE7, DDDDFB, DDDMM7, DDEEE7, DDMBCB, DEEC07, DH000H, DHMEE7, DIIIGN, DK0KDD, DKMKKH, DMBBBB, DMEEE7, DMMM27, E00G27, E07727, E0C707, E0CE77, E0E027, E0EEG7, E0EGE7, E0EL27, E0GE27, E0L207, E0LE27, E0LL27, E2E2E7, E7L2E7, E900IJ, E9EEE5, EAAKK5, EC00E7, EC0G07, EC7007, ECEG07, EE0G07, EE0GE7, EE72E7, EE7L27, EECE07, EECEG7, EEEEE5, EEEK55, EEEKA5, EEEL27, EEGLL7, EELE27, EGLLL7, EKK595, EKKA55, EKKAK5, EKKKK5, F000E5, F0AA35, F0F035, F0FFFH, F0HKK5, F0KKE5, F16BB1, F16MM1, F1BBBB, F1MC6B, F666BB, F66BBB, F6GMM1, FB0BBB, FB1BBB, FBBB0B, FBMMG1, FC0FFH, FCFCCH, FEEE55, FEEEA5, FF03F5, FF0FFH, FF3F35, FFEE35, FFF2CH, FFFCCH, FFFFE5, FFI335, FFKFE5, FGLMMB, FK55I5, FKFE55, FLM8CB, FMC66B, FMMC6B, G0AA4J, G0CCC1, G0LE07, G666F1, GG0007, GG00G7, GG0L07, GGLLL7, GGLMM7, GI444J, GJJ33J, GLLE27, GLMMCB, GM0661, GMMM61, H00G07, H05555, H09FF5, H0C0E7, H0CE07, H0CEE7, H0E227, H0H007, H0H5F5, H0H995, H0HHE7, H0HHH7, H55505, H55II5, H5FII5, H99FF5, HEG007, HFFK55, HH0007, HH02M7, HH0C0H, HH7AEJ, HHC0E7, HHE227, HHH0C7, HHH0M7, HHH995, HHHC0H, HHHE27, HHHEAJ, HHHH07, HHHH8H, HHHHE7, HHHHI5, HHHHJH, HHHJ8H, HHHJCH, HHJ00H, HHK095, HHKKM5, HKK0F5, HKK5F5, HKKK55, HKKKK5, HKM555, HMEEE7, I00555, I05555, I0I94J, I333I5, I33555, I444IJ, I55055, I55505, I55555, IAAC8D, ID000D, IDD0LD, II9I4J, III4IJ, III505, IIIC8D, IIJIJJ, IJIIIJ, IM8LLN, IN00AD, INAACD, INCAAD, ININGN, J00CCH, J0IJJJ, J0J09J, J3333D, JIJIIJ, JJ68KH, JJIJIJ, JJJAJJ, JJJHGJ, JJJJAJ, JJJJGJ, JJJJIJ, K0008H, K00161, K001G1, K001L1, K002CH, K002HH, K00521, K00AKD, K00C0H, K00GF1, K00I0D, K00K95, K00M05, K01621, K05021, K0505D, K051G1, K059F5, K05K95, K0C0C1, K0L291, K0M005, K0M505, K1K661, K2CK0H, K33IAD, K3IIID, K5550D, K56121, K59AA5, K612K1, K61CC1, K66661, K6K611, K900C1, K99661, K9AFF5, K9C001, KAKI8D, KC00C1, KDK00D, KF9991, KI0IID, KK000D, KK01L1, KK0661, KK0I8D, KK0L11, KK0M2H, KK5661, KK59F5, KK61C1, KK9995, KK9EE5, KKA3ID, KKA83D, KKAI8D, KKC001, KKC0C1, KKC1C1, KKCCC1, KKD2HH, KKK595, KKK9A5, KKKK95, KKKKKH, KKKMCH, KKM505, KKMEE5, KKMKCH, KM0005, L222E7, L33AAD, L38I8N, LCCC11, LDFBCB, LEL2E7, LELE27, LELL27, LGMMM1, LLE2E7, LM2ME7, M000M5, M006MB, M00E27, M00MM1, M02227, M06M61, M06MM1, M0E227, M0EE27, M0KME5, M0M5GN, M0MM61, M0MMCB, M0MNNN, M0NNM1, M38LLN, M5K505, M77707, M7E227, M7E727, M8CHHH, MBMMCB, MBMMM1, MEEE77, MHH027, MHH505, MHHC6H, MHHH6H, MHHK05, MKK001, MM2ME7, MM7707, MM7E77, MMBMK1, MMM2E7, MMMC0B, MMMK61, N0003D, N0008D, N0030N, N030NN, N0C0AD, N0CKAD, N0DKDD, N0N3GN, N0NN3N, N333AD, N777A7, N77A77, NAACKD, NAAKDD, NACAKD, NACKAD, NC0AKD, NC0KAD, NCA0KD, NCKAKD, NDNNLN, NNNLAD, NNNNLD, 1BBBBBB, 1BBBBG1, 1M6MMMB, 1MMBBBB, 1MMMMK1, 2000227, 2000EE7, 20EEEE7, 2C0FFFH, 2E2EEE7, 2KKKHCH, 2MEE227, 2MEEE27, 333333D, 3333355, 3335555, 333FFF5, 333IIID, 388NNNN, 38INNNN, 3INNNNN, 4000IMN, 4000JJJ, 400IIIN, 444444J, 44JJJJJ, 488888N, 4IIJIIJ, 4JJJ33J, 50002M1, 5001G21, 5006621, 500LGM1, 555083D, 55555I5, 5616G21, 59MMMM1, 5K999A5, 61CCCC1, 66666K1, 6K0000H, 6K0080H, 70000A7, 70077A7, 70700A7, 7070A77, 77700A7, 77770A7, 7777227, 7777E27, 77L2227, 7LE22E7, 888888B, 888888N, 8888BBN, 8888IIN, 888B88B, 888I8IN, 88IINNN, 88NIINN, 88NNIIN, 8INNNNN, 90444IJ, 904I44J, 9666661, 9666FK1, 9666K61, 9966FK1, A00KK0D, A0K000D, AAAAA35, AAKKI8D, BB8888B, BBB0BLB, BBBB1BB, BBBBBB1, BBBBBGB, BBBBBLB, BBBLMBB, C0007KH, C000F11, C00FFFH, C00HH0H, C00K00H, C0C0HHH, C0CCHHH, C0CHH0H, C0CHHCH, C0FFFFH, C0H0H0H, C0KKC0H, CC0HH0H, CCCCC11, CCCCCC1, CCHHHHH, CDKKKKH, CEL7777, CGGG0G7, CGGGGG7, CH00HHH, CHGGGG7, CHHHH0H, CHHHHCH, CHHHHHH, CK0000H, CKDKKKH, D00DDKD, DD0DDD7, DDBBBLB, DDD2EE7, DDDBBLB, DDDDD27, DDDDDBB, DDDDDC7, DDDDDKD, DDDDDMB, DDDDEE7, DDDDKKD, DDDDLDB, DDDFBBB, DDDLFCB, DDDMEE7, DDM2227, DHHEEE7, DK000KD, DK00D0D, DK0D00D, DNN000N, E000CL7, E000EG7, E000GE7, E00C0G7, E00CE07, E00EE27, E0C00G7, E0C0EG7, E0CE007, E0EC0G7, E0EE207, E0G0007, E20EE27, E22EEE7, E2EE227, E2EEE27, E772227, E77LL27, E7L2227, E9IIIIJ, EAKKKA5, EC000G7, ECG00G7, EE00L27, EE0E0G7, EE20EE7, EEE0EG7, EEE22E7, EEEE727, EEEEE27, EEEEG07, EEEEGE7, EEEKKK5, EELLL27, EI0IIIJ, EKKKAA5, ELLLE27, F00FA35, F0333F5, F0F0FE5, F333335, FAAFF35, FCF0FCH, FEEEE35, FF03335, FF0FA35, FF0FE35, FF0FMCH, FFF0A35, FFF0F35, FFFAF35, FFFF5I5, FFFFM2H, FFFI3I5, FFFIII5, FFH5555, FH55555, FL1MMM1, G0000G7, G0000L7, GGGG007, GLE2227, GLLLLE7, GLLLLL7, H000007, H0000C7, H000HCH, H000HM7, H000M27, H000ME7, H00G227, H00HHM7, H02M227, H0C0HHH, H0CH00H, H0E0007, H0FFF35, H0FFFF5, H0H0ME7, H0HFII5, H0HHHCH, H0M0227, H555555, H5F5FF5, HC000G7, HC00H0H, HCCHHCH, HCHHHCH, HCHHHHH, HEEEE27, HFF5FF5, HFF5FI5, HFKKK05, HG00007, HH00E27, HH0G227, HHH2MM7, HHH55F5, HHH9FF5, HHHFFK5, HHHFK55, HHHH7EJ, HHHHCM7, HHHHHAJ, HHHHHF5, HHHHHHJ, HKK5505, I000055, I00A0ID, I0I4IIJ, I0IIIIJ, I88NIIN, III0055, III0I55, III444J, IINNLIN, J000IJJ, K0000DH, K0000KD, K00033D, K000A5D, K000K5D, K00555D, K009995, K00K00D, K00K8ID, K00KI8D, K00KIAD, K01GCC1, K05033D, K0999F5, K2KKKCH, K53333D, K956661, K999991, KCCC1C1, KFFFE55, KFFKKE5, KFKFKE5, KK009A5, KK00C11, KK01GC1, KK99001, KKIII05, KKK09F5, KKKE9E5, KKKEAK5, KKKKI05, KKKKKE5, KMMEEE5, L1BBBG1, LBMMMCB, LBMMMMB, LDEEE07, LEE22E7, LEE2E27, LEEE2E7, LLLLE27, M0000CB, M000C6B, M02EEE7, M0K0005, M0M0005, M2CHHHH, M2HHHHH, M6MMMM1, MC0000B, MCHHHHH, ME7E777, MEE7777, MEEE2E7, MG06661, MHHHCCH, MHHHH27, MHHHHH7, MHM0027, MM6666B, MM77777, MMC000B, MMM7727, MMNM777, N000NLN, N00333D, N003AAD, N0A00DD, N0NN33D, N0NNLLN, N30000N, N777777, NDNNNNN, NN0N0GN, NN0N30N, NNN300N, NNN333D, NNN3LLN, NNNDDDD, NNNNN3N, NNNNNND, 33333F35, 33FFFF35, 3555FFF5, 3FFFFF55, 3NNNNNLN, 40000I0J, 40I0IIIJ, 444440IJ, 4J0000IJ, 500006G1, 5D00DDDD, 5L1MMMM1, 5MMMMMG1, 5NNDDDDD, 5NNNNDDD, 5NNNNN8D, 6000080H, 777777A7, 77777A77, 7944444J, 800000IN, 996666K1, 999999I5, 9999FEA5, A00003ID, AAAAFF35, BBBGMMMB, C0000011, C000007H, C0CC0H0H, C666666B, CCCH0HHH, CCHH0HCH, CE777777, CEEEEE07, CHH0H00H, D00000GN, D000D0LD, D000IIGN, D0DDDDD7, DDD0E2E7, DDDDDDDB, DDDDDME7, DEEEELE7, DEEELEE7, DEELEEE7, DELEE0E7, E00000C7, E00000G7, E0000CG7, E000C0E7, E000G007, E00CG007, E00E0CG7, E0C00007, E0CGGGG7, E0GGGGG7, E20000E7, EAAKAAA5, EAKKAAA5, EE00E727, EE020007, EEEE2027, ELEE2227, F00003F5, F0000A35, F0003335, F0FFFA35, F1999991, F1999MM1, FAAAAF35, FBBBBBG1, FEAAAAA5, FF000A35, FF00FF35, FF0KEEE5, FFAAAF35, FFF555I5, FFFF33I5, FFFFF035, FFFFF3F5, FFFFFKI5, FFFFFMHH, FKFKEEE5, FKKFEEE5, FMMMMMCB, FMMMMMM1, G2000007, GGGGGMM7, GJJJJJ0J, GJJJJJ3J, H0000E27, H0000G27, H000C0G7, H000CEG7, H000CHH7, H000E0E7, H000EE27, H00CHHG7, H00EEE27, H00M0EE7, H05FF5F5, H0E00EE7, H55FF5F5, HCHH0H0H, HE000EE7, HFFIIII5, HGGG2227, HH00CEG7, HH00H0CH, HH0EEEE7, HH0FFFI5, HHEEEEE7, HHH000CH, HHH00EG7, HHHC00G7, HHHFFFF5, HHHHHKK5, HHHK5F55, HK5555F5, I4IIIIIJ, IA0000ID, II0005I5, III000I5, III055I5, III5NNNN, IIIII9IJ, IIIIIII5, IIINNNGN, JAJJJJJJ, JJAJJJJJ, K000005D, K00009A5, K0000M55, K000M555, K008IIID, K00D0K0D, K00III8D, K00K550D, K00LCC11, K0999951, K0D0000H, K0K00595, K0K9AAF5, K0KK0095, K0KK9FF5, K3333IID, KFKFEEE5, KFKKEEE5, KK00000H, KK0000M5, KK099991, KK55583D, KKKEEEA5, KKKKKKI5, LEEEE227, LLLEEE27, LLLL2E07, M000006B, M000M6CB, M0MMMMM1, M222EEE7, M777E777, M7E77777, ME222EE7, ME2EEEE7, MEE222E7, MM0NNNNN, MME77727, MMM6MMM1, N00003GN, N0000ADD, N000N0GN, N000NNND, N033333D, N0NN0NGN, NDDDDKDD, NN000N3N, NN00N03N, NN03000N, NNN003GN, NNNNDNLN, NNNNNADD, 199999MM1, 200FFFFFH, 222MEEEE7, 2FFFFFFCH, 30000000N, 30N00000N, 400000J3J, 500000M01, 5000166G1, 5000666G1, 50DDDDDDD, 8NN33333D, 999999991, C000000FH, C00000K0H, C000H00HH, C77777707, CH00H000H, CHH0000HH, D00000DKD, D00000DLD, D0000200H, D0000KK0D, D000KK00D, D00D0DDLD, D0D00DDLD, D0LEEEEE7, DDBBBBBBB, DDD000KDD, DDDDDDDE7, DK00000DD, DNNNNNNNN, E00000E27, E00007L27, E0000E727, E0E000C07, E20000027, EAAAAKAA5, EAKAAAAK5, EE0000C07, EEE000E27, EEEEEEGL7, F00FFFF35, F0FFFFF35, FF0000035, FF0FFFF35, FF5555FI5, FFFFFFA35, FFFFFFF35, FFFFFFFI5, FKKKKEEE5, FMMMMMMMB, GGGGG2227, GGGGGG207, GJJJJJJJJ, GLMMMMMMB, H000022M7, H000222M7, H000EEEE7, H0EEEEEE7, H0H0000CH, H0IIIIII5, HCH00000H, HE0EEEEE7, HFFFFFI35, HFFFFKKK5, HHHHHHG27, HHHHHHH55, HHHHHHHH7, HHHHHHM55, HHHKK5555, HHIIIII05, HIIIIII05, HKK5555I5, I000000AD, I000000ID, I000A000D, I00A0000D, IIIII0555, IIIIIII9J, K000000AD, K00000595, K000009F5, K0000550D, K099999A5, K0C00000H, K0I00000D, KK0000595, KK0000HCH, KKK000095, KKKFKFFE5, KKKIIIII5, M77777777, MEEEE2227, MMMMMMMM1, N0000000D, N0000003N, N000003NN, N00000N3N, N00000NGN, N00N000GN, N00NNNN8D, N0NNN00GN, N0NNNN3AD, N0NNNNNGN, N999999M1, NN0NNNNGN, NNN000NGN, NNNNNDD8D, NNNNNN0GN, 16MMMMMMMB, 1MMMMMMBCB, 3333333335, 33333333I5, 400000000N, 40IIIIIIJJ, 4IIIIIIIJJ, 4IIIIIIJIJ, 50000000M1, 70F9999991, 777E777727, 9999995MM1, ADD000000D, C00000088B, C000000CF1, C00000F0HH, CH00000H0H, D00KD0000D, D0D0DDDDLD, D0E2EEEEE7, D2EEEEEEE7, DBBBBBBBBB, DD000000KD, DD0000DDLD, DLE0EEEEE7, EEE0000727, EEEAAAAAA5, EEEEEE00G7, EEEEEEE0G7, F000000F35, F00FFKEEE5, F00KFFEEE5, F0M666666B, FCFFFFFFFH, FFFFFFF2HH, FFFKKKEEE5, GGGGGGG227, H00000C06H, H0000HHH6H, H555FFFFF5, H55FFFFF55, H5FFFFFF55, HF5FFFFFF5, HHHH0H0HCH, HHHHH0HHCH, HHHHHH0HCH, HHHHHHHHM5, HHHIIIIII5, IIDNNNNNLN, IIIIIJJIIJ, IIINNNNNLN, IINNNNNNGN, INNNNNNNLN, J0000000IJ, K0000II8ID, K099999995, K0I0000AID, K0K0009FF5, K9999999F5, KK00000095, KKFFFKEEE5, LLLLLLLME7, LLLMEEEEE7, LMEEEEEEE7, M000000005, MHHHHHHHH5, MK00000005, MMMMMMMBCB, NN000000GN, NN0000NNGN, NN99999991, 2HHHHHHHHHH, 38NNNNNNNNN, 3MNNNNNNNNN, 40IIIIIIIIJ, 4AJJJJJJJJJ, 4J000000J0J, 4JJJJJJJJJJ, 506666666G1, 5DDDDDDDDLD, 999999999F5, 99999999EA5, 99999999FE5, A0000000035, C0000000007, C00000000G7, C00000000KH, CEEEEEEEEL7, D0000000FMH, D000DDDDDLD, D0KD000000D, DDDD00000KD, DEEE0EEEEE7, DEL0EEEEEE7, DELEEEEEE07, E0000E20007, E7777777727, EE000000207, EEE20000007, FFFFFFFFMCH, FM66666666B, GGGGGGGG2M7, HFFFFFFFF55, HHHHHHHHH6H, HHHHHHHHHCH, HHHHHK55555, I9IIIIIIIIJ, IIIIIIII44J, IIIIIIIJJIJ, IINNNNNNNNN, JDDDDDDDDDD, JJIIIIIIIIJ, K00000I8IID, LLLLLLLLL27, 9999999EEAA5, AI000000000D, C77700000007, CH0HH000000H, D00D000000LD, DEEEEEEE0EE7, DN000000000N, EAKAAAAAAAA5, EKAAAAAAAAK5, F6666666666B, H000HHHHHH6H, H55FFFFFFFF5, HFFFFFFFFKK5, K00000000I8D, K999999999A5, 3555555555FF5, 5000000000001, 6G66666666661, 99999999999A5, C00000000000H, CFFFFFFFFFFCH, CHHH00000000H, D00000000K0KD, D0000000K00KD, D0D00000000LD, DEEEEEEEEEL07, E000E20000007, E00E200000007, EEE0000000027, GGGGGGGGGGGM7, GGGGGGGGGGM07, H00000000CHHH, H00HC0000000H, HFFFFFFFFFFK5, I0A000000000D, J000000000J9J, K000000000095, K000000000M2H, M0000000000M1, M0EEEEEEEEEE7, MHHHHHHHHHHHH, MMNNNNNNNNNNN, MNNNNNNNNNNNN, N000000DDDDDD, N000DDDDDDDDD, NNDDDDDDDDDDD, 22EEEEEEEEEEE7, 35FFFFFFFFFFF5, 400000000000JJ, 800000000000GN, DDDDDDDDDDD077, DDDDDDDDDDDDD7, E0000000000L27, EAAAAAAAAAAKA5, EEG00000000007, H0000000000C6H, I5500000000005, II0000000000I5, M0666666666661, M6MMMMMMMMMMMB, 4JJ00000000000J, 506666666666661, BGMMMMMMMMMMMCB, CFFFFFFFFFFFFFH, D0000000000KD0D, D0HEEEEEEEEEEE7, F0BBBBBBBBBBBBB, HGGGGGGGGGGGGG7, K0000000000000D, K00000000000MCH, M0M6MMMMMMMMMMB, 5DDDDDDDDDDDDDDD, C00000000000008B, D00000000000000H, DEEEEEEEEEEEE0L7, DEEEEEEEEEEEEEL7, EEE2EEEEEEEEEEE7, GM66666666666661, H5FFFFFFFFFFFFF5, IIIIIIIIIIIIIJJJ, BGMMMMMMMMMMMMMMB, DLEEEEEEEEEEEEEE7, H0000000000000CHH, H000000000C0000HH, I000000000000000D, IIIIIIIIIIIIIIIJJ, INNNNNNNNNNNNNNNN, J000000000000009J, M666666666666666B, N0000000000000LLN, N00DDDDDDDDDDDDDD, 355555555555555555, 60000000000000008H, 6M6666666666666661, C000000000000000F1, N0DDDDDDDDDDDDDDDD, 666666666666666666B, 800000000000000000N, AD000000000000000DD, DEEEEEEEEEEEEEEEEE7, I500000000000000005, 20000000000000000027, 4000000000000000003J, 400000000000000000IJ, 99999999999999999995, DD00DDDDDDDDDDDDDDLD, E2EEEEEEEEEEEEEEEEE7, N00000000000000000LN, 500000000066666666661, EE0000000000000000727, GGGGGGGGGGGGGGGGGGG07, H0000000000000000006H, 40000000000000IIIIIIIJ, AD0000000000000000000D, K0000000000000000000M5, CL777777777777777777777, D000000000000000000000N, D0000000000000000000IIN, HHHHHHHHHHHHHHHHHHHHHK5, NDDDDDDDDDDDDDDDDDDDDDD, 1MMMMMMMMMMMMMMMMMMMMMBB, D00DDDDDDDDDDDDDDDDDDDLD, FFFFFFFFFFFFFFFFFFFFFFFH, 4J0000000000000000000000J, 566666666666666666666666G1, EKKAAAAAAAAAAAAAAAAAAAAAA5, 6666666666666666666666666G1, AJJJJJJJJJJJJJJJJJJJJJJJJJJJ, H00000000000000000000000008H, N0000000000000000000000000GN, DD0000000000000000000000000LD, IIIIIIIIIIIIIIIIIIIIIIIIIIIIJ, G0666666666666666666666666666661, GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG7, K000000000000000000000000000000000H, EAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA5, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLM7, M2EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE7, C000000000000000000000000000000000000000001, MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMCB, E00000000000000000000000000000000000000000000727, 777777777777777777777777777777777777777777777777727, EG000000000000000000000000000000000000000000000000000007, D000000000000000000000000000000000000000000000000000000000LD, 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===Base 36=== 11, 15, 17, 1B, 1H, 1N, 1P, 1V, 1Z, 21, 27, 2B, 2H, 2P, 2T, 2V, 2Z, 31, 35, 3J, 3N, 3T, 3V, 45, 47, 4D, 4J, 4N, 4T, 4Z, 51, 5B, 5D, 5H, 5J, 5V, 67, 6B, 6D, 6H, 6N, 6P, 6Z, 75, 7B, 7H, 7J, 7P, 7T, 7V, 85, 8J, 8N, 8P, 8T, 97, 9D, 9N, 9P, 9T, 9Z, A7, AD, AJ, AN, AT, B1, B5, BD, BN, BP, BZ, C1, C7, CB, CH, CP, CT, CV, CZ, DB, DJ, DN, DV, DZ, E5, EH, EJ, F1, F7, FH, FN, FT, FV, G1, GB, GH, GN, GP, GV, H1, H5, H7, HJ, HT, HV, HZ, I5, IB, ID, IP, IT, IZ, J7, JH, JP, JZ, K7, KD, KJ, KN, KV, L1, L5, LD, LH, LV, M5, MH, MJ, MT, MV, MZ, N1, NB, NP, NT, NV, NZ, OD, OH, OJ, ON, P7, PB, PJ, PT, Q1, Q5, QB, QH, QV, QZ, R5, RB, RJ, RP, S1, S5, SB, SD, SN, SP, SV, T5, T7, TH, TJ, TP, U7, UB, UD, UH, UN, UT, V1, V7, VD, VZ, W1, WB, WJ, WT, WZ, X5, XD, XP, XT, XZ, Y5, Y7, YD, YP, YZ, ZH, ZJ, ZN, ZT, ZV, 12D, 16J, 18D, 1CD, 1CJ, 1DD, 1GT, 1JD, 1JJ, 1LT, 1QJ, 1RD, 1RT, 1ST, 1XJ, 1YJ, 1YT, 22D, 22J, 22N, 23D, 255, 2A5, 2CD, 2ED, 2EN, 2F5, 2GD, 2GJ, 2IJ, 2JN, 2LN, 2MN, 2ND, 2O5, 2QD, 2QJ, 2U5, 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MYMU1, MYRK1, MYUU1, MYXR1, N000D, N0055, N006J, N007N, N00CN, N00IH, N00O7, N00S7, N029J, N02DD, N0307, N044H, N049H, N0505, N05M7, N05Q7, N06LJ, N070N, N0A55, N0CEN, N0CNN, N0CYN, N0DMD, N0DRD, N0F05, N0FF5, N0FMD, N0GED, N0HNN, N0I4H, N0IS7, N0IXJ, N0JLN, N0JNN, N0K4H, N0KIH, N0L6J, N0LI7, N0LLN, N0M0N, N0MQ7, N0NEN, N0O07, N0O77, N0Q0N, N0QEN, N0RRH, N0S07, N0SS7, N0U0J, N0ULJ, N0W77, N0WC5, N0WL7, N0XO7, N26JJ, N29XJ, N2CNN, N2CYN, N2MMD, N2N6J, N2N9J, N2NLJ, N2YYN, N30ED, N32RD, N33GD, N33MD, N3GGD, N3IW7, N3MR7, N3MX7, N3RGD, N3RI7, N3W2D, N3W77, N3X07, N3X77, N400H, N409H, N44YH, N494H, N49YH, N4AYH, N50K5, N5307, N5537, N5577, N57Q7, N5FK5, N5KK5, N5Q77, N5SQ7, N69LJ, N6L9J, N6U0J, N700D, N7077, N70EN, N70FD, N73RD, N7777, N777N, N7DMD, N7E2N, N7FED, N7M3D, N7MWD, N7SR7, N7WMD, N92LJ, N9XXJ, NA505, NA5F5, NAFK5, NAK05, NAKF5, NARYH, NAWW5, NC00N, NC0EN, NC0NN, NCE0N, NCN0N, NCNQN, NCW05, ND3GD, ND7DD, NDAW5, NDDGD, NDDRD, NDE3D, NDWM7, NDWW7, NE0GD, NED3D, NEDDD, NEL2N, NEMMD, NEMRD, NF0F5, NFEMD, NFKF5, NFWC5, NFWW5, NGDED, NGW2D, NH00H, NH2NN, NHH0H, NHMCN, NHMMD, NHNNN, NI00H, NI0EN, NI0MN, NI0NH, NI0NJ, NI0XJ, NI44H, NI7R7, NI7S7, NII77, NIIM7, NIIMN, NIIYN, NIM0N, NIN0H, NIN0J, NINJJ, NIW37, NIWM7, NIX77, NIXW7, NJ0EN, NJ0NN, NJ30D, NJ33D, NJFMD, NJIEN, NJJ2N, NJJIN, NJJJJ, NJJJN, NJL0J, NJLNN, NJN0N, NJSJJ, NK005, NK04H, NK3KH, NK5F5, NKA05, NKFK5, NKK55, NKKF5, NL09J, NL2IN, NLCNN, NLII7, NLIIN, NLJLN, NLL2J, NLLNJ, NLLNN, NLNCN, NLNIN, NLNLJ, NLNLN, NM00N, NM73D, NM7DD, NM7RD, NME0N, NMELN, NMLL7, NMLQ7, NMMRD, NMQEN, NMRD7, NMRR7, NMWWD, NN02J, NN0AH, NN0EN, NN30H, NN525, NN6UJ, NN92J, NNA55, NNASH, NNCLN, NNCMN, NNELN, NNF25, NNHHN, NNJ2J, NNJJJ, NNLNN, NNN55, NNN9J, NNNMN, NNNNJ, NNNSH, NNNXJ, NNQYN, NNS3H, NNU9J, NNW55, NNXJJ, NO077, NO0W7, NO777, NOI77, NOIM7, NOWL7, NQ00N, NQ0YN, NQ77N, NQ7EN, NQLLN, NQLNN, NQQQN, NQQYN, NRARH, NRAYH, NRDI7, NRHYH, NRQL7, NRRHH, NRW2D, NS0L7, NSA3H, NSHHH, NSIIJ, NSIJJ, NSS07, NSSJJ, NSSL7, NSSS7, NSULJ, NU00J, NUL6J, NUSIJ, NW0C5, NW3W7, NW5W5, NWI77, NWIM7, NWIO7, NWW55, NWWM7, NX0I7, NX0O7, NX3R7, NX9XJ, NXIM7, NXRL7, NXSSJ, NXWI7, NXWW7, NXXSJ, NY44H, NY94H, NYYIN, O0057, O005T, O0077, O0091, O009B, O00BT, O00EV, O00OV, O00TT, O00WP, O00WV, O040V, O0507, O056T, O065T, O08AV, O08E7, O0901, O09AB, O09I1, O09K1, O09WV, O0A4V, O0AI1, O0E9V, O0EAB, O0EAV, O0EFP, O0EKB, O0EOB, O0EU1, O0FMP, O0K6T, O0K91, O0MM1, O0O9B, O0OAB, O0OI1, O0OOB, O0OT1, O0OUV, O0OW7, O0OYT, O0T8V, O0TOT, O0TOV, O0TTV, O0TUV, O0TX1, O0U91, O0UE1, O0W87, O0WOV, O0YOT, O100T, O1EET, O4PAV, O4U0V, O5005, O50EP, O50I7, O50K5, O555P, O55FP, O55UP, O5EFP, O5I07, O5K05, O5KK5, O5OYT, O6LGT, O77E7, O7Q77, O80E7, O8AAZ, O8AFZ, O8AOV, O8FFZ, O8FKB, O8FZB, O8IM7, O8ZRZ, O8ZZB, O900V, O90WV, O96OV, O99LB, O9K01, O9OU1, O9R6V, O9U01, O9UI1, OA091, OA0O1, OA4AV, OA4IV, OA68V, OA8AZ, OA9I1, OAA4V, OAAFZ, OAAO1, OAAOV, OAAZZ, OAFAZ, OAFYB, OAI0V, OAI91, OAOAV, OAOE1, OAOU1, OAUE1, OAUI1, OAY01, OAY0B, OAYI1, OAZZZ, OB66T, OBE0T, OBEET, OE0BT, OE0FB, OE4AV, OE6ET, OE8AV, OE9EB, OEA8V, OEAA1, OEAFB, OEBET, OEBI7, OEEO7, OEFFP, OEFLP, OEKA1, OEKAB, OEKU1, OELMP, OEML7, OEO07, OEO0B, OEO4V, OEO6V, OEOAV, OEOO7, OEOOB, OEOVB, OEPU1, OERET, OERRV, OETY1, OEUK1, OEUU1, OEV9B, OEVFP, OEW8V, OEWW7, OF0FP, OF0OB, OFFFP, OFFMP, OFFZP, OFFZZ, OFMMP, OFMWP, OFOQP, OFPFZ, OFPUZ, OFZ0P, OFZFZ, OFZOB, OFZZZ, OG065, OG0YT, OI0EV, OI7Q7, OI9WV, OIIE7, OIQ77, OIQW7, OIR6V, OIYO1, OK8KB, OKAE1, OKAY1, OKE6T, OKEU1, OKFYB, OKIY1, OKYA1, OKYI1, OL08B, OL4EP, OL777, OL8E7, OLBI7, OLEL7, OLFOP, OLGYT, OLIW7, OLLSZ, OLLZZ, OLOUP, OLR6T, OLUMP, OMAU1, OMOU1, OMXM1, OO0AB, OO0I7, OO0T1, OO1ET, OO4AV, OO4MP, OO4QP, OO6KT, OO6LT, OO6OV, OO7O7, OOA0B, OOAIV, OOAOV, OOBGT, OOEMP, OOERT, OOET1, OOEU1, OOEY1, OOGTT, OOI4V, OOIRV, OOKET, OOLBT, OOMMP, OOMQ7, OOMU1, OOMWP, OOO0B, OOO6T, OOO77, OOOE7, OOOKT, OOOOV, OOORT, OOOW7, OOPY1, OOQ77, OOT01, OOT91, OOTAV, OOTOV, OOTUV, OOU91, OOUUV, OOWIV, OOWMP, OOWVP, OOY01, OOY0B, OOYKB, OOYU1, OOYYB, OP0EV, OP44V, OP8FZ, OPA91, OPAAZ, OPAOV, OPEAV, OPI4V, OPO4V, OPOIV, OPOUV, OPOY1, OPUAZ, OPUEV, OPWUV, OPYU1, OQ7W7, OQ8E7, OQLE7, OQQL7, OR8AZ, ORA8Z, ORKLT, ORS8Z, OS8RZ, OSR8Z, OSS8Z, OT09V, OT0K1, OT66T, OTT8V, OTUOV, OU091, OU0E1, OU0MP, OU4LP, OUAAZ, OUASZ, OUL4P, OUP8Z, OUP91, OUPAZ, OUPY1, OUU0P, OUUPZ, OUYM1, OUZ91, OV0FB, OV0LB, OV54P, OV5WP, OV9K5, OVALB, OVB6T, OVKFB, OVLEB, OVLFP, OVQWP, OW0EV, OW7O7, OW7Q7, OW7W7, OW90V, OWEEP, OWIWV, OWLW7, OWOM7, OWOVP, OWQ87, OWU0V, OWW9V, OWWL7, OWWOV, OXAA1, OXU01, OXUI1, OXXI1, OXXX1, OY00T, OY0E1, OY0RT, OY0U1, OY0YB, OY98B, OYE0T, OYFKB, OYKFB, OYLYB, OYYYB, OZ0OB, OZ0WP, OZ0YB, OZ9EB, OZ9K1, OZ9O1, OZALB, OZAYB, OZEAB, OZEOB, OZFEB, OZFFP, OZFFZ, OZFRZ, OZFZP, OZO0B, OZO91, OZOYB, OZPRZ, OZR8Z, OZUUP, OZZYB, P00W5, P02G5, P049V, P04YV, P0A0V, P0AW5, P0CC5, P0CU5, P0E9V, P0EWV, P0F25, P0FC5, P0G25, P0K3Z, P0KOZ, P0KRH, P0O55, P0OEV, P0PG5, P0PO5, P0PW5, P0R0H, P0ROZ, P0U25, P0UIV, P0UVP, P0V0P, P0VKP, P0W9V, P0YIV, P13WD, P22K5, P2D95, P2NC5, P33ZZ, P38DD, P3GGD, P3SAH, P409V, P444V, P44O1, P44Y1, P4AIV, P4AYV, P4E0V, P4OA1, P4OK1, P4OO1, P4U81, P4UEV, P4UR1, P55EP, P5GZ5, P6A41, P803H, P80AZ, P838D, P8AAZ, P8D8D, P8FKZ, P8K0Z, P8KKZ, P8KXH, P8OE1, P8UE1, P8X0H, P8X3H, P8XKH, P909V, P9941, P99EV, P99U1, P9E9V, P9K91, P9OE1, P9OK1, P9W9V, P9X9H, PA00H, PA441, PA5G5, PA8FZ, PA9O1, PAA8Z, PAI4V, PAI9V, PAMKP, PAOEV, PAR8H, PAUM1, PAVKP, PAZEP, PC0W5, PCAG5, PCAU5, PCCA5, PCWN5, PD0QP, PD0SH, PD33D, PD3AH, PD3MD, PDGC5, PDK3H, PDK8H, PDSAH, PDUAP, PE99V, PEA91, PEAEP, PEEA1, PEEE1, PEER1, PEI9V, PEK61, PEQQP, PERE1, PERK1, PEUY1, PEY0V, PF2K5, PF3CD, PF83D, PFC2D, PFCWD, PFDAP, PFEKP, PFMCD, PFMMD, PFO55, PFR2D, PFZEP, PG295, PG2WD, PG8CD, PGCD5, PGDC5, PGF5Z, PGG8D, PGKF5, PGKK5, PGO95, PGW8D, PH2MD, PHD8D, PK0O5, PK441, PK53Z, PK5GZ, PK83H, PK8KZ, PKAA1, PKAU1, PKKO5, PKP3Z, PKRXH, PKSIH, PMAA1, PMAAP, PMAU1, PMF3D, PMFCD, PMMAP, PMMMD, PMMMP, PMMPD, PMRR1, PMUO1, PN2D5, PN2W5, PNA3H, PNAF5, PNFK5, PNHFD, PNK55, PNKF5, PNSKH, PO09V, PO0EV, PO4AV, PO90V, POAY1, POI4V, POOE1, POUEV, POUY1, POWUV, POWWV, POYA1, POZAZ, POZFZ, PP0QP, PP2CN, PP2MD, PP33D, PP595, PP5W5, PP8CD, PPCNN, PPCRD, PPCU5, PPCW5, PPCWD, PPD05, PPDUP, PPECN, PPF8D, PPFC5, PPFW5, PPGC5, PPGF5, PPGFZ, PPKGZ, PPMCD, PPMCN, PPMEN, PPMND, PPNF5, PPNMN, PPNWD, PPP3Z, PPP95, PPPC5, PPPRZ, PPPZD, PPQKP, PPS0Z, PPW2N, PPWND, PPWWN, PPZC5, PPZSZ, PQQQP, PR1GD, PR8AH, PRA0H, PREK1, PRKO1, PRMM1, PRNFD, PRRM1, PRRRH, PRW2N, PS0ZZ, PS80H, PS80Z, PS8AH, PS8ZZ, PSIKH, PSR3Z, PU025, PU0IV, PU49V, PU4YV, PU8AZ, PU9EV, PUE91, PUE9V, PUI9V, PURK1, PUUM1, PUUR1, PUVAP, PUYIV, PUYO1, PUZMP, PV00P, PV2CN, PV2RN, PVAKP, PVFAP, PVFFP, PVFKP, PVK0P, PVKAP, PVNCN, PVNEN, PW005, PW0C5, PWCWD, PWF8D, PWNC5, PWO9V, PWOWV, PWWOV, PWWWN, PWWWV, PX0IH, PXAIH, PXIAH, PXIKH, PYKE1, PYKO1, PYRK1, PYRM1, PZ0G5, PZ1WD, PZ295, PZ2C5, PZ2G5, PZ481, PZ641, PZE61, PZEAP, PZG25, PZK61, PZKOZ, PZOFZ, PZOO1, PZR41, PZRZZ, PZS0Z, PZSUZ, PZUMP, Q00DT, Q00JD, Q00QT, Q00YN, Q020J, Q03RD, Q0CQN, Q0DRD, Q0FRD, Q0JTT, Q0Q3D, Q0QRN, Q0QXN, Q0XEN, Q0XIJ, Q0XLN, Q0XQN, Q0YYN, Q20RD, Q2D0D, Q32WD, Q3D3D, Q4EKP, Q4PQP, Q7777, Q77Q7, Q7ER7, Q7EW7, Q7LR7, Q7LS7, Q7LW7, Q7QQN, Q7QR7, Q7QYN, Q7RE7, Q7SS7, QAAPP, QAOOP, QAOWP, QAUAP, QAUWP, QAWEP, QAWLP, QCEQN, QCNYN, QCQEN, QD0GD, QD0RD, QD20D, QD33D, QD3DD, QDAWP, QDRET, QDWKP, QE7QN, QECLN, QEEL7, QEES7, QELCN, QEQCN, QEQTT, QERE7, QESR7, QF2WD, QFGUJ, QFR2D, QG00J, QG0IJ, QG7Q7, QG7R7, QGDDD, QGGGD, QGGQD, QGJ2J, QGMQ7, QGOO7, QGQQ7, QGRR7, QJ02J, QJ03D, QJ0SJ, QJ33D, QJ3WD, QJD0D, QJI0J, QK44P, QKPKP, QL7W7, QLES7, QLLL7, QLNNN, QLQ77, QLQL7, QLQM7, QLQPN, QLR87, QLSRT, QMLR7, QMQQ7, QN0YN, QN70N, QNCYN, QNLW7, QNNQN, QNNYN, QNQNN, QO777, QO7E7, QO8M7, QOUWP, QP3RD, QPDAP, QPEEP, QPFRD, QPGGD, QPPQP, QPR2D, QPRGD, QQ0RN, QQ0YN, QQ4LP, QQ4QP, QQAEP, QQCYN, QQELT, QQEPN, QQEQP, QQGDD, QQGGD, QQGO7, QQGQD, QQKKP, QQLAP, QQLPN, QQLRT, QQMR7, QQN77, QQNNN, QQNQN, QQPDD, QQPEN, QQPGD, QQPQP, QQPWD, QQQ0N, QQQ3D, QQQ87, QQQKP, QQQL7, QQQM7, QQQPD, QQQQN, QQQWD, QQRL7, QQRN7, QQRYN, QQSST, QQTDT, QQTQT, QQU4P, QQUAP, QQUUP, QQWPD, QQXEN, QQXQN, QQYXN, QR00N, QR0XN, QR2FD, QR32D, QREKT, QRN0N, QRNL7, QRQQD, QRRE7, QRXXN, QSES7, QSS2J, QSSE7, QSSS7, QSSSJ, QTDET, QTQ2D, QTQLT, QUAAP, QUAWP, QUPPP, QUUAP, QW7G7, QWG77, QWGM7, QWLEP, QWOEP, QWOLP, QX77N, QX7EN, QXELN, QXEQN, QXLLN, QXLQN, QXNQN, QXOE7, QXQXN, QXXNN, R0061, R008D, R00IH, R00NN, R00XN, R013D, R02YN, R030D, R04R1, R06E1, R06ET, R07M1, R08AH, R0A8H, R0AYH, R0D07, R0DDT, R0EK1, R0EYT, R0F3D, R0G8D, R0GDD, R0GGD, R0GQ7, R0H0N, R0KE1, R0LQ7, R0N0N, R0RYN, R0T2N, R0TLT, R0TNN, R0TYN, R0X0N, R0YST, R100T, R130D, R13WD, R166T, R1E0T, R1F3D, R1GFD, R1TGD, R200N, R28WD, R2W8D, R2WFD, R32DD, R330D, R332D, R3W8D, R44XV, R4AAV, R4AYV, R4MO1, R4OI1, R4RO1, R4U4V, R4X9V, R6001, R60K1, R66LT, R66YV, R6E01, R6EE1, R6MK1, R6SST, R6X01, R6X9V, R6XE1, R6XYV, R6Y6V, R7OO1, R7XE1, R80G7, R88FD, R8F8D, R8IE7, R8X8H, R9OAV, R9UVV, R9VOV, R9X6V, RA08H, RAA9V, RAS8Z, RAX0H, RD00D, RD8I7, RD8IH, RDA0H, RDDQT, RDQDT, RE061, RE6KT, RE6LT, REEK1, REEKT, RER01, RERK1, RERR1, RF0DD, RFD0D, RG28D, RGGFD, RGGWD, RGRM7, RGWQD, RH00H, RH0EN, RH8HH, RH9HH, RH9XH, RHH9H, RHHIN, RHHNN, RHIWN, RHNEN, RHWIN, RHWWN, RHX8H, RHXEN, RHY9H, RI00H, RI2IN, RIHEN, RIIWN, RIMRN, RK00T, RKEE1, RKELT, RKL0T, RKQLT, RLEI7, RLQM7, RLR07, RLRE7, RLSSZ, RLSYT, RLZE7, RLZI7, RLZLZ, RM0G7, RM0Q7, RMIWN, RMKO1, RMLQ7, RMOX1, RMR07, RMRR1, RN2YN, RN30D, RND3D, RNI2N, RNIMN, RNMEN, RNW2D, RO0TT, RO66T, RO6KT, ROAAV, ROOKT, ROOOT, ROU9V, ROXM1, ROZK1, RQK0T, RQLKT, RQLST, RQMR7, RQQLT, RQQQT, RQQWD, RQRE7, RQTQD, RR44V, RR4YV, RR601, RR66V, RR6WV, RR9HH, RR9UV, RRA9V, RRAYV, RRE01, RREE1, RRGM7, RRGR7, RRI2N, RRIIN, RRIN7, RRIXN, RRKE1, RRL07, RRLM7, RRM07, RRMG7, RRML7, RRMM1, RRMO1, RRMWN, RRNR7, RRO01, RROI1, RROK1, RRR0N, RRRE1, RRRG7, RRRIN, RRRL7, RRRM1, RRRO1, RRROV, RRRWV, RRU6V, RRV9H, RRVUV, RRVYH, RRWWV, RRXRN, RRY4V, RRY9H, RRYHH, RRYIN, RRYRN, RS8FZ, RSSSZ, RSSYT, RSYST, RT3ZZ, 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W0M0000D, W0W9000V, W404000H, W50000MN, W5500055, W55KFFFP, W700000N, WCWCCCQD, WFFFF0FP, WHEEEEEP, WI0000RV, WIIWIIIN, WKFFFF0P, WMCCCCCD, WMW0000D, WN000WA5, WOOOOOO7, WV0EEELP, WVEEEEEP, WVG000O5, WW00WO0V, WWCCCCQD, WWW0W00V, WWWW90VV, WWWWQQM7, WWWWWWM7, X00000AB, X000099H, X0000B07, X0000BL7, X0000IIN, X000A0YH, X000BRL7, X000N0O7, X000X3LB, X009999H, X00IIYIN, X00IQQQN, X00K0KRH, X00KKKYH, X00LX0XB, X00OOOO7, X00XLX0B, X00YYYBB, X0GOOOO7, X0R00A0H, X0XB00LB, X0Y000BB, X0YIIIIN, X3KKKKKH, X8XXXEE1, XBWOOOO7, XGWWOOO7, XIJSSSSJ, XJXXXA01, XMRRRRR7, XNIIIIIN, XO7OOOO7, XOOWOOO7, XQXXJXXJ, XRMRRRR7, XSGSSSSJ, XSSSS9SJ, XV00X00B, XX00000N, XX0000AB, XX0B00LB, XX0XX0AB, XXIIIIIN, XXJJIIIN, XXJXXXRN, XXQNNNNN, XXRIIIIN, XXSSSGGJ, XXXJ000B, XXXMLLIN, XXXX0XAB, XXXXBJCJ, XXXXE0K1, XXXXIIIN, XXXXXIXN, XXXXXMAB, XXXXXNIN, XXXXXXXN, XXXXYMM1, XXXXYMR1, XYEXXXXN, XYQQQRXN, XYY0000H, XYYK000H, XYYYY3BB, Y00000VV, Y00004K1, Y00004XV, Y00004YB, Y000088H, Y0000B3B, Y0000K41, Y0000RO1, Y0000UGJ, 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JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJ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====Additional known quasi-minimal primes (not necessarily the next)==== P₈₁₉₉₃SZ ==Unsolved families== Families for which not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers > base (''b'')). {|class=wikitable |base (''b'')||unsolved family (base-''b'' form)||unsolved family (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||current search limit of length||factorization of numbers in this family |- |13||9{5}||(113×13^''n''−5)/12||88000||[http://factordb.com/index.php?query=%28113*13%5En-5%29%2F12&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |13||A{3}A||(41×13^''n''+1+27)/4||82000||[http://factordb.com/index.php?query=%2841*13%5E%28n%2B1%29%2B27%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |16||{3}AF||(16^''n''+2+619)/5||76000||[http://factordb.com/index.php?query=%2816%5E%28n%2B2%29%2B619%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |} (If these three families contain primes (and they are excepted to contain primes), then the smallest prime in families 9{5} and A{3}A in base ''b'' = 13 will be index 3196 and 3197 quasi-minimal prime in base ''b'' = 13, and the smallest prime in families {3}AF in base ''b'' = 16 will be index 2347 quasi-minimal prime in base ''b'' = 16) === Base 17 === * 15{0}D * 1{7} * 1F{0}7 * 4{7}A * 51{0}D * 70F{0}D * 8{B}9 * 9{5}9 * 95{F} * A{D}F * B{0}B3 * B{0}DB * {B}2BE * {B}2E * {B}E9 * {B}EE * D0G{D} * E9{B} * F1{9} * FD0{D} * G{7}F === Base 21 === * 2{7}9D * 2F{C}A * 4{3}B * 5{0}DJ * {5}FEK * {7}ID * 99{0}99H * {9}0D * {9}D * B0{H}6H * B3{0}EB * B9{0}E5 * B{D}B * B{H}6H * DH{D} * F{9}D * {F}35 * G{0}FK * H{0}7771 * H{D} * {J}6J === Base 36 === * 7{K}Z * B{0}EUV * HM{0}N * N{0}YYN * O{L}Z * S{0}8H ==Primality certificates for the proven primes > 10²⁹⁹== See also: [[w:Primality certificate|Primality certificate]] and [[w:Elliptic curve primality|Elliptic curve primality]] {|class=wikitable |base (''b'')||index of this quasi-minimal prime in base ''b''||quasi-minimal prime (base-''b'' form)||quasi-minimal prime (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||factordb entry of this prime||primality certificate of this prime |- |9||149||76₃₂₉2||(31×9³³⁰−19)/4||[http://factordb.com/index.php?id=1100000002359003642]||[http://factordb.com/cert.php?id=1100000002359003642] |- |9||150||27₆₈₆07||(23×9⁶⁸⁸−511)/8||[http://factordb.com/index.php?id=1100000002495467486]||[http://factordb.com/cert.php?id=1100000002495467486] |- |9||151||30₁₁₅₈11||3×9¹¹⁶⁰+10||[http://factordb.com/index.php?id=1100000002376318423]||[http://factordb.com/cert.php?id=1100000002376318423] |- |11||1065||A₇₁₃58||11⁷¹⁵−58||[http://factordb.com/index.php?id=1100000003576826487]||[http://factordb.com/cert.php?id=1100000003576826487] |- |11||1066||7₇₅₉44||(7×11⁷⁶¹−367)/10||[http://factordb.com/index.php?id=1100000002505568840]||[http://factordb.com/cert.php?id=1100000002505568840] |- |11||1067||557₁₀₁₁||(607×11¹⁰¹¹−7)/10||[http://factordb.com/index.php?id=1100000002361376522]||[http://factordb.com/cert.php?id=1100000002361376522] |- |13||3165||50₂₇₀44||5×13²⁷²+56||[http://factordb.com/index.php?id=1100000002632397005]||[http://factordb.com/cert.php?id=1100000002632397005] |- |13||3166||9₂₇₁095||(3×13²⁷⁴−6103)/4||[http://factordb.com/index.php?id=1100000003590431654]||[http://factordb.com/cert.php?id=1100000003590431654] |- |13||3167||10₂₈₆7771||13²⁹⁰+16654||[http://factordb.com/index.php?id=1100000003590431633]||[http://factordb.com/cert.php?id=1100000003590431633] |- |13||3168||9₃₀₈1||(3×13³⁰⁹−35)/4||[http://factordb.com/index.php?id=1100000000840126705]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=308&c0=-&EN= 13³⁰⁸−1] |- |13||3169||B₃₄₁C4||(11×13³⁴³+61)/12||[http://factordb.com/index.php?id=1100000003590431618]||[http://factordb.com/cert.php?id=1100000003590431618] |- |13||3170||8B₃₄₃||(107×13³⁴³−11)/12||[http://factordb.com/index.php?id=1100000002321018736]||[http://factordb.com/cert.php?id=1100000002321018736] |- |13||3171||710₃₇₁111||92×13³⁷⁴+183||[http://factordb.com/index.php?id=1100000003590431609]||[http://factordb.com/cert.php?id=1100000003590431609] |- |13||3172||75₃₇₅7||(89×13³⁷⁶+19)/12||[http://factordb.com/index.php?id=1100000003590431596]||[http://factordb.com/cert.php?id=1100000003590431596] |- |13||3173||9B0₃₉₁9||128×13³⁹²+9||[http://factordb.com/index.php?id=1100000002632396790]||[http://factordb.com/cert.php?id=1100000002632396790] |- |13||3174||7B0B₃₉₇||(15923×13³⁹⁷−11)/12||[http://factordb.com/index.php?id=1100000003590431574]||[http://factordb.com/cert.php?id=1100000003590431574] |- |13||3175||10₄₁₄93||13⁴¹⁶+120||[http://factordb.com/index.php?id=1100000002523249240]||[http://factordb.com/cert.php?id=1100000002523249240] |- |13||3176||81010₄₁₅1||17746×13⁴¹⁶+1||[http://factordb.com/index.php?id=1100000003590431555]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3177||8110₄₃₅1||1366×13⁴³⁶+1||[http://factordb.com/index.php?id=1100000002373259109]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3178||B7₄₈₆||(139×13⁴⁸⁶−7)/12||[http://factordb.com/index.php?id=1100000002321015892]||[http://factordb.com/cert.php?id=1100000002321015892] |- |13||3179||B₅₆₃C||(11×13⁵⁶⁴+1)/12||[http://factordb.com/index.php?id=1100000000000217927]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=564&c0=-&EN= 13⁵⁶⁴−1] |- |13||3180||1B₅₇₆||(23×13⁵⁷⁶−11)/12||[http://factordb.com/index.php?id=1100000002321021456]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=576&c0=-&EN= 13⁵⁷⁶−1] |- |13||3181||80₆₉₃87||8×13⁶⁹⁵+111||[http://factordb.com/index.php?id=1100000002615636527]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000002615636532], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000002615636532] |- |13||3182||CC5₇₁₃||(2021×13⁷¹³−5)/12||[http://factordb.com/index.php?id=1100000002615627353]||[http://factordb.com/cert.php?id=1100000002615627353] |- |13||3183||B₈₃₄74||(11×13⁸³⁶−719)/12||[http://factordb.com/index.php?id=1100000003590430871]||[http://factordb.com/cert.php?id=1100000003590430871] |- |13||3184||9₉₆₈B||(3×13⁹⁶⁹+5)/4||[http://factordb.com/index.php?id=1100000000258566244]||[http://factordb.com/cert.php?id=1100000000258566244] |- |13||3185||10₁₂₉₅181||13¹²⁹⁸+274||[http://factordb.com/index.php?id=1100000002615445013]||[http://factordb.com/cert.php?id=1100000002615445013] |- |13||3186||9₁₃₆₂5||(3×13¹³⁶³−19)/4||[http://factordb.com/index.php?id=1100000002321017776]||[http://factordb.com/cert.php?id=1100000002321017776] |- |13||3187||7₁₅₀₄1||(7×13¹⁵⁰⁵−79)/12||[http://factordb.com/index.php?id=1100000002320890755]||[http://factordb.com/cert.php?id=1100000002320890755] |- |13||3188||930₁₅₅₁1||120×13¹⁵⁵²+1||[http://factordb.com/index.php?id=1100000000765961452]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3189||720₂₂₉₇2||93×13²²⁹⁸+2||[http://factordb.com/index.php?id=1100000002632396910]||[http://factordb.com/cert.php?id=1100000002632396910] |- |13||3190||1770₂₇₀₃17||267×13²⁷⁰⁵+20||[http://factordb.com/index.php?id=1100000003590430825]||[http://factordb.com/cert.php?id=1100000003590430825] |- |13||3191||390₆₂₆₆1||48×13⁶²⁶⁷+1||[http://factordb.com/index.php?id=1100000000765961441]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3192||B0₆₅₄₀BBA||11×13⁶⁵⁴³+2012||[http://factordb.com/index.php?id=1100000002616382906]||[http://factordb.com/cert.php?id=1100000002616382906] |- |13||3193||C₁₀₆₃₁92||13¹⁰⁶³³−50||[http://factordb.com/index.php?id=1100000003590493750]||[http://factordb.com/cert.php?id=1100000003590493750] |- |14||649||34D₇₀₈||47×14⁷⁰⁸−1||[http://factordb.com/index.php?id=1100000001540144903]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |14||650||4D₁₉₆₉₈||5×14¹⁹⁶⁹⁸−1||[http://factordb.com/index.php?id=1100000000884560233]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |16||2328||880₂₄₆7||136×16²⁴⁷+7||[http://factordb.com/index.php?id=1100000002468140199]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, and this prime factor is < 10²⁹⁹ |- |16||2329||D4₂₆₃D||(199×16²⁶⁴+131)/15||[http://factordb.com/index.php?id=1100000002468170238]||[http://factordb.com/cert.php?id=1100000002468170238] |- |16||2330||E0₂₆₁4DD||14×16²⁶⁴+1245||[http://factordb.com/index.php?id=1100000003588388352]||[http://factordb.com/cert.php?id=1100000003588388352] |- |16||2331||8C0₂₉₀ED||140×16²⁹²+237||[http://factordb.com/index.php?id=1100000003588388307]||[http://factordb.com/cert.php?id=1100000003588388307] |- |16||2332||DA₃₀₅5||(41×16³⁰⁶−17)/3||[http://factordb.com/index.php?id=1100000003588388284]||[http://factordb.com/cert.php?id=1100000003588388284] |- |16||2333||CE80₄₂₂D||3304×16⁴²³+13||[http://factordb.com/index.php?id=1100000003588388257]||[http://factordb.com/cert.php?id=1100000003588388257] |- |16||2334||5F₅₄₄6F||6×16⁵⁴⁶−145||[http://factordb.com/index.php?id=1100000002604723967]||[http://factordb.com/cert.php?id=1100000002604723967] |- |16||2335||88F₅₄₅||137×16⁵⁴⁵−1||[http://factordb.com/index.php?id=1100000000413679658]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |16||2336||BE0₇₉₂BB||190×16⁷⁹⁴+187||[http://factordb.com/index.php?id=1100000003588387938]||[http://factordb.com/cert.php?id=1100000003588387938] |- |16||2337||D9₁₀₅₂||(68×16¹⁰⁵²−3)/5||[http://factordb.com/index.php?id=1100000002321036020]||[http://factordb.com/cert.php?id=1100000002321036020] |- |16||2338||FAF₁₀₆₂45||251×16¹⁰⁶⁴−187||[http://factordb.com/index.php?id=1100000003588387610]||[http://factordb.com/cert.php?id=1100000003588387610] |- |16||2339||F8₁₅₁₇F||(233×16¹⁵¹⁸+97)/15||[http://factordb.com/index.php?id=1100000000633744824]||[http://factordb.com/cert.php?id=1100000000633744824] |- |16||2340||20₁₇₁₃321||2×16¹⁷¹⁶+801||[http://factordb.com/index.php?id=1100000003588386735]||[http://factordb.com/cert.php?id=1100000003588386735] |- |16||2341||300F₁₉₆₀AF||769×16¹⁹⁶²−81||[http://factordb.com/index.php?id=1100000003588368750]||[http://factordb.com/cert.php?id=1100000003588368750] |- |16||2342||90₃₅₄₂91||9×16³⁵⁴⁴+145||[http://factordb.com/index.php?id=1100000000633424191]||[http://factordb.com/cert.php?id=1100000000633424191] |- |16||2343||5BC₃₇₀₀D||(459×16³⁷⁰¹+1)/5||[http://factordb.com/index.php?id=1100000000993764322]||[http://factordb.com/cert.php?id=1100000000993764322] |- |16||2344||D0B₁₇₈₀₄||(3131×16¹⁷⁸⁰⁴−11)/15||[http://factordb.com/index.php?id=1100000003589278511]||[http://factordb.com/cert.php?id=1100000003589278511] |- |18||547||80₂₉₈B||8×18²⁹⁹+11||[http://factordb.com/index.php?id=1100000002355574745]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has sum-of-two-cubes algebraic factorization, 6×18⁹⁹+1 is an algebraic factor of ''N''+1, factordb entry of 6×18⁹⁹+1 is [http://factordb.com/index.php?id=1100000000900149167] |- |18||548||H₇₆₆FH||18⁷⁶⁸−37||[http://factordb.com/index.php?id=1100000003590430490]||[http://factordb.com/cert.php?id=1100000003590430490] |- |18||549||C0₆₂₆₈C5||12×18⁶²⁷⁰+221||[http://factordb.com/index.php?id=1100000003590442437]||[http://factordb.com/cert.php?id=1100000003590442437] |- |20||3301||H₂₄₇A0H||(17×20²⁵⁰−59677)/19||[http://factordb.com/index.php?id=1100000003590502619]||[http://factordb.com/cert.php?id=1100000003590502619] |- |20||3302||7₂₄₉A7||(7×20²⁵¹+1133)/19||[http://factordb.com/index.php?id=1100000003590502602]||[http://factordb.com/cert.php?id=1100000003590502602] |- |20||3303||J7₂₇₀||(368×20²⁷⁰−7)/19||[http://factordb.com/index.php?id=1100000002325395462]||[http://factordb.com/cert.php?id=1100000002325395462] |- |20||3304||J₃₃₀CCC7||20³³⁴−58953||[http://factordb.com/index.php?id=1100000003590502572]||[http://factordb.com/cert.php?id=1100000003590502572] |- |20||3305||40₃₈₇404B||4×20³⁹¹+32091||[http://factordb.com/index.php?id=1100000003590502563]||[http://factordb.com/cert.php?id=1100000003590502563] |- |20||3306||EC0₄₂₉7||292×20⁴³⁰+7||[http://factordb.com/index.php?id=1100000002633348702]||[http://factordb.com/cert.php?id=1100000002633348702] |- |20||3307||G₄₄₇99||(16×20⁴⁴⁹−2809)/19||[http://factordb.com/index.php?id=1100000000840126753]||[http://factordb.com/cert.php?id=1100000000840126753] |- |20||3308||3A₅₂₇3||(67×20⁵²⁸−143)/19||[http://factordb.com/index.php?id=1100000003590502531]||[http://factordb.com/cert.php?id=1100000003590502531] |- |20||3309||E₅₆₆C7||(14×20⁵⁶⁸−907)/19||[http://factordb.com/index.php?id=1100000003590502516]||[http://factordb.com/cert.php?id=1100000003590502516] |- |20||3310||JCJ₆₂₉||393×20⁶²⁹−1||[http://factordb.com/index.php?id=1100000001559454258]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |20||3311||J₆₅₅05J||20⁶⁵⁸−7881||[http://factordb.com/index.php?id=1100000003590502490]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000003591067052], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000003591067052] |- |20||3312||50₁₁₆₃AJ||5×20¹¹⁶⁵+219||[http://factordb.com/index.php?id=1100000003590502412]||[http://factordb.com/cert.php?id=1100000003590502412] |- |20||3313||CD₂₄₄₉||(241×20²⁴⁴⁹−13)/19||[http://factordb.com/index.php?id=1100000002325393915]||[http://factordb.com/cert.php?id=1100000002325393915] |- |20||3314||G0₆₂₆₉D||16×20⁶²⁷⁰+13||[http://factordb.com/index.php?id=1100000003590539457]||[http://factordb.com/cert.php?id=1100000003590539457] |- |22||7984||I7G0₂₅₄H||8882×22²⁵⁵+17||[http://factordb.com/index.php?id=1100000003591372788]||[http://factordb.com/cert.php?id=1100000003591372788] |- |22||7985||D0₂₅₅5EEF||13×22²⁵⁹+60339||[http://factordb.com/index.php?id=1100000003591371932]||[http://factordb.com/cert.php?id=1100000003591371932] |- |22||7986||IK₃₂₂F||(398×22³²³−125)/21||[http://factordb.com/index.php?id=1100000000840384145]||[http://factordb.com/cert.php?id=1100000000840384145] |- |22||7987||C0₃₄₀G9||12×22³⁴²+361||[http://factordb.com/index.php?id=1100000000840384159]||[http://factordb.com/cert.php?id=1100000000840384159] |- |22||7988||77E₃₄₈K7||(485×22³⁵⁰+373)/3||[http://factordb.com/index.php?id=1100000003591369779]||[http://factordb.com/cert.php?id=1100000003591369779] |- |22||7989||J₃₇₉KJ||(19×22³⁸¹+443)/21||[http://factordb.com/index.php?id=1100000003591369027]||[http://factordb.com/cert.php?id=1100000003591369027] |- |22||7990||J₃₈₈EJ||(19×22³⁹⁰−2329)/21||[http://factordb.com/index.php?id=1100000003591367729]||[http://factordb.com/cert.php?id=1100000003591367729] |- |22||7991||DJ₄₀₀||(292×22⁴⁰⁰−19)/21||[http://factordb.com/index.php?id=1100000002325880110]||[http://factordb.com/cert.php?id=1100000002325880110] |- |22||7992||E₄₀₄K7||(2×22⁴⁰⁶+373)/3||[http://factordb.com/index.php?id=1100000003591366298]||[http://factordb.com/cert.php?id=1100000003591366298] |- |22||7993||66F₄₅₃B3||(971×22⁴⁵⁵−705)/7||[http://factordb.com/index.php?id=1100000003591365809]||[http://factordb.com/cert.php?id=1100000003591365809] |- |22||7994||L0₄₅₄B63||21×22⁴⁵⁷+5459||[http://factordb.com/index.php?id=1100000003591365331]||[http://factordb.com/cert.php?id=1100000003591365331] |- |22||7995||L₄₈₃G3||22⁴⁸⁵−129||[http://factordb.com/index.php?id=1100000003591364730]||[http://factordb.com/cert.php?id=1100000003591364730] |- |22||7996||E60₄₉₆L||314×22⁴⁹⁷+21||[http://factordb.com/index.php?id=1100000000632703239]||[http://factordb.com/cert.php?id=1100000000632703239] |- |22||7997||I₆₂₆AF||(6×22⁶²⁸−1259)/7||[http://factordb.com/index.php?id=1100000000632724334]||[http://factordb.com/cert.php?id=1100000000632724334] |- |22||7998||K0₇₆₀EC1||20×22⁷⁶³+7041||[http://factordb.com/index.php?id=1100000000632724415]||[http://factordb.com/cert.php?id=1100000000632724415] |- |22||7999||J0₇₆₇IGGJ||19×22⁷⁷¹+199779||[http://factordb.com/index.php?id=1100000003591362567]||[http://factordb.com/cert.php?id=1100000003591362567] |- |22||8000||7₉₅₉K7||(22⁹⁶¹+857)/3||[http://factordb.com/index.php?id=1100000003591361817]||[http://factordb.com/cert.php?id=1100000003591361817] |- |22||8001||L₂₃₈₅KE7||22²³⁸⁸−653||[http://factordb.com/index.php?id=1100000003591360774]||[http://factordb.com/cert.php?id=1100000003591360774] |- |22||8002||7₃₈₁₅2L||(22³⁸¹⁷−289)/3||[http://factordb.com/index.php?id=1100000003591359839]||[http://factordb.com/cert.php?id=1100000003591359839] |- |24||3400||I0₂₄₁I5||18×24²⁴³+437||[http://factordb.com/index.php?id=1100000002633360037]||[http://factordb.com/cert.php?id=1100000002633360037] |- |24||3401||D0₂₅₉KKD||13×24²⁶²+12013||[http://factordb.com/index.php?id=1100000003593270725]||[http://factordb.com/cert.php?id=1100000003593270725] |- |24||3402||C7₂₉₈||(283×24²⁹⁸−7)/23||[http://factordb.com/index.php?id=1100000002326181235]||[http://factordb.com/cert.php?id=1100000002326181235] |- |24||3403||20₃₁₃7||2×24³¹⁴+7||[http://factordb.com/index.php?id=1100000002355610241]||[http://factordb.com/cert.php?id=1100000002355610241] |- |24||3404||BC0₃₃₁B||276×24³³²+11||[http://factordb.com/index.php?id=1100000002633359842]||[http://factordb.com/cert.php?id=1100000002633359842] |- |24||3405||N₂₆₄₄LLN||24²⁶⁴⁷−1201||[http://factordb.com/index.php?id=1100000003593270089]||[http://factordb.com/cert.php?id=1100000003593270089] |- |24||3406||D₂₆₉₈LD||(13×24²⁷⁰⁰+4403)/23||[http://factordb.com/index.php?id=1100000003593269876]||[http://factordb.com/cert.php?id=1100000003593269876] |- |24||3407||A0₂₉₅₁8ID||10×24²⁹⁵⁴+5053||[http://factordb.com/index.php?id=1100000003593269654]||[http://factordb.com/cert.php?id=1100000003593269654] |- |24||3408||88N₅₉₅₁||201×24⁵⁹⁵¹−1||[http://factordb.com/index.php?id=1100000003593275880]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |24||3409||N00N₈₁₂₉LN||13249×24⁸¹³¹−49||[http://factordb.com/index.php?id=1100000003593391606]||[http://factordb.com/cert.php?id=1100000003593391606] |- |30||2613||AN₂₀₆||(313×30²⁰⁶−23)/29||[http://factordb.com/index.php?id=1100000002327651073]||[http://factordb.com/cert.php?id=1100000002327651073] |- |30||2614||M₂₄₁QB||(22×30²⁴³+3139)/29||[http://factordb.com/index.php?id=1100000003593408295]||[http://factordb.com/cert.php?id=1100000003593408295] |- |30||2615||M0₅₄₇SS7||22×30⁵⁵⁰+26047||[http://factordb.com/index.php?id=1100000003593407988]||[http://factordb.com/cert.php?id=1100000003593407988] |- |30||2616||C0₁₀₂₂1||12×30¹⁰²³+1||[http://factordb.com/index.php?id=1100000000785448736]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |30||2617||5₄₈₈₂J||(5×30⁴⁸⁸³+401)/29||[http://factordb.com/index.php?id=1100000002327649423]||[http://factordb.com/cert.php?id=1100000002327649423] |- |30||2619||OT₃₄₂₀₅||25×30³⁴²⁰⁵−1||[http://factordb.com/index.php?id=1100000000800812865]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |} ==Unproven PRPs== {|class=wikitable |base (''b'')||index of this quasi-minimal prime in base ''b'' (assuming the primality of all PRP in base ''b'')||unproven PRP (base-''b'' form)||unproven PRP (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||factordb entry of this PRP |- |11||1068||57₆₂₆₆₈||(57×11⁶²⁶⁶⁸−7)/10||[http://factordb.com/index.php?id=1100000003573679860] |- |13||3194||C5₂₃₇₅₅C||(149×13²³⁷⁵⁶+79)/12||[http://factordb.com/index.php?id=1100000003590647776] |- |13||3195||80₃₂₀₁₇111||8×13³²⁰²⁰+183||[http://factordb.com/index.php?id=1100000000490878060] |- |16||2345||DB₃₂₂₃₄||(206×16³²²³⁴−11)/15||[http://factordb.com/index.php?id=1100000002383583629] |- |16||2346||4₇₂₇₈₅DD||(4×16⁷²⁷⁸⁷+2291)/15||[http://factordb.com/index.php?id=1100000003615909841] |- |22||8003||BK₂₂₀₀₁5||(251×22²²⁰⁰²−335)/21||[http://factordb.com/index.php?id=1100000003594696838] |- |30||2618||I0₂₄₆₀₈D||18×30²⁴⁶⁰⁹+13||[http://factordb.com/index.php?id=1100000003593967511] |} All these PRPs pass the [[w:Miller–Rabin primality test|Miller–Rabin primality test]] to bases 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59 and 61, and pass the [[w:Lucas pseudoprime#Strong Lucas pseudoprimes|strong Lucas primality test]] with parameters (''P'', ''Q'') defined by Selfridge's Method ''A'', and [[w:Trial division|trial factored]] to 10¹⁶. (Thus, they pass the [[w:Baillie–PSW primality test|Baillie–PSW primality test]]) ==Proof== ===Base 2=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. ===Base 3=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (2,1), (2,2) * Case (1,1): ** Since 12, 21, 111 are primes, we only need to consider the family 1{0}1 (since any digits 1, 2 between them will produce smaller primes) *** All numbers of the form 1{0}1 are divisible by 2, thus cannot be prime. * Case (1,2): ** 12 is prime, and thus the only minimal prime in this family. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,2): ** Since 21, 12 are primes, we only need to consider the family 2{0,2}2 (since any digits 1 between them will produce smaller primes) *** All numbers of the form 2{0,2}2 are divisible by 2, thus cannot be prime. ===Base 4=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (2,1), (2,3), (3,1), (3,3) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 11, 31, 221 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3 between them will produce smaller primes) *** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 31, 13, 23 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2 between them will produce smaller primes) *** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime. ===Base 5=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (2,4), (3,1), (3,2), (3,3), (3,4), (4,1), (4,2), (4,3), (4,4) * Case (1,1): ** Since 12, 21, 111, 131 are primes, we only need to consider the family 1{0,4}1 (since any digits 1, 2, 3 between them will produce smaller primes) *** All numbers of the form 1{0,4}1 are divisible by 2, thus cannot be prime. * Case (1,2): ** 12 is prime, and thus the only minimal prime in this family. * Case (1,3): ** Since 12, 23, 43, 133 are primes, we only need to consider the family 1{0,1}3 (since any digits 2, 3, 4 between them will produce smaller primes) *** Since 111 is prime, we only need to consider the families 1{0}3 and 1{0}1{0}3 (since any digit combo 11 between (1,3) will produce smaller primes) **** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime. **** For the 1{0}1{0}3 family, since 10103 is prime, we only need to consider the families 1{0}13 and 11{0}3 (since any digit combo 010 between (1,3) will produce smaller primes) ***** The smallest prime of the form 1{0}13 is 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013, which can be written as 1(0^93)13 and equal the prime 5^95+8 ([http://factordb.com/index.php?id=1100000000034686071 factordb]) ***** All numbers of the form 11{0}3 are divisible by 3, thus cannot be prime. * Case (1,4): ** Since 12, 34, 104 are primes, we only need to consider the family 1{1,4}4 (since any digits 0, 2, 3 between them will produce smaller primes) *** Since 111, 414 are primes, we only need to consider the families 1{4}4 and 11{4}4 (since any digit combo 11 or 41 between them will produce smaller primes) **** The smallest prime of the form 1{4}4 is 14444. **** All numbers of the form 11{4}4 are divisible by 2, thus cannot be prime. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,2): ** Since 21, 23, 12, 32 are primes, we only need to consider the family 2{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 2{0,2,4}2 are divisible by 2, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,4): ** Since 21, 23, 34 are primes, we only need to consider the family 2{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 2{0,2,4}4 are divisible by 2, thus cannot be prime. * Case (3,1): ** Since 32, 34, 21 are primes, we only need to consider the family 3{0,1,3}1 (since any digits 2, 4 between them will produce smaller primes) *** Since 313, 111, 131, 3101 are primes, we only need to consider the families 3{0,3}1 and 3{0,3}11 (since any digit combo 10, 11, 13 between (3,1) will produce smaller primes) **** For the 3{0,3}1 family, we can separate this family to four families: ***** For the 30{0,3}01 family, we have the prime 30301, and the remain case is the family 30{0}01. ****** All numbers of the form 30{0}01 are divisible by 2, thus cannot be prime. ***** For the 30{0,3}31 family, note that there must be an even number of 3's between (30,31), or the result number will be divisible by 2 and cannot be prime. ****** Since 33331 is prime, any digit combo 33 between (30,31) will produce smaller primes. ******* Thus, the only possible prime is the smallest prime in the family 30{0}31, and this prime is 300031. ***** For the 33{0,3}01 family, note that there must be an even number of 3's between (33,01), or the result number will be divisible by 2 and cannot be prime. ****** Since 33331 is prime, any digit combo 33 between (33,01) will produce smaller primes. ******* Thus, the only possible prime is the smallest prime in the family 33{0}01, and this prime is 33001. ***** For the 33{0,3}31 family, we have the prime 33331, and the remain case is the family 33{0}31. ****** All numbers of the form 33{0}31 are divisible by 2, thus cannot be prime. **** All numbers of the form 3{0,3}11 are divisible by 3, thus cannot be prime. * Case (3,2): ** 32 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 32, 34, 23, 43, 313 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2, 4 between them will produce smaller primes) *** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime. * Case (3,4): ** 34 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 43, 21, 401 are primes, we only need to consider the family 4{1,4}1 (since any digits 0, 2, 3 between them will produce smaller primes) *** Since 414, 111 are primes, we only need to consider the families 4{4}1 and 4{4}11 (since any digit combo 14 or 11 between them will produce smaller primes) **** The smallest prime of the form 4{4}1 is 44441. **** All numbers of the form 4{4}11 are divisible by 2, thus cannot be prime. * Case (4,2): ** Since 43, 12, 32 are primes, we only need to consider the family 4{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 4{0,2,4}2 are divisible by 2, thus cannot be prime. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,4): ** Since 43, 34, 414 are primes, we only need to consider the family 4{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 4{0,2,4}4 are divisible by 2, thus cannot be prime. ===Base 6=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,5), (2,1), (2,5), (3,1), (3,5), (4,1), (4,5), (5,1), (5,5) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 11, 21, 31, 51 are primes, we only need to consider the family 4{0,4}1 (since any digits 1, 2, 3, 5 between them will produce smaller primes) *** Since 4401 and 4441 are primes, we only need to consider the families 4{0}1 and 4{0}41 (since any digits combo 40 and 44 between them will produce smaller primes) **** All numbers of the form 4{0}1 are divisible by 5, thus cannot be prime. **** The smallest prime of the form 4{0}41 is 40041 * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 15, 25, 35, 45 are primes, we only need to consider the family 5{0,5}5 (since any digits 1, 2, 3, 4 between them will produce smaller primes) *** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. ===Base 7=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6) * Case (1,1): ** Since 14, 16, 41, 61, 131 are primes, we only need to consider the family 1{0,1,2,5}1 (since any digits 3, 4, 6 between them will produce smaller primes) *** Since the digit sum of primes must be odd (otherwise the number will be divisible by 2, thus cannot be prime), there is an odd total number of 1 and 5 in the {} **** If there are >=3 number of 1 and 5 in the {}: ***** If there is 111 in the {}, then we have the prime 11111 ***** If there is 115 in the {}, then the prime 115 is a subsequence ***** If there is 151 in the {}, then the prime 115 is a subsequence ***** If there is 155 in the {}, then the prime 155 is a subsequence ***** If there is 511 in the {}, then the current number is 15111, which has digit sum = 12, but digit sum divisible by 3 will cause the number divisible by 3 and cannot be prime, and we cannot add more 1 or 5 to this number (to avoid 11111, 155, 515, 551 as subsequence), thus we must add at least one 2 to this number, but then the number has both 2 and 5, and will have either 25 or 52 as subsequence, thus cannot be minimal prime ***** If there is 515 in the {}, then the prime 515 is a subsequence ***** If there is 551 in the {}, then the prime 551 is a subsequence ***** If there is 555 in the {}, then the prime 551 is a subsequence **** Thus there is only one 1 (and no 5) or only one 5 (and no 1) in the {}, i.e. we only need to consider the families 1{0,2}1{0,2}1 and 1{0,2}5{0,2}1 ***** For the 1{0,2}1{0,2}1 family, since 1211 is prime, we only need to consider the family 1{0}1{0,2}1 ****** Since all numbers of the form 1{0}1{0}1 are divisible by 3 and cannot be prime, we only need to consider the family 1{0}1{0}2{0}1 ******* Since 11201 is prime, we only need to consider the family 1{0}1{0}21 ******** The smallest prime of the form 11{0}21 is 1100021 ******** All numbers of the form 101{0}21 are divisible by 5, thus cannot be prime ******** The smallest prime of the form 1001{0}21 is 100121 ********* Since this prime has no 0 between 1{0}1 and 21, we do not need to consider more families ***** For the 1{0,2}5{0,2}1 family, since 25 and 52 are primes, we only need to consider the family 1{0}5{0}1 ****** Since 1051 is prime, we only need to consider the family 15{0}1 ******* The smallest prime of the form 15{0}1 is 150001 * Case (1,2): ** Since 14, 16, 32, 52 are primes, we only need to consider the family 1{0,1,2}2 (since any digits 3, 4, 5, 6 between them will produce smaller primes) *** Since 1112 and 1222 are primes, there is at most one 1 and at most one 2 in {} **** If there are one 1 and one 2 in {}, then the digit sum is 6, and the number will be divisible by 6 and cannot be prime. **** If there is one 1 but no 2 in {}, then the digit sum is 4, and the number will be divisible by 2 and cannot be prime. **** If there is no 1 but one 2 in {}, then the form is 1{0}2{0}2 ***** Since 1022 and 1202 are primes, we only need to consider the number 122 ****** 122 is not prime. **** If there is no 1 and no 2 in {}, then the digit sum is 3, and the number will be divisible by 3 and cannot be prime. * Case (1,3): ** Since 14, 16, 23, 43, 113, 133 are primes, we only need to consider the family 1{0,5}3 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes) *** Since 155 is prime, we only need to consider the family 1{0}3 and 1{0}5{0}3 **** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime. **** All numbers of the form 1{0}5{0}3 are divisible by 3, thus cannot be prime. * Case (1,4): ** 14 is prime, and thus the only minimal prime in this family. * Case (1,5): ** Since 14, 16, 25, 65, 115, 155 are primes, we only need to consider the family 1{0,3}5 (since any digits 1, 2, 4, 5, 6 between them will produce smaller primes) *** All numbers of the form 1{0,3}5 are divisible by 3, thus cannot be prime. * Case (1,6): ** 16 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 25, 41, 61, 221 are primes, we only need to consider the family 2{0,1}1 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes) *** Since 2111 is prime, we only need to consider the families 2{0}1 and 2{0}1{0}1 **** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. **** All numbers of the form 2{0}1{0}1 are divisible by 2, thus cannot be prime. * Case (2,2): ** Since 23, 25, 32, 52, 212 are primes, we only need to consider the family 2{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,4): ** Since 23, 25, 14 are primes, we only need to consider the family 2{0,2,4,6}4 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}4 are divisible by 2, thus cannot be prime. * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (2,6): ** Since 23, 25, 16, 56 are primes, we only need to consider the family 2{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}6 are divisible by 2, thus cannot be prime. * Case (3,1): ** Since 32, 41, 61 are primes, we only need to consider the family 3{0,1,3,5}1 (since any digits 2, 4, 6 between them will produce smaller primes) *** Since 551 is prime, we only need to consider the family 3{0,1,3}1 and 3{0,1,3}5{0,1,3}1 (since any digits combo 55 between (3,1) will produce smaller primes) **** For the 3{0,1,3}1 family, since 3031 and 131 are primes, we only need to consider the families 3{0,1}1 and 3{3}3{0,1}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes, thus for the digits between (3,1), all 3's must be before all 0's and 1's, and thus we can let the red 3 in 3{3}3{0,1}1 be the rightmost 3 between (3,1), all digits before this 3 must be 3's, and all digits after this 3 must be either 0's or 1's) ***** For the 3{0,1}1 family: ****** If there are >=2 0's and >=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence. ****** If there are no 1's between (3,1), then the form will be 3{0}1 ******* All numbers of the form 3{0}1 are divisible by 2, thus cannot be prime. ****** If there are no 0's between (3,1), then the form will be 3{1}1 ******* The smallest prime of the form 3{1}1 is 31111 ****** If there are exactly 1 0's between (3,1), then there must be <3 1's between (3,1), or 31111 will be a subsequence. ******* If there are 2 1's between (3,1), then the digit sum is 6, thus the number is divisible by 6 and cannot be prime. ******* If there are 1 1's between (3,1), then the number can only be either 3101 or 3011 ******** Neither 3101 nor 3011 is prime. ******* If there are no 1's between (3,1), then the number must be 301 ******** 301 is not prime. ***** For the 3{3}3{0,1}1 family: ****** If there are at least one 3 between (3,3{0,1}1) and at least one 1 between (3{3}3,1), then 33311 will be a subsequence. ****** If there are no 3 between (3,3{0,1}1), then the form will be 33{0,1}1 ******* If there are at least 3 1's between (33,1), then 31111 will be a subsequence. ******* If there are exactly 2 1's between (33,1), then the digit sum is 12, thus the number is divisible by 3 and cannot be prime. ******* If there are exactly 1 1's between (33,1), then the digit sum is 11, thus the number is divisible by 2 and cannot be prime. ******* If there are no 1's between (33,1), then the form will be 33{0}1 ******** The smallest prime of the form 33{0}1 is 33001 ****** If there are no 1 between (3{3}3,1), then the form will be 3{3}3{0}1 ******* If there are at least 2 0's between (3{3}3,1), then 33001 will be a subsequence. ******* If there are exactly 1 0's between (3{3}3,1), then the form is 3{3}301 ******** The smallest prime of the form 3{3}301 is 33333301 ******* If there are no 0's between (3{3}3,1), then the form is 3{3}31 ******** The smallest prime of the form 3{3}31 is 33333333333333331 **** For the 3{0,1,3}5{0,1,3}1 family, since 335 is prime, we only need to consider the family 3{0,1}5{0,1,3}1 ***** Numbers containing 3 between (3{0,1}5,1): ****** The form is 3{0,1}5{0,1,3}3{0,1,3}1 ******* Since 3031 and 131 are primes, we only need to consider the family 35{3}3{0,1,3}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes) ******** Since 533 is prime, we only need to consider the family 353{0,1}1 (since any digits combo 33 between (35,1) will produce smaller primes) ********* Since 5011 is prime, we only need to consider the family 353{1}{0}1 (since any digits combo 01 between (353,1) will produce smaller primes) ********** If there are at least 3 1's between (353,{0}1), then 31111 will be a subsequence. ********** If there are exactly 2 1's between (353,{0}1), then the digit sum is 20, thus the number is divisible by 2 and cannot be prime. ********** If there are exactly 1 1's between (353,{0}1), then the form is 3531{0}1 *********** The smallest prime of the form 3531{0}1 is 3531001, but it is not minimal prime since 31001 is prime. ********** If there are no 1's between (353,{0}1), then the digit sum is 15, thus the number is divisible by 6 and cannot be prime. ***** Numbers not containing 3 between (3{0,1}5,1): ****** The form is 3{0,1}5{0,1}1 ******* If there are >=2 0's and >=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence. ******* If there are no 1's between (3,1), then the form will be 3{0}5{0}1 ******** All numbers of the form 3{0}5{0}1 are divisible by 3, thus cannot be prime. ******* If there are no 0's between (3,1), then the form will be 3{1}5{1}1 ******** If there are >=3 1's between (3,1), then 31111 will be a subsequence. ******** If there are exactly 2 1's between (3,1), then the number can only be 31151, 31511, 35111 ********* None of 31151, 31511, 35111 are primes. ******** If there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime. ******** If there are no 1's between (3,1), then the number is 351 ********* 351 is not prime. ******* If there are exactly 1 0's between (3,1), then the form will be 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1 ******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are >=3 1's between (3,1), then 31111 will be a subsequence. ******** If there are exactly 2 1's between (3,1), then the number can only be 311051, 310151, 310511, 301151, 301511, 305111, 311501, 315101, 315011, 351101, 351011, 350111 ********* Of these numbers, 311051, 301151, 311501, 351101, 350111 are primes. ********** However, 311051, 301151, 311501 have 115 as subsequence, and 350111 has 5011 as subsequence, thus only 351101 is minimal prime. ******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime. ******** If there are no 1's between (3,1), then the number is 3051 for 3{1}0{1}5{1}1 or 3501 for 3{1}5{1}0{1}1 ********* Neither 3051 nor 3501 is prime. * Case (3,2): ** 32 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 32, 23, 43, 313 are primes, we only need to consider the family 3{0,3,5,6}3 (since any digits 1, 2, 4 between them will produce smaller primes) *** If there are >=2 5's in {}, then 553 will be a subsequence. *** If there are no 5's in {}, then the family will be 3{0,3,6}3 **** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime. *** If there are exactly 1 5's in {}, then the family will be 3{0,3,6}5{0,3,6}3 **** Since 335, 65, 3503, 533, 56 are primes, we only need to consider the family 3{0}53 (since any digit 3, 6 between (3,5{0,3,6}3) and any digit 0, 3, 6 between (3{0,3,6}5,3) will produce smaller primes) ***** The smallest prime of the form 3{0}53 is 300053 * Case (3,4): ** Since 32, 14, 304, 344, 364 are primes, we only need to consider the family 3{3,5}4 (since any digits 0, 1, 2, 4, 6 between them will produce smaller primes) *** Since 3334 and 335 are primes, we only need to consider the family 3{5}4 and 3{5}34 (since any digits combo 33, 35 between them will produce smaller primes) **** The smallest prime of the form 3{5}4 is 35555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555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with 9234 5's, which can be written as 3(5^9234)4 and equal the prime (23*7^9235-11)/6 ([http://factordb.com/index.php?id=1100000002766595757 factordb]) ([http://factordb.com/cert.php?id=1100000002766595757 primality certificate]) (not minimal prime, since 35555 and 5554 are primes) **** The smallest prime of the form 3{5}34 is 355555555555555555555555555555555555555555555555555555555555555534 (not minimal prime, since 35555, 553, and 5554 are primes) * Case (3,5): ** Since 32, 25, 65, 335 are primes, we only need to consider the family 3{0,1,4,5}5 (since any digits 2, 3, 6 between them will produce smaller primes) *** If there are at least one 1's and at least one 5's in {}, then either 155 or 515 will be a subsequence. *** If there are at least one 1's and at least one 4's in {}, then either 14 or 41 will be a subsequence. *** If there are at least two 1's in {}, then 115 will be a subsequence. *** If there are exactly one 1's and no 4's or 5's in {}, then the family will be 3{0}1{0}5 **** All numbers of the form 3{0}1{0}5 are divisible by 3, thus cannot be prime. *** If there is no 1's in {}, then the family will be 3{0,4,5}5 **** If there are at least to 4's in {}, then 344 and 445 will be subsequences. **** If there is no 4's in {}, then the family will be 3{0,5}5 ***** Since 3055 and 3505 are primes, we only need to consider the families 3{0}5 and 3{5}5 ****** All numbers of the form 3{0}5 are divisible by 2, thus cannot be prime. ****** The smallest prime of the form 3{5}5 is 35555 **** If there is exactly one 4's in {}, then the family will be 3{0,5}4{0,5}5 ***** Since 304, 3545 are primes, we only need to consider the families 34{0,5}5 (since any digits 0 or 5 between (3,4{0,5}5) will produce small primes) ****** All numbers of the form 34{0,5}5 are divisible by 5, thus cannot be prime. * Case (3,6): ** Since 32, 16, 56, 346 are primes, we only need to consider the family 3{0,3,6}6 (since any digits 1, 2, 4, 5 between them will produce smaller primes) *** All numbers of the form 3{0,3,6}6 are divisible by 3, thus cannot be prime. * Case (4,1): ** 41 is prime, and thus the only minimal prime in this family. * Case (4,2): ** Since 41, 43, 32, 52 are primes, we only need to consider the family 4{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 4{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,4): ** Since 41, 43, 14 are primes, we only need to consider the family 4{0,2,4,5,6}4 (since any digits 1, 3 between them will produce smaller primes) *** If there is no 5's in {}, then the family will be 4{0,2,4,6}4 **** All numbers of the form 4{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there is at least one 5's in {}, then there cannot be 2 in {} (since if so, then either 25 or 52 will be a subsequence) and there cannot be 6 in {} (since if so, then either 65 or 56 will be a subsequence), thus the family is 4{0,4,5}5{0,4,5}4 **** Since 445, 4504, 544 are primes, we only need to consider the family 4{0,5}5{5}4 (since any digit 4 between (4,5{0,4,5}4) and any digit 0, 4 between (4{0,4,5}5,4) will produce smaller primes) ***** If there are at least two 0's between (4,5{0,4,5}4), then 40054 will be a subsequence. ***** If there is no 0's between (4,5{0,4,5}4), then the family will be 4{5}5{5}4, which is equivalent to 4{5}4 ****** The smallest prime of the form 4{5}4 is 45555555555555554 (not minimal prime, since 4555 and 5554 are primes) ***** If there is exactly one 0's between (4,5{0,4,5}4), then the family will be 4{5}0{5}5{5}4 ****** Since 4504 is prime, we only need to consider the family 40{5}5{5}4 (since any digit 5 between (4,0{5}5{5}4) will produce small primes), which is equivalent to 40{5}4 ******* The smallest prime of the form 40{5}4 is 405555555555555554 (not minimal prime, since 4555 and 5554 are primes) * Case (4,5): ** Since 41, 43, 25, 65, 445 are primes, we only need to consider the family 4{0,5}5 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes) *** If there are at least two 5's in {}, then 4555 will be a subsequence. *** If there is exactly one 5's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime. *** If there is no 5's in {}, then the family will be 4{0}5 **** All numbers of the form 4{0}5 are divisible by 3, thus cannot be prime. * Case (4,6): ** Since 41, 43, 16, 56 are primes, we only need to consider the family 4{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 4{0,2,4,6}6 are divisible by 2, thus cannot be prime. * Case (5,1): ** Since 52, 56, 41, 61, 551 are primes, we only need to consider the family 5{0,1,3}1 (since any digits 2, 4, 5, 6 between them will produce smaller primes) *** If there are at least two 3's in {}, then 533 will be a subsequence. *** If there is no 3's in {}, then the family will be 5{0,1}1 **** Since 5011 is prime, we only need to consider the family 5{1}{0}1 ***** Since 11111 is prime, we only need to consider the families 5{0}1, 51{0}1, 511{0}1, 5111{0}1 (since any digits combo 1111 between (5,1) will produce small primes) ****** All numbers of the form 5{0}1 are divisible by 6, thus cannot be prime. ****** The smallest prime of the form 51{0}1 is 5100000001 ****** All numbers of the form 511{0}1 are divisible by 2, thus cannot be prime. ****** All numbers of the form 5111{0}1 are divisible by 3, thus cannot be prime. *** If there is exactly one 3's in {}, then the family will be 5{0,1}3{0,1}1 **** If there is at least one 1's between (5,3{0,1}1), then 131 will be a subsequence. ***** Thus we only need to consider the family 5{0}3{0,1}1 ****** If there are no 1's between (5{0}3,1), then the digit sum is 12, and the number will be divisible by 3 and cannot be prime. ****** If there are exactly one 1's between (5{0}3,1), then the digit sum is 13, and the number will be divisible by 2 and cannot be prime. ****** If there are exactly three 1's between (5{0}3,1), then the digit sum is 15, and the number will be divisible by 6 and cannot be prime. ****** If there are at least four 1's between (5{0}3,1), then 11111 will be a subsequence. ****** If there are exactly two 1's between (5{0}3,1), then the family will be 5{0}3{0}1{0}1{0}1 ******* Since 5011 is prime, we only need to consider the family 5311{0}1 (since any digit 0 between (5,1{0}1) will produce small primes, this includes the leftmost three {} in 5{0}3{0}1{0}1{0}1, and thus only the rightmost {} can contain 0) ******** The smallest prime of the form 5311{0}1 is 531101 * Case (5,2): ** 52 is prime, and thus the only minimal prime in this family. * Case (5,3): ** Since 52, 56, 23, 43, 533, 553 are primes, we only need to consider the family 5{0,1}3 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes) *** If there are at least two 1's in {}, then 113 will be a subsequence. *** If there is exactly one 1's in {}, then the digit sum is 12, and the number will be divisible by 3 and cannot be prime. *** If there is no 1's in {}, then the digit sum is 11, and the number will be divisible by 2 and cannot be prime. * Case (5,4): ** Since 52, 56, 14, 544 are primes, we only need to consider the family 5{0,3,5}4 (since any digits 1, 2, 4, 6 between them will produce smaller primes) *** If there are no 5's in {}, then the family will be 5{0,3}4 **** All numbers of the form 5{0,3}4 are divisible by 3, thus cannot be prime. *** If there are at least one 5's and at least one 3's in {}, then either 535 or 553 will be a subsequence. *** If there are exactly one 5's and no 3's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime. *** If there are at least two 5's in {}, then 5554 will be a subsequence. * Case (5,5): ** Since 52, 56, 25, 65, 515, 535 are primes, we only need to consider the family 5{0,4,5}5 (since any digits 1, 2, 3, 6 between them will produce smaller primes) *** If there are no 4's in {}, then the family will be 5{0,5}5 **** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. *** If there are no 5's in {}, then the family will be 5{0,4}5 **** All numbers of the form 5{0,4}5 are divisible by 2, thus cannot be prime. *** If there are at least one 4's and at least one 5's in {}, then either 5455 or 5545 will be a subsequence. * Case (5,6): ** 56 is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,2): ** Since 61, 65, 32, 52 are primes, we only need to consider the family 6{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 6{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (6,3): ** Since 61, 65, 23, 43 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5 between them will produce smaller primes) *** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (6,4): ** Since 61, 65, 14 are primes, we only need to consider the family 6{0,2,3,4,6}4 (since any digits 1, 5 between them will produce smaller primes) *** If there is no 3's in {}, then the family will be 6{0,2,4,6}4 **** All numbers of the form 6{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there are exactly two 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4 **** All numbers of the form 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there are at least three 3's in {}, then 3334 will be a subsequence. *** If there is exactly one 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}4 **** If there is 0 between (6,3{0,2,4,6}4), then 6034 will be a subsequence. **** If there is 2 between (6,3{0,2,4,6}4), then 23 will be a subsequence. **** If there is 4 between (6,3{0,2,4,6}4), then 43 will be a subsequence. **** If there is 6 between (6,3{0,2,4,6}4), then 6634 will be a subsequence. **** If there is 0 between (6{0,2,4,6}3,4), then 304 will be a subsequence. **** If there is 2 between (6{0,2,4,6}3,4), then 32 will be a subsequence. **** If there is 4 between (6{0,2,4,6}3,4), then 344 will be a subsequence. **** If there is 6 between (6{0,2,4,6}3,4), then 364 will be a subsequence. **** Thus the number can only be 634 ***** 634 is not prime. * Case (6,5): ** 65 is prime, and thus the only minimal prime in this family. * Case (6,6): ** Since 61, 65, 16, 56 are primes, we only need to consider the family 6{0,2,3,4,6}6 (since any digits 1, 5 between them will produce smaller primes) *** If there is no 3's in {}, then the family will be 6{0,2,4,6}6 **** All numbers of the form 6{0,2,4,6}6 are divisible by 2, thus cannot be prime. *** If there is no 2's and no 4's in {}, then the family will be 6{0,3,6}6 **** All numbers of the form 6{0,3,6}6 are divisible by 3, thus cannot be prime. *** If there is at least one 3's and at least one 2's in {}, then either 32 or 23 will be a subsequence. *** If there is at least one 3's and at least one 4's in {}, then either 346 or 43 will be a subsequence. ===Base 8=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (1,5), (1,7), (2,1), (2,3), (2,5), (2,7), (3,1), (3,3), (3,5), (3,7), (4,1), (4,3), (4,5), (4,7), (5,1), (5,3), (5,5), (5,7), (6,1), (6,3), (6,5), (6,7), (7,1), (7,3), (7,5), (7,7) * Case (1,1): ** Since 13, 15, 21, 51, 111, 141, 161 are primes, we only need to consider the family 1{0,7}1 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes) *** Since 107, 177, 701 are primes, we only need to consider the number 171 and the family 1{0}1 (since any digits combo 07, 70, 77 between them will produce smaller primes) **** 171 is not prime. **** All numbers of the form 1{0}1 factored as 10^n+1 = (2^n+1) * (4^n-2^n+1) (n≥1) (and since if n≥1, 2^n+1 ≥ 2^1+1 = 3 > 1, 4^n-2^n+1 ≥ 4^1-2^1+1 = 3 > 1, this factorization is nontrivial), thus cannot be prime. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (1,7): ** Since 13, 15, 27, 37, 57, 107, 117, 147, 177 are primes, we only need to consider the family 1{6}7 (since any digits 0, 1, 2, 3, 4, 5, 7 between them will produce smaller primes) *** The smallest prime of the form 1{6}7 is 16667 (not minimal prime, since 667 is prime) * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,5): ** Since 21, 23, 27, 15, 35, 45, 65, 75, 225, 255 are primes, we only need to consider the family 2{0}5 (since any digits 1, 2, 3, 4, 5, 6, 7 between them will produce smaller primes) *** All numbers of the form 2{0}5 are divisible by 7, thus cannot be prime. * Case (2,7): ** 27 is prime, and thus the only minimal prime in this family. * Case (3,1): ** Since 35, 37, 21, 51, 301, 361 are primes, we only need to consider the family 3{1,3,4}1 (since any digits 0, 2, 5, 6, 7 between them will produce smaller primes) *** Since 13, 343, 111, 131, 141, 431, 3331, 3411 are primes, we only need to consider the families 3{3}11, 33{1,4}1, 3{3,4}4{4}1 (since any digits combo 11, 13, 14, 33, 41, 43 between them will produce smaller primes) **** All numbers of the form 3{3}11 are divisible by 3, thus cannot be prime. **** For the 33{1,4}1 family, since 111 and 141 are primes, we only need to consider the families 33{4}1 and 33{4}11 (since any digits combo 11, 14 between them will produce smaller primes) ***** The smallest prime of the form 33{4}1 is 3344441 ***** All numbers of the form 33{4}11 are divisible by 301, thus cannot be prime. **** For the 3{3,4}4{4}1 family, since 3331 and 3344441 are primes, we only need to consider the families 3{4}1, 3{4}31, 3{4}341, 3{4}3441, 3{4}34441 (since any digits combo 33 or 34444 between (3,1) will produce smaller primes) ***** All numbers of the form 3{4}1 are divisible by 31, thus cannot be prime. ***** Since 4443 is prime, we only need to consider the numbers 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 (since any digit combo 444 between (3,3{4}1) will produce smaller primes) ****** None of 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 are primes. * Case (3,3): ** Since 35, 37, 13, 23, 53, 73, 343 are primes, we only need to consider the family 3{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes) *** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 21, 51, 401, 431, 471 are primes, we only need to consider the family 4{1,4,6}1 (since any digits 0, 2, 3, 5, 7 between them will produce smaller primes) *** Since 111, 141, 161, 661, 4611 are primes, we only need to consider the families 4{4}11, 4{4,6}4{1,4,6}1, 4{4}6{4}1 (since any digits combo 11, 14, 16, 61, 66 between them will produce smaller primes) **** The smallest prime of the form 4{4}11 is 44444444444444411 (not minimal prime, since 444444441 is prime) **** For the 4{4,6}4{1,4,6}1 family, we can separate this family to 4{4,6}41, 4{4,6}411, 4{4,6}461 ***** For the 4{4,6}41 family, since 661 and 6441 are primes, we only need to consider the families 4{4}41 and 4{4}641 (since any digits combo 64 or 66 between (4,41) will produce smaller primes) ****** The smallest prime of the form 4{4}41 is 444444441 ****** The smallest prime of the form 4{4}641 is 444641 ***** For the 4{4,6}411 family, since 661 and 6441 are primes, we only need to consider the families 4{4}411 and 4{4}6411 (since any digits combo 64 or 66 between (4,411) will produce smaller primes) ****** The smallest prime of the form 4{4}411 is 444444441 ****** The smallest prime of the form 4{4}6411 is 4444444444444446411 (not minimal prime, since 444444441 and 444641 are primes) ***** For the 4{4,6}461 family, since 661 is prime, we only need to consider the family 4{4}461 ****** The smallest prime of the form 4{4}461 is 4444444461 (not minimal prime, since 444444441 is prime) **** For the 4{4}6{4}1 family, since 6441 is prime, we only need to consider the families 4{4}61 and 4{4}641 (since any digits combo 44 between (4{4}6,1) will produce smaller primes) ***** The smallest prime of the form 4{4}61 is 4444444461 (not minimal prime, since 444444441 is prime) ***** The smallest prime of the form 4{4}641 is 444641 * Case (4,3): ** Since 45, 13, 23, 53, 73, 433, 463 are primes, we only need to consider the family 4{0,4}3 (since any digits 1, 2, 3, 5, 6, 7 between them will produce smaller primes) *** Since 4043 and 4443 are primes, we only need to consider the families 4{0}3 and 44{0}3 (since any digits combo 04, 44 between them will produce smaller primes) **** All numbers of the form 4{0}3 are divisible by 7, thus cannot be prime. **** All numbers of the form 44{0}3 are divisible by 3, thus cannot be prime. * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (4,7): ** Since 45, 27, 37, 57, 407, 417, 467 are primes, we only need to consider the family 4{4,7}7 (since any digits 0, 1, 2, 3, 5, 6 between them will produce smaller primes) *** Since 747 is prime, we only need to consider the families 4{4}7, 4{4}77, 4{7}7, 44{7}7 (since any digits combo 74 between (4,7) will produce smaller primes) **** The smallest prime of the form 4{4}7 is 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447, with 220 4's, which can be written as (4^220)7 and equal the prime (4*8^221+17)/7 ([http://factordb.com/index.php?id=1100000000416605822 factordb]) **** The smallest prime of the form 4{4}77 is 4444477 **** The smallest prime of the form 4{7}7 is 47777 **** The smallest prime of the form 44{7}7 is 4477777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777, with 851 7's, which can be written as 44(7^851) and equal the prime 37*8^851-1 ([http://factordb.com/index.php?id=1100000000413677646 factordb]) (not minimal prime, since 47777 is prime) * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,3): ** 53 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 53, 57, 15, 35, 45, 65, 75 are primes, we only need to consider the family 5{0,2,5}5 (since any digits 1, 3, 4, 6, 7 between them will produce smaller primes) *** Since 225, 255, 5205 are primes, we only need to consider the families 5{0,5}5 and 5{0,5}25 (since any digits combo 20, 22, 25 between them will produce smaller primes) **** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. **** For the 5{0,5}25 family, since 500025 and 505525 are primes, we only need to consider the number 500525 the families 5{5}25, 5{5}025, 5{5}0025, 5{5}0525, 5{5}00525, 5{5}05025 (since any digits combo 000, 055 between (5,25) will produce smaller primes) ***** 500525 is not prime. ***** The smallest prime of the form 5{5}25 is 555555555555525 ***** The smallest prime of the form 5{5}025 is 55555025 ***** The smallest prime of the form 5{5}0025 is 5555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555550025, with 184 5's, which can be written as (5^183)0025 and equal the prime (5*8^187-20333)/7 ([http://factordb.com/index.php?id=1100000002350205912 factordb]) (not minimal prime, since 55555025 and 555555555555525 are primes) ***** The smallest prime of the form 5{5}0525 is 5550525 ***** The smallest prime of the form 5{5}00525 is 5500525 ***** The smallest prime of the form 5{5}05025 is 5555555555555555555555505025 (not minimal prime, since 5550525, 55555025, and 555555555555525 are primes) * Case (5,7): ** 57 is prime, and thus the only minimal prime in this family. * Case (6,1): ** Since 65, 21, 51, 631, 661 are primes, we only need to consider the family 6{0,1,4,7}1 (since any digits 2, 3, 5, 6 between them will produce smaller primes) *** Numbers containing 4: (note that the number cannot contain two or more 4's, or 6441 will be a subsequence) **** The form is 6{0,1,7}4{0,1,7}1 ***** Since 141, 401, 471 are primes, we only need to consider the family 6{0,7}4{1}1 ****** Since 111 is prime, we only need to consider the families 6{0,7}41 and 6{0,7}411 ******* For the 6{0,7}41 family, since 60741 is prime, we only need to consider the family 6{7}{0}41 ******** Since 6777 is prime, we only need to consider the families 6{0}41, 67{0}41, 677{0}41 ********* All numbers of the form 6{0}41 are divisible by 3, thus cannot be prime. ********* All numbers of the form 67{0}41 are divisible by 13, thus cannot be prime. ********* All numbers of the form 677{0}41 are divisible by 3, thus cannot be prime. ******* For the 6{0,7}411 family, since 60411 is prime, we only need to consider the family 6{7}411 ******** The smallest prime of the form 6{7}411 is 67777411 (not minimal prime, since 6777 is prime) *** Numbers not containing 4: **** The form is 6{0,1,7}1 ***** Since 111 is prime, we only need to consider the families 6{0,7}1 and 6{0,7}1{0,7}1 ****** All numbers of the form 6{0,7}1 are divisible by 7, thus cannot be prime. ****** For the 6{0,7}1{0,7}1 family, since 711 and 6101 are primes, we only need to consider the family 6{0}1{7}1 ******* Since 60171 is prime, we only need to consider the families 6{0}11 and 61{7}1 ******** All numbers of the form 6{0}11 are divisible by 3, thus cannot be prime. ******** The smallest prime of the form 61{7}1 is 617771 (not minimal prime, since 6777 is prime) * Case (6,3): ** Since 65, 13, 23, 53, 73, 643 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes) *** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (6,5): ** 65 is prime, and thus the only minimal prime in this family. * Case (6,7): ** Since 65, 27, 37, 57, 667 are primes, we only need to consider the family 6{0,1,4,7}7 (since any digits 2, 3, 5, 6 between them will produce smaller primes) *** Since 107, 117, 147, 177, 407, 417, 717, 747, 6007, 6477, 6707, 6777 are primes, there cannot be digits combo 00, 10, 11, 14, 17, 40, 41, 47, 70, 71, 74, 77 between them **** If there is 1 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}17 ***** All numbers of the form 6{0}17 are divisible by 3, thus cannot be prime. **** If there is 7 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}77 ***** All numbers of the form 6{0}77 are divisible by 3, thus cannot be prime. **** If there is neither 1 nor 7 between them, then the form is 6{0,4}7 ***** Since 6007, 407 at primes, we only need to consider the families 6{4}7 and 60{4}7 (since any digits combo 00, 40 between them will produce smaller primes) ****** All numbers of the form 6{4}7 are divisible by 3, 5, or 15, thus cannot be prime. ****** All numbers of the form 60{4}7 are divisible by 21, thus cannot be prime. * Case (7,1): ** Since 73, 75, 21, 51, 701, 711 are primes, we only need to consider the family 7{4,6,7}1 (since any digits 0, 1, 2, 3, 5 between them will produce smaller primes) *** Since 747, 767, 471, 661, 7461, 7641 are primes, we only need to consider the families 7{4,7}4{4}1, 7{7}61, 7{7}7{4,6,7}1 (since any digits combo 46, 47, 64, 66, 67 between them will produce smaller primes) **** For the 7{4,7}4{4}1 family, since 747, 471 are primes, we only need to consider the family 7{7}{4}1 (since any digits combo 47 between (7,4{4}1) will produce smaller primes) ***** The smallest prime of the form 7{7}1 is 7777777777771 ***** The smallest prime of the form 7{7}41 is 777777777777777777777777777777777777777777777777777777777777777777777777777777741, with 79 7's, which can be written as (7^79)41 and equal the prime 8^81-31 ([http://factordb.com/index.php?id=1100000000294462449 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777441, with 84 7's, which can be written as (7^84)441 and equal the prime 8^87-223 ([http://factordb.com/index.php?id=1100000000294462776 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}4441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777774441, with 233 7's, which can be written as (7^233)4441 and equal the prime 8^237-1759 ([http://factordb.com/index.php?id=1100000002352073382 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}44441 is 7777777777777777777777777777777777777777777777777777777744441, with 56 7's, which can be written as (7^56)44441 and equal the prime 8^61-14047 ([http://factordb.com/index.php?id=1100000002350250002 factordb]) (not minimal prime, since 7777777777771 is prime) ***** All numbers of the form 7{7}444441 are divisible by 7, thus cannot be prime. ***** The smallest prime of the form 7{7}4444441 is 77774444441 ****** Since this prime has just 4 7's, we only need to consider the families with <=3 7's ******* The smallest prime of the form 7{4}1 is 744444441 ******* All numbers of the form 77{4}1 are divisible by 5, thus cannot be prime. ******* The smallest prime of the form 777{4}1 is 777444444444441 (not minimal prime, since 444444441 and 744444441 are primes) * Case (7,3): ** 73 is prime, and thus the only minimal prime in this family. * Case (7,5): ** 75 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 73, 75, 27, 37, 57, 717, 747, 767 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. ===Base 10=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (1,7), (1,9), (2,1), (2,3), (2,7), (2,9), (3,1), (3,3), (3,7), (3,9), (4,1), (4,3), (4,7), (4,9), (5,1), (5,3), (5,7), (5,9), (6,1), (6,3), (6,7), (6,9), (7,1), (7,3), (7,7), (7,9), (8,1), (8,3), (8,7), (8,9), (9,1), (9,3), (9,7), (9,9) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (1,7): ** 17 is prime, and thus the only minimal prime in this family. * Case (1,9): ** 19 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 29, 11, 31, 41, 61, 71, 251, 281 are primes, we only need to consider the family 2{0,2}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 2221 and 20201 are primes, we only need to consider the families 2{0}1, 2{0}21, 22{0}1 (since any digits combo 22 or 020 between them will produce smaller primes) **** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. **** The smallest prime of the form 2{0}21 is 20021 **** The smallest prime of the form 22{0}1 is 22000001 * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,7): ** Since 23, 29, 17, 37, 47, 67, 97, 227, 257, 277 are primes, we only need to consider the family 2{0,8}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 9 between them will produce smaller primes) *** Since 887 and 2087 are primes, we only need to consider the families 2{0}7 and 28{0}7 (since any digit combo 08 or 88 between them will produce smaller primes) **** All numbers of the form 2{0}7 are divisible by 3, thus cannot be prime. **** All numbers of the form 28{0}7 are divisible by 7, thus cannot be prime. * Case (2,9): ** 29 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 31, 37, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 3{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 3{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (3,9): ** Since 31, 37, 19, 29, 59, 79, 89, 349 are primes, we only need to consider the family 3{0,3,6,9}9 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 3{0,3,6,9}9 are divisible by 3, thus cannot be prime. * Case (4,1): ** 41 is prime, and thus the only minimal prime in this family. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,7): ** 47 is prime, and thus the only minimal prime in this family. * Case (4,9): ** Since 41, 43, 47, 19, 29, 59, 79, 89, 409, 449, 499 are primes, we only need to consider the family 4{6}9 (since any digits 0, 1, 2, 3, 4, 5, 7, 8, 9 between them will produce smaller primes) *** All numbers of the form 4{6}9 are divisible by 7, thus cannot be prime. * Case (5,1): ** Since 53, 59, 11, 31, 41, 61, 71, 521 are primes, we only need to consider the family 5{0,5,8}1 (since any digits 1, 2, 3, 4, 6, 7, 9 between them will produce smaller primes) *** Since 881 is prime, we only need to consider the families 5{0,5}1 and 5{0,5}8{0,5}1 (since any digit combo 88 between them will produce smaller primes) **** For the 5{0,5}1 family, since 5051 and 5501 are primes, we only need to consider the families 5{0}1 and 5{5}1 (since any digit combo 05 or 50 between them will produce smaller primes) ***** All numbers of the form 5{0}1 are divisible by 3, thus cannot be prime. ***** The smallest prime of the form 5{5}1 is 555555555551 **** For the 5{0,5}8{0,5}1 family, since 5081, 5581, 5801, 5851 are primes, we only need to consider the number 581 ***** 581 is not prime. * Case (5,3): ** 53 is prime, and thus the only minimal prime in this family. * Case (5,7): ** Since 53, 59, 17, 37, 47, 67, 97, 557, 577, 587 are primes, we only need to consider the family 5{0,2}7 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 227 and 50207 are primes, we only need to consider the families 5{0}7, 5{0}27, 52{0}7 (since any digits combo 22 or 020 between them will produce smaller primes) **** All numbers of the form 5{0}7 are divisible by 3, thus cannot be prime. **** The smallest prime of the form 5{0}27 is 5000000000000000000000000000027 **** The smallest prime of the form 52{0}7 is 5200007 * Case (5,9): ** 59 is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,3): ** Since 61, 67, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 6{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 6{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (6,7): ** 67 is prime, and thus the only minimal prime in this family. * Case (6,9): ** Since 61, 67, 19, 29, 59, 79, 89 are primes, we only need to consider the family 6{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes) *** Since 449 is prime, we only need to consider the families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes) **** All numbers of the form 6{0,3,6,9}9 are divisible by 3, thus cannot be prime. **** For the 6{0,3,6,9}4{0,3,6,9}9 family, since 409, 43, 6469, 499 are primes, we only need to consider the family 6{0,3,6,9}49 ***** Since 349, 6949 are primes, we only need to consider the family 6{0,6}49 ****** Since 60649 is prime, we only need to consider the family 6{6}{0}49 (since any digits combo 06 between {6,49} will produce smaller primes) ******* The smallest prime of the form 6{6}49 is 666649 ******** Since this prime has just 4 6's, we only need to consider the families with <=3 6's ********* The smallest prime of the form 6{0}49 is 60000049 ********* The smallest prime of the form 66{0}49 is 66000049 ********* The smallest prime of the form 666{0}49 is 66600049 * Case (7,1): ** 71 is prime, and thus the only minimal prime in this family. * Case (7,3): ** 73 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 71, 73, 79, 17, 37, 47, 67, 97, 727, 757, 787 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9 between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. * Case (7,9): ** 79 is prime, and thus the only minimal prime in this family. * Case (8,1): ** Since 83, 89, 11, 31, 41, 61, 71, 821, 881 are primes, we only need to consider the family 8{0,5}1 (since any digits 1, 2, 3, 4, 6, 7, 8, 9 between them will produce smaller primes) *** Since 8501 is prime, we only need to consider the family 8{0}{5}1 (since any digits combo 50 between them will produce smaller primes) **** Since 80051 is prime, we only need to consider the families 8{0}1, 8{5}1, 80{5}1 (since any digits combo 005 between them will produce smaller primes) ***** All numbers of the form 8{0}1 are divisible by 3, thus cannot be prime. ***** The smallest prime of the form 8{5}1 is 8555555555555555555551 (not minimal prime, since 555555555551 is prime) ***** The smallest prime of the form 80{5}1 is 80555551 * Case (8,3): ** 83 is prime, and thus the only minimal prime in this family. * Case (8,7): ** Since 83, 89, 17, 37, 47, 67, 97, 827, 857, 877, 887 are primes, we only need to consider the family 8{0}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** All numbers of the form 8{0}7 are divisible by 3, thus cannot be prime. * Case (8,9): ** 89 is prime, and thus the only minimal prime in this family. * Case (9,1): ** Since 97, 11, 31, 41, 61, 71, 991 are primes, we only need to consider the family 9{0,2,5,8}1 (since any digits 1, 3, 4, 6, 7, 9 between them will produce smaller primes) *** Since 251, 281, 521, 821, 881, 9001, 9221, 9551, 9851 are primes, we only need to consider the families 9{2,5,8}0{2,5,8}1, 9{0}2{0}1, 9{0}5{0,8}1, 9{0,5}8{0}1 (since any digits combo 00, 22, 25, 28, 52, 55, 82, 85, 88 between them will produce smaller primes) **** For the 9{2,5,8}0{2,5,8}1 family, since any digits combo 22, 25, 28, 52, 55, 82, 85, 88 between (9,1) will produce smaller primes, we only need to consider the numbers 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801 ***** 95801 is the only prime among 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. **** For the 9{0}2{0}1 family, since 9001 is prime, we only need to consider the numbers 921, 9201, 9021 ***** None of 921, 9201, 9021 are primes. **** For the 9{0}5{0,8}1 family, since 9001 and 881 are primes, we only need to consider the numbers 951, 9051, 9501, 9581, 90581, 95081, 95801 ***** 95801 is the only prime among 951, 9051, 9501, 9581, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. **** For the 9{0,5}8{0}1 family, since 9001 and 5581 are primes, we only need to consider the numbers 981, 9081, 9581, 9801, 90581, 95081, 95801 ***** 95801 is the only prime among 981, 9081, 9581, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. * Case (9,3): ** Since 97, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 9{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 9{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (9,7): ** 97 is prime, and thus the only minimal prime in this family. * Case (9,9): ** Since 97, 19, 29, 59, 79, 89 are primes, we only need to consider the family 9{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes) *** Since 449 is prime, we only need to consider the families 9{0,3,6,9}9 and 9{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes) **** All numbers of the form 9{0,3,6,9}9 are divisible by 3, thus cannot be prime. **** For the 9{0,3,6,9}4{0,3,6,9}9 family, since 9049, 349, 9649, 9949 are primes, we only need to consider the family 94{0,3,6,9}9 ***** Since 409, 43, 499 are primes, we only need to consider the family 94{6}9 (since any digits 0, 3, 9 between (94,9) will produce smaller primes) ****** The smallest prime of the form 94{6}9 is 946669 ===Base 12=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,5), (1,7), (1,B), (2,1), (2,5), (2,7), (2,B), (3,1), (3,5), (3,7), (3,B), (4,1), (4,5), (4,7), (4,B), (5,1), (5,5), (5,7), (5,B), (6,1), (6,5), (6,7), (6,B), (7,1), (7,5), (7,7), (7,B), (8,1), (8,5), (8,7), (8,B), (9,1), (9,5), (9,7), (9,B), (A,1), (A,5), (A,7), (A,B), (B,1), (B,5), (B,7), (B,B) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (1,7): ** 17 is prime, and thus the only minimal prime in this family. * Case (1,B): ** 1B is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 25, 27, 11, 31, 51, 61, 81, 91, 221, 241, 2A1, 2B1 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B between them will produce smaller primes) *** The smallest prime of the form 2{0}1 is 2001 * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (2,7): ** 27 is prime, and thus the only minimal prime in this family. * Case (2,B): ** Since 25, 27, 1B, 3B, 4B, 5B, 6B, 8B, AB, 2BB are primes, we only need to consider the family 2{0,2,9}B (since any digits 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes) *** Since 90B, 200B, 202B, 222B, 229B, 292B, 299B are primes, we only need to consider the numbers 20B, 22B, 29B, 209B, 220B (since any digits combo 00, 02, 22, 29, 90, 92, 99 between them will produce smaller primes) **** None of 20B, 22B, 29B, 209B, 220B are primes. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (3,B): ** 3B is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 4B, 11, 31, 51, 61, 81, 91, 401, 421, 471 are primes, we only need to consider the family 4{4,A}1 (since any digit 0, 1, 2, 3, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since A41 and 4441 are primes, we only need to consider the families 4{A}1 and 44{A}1 (since any digit combo 44, A4 between them will produce smaller primes) **** All numbers of the form 4{A}1 are divisible by 5, thus cannot be prime. **** The smallest prime of the form 44{A}1 is 44AAA1 * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (4,7): ** Since 45, 4B, 17, 27, 37, 57, 67, 87, A7, B7, 447, 497 are primes, we only need to consider the family 4{0,7}7 (since any digit 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes) *** Since 4707 and 4777 are primes, we only need to consider the families 4{0}7 and 4{0}77 (since any digit combo 70, 77 between them will produce smaller primes) **** All numbers of the form 4{0}7 are divisible by B, thus cannot be prime. **** The smallest prime of the form 4{0}77 is 400000000000000000000000000000000000000077 * Case (4,B): ** 4B is prime, and thus the only minimal prime in this family. * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 57, 5B, 15, 25, 35, 45, 75, 85, 95, B5, 565 are primes, we only need to consider the family 5{0,5,A}5 (since any digits 1, 2, 3, 4, 6, 7, 8, 9, B between them will produce smaller primes) *** All numbers of the form 5{0,5,A}5 are divisible by 5, thus cannot be prime. * Case (5,7): ** 57 is prime, and thus the only minimal prime in this family. * Case (5,B): ** 5B is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,5): ** Since 61, 67, 6B, 15, 25, 35, 45, 75, 85, 95, B5, 655, 665 are primes, we only need to consider the family 6{0,A}5 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since 6A05 and 6AA5 are primes, we only need to consider the families 6{0}5 and 6{0}A5 (since any digit combo A0, AA between them will produce smaller primes) **** All numbers of the form 6{0}5 are divisible by B, thus cannot be prime. **** The smallest prime of the form 6{0}A5 is 600A5 * Case (6,7): ** 67 is prime, and thus the only minimal prime in this family. * Case (6,B): ** 6B is prime, and thus the only minimal prime in this family. * Case (7,1): ** Since 75, 11, 31, 51, 61, 81, 91, 701, 721, 771, 7A1 are primes, we only need to consider the family 7{4,B}1 (since any digits 0, 1, 2, 3, 5, 6, 7, 8, 9, A between them will produce smaller primes) *** Since 7BB, 7441 and 7B41 are primes, we only need to consider the numbers 741, 7B1, 74B1 **** None of 741, 7B1, 74B1 are primes. * Case (7,5): ** 75 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 75, 17, 27, 37, 57, 67, 87, A7, B7, 747, 797 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. * Case (7,B): ** Since 75, 1B, 3B, 4B, 5B, 6B, 8B, AB, 70B, 77B, 7BB are primes, we only need to consider the family 7{2,9}B (since any digits 0, 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes) *** Since 222B, 729B is prime, we only need to consider the families 7{9}B, 7{9}2B, 7{9}22B (since any digits combo 222, 29 between them will produce smaller primes) **** The smallest prime of the form 7{9}B is 7999B **** The smallest prime of the form 7{9}2B is 79992B (not minimal prime, since 992B and 7999B are primes) **** The smallest prime of the form 7{9}22B is 79922B (not minimal prime, since 992B is prime) * Case (8,1): ** 81 is prime, and thus the only minimal prime in this family. * Case (8,5): ** 85 is prime, and thus the only minimal prime in this family. * Case (8,7): ** 87 is prime, and thus the only minimal prime in this family. * Case (8,B): ** 8B is prime, and thus the only minimal prime in this family. * Case (9,1): ** 91 is prime, and thus the only minimal prime in this family. * Case (9,5): ** 95 is prime, and thus the only minimal prime in this family. * Case (9,7): ** Since 91, 95, 17, 27, 37, 57, 67, 87, A7, B7, 907 are primes, we only need to consider the family 9{4,7,9}7 (since any digit 0, 1, 2, 3, 5, 6, 8, A, B between them will produce smaller primes) *** Since 447, 497, 747, 797, 9777, 9947, 9997 are primes, we only need to consider the numbers 947, 977, 997, 9477, 9977 (since any digits combo 44, 49, 74, 77, 79, 94, 99 between them will produce smaller primes) **** None of 947, 977, 997, 9477, 9977 are primes. * Case (9,B): ** Since 91, 95, 1B, 3B, 4B, 5B, 6B, 8B, AB, 90B, 9BB are primes, we only need to consider the family 9{2,7,9}B (since any digit 0, 1, 3, 4, 5, 6, 8, A, B between them will produce smaller primes) *** Since 27, 77B, 929B, 992B, 997B are primes, we only need to consider the families 9{2,7}2{2}B, 97{2,9}B, 9{7,9}9{9}B (since any digits combo 27, 29, 77, 92, 97 between them will produce smaller primes) **** For the 9{2,7}2{2}B family, since 27 and 77B are primes, we only need to consider the families 9{2}2{2}B and 97{2}2{2}B (since any digits combo 27, 77 between (9,2{2}B) will produce smaller primes) ***** The smallest prime of the form 9{2}2{2}B is 9222B (not minimal prime, since 222B is prime) ***** The smallest prime of the form 97{2}2{2}B is 9722222222222B (not minimal prime, since 222B is prime) **** For the 97{2,9}B family, since 729B and 929B are primes, we only need to consider the family 97{9}{2}B (since any digits combo 29 between (97,B) will produce smaller primes) ***** Since 222B is prime, we only need to consider the families 97{9}B, 97{9}2B, 97{9}22B (since any digit combo 222 between (97,B) will produce smaller primes) ****** All numbers of the form 97{9}B are divisible by 11, thus cannot be prime. ****** The smallest prime of the form 97{9}2B is 979999992B (not minimal prime, since 9999B is prime) ****** All numbers of the form 97{9}22B are divisible by 11, thus cannot be prime. **** For the 9{7,9}9{9}B family, since 77B and 9999B are primes, we only need to consider the numbers 99B, 999B, 979B, 9799B, 9979B ***** None of 99B, 999B, 979B, 9799B, 9979B are primes. * Case (A,1): ** Since A7, AB, 11, 31, 51, 61, 81, 91, A41 are primes, we only need to consider the family A{0,2,A}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since 221, 2A1, A0A1, A201 are primes, we only need to consider the families A{A}{0}1 and A{A}{0}21 (since any digits combo 0A, 20, 22, 2A between them will produce smaller primes) **** For the A{A}{0}1 family: ***** All numbers of the form A{0}1 are divisible by B, thus cannot be prime. ***** The smallest prime of the form AA{0}1 is AA000001 ***** The smallest prime of the form AAA{0}1 is AAA0001 ***** The smallest prime of the form AAAA{0}1 is AAAA1 ****** Since this prime has no 0's, we do not need to consider the families {A}1, {A}01, {A}001, etc. **** All numbers of the form A{A}{0}21 are divisible by 5, thus cannot be prime. * Case (A,5): ** Since A7, AB, 15, 25, 35, 45, 75, 85, 95, B5 are primes, we only need to consider the family A{0,5,6,A}5 (since any digits 1, 2, 3, 4, 7, 8, 9, B between them will produce smaller primes) *** Since 565, 655, 665, A605, A6A5, AA65 are primes, we only need to consider the families A{0,5,A}5 and A{0}65 (since any digits combo 56, 60, 65, 66, 6A, A6 between them will produce smaller primes) **** All numbers of the form A{0,5,A}5 are divisible by 5, thus cannot be prime. **** The smallest prime of the form A{0}65 is A00065 * Case (A,7): ** A7 is prime, and thus the only minimal prime in this family. * Case (A,B): ** AB is prime, and thus the only minimal prime in this family. * Case (B,1): ** Since B5, B7, 11, 31, 51, 61, 81, 91, B21 are primes, we only need to consider the family B{0,4,A,B}1 (since any digits 1, 2, 3, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 4B, AB, 401, A41, B001, B0B1, BB01, BB41 are primes, we only need to consider the families B{A}0{4,A}1, B{0,4}4{4,A}1, B{0,4,A,B}A{0,A}1, B{B}B{A,B}1 (since any digits combo 00, 0B, 40, 4B, A4, AB, B0, B4 between them will produce smaller primes) **** For the B{A}0{4,A}1 family, since A41 is prime, we only need consider the families B0{4}{A}1 and B{A}0{A}1 ***** For the B0{4}{A}1 family, since B04A1 is prime, we only need to consider the families B0{4}1 and B0{A}1 ****** The smallest prime of the form B0{4}1 is B04441 (not minimal prime, since 4441 is prime) ****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime) ***** For the B{A}0{A}1 family, since A0A1 is prime, we only need to consider the families B{A}01 and B0{A}1 ****** The smallest prime of the form B{A}01 is BAA01 ****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime) **** For the B{0,4}4{4,A}1 family, since 4441 is prime, we only need to consider the families B{0}4{4,A}1 and B{0,4}4{A}1 ***** For the B{0}4{4,A}1 family, since B001 is prime, we only need to consider the families B4{4,A}1 and B04{4,A}1 ****** For the B4{4,A}1 family, since A41 is prime, we only need to consider the family B4{4}{A}1 ******* Since 4441 and BAAA1 are primes, we only need to consider the numbers B41, B441, B4A1, B44A1, B4AA1, B44AA1 ******** None of B41, B441, B4A1, B44A1, B4AA1, B44AA1 are primes. ****** For the B04{4,A}1 family, since B04A1 is prime, we only need to consider the family B04{4}1 ******* The smallest prime of the form B04{4}1 is B04441 (not minimal prime, since 4441 is prime) ***** For the B{0,4}4{A}1 family, since 401, 4441, B001 are primes, we only need to consider the families B4{A}1, B04{A}1, B44{A}1, B044{A}1 (since any digits combo 00, 40, 44 between (B,4{A}1) will produce smaller primes) ****** The smallest prime of the form B4{A}1 is B4AAA1 (not minimal prime, since BAAA1 is prime) ****** The smallest prime of the form B04{A}1 is B04A1 ****** The smallest prime of the form B44{A}1 is B44AAAAAAA1 (not minimal prime, since BAAA1 is prime) ****** The smallest prime of the form B044{A}1 is B044A1 (not minimal prime, since B04A1 is prime) **** For the B{0,4,A,B}A{0,A}1 family, since all numbers in this family with 0 between (B,1) are in the B{A}0{4,A}1 family, and all numbers in this family with 4 between (B,1) are in the B{0,4}4{4,A}1 family, we only need to consider the family B{A,B}A{A}1 ***** Since BAAA1 is prime, we only need to consider the families B{A,B}A1 and B{A,B}AA1 ****** For the B{A,B}A1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}A1 and B{B}AA1 ******* All numbers of the form B{B}A1 are divisible by B, thus cannot be prime. ******* The smallest prime of the form B{B}AA1 is BBBAA1 ****** For the B{A,B}AA1 family, since BAAA1 is prime, we only need to consider the families B{B}AA1 ******* The smallest prime of the form B{B}AA1 is BBBAA1 **** For the B{B}B{A,B}1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}B1, B{B}BA1, B{B}BAA1 (since any digits combo AB or AAA between (B{B}B,1) will produce smaller primes) ***** The smallest prime of the form B{B}B1 is BBBB1 ***** All numbers of the form B{B}BA1 are divisible by B, thus cannot be prime. ***** The smallest prime of the form B{B}BAA1 is BBBAA1 * Case (B,5): ** B5 is prime, and thus the only minimal prime in this family. * Case (B,7): ** B7 is prime, and thus the only minimal prime in this family. * Case (B,B): ** Since B5, B7, 1B, 3B, 4B, 5B, 6B, 8B, AB, B2B are primes, we only need to consider the family B{0,9,B}B (since any digits 1, 2, 3, 4, 5, 6, 7, 8, A between them will produce smaller primes) *** Since 90B and 9BB are primes, we only need to consider the families B{0,B}{9}B **** Since 9999B is prime, we only need to consider the families B{0,B}B, B{0,B}9B, B{0,B}99B, B{0,B}999B ***** All numbers of the form B{0,B}B are divisible by B, thus cannot be prime. ***** For the B{0,B}9B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}9B and B{B}9B (since any digits combo 0B, B0 between (B,9B) will produce smaller primes) ******* The smallest prime of the form B{0}9B is B0000000000000000000000000009B ******* All numbers of the from B{B}9B is either divisible by 11 (if totally number of B's is even) or factored as 10^(2*n)-21 = (10^n-5) * (10^n+5) (if totally number of B's is odd number 2*n-1 (n≥1)) (and since if n≥1, 10^n-5 ≥ 10^1-5 = 7 > 1, 10^n+5 ≥ 10^1+5 = 15 > 1, this factorization is nontrivial), thus cannot be prime. ***** For the B{0,B}99B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}99B and B{B}99B (since any digits combo 0B, B0 between (B,99B) will produce smaller primes) ******* The smallest prime of the form B{0}99B is B00099B ******* The smallest prime of the form B{B}99B is BBBBBB99B ***** For the B{0,B}999B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}999B and B{B}999B (since any digits combo 0B, B0 between (B,999B) will produce smaller primes) ******* The smallest prime of the form B{0}999B is B0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000999B, with 1765 0's, which can be written as B(0^1765)999B and equal the prime 11*12^1769+16967 ([http://factordb.com/index.php?id=1100000002378273165 factordb]) ([http://factordb.com/cert.php?id=1100000002378273165 primality certificate]) (not minimal prime, since B00099B and B0000000000000000000000000009B are primes) ******* The smallest prime of the form B{B}999B is BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB999B, with 245 B's, which can be written as (B^244)999B and equal the prime 12^248-3769 ([http://factordb.com/index.php?id=1100000002378270237 factordb]) (not minimal prime, since BBBBBB99B is prime) == Examples of families which can be ruled out as contain no primes > ''b'' == It is not known if this problem is solvable: Problem: Given strings ''x'', ''y'', ''z'', and a base ''b'', does there exist a prime number whose base-''b'' expansion is of the form ''x''{''y''}''z''? It will be necessary for our algorithm to determine if families of the form ''x''{''y''}''z'' contain a prime > ''b'' or not. We use two different heuristic strategies to show that such families contain no primes > ''b''. In the first strategy, we mimic the well-known technique of “covering congruences”, by finding some finite set ''S'' of primes ''p'' such that every number in a given family is divisible by some element of ''S''. In the second strategy, we attempt to find an algebraic factorization, such as difference-of-squares factorization, difference-of-cubes factorization, and Aurifeuillian factorization for numbers of the form ''x''⁴+4''y''⁴. Examples of first strategy: (we can show that the corresponding numbers are > all elements in ''S'', if ''n'' makes corresponding numbers > ''b'' (i.e. ''n''≥1 for 5{1} in base 9 and 2{5} in base 11 and {4}D in base 16 and {8}F in base 16, ''n''≥0 for other examples), thus these factorizations are nontrivial) * In base 10, all numbers of the form 4{6}9 are divisible by 7 * In base 6, all numbers of the form 4{0}1 are divisible by 5 * In base 15, all numbers of the form 9{6}8 are divisible by 11 * In base 9, all numbers of the form 5{1} are divisible by some element of {2, 5} * In base 11, all numbers of the form 2{5} are divisible by some element of {2, 3} * In base 14, all numbers of the form B{0}1 are divisible by some element of {3, 5} * In base 8, all numbers of the form 6{4}7 are divisible by some element of {3, 5, 13} * In base 13, all numbers of the form 3{0}95 are divisible by some element of {5, 7, 17} * In base 16, all numbers of the form {4}D are divisible by some element of {3, 7, 13} * In base 16, all numbers of the form {8}F are divisible by some element of {3, 7, 13} Examples of second strategy: (we can show that both factors are > 1, if ''n'' makes corresponding numbers > ''b'' (i.e. ''n''≥2 for {1} in base 9, ''n''≥0 for 1{0}1 in base 8 and B{4}1 in base 16, ''n''≥1 for other examples), thus these factorizations are nontrivial) * In base 9, all numbers of the form {1} factored as difference of squares * In base 8, all numbers of the form 1{0}1 factored as sum of cubes * In base 9, all numbers of the form 3{8} factored as difference of squares * In base 16, all numbers of the form 8{F} factored as difference of squares * In base 16, all numbers of the form {F}7 factored as difference of squares * In base 9, all numbers of the form 3{1} factored as difference of squares * In base 16, all numbers of the form {4}1 factored as difference of squares * In base 16, all numbers of the form 1{5} factored as difference of squares * In base 16, all numbers of the from {C}D factored as ''x''⁴+4''y''⁴ * In base 16, all numbers of the form B{4}1 factored as difference of squares Examples of combine of the two strategies: (we can show that for the part of the first strategy, the corresponding numbers are > all elements in S, and for the part of the second strategy, both factors are > 1, if n makes corresponding numbers > b, thus these factorizations are nontrivial) * In base 14, numbers of the form 8{D} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 12, numbers of the form {B}9B are divisible by 13 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 14, numbers of the form {D}5 are divisible by 5 if ''n'' is even and factored as difference of squares if ''n'' is odd * In base 17, numbers of the form 1{9} are divisible by 2 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 19, numbers of the form 1{6} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even == Bases 2≤''b''≤1024 such that these families can be ruled out as contain no primes > ''b'' == (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) === 1{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-powers factorization === 1{0}2 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 1{0}3 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} === 1{0}4 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ === 1{0}5 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 0 mod 5: Finite covering set {5} === 1{0}6 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 7: Finite covering set {7} === 1{0}7 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 7: Finite covering set {7} === 1{0}z === (none) === 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === * ''b'' == 1 mod 3: Finite covering set {3} === 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) === (none) === 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === * ''b'' == 1 mod 3: Finite covering set {3} === {1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z) === * ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841) === {1} === * ''b'' = ''m''^''r'' with ''r''>1: Difference-of-''r''th-powers factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 4 (length 2), 8 (length 3), 16 (length 2), 27 (length 3), 36 (length 2), 100 (length 2), 128 (length 7), 196 (length 2), 256 (length 2), 400 (length 2), 512 (length 3), 576 (length 2), 676 (length 2)) === {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) === * ''b'' == 0 mod 2: Finite covering set {2} === 1{2} === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49) === 1{3} === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' such that ''b'' and 3(''b''+2) are both squares: Difference-of-squares factorization (such bases are 25, 361) * ''b'' == 1 mod 2 such that 3(''b''+2) is square: Combine of finite covering set {2} (when length is even) and difference-of-squares factorization (when length is odd) (such bases are 25, 73, 145, 241, 361, 505, 673, 865) === 1{4} === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' such that ''b'' and 4(''b''+3) are both squares: Difference-of-squares factorization === 1{z} === (none) === 2{0}1 === * ''b'' == 1 mod 3: Finite covering set {3} === 2{0}3 === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 5: Finite covering set {5} === 2{1} (not quasi-minimal prime if there is smaller prime of the form {1}) === * ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841) === {2}1 === * ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49) === 2{z} === * ''b'' == 1 mod 2: Finite covering set {2} === 3{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} === 3{0}2 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} === 3{0}4 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 7: Finite covering set {7} === {3}1 === * ''b'' such that ''b'' and 3(2''b''+1) are both squares: Difference-of-squares factorization (such bases are 121) === 3{z} === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === 4{0}1 === * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ === 4{0}3 === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 7: Finite covering set {7} === {4}1 === * ''b'' such that ''b'' and 4(3''b''+1) are both squares: Difference-of-squares factorization (such bases are 16, 225) === 4{z} === * ''b'' == 1 mod 2: Finite covering set {2} === 5{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 5{z} === * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 34 mod 35: Finite covering set {5, 7} * ''b'' = 6''m''² with ''m'' == 2 or 3 mod 5: Combine of finite covering set {5} (when length is odd) and difference-of-squares factorization (when length is even) (such bases are 24, 54, 294, 384, 864, 1014) === 6{0}1 === * ''b'' == 1 mod 7: Finite covering set {7} * ''b'' == 34 mod 35: Finite covering set {5, 7} === 6{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 7{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} === 7{z} === * ''b'' == 1 mod 7: Finite covering set {7} * ''b'' == 20 mod 21: Finite covering set {3, 7} * ''b'' == 83, 307 mod 455: Finite covering set {5, 7, 13} (such bases are 83, 307, 538, 762, 993) * ''b'' = ''m''³: Difference-of-cubes factorization === 8{0}1 === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 20 mod 21: Finite covering set {3, 7} * ''b'' == 47, 83 mod 195: Finite covering set {3, 5, 13} (such bases are 47, 83, 242, 278, 437, 473, 632, 668, 827, 863, 1022) * ''b'' = 467: Finite covering set {3, 5, 7, 19, 37} * ''b'' = 722: Finite covering set {3, 5, 13, 73, 109} * ''b'' = ''m''³: Sum-of-cubes factorization * ''b'' = 128: Cannot have primes since 7''n''+3 cannot be power of 2 === 8{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === 9{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} === 9{z} === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 32 mod 33: Finite covering set {3, 11} === A{0}1 === * ''b'' == 1 mod 11: Finite covering set {11} * ''b'' == 32 mod 33: Finite covering set {3, 11} === A{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} === B{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} === B{z} === * ''b'' == 1 mod 11: Finite covering set {11} * ''b'' == 142 mod 143: Finite covering set {11, 13} * ''b'' = 307: Finite covering set {5, 11, 29} * ''b'' = 901: Finite covering set {7, 11, 13, 19} === C{0}1 === * ''b'' == 1 mod 13: Finite covering set {13} * ''b'' == 142 mod 143: Finite covering set {11, 13} * ''b'' = 296, 901: Finite covering set {7, 11, 13, 19} * ''b'' = 562, 828, 900: Finite covering set {7, 13, 19} * ''b'' = 563: Finite covering set {5, 7, 13, 19, 29} * ''b'' = 597: Finite covering set {5, 13, 29} === {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) === (none) === {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) === * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-power factorization === #{z} (for even bases b, # = b/2−1) === (none) === y{z} === (none) === {y}z === (none) === z{0}1 === (none) === {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) === * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-power factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 128 (length 7), 216 (length 3), 343 (length 3), 729 (length 3)) * ''b'' = 4''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ (base 4 still have primes, since for the corresponding length this factorization is trivial, but it only have this prime, at length 2) === {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) === (none) === {z}1 === (none) === {z}t === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 0 mod 7: Finite covering set {7} === {z}u === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 34 mod 35: Finite covering set {5, 7} === {z}v === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 5: Finite covering set {5} === {z}w === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === {z}x === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} === {z}y === * ''b'' == 0 mod 2: Finite covering set {2} == Large known (probable) primes (length ≥10000) in these families (for bases 2≤''b''≤1024) == Format: base (length) (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) === 1{0}1 === (none) === 1{0}2 === (none) === 1{0}3 === (none) === 1{0}4 === 53 (13403) 113 (10647) === 1{0}z === 113 (20089) 123 (64371) === 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === (none) === 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) === 208 (26682) 607 (11032) 828 (19659) === 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === 201 (31276) 222 (52727) 227 (36323) 327 (135983) 425 (11231) 710 (24112) 717 (37508) 719 (13420) === {1} === 152 (270217) 184 (16703) 200 (17807) 311 (36497) 326 (26713) 331 (25033) 371 (15527) 485 (99523) 629 (32233) 649 (43987) 670 (18617) 684 (22573) 691 (62903) 693 (41189) 731 (15427) 752 (32833) 872 (10093) 932 (20431) === {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) === (none) === 1{z} === 107 (21911) 170 (166429) 278 (43909) 303 (40175) 383 (20957) 515 (58467) 522 (62289) 578 (129469) 590 (15527) 647 (21577) 662 (16591) 698 (127559) 704 (62035) 845 (39407) 938 (40423) 969 (24097) 989 (26869) === 2{0}1 === 101 (192276) 206 (46206) 218 (333926) 236 (161230) 257 (12184) 305 (16808) 467 (126776) 578 (44166) 626 (174204) 695 (94626) 752 (26164) 788 (72918) 869 (49150) 887 (27772) 899 (15732) 932 (13644) === 2{z} === 432 (16003) === 3{0}1 === (none) === 3{z} === 72 (1119850) 212 (34414) 218 (23050) 270 (89662) 303 (198358) 312 (51566) 422 (21738) 480 (93610) 513 (38032) 527 (46074) 566 (23874) 650 (498102) 686 (16584) 758 (15574) 783 (12508) 800 (33838) 921 (98668) 947 (10056) === 4{0}1 === 107 (32587) 227 (13347) 257 (160423) 355 (10990) 410 (144079) 440 (56087) 452 (14155) 482 (30691) 542 (15983) 579 (67776) 608 (20707) 635 (11723) 650 (96223) 679 (69450) 737 (269303) 740 (58043) 789 (149140) 797 (468703) 920 (103687) 934 (101404) 962 (84235) === 4{z} === 14 (19699) 68 (13575) 254 (15451) 800 (20509) === 5{0}1 === 326 (400786) 350 (20392) 554 (10630) 662 (13390) 926 (40036) === 5{z} === 258 (212135) 272 (148427) 299 (64898) 307 (26263) 354 (25566) 433 (283919) 635 (36163) 678 (40859) 692 (45447) 719 (20552) 768 (70214) 857 (23083) 867 (61411) 972 (36703) === 6{0}1 === 108 (16318) 129 (16797) 409 (369833) 522 (52604) 587 (24120) 643 (164916) 762 (11152) 789 (27297) 986 (21634) === 6{z} === 68 (25396) 332 (15222) 338 (42868) 362 (146342) 488 (33164) 566 (164828) 980 (50878) 986 (12506) 1016 (23336) === 7{0}1 === 398 (17473) 1004 (54849) === 7{z} === 97 (192336) 170 (15423) 194 (38361) 202 (155772) 282 (21413) 283 (164769) 332 (13205) 412 (29792) 560 (19905) 639 (10668) 655 (53009) 811 (31784) 814 (17366) 866 (108591) 908 (61797) 962 (31841) 992 (10605) 997 (15815) === 8{0}1 === 23 (119216) 53 (227184) 158 (123476) 254 (67716) 320 (52004) 410 (279992) 425 (94662) 513 (19076) 518 (11768) 596 (148446) 641 (87702) 684 (23387) 695 (39626) 785 (900326) 788 (11408) 893 (86772) 908 (243440) 920 (107822) 962 (47222) 998 (81240) 1013 (43872) === 8{z} === 138 (35686) 412 (12154) 788 (11326) 990 (23032) === 9{0}1 === 248 (39511) 592 (96870) === 9{z} === 431 (43574) 446 (152028) 458 (126262) 599 (11776) 846 (12781) === A{0}1 === 173 (264235) 198 (47665) 311 (314807) 341 (106009) 449 (18507) 492 (42843) 605 (12395) 708 (17563) 710 (31039) 743 (285479) 744 (137056) 786 (68169) 800 (15105) 802 (149320) 879 (25004) 929 (13065) 977 (125873) 986 (48279) 1004 (10645) === A{z} === 368 (10867) 488 (10231) 534 (80328) 662 (13307) 978 (14066) === B{0}1 === 710 (15272) 740 (33520) 878 (227482) === B{z} === 153 (21660) 186 (112718) 439 (18752) 593 (16064) 602 (36518) 707 (10573) 717 (67707) === C{0}1 === 68 (656922) 219 (29231) 230 (94751) 312 (21163) 334 (83334) 353 (20262) 359 (61295) 457 (10024) 481 (45941) 501 (20140) 593 (42779) 600 (11242) 604 (17371) 641 (26422) 700 (91953) 887 (13961) 919 (45359) 923 (64365) 992 (10300) === {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) === (none) === {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) === (none) === #{z} (for even bases b, # = b/2−1) === (none) === y{z} === 38 (136212) 83 (21496) 113 (286644) 188 (13508) 401 (103670) 417 (21003) 458 (46900) 494 (21580) 518 (129372) 527 (65822) 602 (17644) 608 (36228) 638 (74528) 663 (47557) 723 (24536) 758 (50564) 833 (12220) 904 (13430) 938 (50008) 950 (16248) === z{0}1 === 202 (46774) 251 (102979) 272 (16681) 297 (14314) 298 (60671) 326 (64757) 347 (69661) 363 (142877) 452 (71941) 543 (10042) 564 (38065) 634 (84823) 788 (13541) 869 (12289) 890 (37377) 953 (60995) 1004 (29685) === {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) === 53 (21942) 124 (16426) 175 (31626) 188 (22036) 316 (48538) 365 (25578) 373 (24006) 434 (10090) 530 (11086) 545 (12346) 560 (15072) 596 (12762) 701 (12576) 706 (10656) 821 (13536) 833 (17116) 966 (14820) 983 (11272) === {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) === (none) === {z}1 === (none) === {z}y === 317 (13896) == Bases 2≤''b''≤1024 which have these families as unsolved families == Unsolved families are families which are neither primes (>''b'') found nor can be ruled out as contain no primes > ''b'' (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) 1{0}1: 38, 50, 62, 68, 86, 92, 98, 104, 122, 144, 168, 182, 186, 200, 202, 212, 214, 218, 244, 246, 252, 258, 286, 294, 298, 302, 304, 308, 322, 324, 338, 344, 354, 356, 362, 368, 380, 390, 394, 398, 402, 404, 410, 416, 422, 424, 446, 450, 454, 458, 468, 480, 482, 484, 500, 514, 518, 524, 528, 530, 534, 538, 552, 558, 564, 572, 574, 578, 580, 590, 602, 604, 608, 620, 622, 626, 632, 638, 648, 650, 662, 666, 668, 670, 678, 684, 692, 694, 698, 706, 712, 720, 722, 724, 734, 744, 746, 752, 754, 762, 766, 770, 792, 794, 802, 806, 812, 814, 818, 836, 840, 842, 844, 848, 854, 868, 870, 872, 878, 888, 896, 902, 904, 908, 922, 924, 926, 932, 938, 942, 944, 948, 954, 958, 964, 968, 974, 978, 980, 988, 994, 998, 1002, 1006, 1014, 1016 (length limit: ≥8388608) 1{0}2: 167, 257, 323, 353, 383, 527, 557, 563, 623, 635, 647, 677, 713, 719, 803, 815, 947, 971, 1013 (length limit: 2000) 1{0}3: 646, 718, 998 (length limit: 2000) 1{0}4: 139, 227, 263, 315, 335, 365, 485, 515, 647, 653, 683, 773, 789, 797, 815, 857, 875, 893, 939, 995, 1007 (length limit: 2000) 1{0}5 1{0}6 1{0}7 1{0}8 1{0}9 1{0}A 1{0}B 1{0}C 1{0}D 1{0}E 1{0}F 1{0}G 1{0}z: 173, 179, 257, 277, 302, 333, 362, 392, 422, 452, 467, 488, 512, 527, 545, 570, 575, 614, 622, 650, 677, 680, 704, 707, 734, 740, 827, 830, 851, 872, 886, 887, 902, 904, 908, 929, 932, 942, 947, 949, 962, 973, 1022 (length limit: 2000) 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 198, 213, 318, 327, 353, 375, 513, 591, 647, 732, 734, 738, 759, 948, 951, 957, 1013, 1014 (length limit: 2000) 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}): 575 (length limit: 247000) 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 813, 863, 962, 1017 (length limit: ≥100000) {1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z): 137, 161, 167, 217, 229, 232, 253, 261, 317, 325, 337, 347, 355, 375, 403, 411, 421, 427, 457, 479, 483, 505, 507, 537, 547, 577, 597, 599, 601, 613, 627, 631, 632, 641, 643, 649, 657, 679, 688, 697, 707, 711, 729, 733, 737, 742, 762, 773, 787, 793, 797, 817, 819, 841, 843, 853, 859, 861, 874, 877, 895, 899, 907, 913, 916, 917, 927, 957, 959, 997, 1003, 1009, 1015, 1017 (length limit: 2000) {1}: 185, 269, 281, 380, 384, 385, 394, 452, 465, 511, 574, 601, 631, 632, 636, 711, 713, 759, 771, 795, 861, 866, 881, 938, 948, 951, 956, 963, 1005, 1015 (length limit: ≥100000) 11{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}): 31, 61, 91, 93, 143, 247, 253, 293, 313, 329, 371, 383, 391, 393, 403, 415, 435, 443, 451, 491, 493, 513, 523, 527, 537, 541, 553, 565, 581, 587, 601, 613, 615, 623, 627, 635, 663, 729, 735, 757, 763, 775, 783, 823, 843, 865, 873, 877, 883, 897, 931, 941, 943, 955, 983, 1013, 1015, 1021, 1023 (length limit: 2000) {1}z 1{2}: 265, 355, 379, 391, 481, 649, 661, 709, 745, 811, 877, 977 (length limit: 2000) 1{3}: 107, 133, 179, 281, 305, 365, 473, 485, 487, 491, 535, 541, 601, 617, 665, 737, 775, 787, 802, 827, 905, 911, 928, 953, 955, 995 1{4}: 83, 143, 185, 239, 269, 293, 299, 305, 319, 325, 373, 383, 395, 431, 471, 503, 551, 577, 581, 593, 605, 617, 631, 659, 743, 761, 773, 781, 803, 821, 857, 869, 897, 911, 917, 923, 935, 983, 1019 (length limit: 2000) 1{z}: 581, 992, 1019 (length limit: ≥100000) 2{0}1: 365, 383, 461, 512, 542, 647, 773, 801, 836, 878, 908, 914, 917, 947, 1004 (length limit: ≥100000) 2{0}3: 79, 149, 179, 254, 359, 394, 424, 434, 449, 488, 499, 532, 554, 578, 664, 683, 694, 749, 794, 839, 908, 944, 982 (length limit: 2000) 2{1} (not quasi-minimal prime if there is smaller prime of the form {1}): 109, 117, 137, 147, 157, 175, 177, 201, 227, 235, 256, 269, 271, 297, 310, 331, 335, 397, 417, 427, 430, 437, 442, 451, 465, 467, 481, 502, 517, 547, 557, 567, 572, 577, 591, 597, 607, 627, 649, 654, 655, 667, 679, 687, 691, 697, 715, 727, 739, 759, 766, 782, 787, 796, 797, 808, 817, 821, 829, 841, 852, 877, 881, 899, 903, 907, 937, 947, 955, 1007, 1011, 1021 (length limit: 2000) {2}1: 106, 238, 262, 295, 364, 382, 391, 397, 421, 458, 463, 478, 517, 523, 556, 601, 647, 687, 754, 790, 793, 832, 872, 898, 962, 1002, 1021 (length limit: 2000) 2{z}: 588, 972 (length limit: ≥100000) 3{0}1: 718, 912 (length limit: ≥100000) 3{0}2: 223, 283, 359, 489, 515, 529, 579, 619, 669, 879, 915, 997 (length limit: 2000) 3{0}4: 167, 391, 447, 487, 529, 653, 657, 797, 853, 913, 937 (length limit: 2000) {3}1: 79, 101, 189, 215, 217, 235, 243, 253, 255, 265, 313, 338, 341, 378, 379, 401, 402, 413, 489, 498, 499, 508, 525, 535, 589, 591, 599, 611, 621, 635, 667, 668, 681, 691, 711, 717, 719, 721, 737, 785, 804, 805, 813, 831, 835, 837, 849, 873, 911, 915, 929, 933, 941, 948, 959, 999, 1013, 1019 (length limit: 2000) 3{z}: 275, 438, 647, 653, 812, 927, 968 (length limit: ≥100000) 4{0}1: 32, 53, 155, 174, 204, 212, 230, 332, 334, 335, 395, 467, 512, 593, 767, 803, 848, 875, 1024 (length limit: ≥100000) 4{0}3: 83, 88, 97, 167, 188, 268, 289, 293, 412, 419, 425, 433, 503, 517, 529, 548, 613, 620, 622, 650, 668, 692, 706, 727, 763, 818, 902, 913, 937, 947, 958 (length limit: 2000) {4}1: 46, 77, 103, 107, 119, 152, 198, 203, 211, 217, 229, 257, 263, 291, 296, 305, 332, 371, 374, 407, 413, 416, 440, 445, 446, 464, 467, 500, 542, 545, 548, 557, 566, 586, 587, 605, 611, 614, 632, 638, 641, 653, 659, 698, 701, 731, 733, 736, 755, 786, 812, 820, 821, 827, 830, 887, 896, 899, 901, 922, 923, 935, 941, 953, 977, 983, 991, 1004 (length limit: 2000) 4{z}: 338, 998 (length limit: ≥100000) 5{0}1: 308, 512, 824 (length limit: ≥100000) 5{z}: 234, 412, 549, 553, 573, 619, 750, 878, 894, 954 (length limit: ≥100000) 6{0}1: 212, 509, 579, 625, 774, 794, 993, 999 (length limit: ≥100000) 6{z}: 308, 392, 398, 518, 548, 638, 662, 878 (length limit: ≥100000) 7{0}1: (none) 7{z}: 321, 328, 374, 432, 665, 697, 710, 721, 727, 728, 752, 800, 815, 836, 867, 957, 958, 972 (length limit: ≥100000) 8{0}1: 86, 140, 182, 263, 353, 368, 389, 395, 422, 426, 428, 434, 443, 488, 497, 558, 572, 575, 593, 606, 698, 710, 746, 758, 770, 773, 824, 828, 866, 911, 930, 953, 957, 983, 993, 1014 (length limit: ≥100000) 8{z}: 378, 438, 536, 566, 570, 592, 636, 688, 718, 830, 852, 926, 1010 (length limit: ≥100000) 9{0}1: 724, 884 (length limit: ≥100000) 9{z}: 80, 233, 530, 551, 611, 899, 912, 980 (length limit: ≥100000) A{0}1: 185, 338, 417, 432, 614, 668, 773, 863, 935, 1000 (length limit: ≥100000) A{z}: 214, 422, 444, 452, 458, 542, 638, 668, 804, 872, 950, 962 (length limit: ≥100000) B{0}1: 560, 770, 968 (length limit: ≥100000) B{z}: 263, 615, 912, 978 (length limit: ≥100000) C{0}1: 163, 207, 354, 362, 368, 480, 620, 692, 697, 736, 753, 792, 978, 998, 1019, 1022 (length limit: ≥100000) C{z} D{0}1 D{z} E{0}1 E{z} F{0}1 F{z} G{0}1 {&#35;}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3): 808, 829, 859, 1006 (length limit: 2000) {&#35;}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2): 31, 37, 55, 63, 67, 77, 83, 89, 91, 93, 97, 99, 107, 109, 117, 123, 127, 133, 135, 137, 143, 147, 149, 151, 155, 161, 177, 179, 183, 189, 193, 197, 207, 211, 213, 215, 217, 223, 225, 227, 233, 235, 241, 247, 249, 255, 257, 263, 265, 269, 273, 277, 281, 283, 285, 287, 291, 293, 297, 303, 307, 311, 319, 327, 347, 351, 355, 357, 359, 361, 367, 369, 377, 381, 383, 385, 387, 389, 393, 397, 401, 407, 411, 413, 417, 421, 423, 437, 439, 443, 447, 457, 465, 467, 469, 473, 475, 481, 483, 489, 493, 495, 497, 509, 511, 515, 533, 541, 547, 549, 555, 563, 591, 593, 597, 601, 603, 611, 615, 619, 621, 625, 627, 629, 633, 635, 637, 645, 647, 651, 653, 655, 659, 663, 667, 671, 673, 675, 679, 683, 687, 691, 693, 697, 707, 709, 717, 731, 733, 735, 737, 741, 743, 749, 753, 755, 757, 759, 765, 767, 771, 773, 775, 777, 783, 785, 787, 793, 797, 801, 807, 809, 813, 817, 823, 825, 849, 851, 853, 865, 867, 873, 877, 887, 889, 893, 897, 899, 903, 907, 911, 915, 923, 927, 933, 937, 939, 941, 943, 945, 947, 953, 957, 961, 967, 975, 977, 983, 987, 993, 999, 1003, 1005, 1009, 1017 (length limit: ≥262143) &#35;{z} (for even bases ''b'', # = ''b''/2−1): 108, 278, 296, 338, 386, 494, 626, 920 (length limit: 2000) ${&#35;} (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) x{z} y{z}: 128, 233, 268, 383, 478, 488, 533, 554, 665, 698, 779, 863, 878, 932, 941, 1010 (length limit: ≥200000) z{0}1: 123, 342, 362, 422, 438, 479, 487, 512, 542, 602, 757, 767, 817, 830, 872, 893, 932, 992, 997, 1005, 1007 (length limit: ≥100000) {y}z: 143, 173, 176, 213, 235, 248, 253, 279, 327, 343, 353, 358, 373, 383, 401, 413, 416, 427, 439, 448, 453, 463, 481, 513, 522, 527, 535, 547, 559, 565, 583, 591, 598, 603, 621, 623, 653, 659, 663, 679, 691, 698, 711, 743, 745, 757, 768, 785, 793, 796, 801, 808, 811, 821, 835, 845, 847, 853, 856, 883, 898, 903, 927, 955, 961, 971, 973, 993, 1005, 1013, 1019, 1021 (length limit: 2000) {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family): 97, 103, 113, 186, 187, 220, 304, 306, 309, 335, 414, 416, 428, 433, 445, 459, 486, 498, 539, 550, 557, 587, 592, 597, 598, 617, 624, 637, 659, 665, 671, 677, 696, 717, 726, 730, 740, 754, 766, 790, 851, 873, 890, 914, 923, 929, 943, 944, 965, 984, 985, 996, 1004, 1005 (length limit: ≥17326) zy{z} (not quasi-minimal prime if there is smaller prime of the form y{z}) {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y): 215, 353, 517, 743, 852, 899, 913 (length limit: 2000) {z}01 (not quasi-minimal prime if there is smaller prime of the form {z}1) {z}1: 93, 113, 152, 158, 188, 217, 218, 226, 227, 228, 233, 240, 275, 278, 293, 312, 338, 350, 353, 383, 404, 438, 464, 471, 500, 533, 576, 614, 641, 653, 704, 723, 728, 730, 758, 779, 788, 791, 830, 878, 881, 899, 908, 918, 929, 944, 953, 965, 968, 978, 983, 986, 1013 (length limit: 2000) {z}k {z}l {z}m {z}n {z}o {z}p {z}q {z}r {z}s {z}t {z}u {z}v {z}w: 207, 221, 293, 375, 387, 533, 633, 647, 653, 687, 701, 747, 761, 785, 863, 897, 905, 965, 1017 (length limit: 2000) {z}x: (none) {z}y: 305, 353, 397, 485, 487, 535, 539, 597, 641, 679, 731, 739, 755 (length limit: 2000) == List of lengths for quasi-minimal primes in some simple families == [https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list of lengths for quasi-minimal primes in some simple families for bases 2≤''b''≤1024] NB: this family is not interpretable in this base (e.g. family 7{0}1 and 7{z} in bases <=7, family {z}x in bases <=3) (including the case which this family has either leading zeros (leading zeros do not count) or ending zeros (numbers ending in zero cannot be prime > base) in this base) RC: this family can be proven to only contain composite numbers (only count numbers > base) unknown: this family has no primes or PRPs found, nor can this family be proven to only contain composite numbers (only count numbers > base) Background color: red for title (bases or families), green for length > 10000, orange for 2500 < length ≤ 10000, white for length ≤ 2500, cyan for "RC", pink for "NB", yellow for "unknown". Search limit for lengths: ≥8388608 for 1{0}1, ≥200000 for y{z}, ≥100000 for ''d''{0}1 (''d'' = one of digits in {2, 3, 4, 5, 6, 7, 8, 9, A, B, C}) and ''d''{z} (''d'' = one of digits in {1, 2, 3, 4, 5, 6, 7, 8, 9, A, B}) and z{0}1 and {1}, ≥5000 for 1{0}2, {z}y, 1{0}z, {z}1, {y}z, ≥2500 for other families. == References == * [https://mersenneforum.org/showthread.php?t=24972 mersenneforum thread of this problem] * [https://docs.google.com/document/d/e/2PACX-1vQct6Hx-IkJd5-iIuDuOKkKdw2teGmmHW-P75MPaxqBXB37u0odFBml5rx0PoLa0odTyuW67N_vn96J/pub Minimal elements for the base ''b'' representations of the primes which are > ''b'' for bases ''b''≤16] * [https://primes.utm.edu/glossary/xpage/MinimalPrime.html article “minimal prime” in The Prime Glossary] * [https://en.wikipedia.org/wiki/Minimal_prime_(recreational_mathematics article “minimal prime” in Wikipedia] * [https://www.primepuzzles.net/puzzles/puzz_178.htm the puzzle of minimal primes (when the restriction of prime>base is not required) in The Prime Puzzles & Problems Connection] * [https://www.primepuzzles.net/problems/prob_083.htm the problem of minimal primes in The Prime Puzzles & Problems Connection] * [https://github.com/xayahrainie4793/non-single-digit-primes my data for these M(Lb) sets for 2 ≤ b ≤ 16] * [http://www.cs.uwaterloo.ca/~shallit/Papers/minimal5.pdf Shallit’s proof of base 10 minimal primes, when the restriction of prime>base is not required] * [https://scholar.colorado.edu/downloads/hh63sw661 proofs of minimal primes in bases b≤10, when the restriction of prime>base is not required] * [https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://cs.uwaterloo.ca/~cbright/talks/minimal-slides.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://doi.org/10.1080/10586458.2015.1064048 the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://github.com/curtisbright/mepn-data data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 30, when the restriction of prime>base is not required, search limits of lengths: 1000000 for b=17, 707000 for b=19, 506000 for b=21, 292000 for b=25, 486000 for b=26, 543000 for b=28, 233000 for b=29] * [https://github.com/RaymondDevillers/primes data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 50, when the restriction of prime>base is not required, search limits of lengths: 10000 for all b] * [http://www.bitman.name/math/article/730 article for minimal primes, when the restriction of prime>base is not required] * [http://www.bitman.name/math/table/497 data for minimal primes in bases 2 ≤ b ≤ 16, when the restriction of prime>base is not required] * [http://www.prothsearch.com/sierp.html the Sierpinski problem] * [http://www.prothsearch.com/rieselprob.html the Riesel problem] * [https://oeis.org/A076336/a076336c.html the dual Sierpinski problem] * [http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm generalized Sierpinski conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &gt; b] * [http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm generalized Riesel conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &gt; b] * [http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm list for the status of the generalized Sierpinski conjectures and the generalized Riesel conjectures in bases b≤1030] * [https://www.utm.edu/staff/caldwell/preprints/2to100.pdf article for generalized Sierpinski conjectures in bases b≤100] * [http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf article for the mixed (original+dual) Sierpinski problem] * [http://www.fermatquotient.com/PrimSerien/GenRepu.txt generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤160, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://jeppesn.dk/generalized-fermat.html generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://www.noprimeleftbehind.net/crus/GFN-primes.htm generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1030, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt list of generalized half Fermat primes (primes of the form (b2^n+1)/2) sorted by n, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://harvey563.tripod.com/wills.txt primes of the form (b−1)*bn−1 for bases b≤2049, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Williams_prime_MM_least the smallest primes of the form (b−1)*bn−1 for bases b≤2049, these primes (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Williams_prime_MP_least the smallest primes of the form (b−1)*bn+1 for bases b≤1024, these primes (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n the smallest primes of the form k*bn−1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b>k] * [https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n the smallest primes of the form k*bn+1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b>k] * [https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list for the smallest primes in given simple family in bases b≤1024] * [https://www.rose-hulman.edu/~rickert/Compositeseq/ a problem related to this project] * [http://www.worldofnumbers.com/Appending%201s%20to%20n.txt a problem related to this project] * [https://stdkmd.net/nrr/prime/primecount.txt near- and quasi- repdigit (probable) primes sorted by count] * [https://stdkmd.net/nrr/prime/primedifficulty.txt near- and quasi- repdigit (probable) primes sorted by difficulty] * [http://www.prothsearch.com/fermat.html factoring status of Fermat numbers] * [http://www.rieselprime.de/dl/CRUS_pack.zip srsieve, sr1sieve, sr2sieve, pfgw, and llr softwares] * [https://www.bc-team.org/app.php/dlext/?cat=3 srsieve, sr1sieve, sr2sieve, sr5sieve software] * [https://sourceforge.net/projects/openpfgw/ pfgw software] * [http://jpenne.free.fr/index2.html llr software] * [http://www.ellipsa.eu/public/primo/primo.html PRIMO software] * [https://primes.utm.edu/prove/index.html website for primality proving] * [https://primes.utm.edu/curios/page.php?number_id=22380 the largest base 10 minimal prime in Prime Curios!] * [https://oeis.org/A071062 OEIS sequence for base 10 minimal primes, when the restriction of prime>base is not required] * [https://oeis.org/A326609 OEIS sequence for the largest base b minimal prime, when the restriction of prime>base is not required] * [https://primes.utm.edu/primes/lists/all.txt top proven primes] * [http://www.primenumbers.net/prptop/prptop.php top PRPs] * [http://factordb.com online factor database, including many primes which are minimal primes in a small base] gvezgqxq2v5fy2fg4mut6l3xtrtlt8n 2 274697 2410238 2397509 2022-07-29T14:11:47Z Platos Cave (physics) 2562653 wikitext text/x-wiki '''Natural Planck units as geometrical objects''' [[w:Planck units |Planck unit]] theories use basic units for mass, length, time and charge, and operate at the Planck scale. In a geometrical Planck theory, these basic units are assigned geometrical objects (''MLTA'') rather than numerical values, the advantage being that the geometries themselves can encode the function of the unit, for example the object for length (''L'') will encode the function of ''length'', the geometrical ''L'' is 1 unit of (Planck) length, such that, unlike numerical models, a dimensioned descriptive (i.e.: ''kg, m, s, A'', ... ) is not required. The ''MLTA'' geometrical objects are selected whereby they may interact with each other (the mass object for example is not independent of the length and the time objects). This permits a mathematical relationship between them, and so a physical universe can be constructed [[w:Lego |Lego-style]] by combining the base (Planck unit) ''MLTA'' objects to form more complex objects such as electrons (i.e.: by embedding ''L'' and ''A'' into the geometry of the electron, the electron can have wavelength and charge). Furthermore, these objects can overlap and cancel in a particular ratio (according to that mathematical relationship), and this ratio occurs in the electron. And so, although the electron has physical parameters (wavelength, charge ...), '''the electron itself is a mathematical particle (units = 1)''', not a physical particle. Furthermore, due to this unit-less ratio, for any system of units '''if we know the numerical values of any 2 Planck units, then we can solve the dimensioned constants''' (''G, h, c, e, m_e, k_B'') ... for that system of units. === Geometrical objects === Base units for mass <math>M</math>, length <math>L</math>, time <math>T</math>, and ampere <math>A</math> can be constructed from the geometry of 2 [[w:dimensionless physical constant | dimensionless physical constants]], the (inverse) [[w:fine-structure constant | fine structure constant '''α''']] = 137.036 and [[v:Simulation_argument_(coding_Planck_units)#Omega | Omega]] '''Ω''' = 2.007 134 949 <ref>Macleod, M.J. {{Cite journal |title= Programming Planck units from a mathematical electron; a Simulation Hypothesis |journal=Eur. Phys. J. Plus |volume=113 |pages=278 |date=22 March 2018 | doi=10.1140/epjp/i2018-12094-x }}</ref>. Being independent of any numerical system and of any system of units, these MLTA units would qualify as "natural units"; {{bq|''...ihre Bedeutung für alle Zeiten und für alle, auch außerirdische und außermenschliche Kulturen notwendig behalten und welche daher als »natürliche Maßeinheiten« bezeichnet werden können...'' ...These necessarily retain their meaning for all times and for all civilizations, even extraterrestrial and non-human ones, and can therefore be designated as "natural units"... -Max Planck <ref>Planck (1899), p. 479.</ref><ref name="TOM">*Tomilin, K. A., 1999, "[http://www.ihst.ru/personal/tomilin/papers/tomil.pdf Natural Systems of Units: To the Centenary Anniversary of the Planck System]", 287–296.</ref>}} ==== Objects ==== Each object is assigned a geometry and a dimensioned attribute (the object function); :<math>\beta = (2\pi\Omega)</math> [[v:Planck_units_(geometrical)#u_as_sqrt(velocity/mass) |sqrt(velocity/mass)]] :<math>M = (1)</math> mass :<math>T = (\pi)</math> time :<math>P = (\Omega)</math> [[v:Sqrt_Planck_momentum | sqrt of momentum]] :<math>V = (2\pi\Omega^2)</math> velocity :<math>L = (2\pi^2\Omega^2)</math> length :<math>A = (\frac{2^7 \pi^3 \Omega^3}{\alpha})</math> ampere ==== Mathematical relationship ==== A relationship between the objects is defined using '''uⁿ''' whereby; :<math>(\beta)\;u\;</math> :<math>(M)\;u^{15}\;</math> :<math>(T)\;u^{-30}\;</math> :<math>(P)\;u^{16}\;</math> :<math>(V)\;u^{17}\;</math> :<math>(L)\;u^{-13}\;</math> :<math>(A) \;u^{3}\;</math> ==== Scalars ==== To translate from geometrical objects to a numerical system of units requires scalars ('''kltpva''') that can be assigned numerical values. For example, scalars for the SI units; :If we use '''k''' to convert '''M''' to the SI Planck mass <math>m_P</math> (M = 1k = <math>m_P</math>), then '''k''' = 0.2176728e-7kg and '''<math>u^{15}</math>''' will equate to '''kg'''. :To convert '''V''' = 2πΩ² = 25.3123819 to '''c''' requires scalar '''v''' = 11843707.905m/s ('''V''' ''v'' = 2πΩ²''v'' = 299792458m/s) with '''<math>u^{17}</math>''' equating to '''m/s'''. {| class="wikitable" |+Geometrical units ! Attribute ! Geometrical object ! Scalar ! Unit |- | [[v:Planck_units_(geometrical)#u_as_sqrt(velocity/mass) |sqrt(velocity/mass)]] | <math>\beta = (2\pi \Omega)</math> | | <math>unit = u</math> |- | mass | <math>M = (1)</math> | k | <math>unit = u^{15}</math> |- | time | <math>T = (\pi)</math> | t | <math>unit = u^{-30}</math> |- | [[v:Sqrt_Planck_momentum | sqrt(momentum)]] | <math>P = \frac{\beta M}{2\pi} = (\Omega)</math> | p | <math>unit = u^{16}</math> |- | velocity | <math>V = \frac{\beta^2 M}{2\pi} = (2\pi\Omega^2)</math> | v | <math>unit = u^{17}</math> |- | length | <math>L = TV = (2\pi^2\Omega^2)</math> | l | <math>unit = u^{-13}</math> |- | ampere | <math>A = \frac{16 V^3}{\alpha P^3} = (\frac{2^7 \pi^3 \Omega^3}{\alpha})</math> | a | <math>unit = u^3</math> |} ===== Scalar relationships ===== The following ''uⁿ'' groups cancel ('''units = scalars = 1'''), as such '''only 2 numerical scalars are required''', for example, if we know '''a''' and '''l''' then we know '''t''' ('''t = a³l³'''), and from '''l''' and '''t''' we know '''k'''. :<math>\frac{u^{3*3} u^{-13*3}}{u^{-30}}\;(\frac{a^3 l^3}{t}) = \frac{u^{-13*15}}{u^{15*9} u^{-30*11}} \;(\frac{l^{15}}{k^9 t^{11}}) = \;...\; =1</math> This means that if we know any 2 constants, then we can solve the scalars for those constants, and from those 2 scalars we can solve all the Planck units, and from these the dimensioned physical constants. This will apply to any set of units. In this example, to maintain integer exponents, scalar ''p'' is defined in terms of a scalar ''r''. :<math>r = \sqrt{p} = \sqrt{\Omega},\; unit \;u^{16/2=8}</math> The SI Planck units are known with a low precision, conversely 2 of the CODATA 2014 physical constants have been assigned exact numerical values; ''c'' and permeability of vacuum ''μ₀''. Scalars ''r'' and ''v'' were chosen as they can be derived directly from ''V = c'' and ''μ₀'' = 4π/10^7 (see table [[v:Planck_units_(geometrical)#Physical_constants_(as_geometrical_formulas) |geometrical physical constants]] below). Using α = 137.035 999 139 (CODATA 2014), Ω = 2.007 134 949 636... :<math>v = \frac{c}{2 \pi \Omega^2}= 11 843 707.905 ...,\; units = m/s</math> :<math>r^7 = \frac{2^{11} \pi^5 \Omega^4 \mu_0}{\alpha};\; r = 0.712 562 514 304 ...,\; units = (\frac{kg.m}{s})^{1/4}</math> This gives scalars ''klta'' ([[v:Planck_units_(geometrical)#u_as_sqrt(velocity/mass)) |for derivation of units kg, m, s, A from r, v]]); :<math>k = \frac{r^4}{v}</math> = 0.217 672 817 580... ''x'' 10^-7kg, <math>\;\;\;u^{15} = \frac{(u^8)^4}{u^{17}}</math> :<math>l = \frac{r^9}{v^5}</math> = 0.203 220 869 487... ''x'' 10^-36m, <math>\;\;\;u^{-13} = \frac{(u^8)^9}{(u^{17})^5}</math> :<math>t = \frac{r^9}{v^6}</math> = 0.171 585 512 841... ''x'' 10^-43s, <math>\;\;\;u^{-30} = \frac{(u^8)^9}{(u^{17})^6}</math> :<math>a = \frac{v^3}{r^6}</math> = 0.126 918 588 592... ''x'' 10²³A, <math>\;\;\;u^{3} = \frac{(u^{17})^3}{(u^8)^6}</math> ===== Natural units MLTPA ===== Regardless of which system of units we use, alien or terrestrial, any combination of constants where '''scalars = 1''' (i.e.: the scalars overlap and cancel) will give the same numerical result, they will default to the MLTPA objects. This implies that these objects are Planck's 'natural' units, i.e.: that '''all possible systems of units''' are based on these objects, and so, given that these are geometrical objects, they can be construed as evidence of a mathematical universe. The following are examples of '''units = scalars = 1''' ratios using SI units <ref>Macleod, Malcolm J. {{Cite journal |title= Do the fundamental constants embed evidence of a mathematical universe at the Planck scale? |journal=RG | doi=10.13140/RG.2.2.15874.15041/1 }}</ref>. Note: the geometry <math>\color{red}(\Omega^{15})^n\color{black}</math> (integer n ≥ 0) is common to these ratios. ====== m_P, l_p, t_p ====== In this ratio, the MLT units and ''klt'' scalars both cancel; units = scalars = 1, reverting to the base MLT objects. Setting the scalars ''klt'' to convert the MLT objects to the SI Planck units; :k = 0.217 672 817 580... ''x'' 10^-7kg :l = 0.203 220 869 487... ''x'' 10^-36m :t = 0.171 585 512 841... ''x'' 10^-43s :<math>\frac{L^{15}}{M^{9} T^{11}} = \frac{(2\pi^2\Omega^2)^{15}}{(1)^{9} (\pi)^{11}} (\frac{l^{15}}{k^9 t^{11}}) = \frac{l_p^{15}}{m_P^{9} t_p^{11}} </math> (CODATA 2018 mean) The ''klt'' scalars cancel, leaving; :<math>\frac{L^{15}}{M^{9} T^{11}} = \frac{(2\pi^2\Omega^2)^{15}}{(1)^{9} (\pi)^{11}} (\frac{l^{15}}{k^9 t^{11}}) = 2^{15} \pi^{19} \color{red}(\Omega^{15})^2\color{black} = </math>{{font color|blue|yellow|'''0.109 293... 10²⁴ '''}}, <math>(\frac{l^{15}}{k^9 t^{11}}) = 1, \;\frac{u^{-13*15}}{u^{15*9} u^{-30*11}} = 1</math> Solving for the SI units; :<math>\frac{l_p^{15}}{m_P^{9} t_p^{11}} = \frac{(1.616255e-35)^{15}}{(2.176434e-8)^{9} (5.391247e-44)^{11}} = </math> {{font color|blue|yellow| '''0.109 485... 10²⁴'''}} ====== A, l_p, t_p ====== :a = 0.126 918 588 592... ''x'' 10²³A :<math>\frac{A^3 L^3}{T} = (\frac{2^7 \pi^3 \Omega^3}{\alpha})^3 \frac{(2\pi^2\Omega^2)^3}{(\pi)} (\frac{a^3 l^3}{t}) = \frac{2^{24} \pi^{14} \color{red}(\Omega^{15})^1\color{black}}{\alpha^3} = </math> {{font color|green|yellow| '''0.205 571... 10¹³'''}}, <math>(\frac{a^3 l^3}{t}) = 1,\; \frac{u^{3*3} u^{-13*3}}{u^{-30}} = 1</math> :<math>\frac{(e / t_p)^3 l_p^3}{t_p} = \frac{(1.602176634e-19/5.391247e-44)^3 (1.616255e-35)^3}{(5.391247e-44)} = </math> {{font color|green|yellow| '''0.205 543... 10¹³'''}}, <math>units = \frac{(C/s)^3 m^3}{s} </math> The Planck units are known with low precision, and so by defining the 3 most accurately known dimensioned constants in [[v:Planck_units_(geometrical)#Physical_constants_(as_geometrical_formulas) |terms of these objects]] (c, R = Rydberg constant, <math>\mu_0</math>; CODATA 2014 mean values), we can test to greater precision; ====== c, μ₀, R ====== :<math>\frac{(c^*)^{35}}{(\mu_0^*)^9 (R^*)^7} = (2 \pi \Omega^2 v)^{35}/(\frac{\alpha r^7}{2^{11} \pi^5 \Omega^4})^9 .(\frac{v^5} {2^{23} 3^3 \pi^{11} \alpha^5 \Omega^{17} r^9})^7 = 2^{295} \pi^{157} 3^{21} \alpha^{26} \color{red}(\Omega^{15})^{15}\color{black} = </math> {{font color|red|yellow| '''0.326 103 528 6170... 10³⁰¹'''}}, <math>\frac{(u^{17})^{35}}{(u^{56})^9 (u^{13})^7} = 1, \;(v^{35})/(r^7)^9 (\frac{v^5}{r^9})^7 = 1</math> :<math>\frac{c^{35}}{\mu_0^9 R^7} = \frac{(299792458)^{35}}{(4 \pi/10^7)^9 (10973731.568160)^7} = </math> {{font color|red|yellow| '''0.326 103 528 6170... 10³⁰¹'''}}, <math>units = \frac{m^{33}A^{18}}{s^{17}kg^9} == \frac{(u^{-13})^{33} (u^{3})^{18}}{(u^{-30})^{17} (u^{15})^9} = 1</math> The [[w:2019 redefinition of SI base units | 2019 SI unit revision]] assigned exact numerical values to 4 constants (c, e, k_B, h). {{see also |Planck units (geometrical)#2019 SI unit revision}} From the table [[v:Planck_units_(geometrical)#Physical_constants_(as_geometrical_formulas) |geometrical physical constants]], we get geometrical formulas and scalars for; :<math>h^* = 2 \pi MVL = 2^3 \pi^4 \Omega^4 \frac{r^{13}}{v^5},\; u^{15+17-13 = 19}</math> :<math>e^* = AT = \frac{2^7 \pi^4 \Omega^3}{\alpha}\frac{r^3}{v^3},\; u^{3-30 = -27}</math> :<math>k_B^*= 2 \pi MV/A = \frac{\alpha}{2^5 \pi \Omega} \frac{r^{10}}{v^3},\; u^{17+15-3 = 29}</math> ====== c, e, k_B, h ====== :<math>\frac{(k_B^*) (e^*) (c^*)}{(h^*)} = (\frac{\alpha}{2^5 \pi \Omega} \frac{r^{10}}{v^3}) (\frac{2^7 \pi^4 \Omega^3}{\alpha} \frac{r^3}{v^3}) (2 \pi \Omega^2 v) / (2^3 \pi^4 \Omega^4 \frac{r^{13}}{v^5}) </math> = {{font color|blue|yellow|'''1.0'''}}, <math>\frac{ (u^{29}) (u^{-27}) (u^{17}) }{ (u^{19}) } = 1,\; (\frac{r^{10}}{v^3}) (\frac{r^3}{v^3}) (v) / (\frac{r^{13}}{v^5}) = 1</math> :<math>\frac{k_B e c}{h} = </math> {{font color|blue|yellow|'''1.000 8254'''}}, <math>units = \frac{m C}{s^2 K} == \frac{(u^{-13}) (u^{-27})}{(u^{-30})^2 (u^{20})} = 1</math> ====== c, h, e ====== :<math>\frac{(h^*)^3}{(e^*)^{13} (c^*)^{24}} = (2^3 \pi^4 \Omega^4 \frac{r^{13}}{v^5})^3/(\frac{2^7 \pi^4 \Omega^3 r^3}{\alpha v^3})^7.(2\pi\Omega^2 v)^{24} = \frac{\alpha^{13}}{2^{106} \pi^{64} (\color{red}\Omega^{15})^5\color{black}} = </math> {{font color|green|yellow| '''0.228 473 759... 10^-58'''}}, <math>\frac{(u^{19})^{3}}{(u^{-27})^{13} (u^{17})^{24}} = 1, \;(\frac{r^{13}}{v^5})^3 / (\frac{r^3}{v^3})^{13} (v^{24}) = 1</math> :<math>\frac{h^3}{e^{13} c^{24}} = </math> {{font color|green|yellow| '''0.228 473 639... 10^-58'''}}, <math>units = \frac{kg^3 s^{21}}{m^{18} C^{13}} == \frac{(u^{15})^3 (u^{-30})^{21}}{(u^{-13})^{18} (u^{-27})^{13}} = 1</math> ====== m_e, λ_e ====== :<math>\sigma_{e} = \frac{3 \alpha^2 A L}{2\pi^2} = {2^7 3 \pi^3 \alpha \Omega^5}\frac{r^3}{v^2},\; u^{-10}</math> :<math>f_e = \frac{\sigma_{e}^3}{2 T} = 2^{20} 3^3 \pi^8 \alpha^3 (\color{red}\Omega^{15})\color{black},\; \frac{(u^{-10})^3}{u^{-30}} =1,\; (\frac{r^3}{v^2})^3 \frac{v^6}{r^9} = 1</math> :<math>(m_e^*) = \frac{M}{f_e} = \color{blue}9.109\;382\;3227 \;10^{-31}\color{black}\;u^{15}</math> :<math>(m_e^*) = \frac{2^3 \pi^5 (h^*)}{3^3 \alpha^6 (e^*)^3 (c^*)^5} = \frac{1}{2^{20} \pi^8 3^3 \alpha^3 (\color{red}\Omega^{15})\color{black}} \frac{r^4 u^{15}}{v} = \color{blue}9.109\;382\;3227 \;10^{-31}\color{black}\;u^{15}</math> :<math>m_e = \color{blue}9.109\;383\;7015... \;10^{-31}\color{black}\;kg</math> :<math>(\lambda_e^*) = 2 \pi L f_e = \color{purple}2.426\;310\;238\;667 \;10^{-12}\color{black}\;u^{-13}</math> :<math>\lambda_e = \frac{h}{m_e c} = \color{purple}2.426 \;310 \;238 \;67 \;10^{-12}\color{black}\;m</math> ====== c, e, m_e ====== :<math>(m_e^*)= \frac{M}{f_e}, \;f_e = 2^{20} 3^3 \pi^8 \alpha^3 (\color{red}\Omega^{15})^1\color{black} </math>, units = scalars = 1 ([[v:Planck_units_(geometrical)#Electron_formula |m_e formula]]) :<math>\frac{(c^*)^9 (e^*)^4}{(m_e^*)^3} = 2^{97} \pi^{49} 3^9 \alpha^5 (\color{red}\Omega^{15})^5\color{black} = </math> {{font color|red|yellow| '''0.170 514 368... 10⁹²'''}}, <math>\frac{(u^{17})^9 (u^{-27})^4}{(u^{15})^3} = 1,\; (v^9) (\frac{r^3}{v^3})^4 / (\frac{r^4}{v})^3 = 1</math> :<math>\frac{c^9 e^4}{m_e^3} = </math> {{font color|red|yellow| '''0.170 514 342... 10⁹²'''}}, <math>units = \frac{m^9 C^4}{s^9 kg^3} == \frac{(u^{-13})^9 (u^{-27})^4}{(u^{-30})^9 (u^{15})^3} = 1</math> ====== k_B, c, e, m_e ====== :<math>\frac{(k_B^*)}{(e^*)^2 (m_e^*) (c^*)^4} = \frac{3^3 \alpha^6}{2^3 \pi^5} = </math> {{font color|blue|yellow| '''73 035 235 897.'''}}, <math>\frac{(u^{29})}{(u^{-27})^2 (u^{15}) (u^{17})^4} = 1,\; (\frac{r^{10}}{v^3}) / (\frac{r^3}{v^3})^2 (\frac{r^4}{v}) (v)^4 = 1</math> :<math>\frac{k_B}{e^2 m_e c^4} = </math> {{font color|blue|yellow| '''73 095 507 858.'''}}, <math>units = \frac{s^2}{m^2 K C^2} == \frac{(u^{-30})^2}{(u^{-13})^2 (u^{20}) (u^{-27})^2} = 1</math> ====== m_P, t_p, ε₀ ====== These 3 constants, Planck mass, Planck time and the vacuum permittivity have no Omega term. :<math>\frac{M^4 (\epsilon_0^*)}{T} = (1) (\frac{2^9 \pi^3}{\alpha}) / (\pi) = \frac{2^9 \pi^2}{\alpha} = </math> {{font color|green|yellow| '''36.875'''}}, <math>\frac{(u^{15})^4 (u^{-90})}{(u^{-30})} = 1,\; (\frac{r^4}{v})^4 (\frac{1}{r^7 v^2}) / (\frac{r^9}{v^6}) = 1</math> :<math>\frac{m_p^4 (\epsilon_0)}{t_p} = </math> {{font color|green|yellow| '''36.850'''}}, <math>units = \frac{kg^4}{s} \frac{s^4 A^2}{m^3 kg} = \frac{kg^3 A^2 s^3}{m^3} == \frac{(u^{15})^3 (u^{3})^2 (u^{-30})^3}{(u^{-13})^3} = 1</math> ====== G, h, c, e, m_e, K_B ====== :<math>\frac{(h^*) (c^*)^2 (e^*) (m_e^*)}{(G^*)^2 (k_B^*)} = (m_e^*) (\frac{2^{11} \pi^3}{\alpha^2}) = </math> {{font color|red|yellow| '''0.1415... 10^-21'''}}, <math>\frac{ (u^{19}) (u^{17})^2 (u^{-27}) (u^{15}) }{ (u^{6})^2 (u^{29}) } = 1,\; (\frac{r^{13}}{v^5}) v^2 (\frac{r^{3}}{v^3})(\frac{r^{4}}{v^1}) / (\frac{r^5}{v^2})^2 (\frac{r^{10}}{v^3}) = 1</math> :<math>\frac{h c^2 e m_e}{G^2 k_B} = </math> {{font color|red|yellow| '''0.1413... 10^-21'''}}, <math>units = \frac{kg^3 s^3 C K}{m^4} == \frac{(u^{15})^3 (u^{-30})^3 (u^{-27}) (u^{20}) }{(u^{-13})^4} = 1</math> ====== α ====== :<math>\frac{2 (h^*)}{(\mu_0^*) (e^*)^2 (c^*)} = 2({2^3 \pi^4 \Omega^4})/(\frac{\alpha}{2^{11} \pi^5 \Omega^4})(\frac{2^{7} \pi^4 \Omega^3}{\alpha})^2(2 \pi \Omega^2) = \color{blue}\alpha \color{black},\; \frac{u^{19}}{u^{56} (u^{-27})^2 u^{17}} = 1,\; (\frac{r^{13}}{v^5})(\frac{1}{r^7})(\frac{v^6}{r^6})(\frac{1}{v}) = 1</math> Note: This will apply to any combinations of constants where '''units = scalars = 1'''. ===== SI Planck unit scalars ===== :<math>M = m_P = (1)k;\; k = m_P = .217\;672\;817\;58... \;10^{-7},\; u^{15}\; (kg)</math> :<math>T = t_p = {\pi}t;\; t = \frac{t_p}{\pi} = .171\;585\;512\;84... 10^{-43},\; u^{-30}\; (s)</math> :<math>L = l_p = {2\pi^2\Omega^2}l;\; l = \frac{l_p}{2\pi^2\Omega^2} = .203\;220\;869\;48... 10^{-36},\; u^{-13}\; (m)</math> :<math>V = c = {2\pi\Omega^2}v;\; v = \frac{c}{2\pi\Omega^2} = 11\;843\;707.905... ,\; u^{17}\; (m/s)</math> :<math>A = e/t_p = (\frac{2^7 \pi^3 \Omega^3}{\alpha})a = .126\;918\;588\;59... 10^{23},\; u^{3}\; (A)</math> Example MLT (units = scalars = 1); :<math>\frac{l^{15}}{k^9 t^{11}} = \frac{(.203...x10^{-36})^{15}}{(.217...x10^{-7})^9 (.171...x10^{-43})^{11}} \frac{u^{- 13*15}}{u^{15*9} u^{-30*11}} = 1</math> Example ALT (units = scalars = 1); :<math>\frac{a^3 l^3}{t} = \frac{(.126...x10^{23})^3 (.203...x10^{-36})^3}{ (.171...x10^{-43})} \frac{u^{3*3} u^{-13*3}} {u^{-30}} = 1</math> ===== MT to LPVA ===== In this example LPVA are derived from MT. The formulas for MT; :<math>M = (1)k,\; unit = u^{15}</math> :<math>T = (\pi) t,\; unit = u^{-30}</math> Replacing scalars ''pvla'' with ''kt'' :<math>P = (\Omega)\;\frac{k^{12/15}}{t^{2/15}},\; unit = u^{12/15*15-2/15*(-30)=16}</math> :<math>V = \frac{2 \pi P^2}{M} = (2 \pi \Omega^2)\; \frac{k^{9/15}}{t^{4/15}},\; unit = u^{9/15*15-4/15*(-30)=17} </math> :<math>L = T V = (2 \pi^2 \Omega^2) \; k^{9/15} t^{11/15},\; unit = u^{9/15*15+11/15*(-30)=-13}</math> :<math>A = \frac{2^4 V^3}{\alpha P^3} = \left(\frac{2^7 \pi^3 \Omega^3}{\alpha}\right)\; \frac{1}{k^{3/5} t^{2/5}},\; unit = u^{9/15*(-15)+6/15*30=3} </math> ===== PV to MTLA ===== In this example MLTA are derived from PV. The formulas for PV; :<math>P = (\Omega)p,\; unit = u^{16}</math> :<math>V = (2\pi\Omega^2)v,\; unit = u^{17}</math> Replacing scalars ''klta'' with ''pv'' :<math>M = \frac{2\pi P^2}{V} = (1)\frac{p^2}{v},\; unit = u^{16*2-17=15} </math> :<math>T = (\pi) \frac{p^{9/2}}{v^6},\; unit = u^{16*9/2-17*6=-30} </math> :<math>L = T V = (2\pi^2\Omega^2)\frac{p^{9/2}}{v^5},\; unit = u^{16*9/2-17*5=-13}</math> :<math>A = \frac{2^4 V^3}{\alpha P^3} = (\frac{2^7 \pi^3 \Omega^3}{\alpha})\frac{v^3}{p^3},\; unit = u^{17*3-16*3=3}</math> ==== Physical constants (as geometrical formulas) ==== note: <math>\color{red}(u^{15})^n\color{black}</math> constants have no Omega term. {| class="wikitable" |+Dimensioned constants; geometrical vs CODATA 2014 ! Constant ! In Planck units ! Geometrical object ! SI calculated (r, v, Ω, α^*) ! SI CODATA 2014 <ref>[http://www.codata.org/] | CODATA, The Committee on Data for Science and Technology | (2014)</ref> |- | [[w:Speed of light | Speed of light]] | V | <math>c^* = (2\pi\Omega^2)v,\;u^{17} </math> | ''c^*'' = 299 792 458, unit = u¹⁷ | ''c'' = 299 792 458 (exact) |- | [[w:Fine structure constant | Fine structure constant]] | | | ''α^*'' = 137.035 999 139 (mean) | ''α'' = 137.035 999 139(31) |- | [[w:Rydberg constant | Rydberg constant]] | <math>R^* = (\frac{m_e}{4 \pi L \alpha^2 M})</math> | <math>R^* = \frac{1}{2^{23} 3^3 \pi^{11} \alpha^5 \Omega^{17}}\frac{v^5}{r^9},\;u^{13} </math> | ''R^*'' = 10 973 731.568 508, unit = u¹³ | ''R'' = 10 973 731.568 508(65) |- | [[w:Vacuum permeability | Vacuum permeability]] | <math>\mu_0^* = \frac{4 \pi V^2 M}{\alpha L A^2}</math> | <math>\mu_0^* = \frac{\alpha}{2^{11} \pi^5 \Omega^4} r^7,\; u^{17*2+15+13-6=7*8=56}</math> | ''μ₀^*'' = 4π/10^7, unit = u⁵⁶ | ''μ₀'' = 4π/10^7 (exact) |- | [[w:Vacuum permittivity | Vacuum permittivity]] | <math>\epsilon_0^* = \frac{1}{\mu_0^* (c^*)^2}</math> | <math>\epsilon_0^* = \frac{2^9 \pi^3}{\alpha}\frac{1}{r^7 v^2},\; \color{red}1/(u^{15})^6\color{black} = u^{-90}</math> | | |- | [[w:Planck constant | Planck constant]] | <math>h^* = 2 \pi M V L</math> | <math>h^* = 2^3 \pi^4 \Omega^4 \frac{r^{13}}{v^5},\; u^{15+17-13 = 8*13-17*5 = 19}</math> | ''h^*'' = 6.626 069 134 e-34, unit = u¹⁹ | ''h'' = 6.626 070 040(81) e-34 |- | [[w:Gravitational constant | Gravitational constant]] | <math>G^* = \frac{V^2 L}{M}</math> | <math>G^* = 2^3 \pi^4 \Omega^6 \frac{r^5}{v^2},\; u^{34-13-15 = 8*5-17*2 = 6}</math> | ''G^*'' = 6.672 497 192 29 e11, unit = u⁶ | ''G'' = 6.674 08(31) e-11 |- | [[w:Elementary charge | Elementary charge]] | <math>e^* = A T</math> | <math>e^* = \frac{2^7 \pi^4 \Omega^3}{\alpha}\frac{r^3}{v^3},\; u^{3-30=3*8-17*3=-27}</math> | ''e^*'' = 1.602 176 511 30 e-19, unit = u^-19 | ''e'' = 1.602 176 620 8(98) e-19 |- | [[w:Boltzmann constant | Boltzmann constant]] | <math>k_B^* = \frac{2 \pi V M}{A}</math> | <math>k_B^* = \frac{\alpha}{2^5 \pi \Omega} \frac{r^{10}}{v^3},\; u^{17+15-3=10*8-17*3=29}</math> | ''k_B^*'' = 1.379 510 147 52 e-23, unit = u²⁹ | ''k_B'' = 1.380 648 52(79) e-23 |- | [[w:Electron mass | Electron mass]] | | <math>m_e^* = \frac{M}{f_e},\; u^{15}</math> | ''m_e^*'' = 9.109 382 312 56 e-31, unit = u¹⁵ | ''m_e'' = 9.109 383 56(11) e-31 |- | [[w:Classical electron radius | Classical electron radius]] | | <math>\lambda_e^* = 2\pi L f_e,\; u^{-13}</math> | ''λ_e^*'' = 2.426 310 2366 e-12, unit = u^-13 | ''λ_e'' = 2.426 310 236 7(11) e-12 |- | [[w:Planck temperature | Planck temperature]] | <math>T_p^* = \frac{A V}{\pi}</math> | <math>T_p^* = \frac{2^7 \pi^3 \Omega^5}{\alpha} \frac{v^4}{r^6} ,\; u^{3+17=17*4-6*8=20} </math> | ''T_p^*'' = 1.418 145 219 e32, unit = u²⁰ | ''T_p'' = 1.416 784(16) e32 |- | [[w:Planck mass | Planck mass]] | M | <math>m_P^* = (1)\frac{r^4}{v} ,\; \color{red}(u^{15})^1\color{black}</math> | ''m_P^*'' = .217 672 817 580 e-7, unit = u¹⁵ | ''m_P'' = .217 647 0(51) e-7 |- | [[w:Planck length | Planck length]] | L = TV | <math>l_p^* = (2\pi^2\Omega^2)\frac{r^9}{v^5},\;u^{-13} </math> | ''l_p^*'' = .161 603 660 096 e-34, unit = u^-13 | ''l_p'' = .161 622 9(38) e-34 |- | [[w:Planck time | Planck time]] | T | <math>t_p^* = (\pi)\frac{r^9}{v^6} ,\; \color{red}1/(u^{15})^2\color{black} </math> | ''t_p^*'' = 5.390 517 866 e-44, unit = u^-30 | ''t_p'' = 5.391 247(60) e-44 |- | [[w:Ampere | Ampere]] | <math>A = \frac{16 V^3}{\alpha P^3}</math> | <math>A^* = \frac{2^7\pi^3\Omega^3}{\alpha}\frac{v^3}{r^6} ,\; u^3 </math> | A^* = 0.297 221 e25, unit = u³ | ''e/t_p'' = 0.297 181 e25 |- | [[w:Quantum Hall effect | Von Klitzing constant ]] | <math>R_K^* = (\frac{h}{e^2})^*</math> | <math>R_K^* = \frac{\alpha^2}{2^{11} \pi^4 \Omega^2} r^7 v ,\; u^{73}</math> | ''R_K^*'' = 25812.807 455 59, unit = u⁷³ | ''R_K'' = 25812.807 455 5(59) |- | [[w:Gyromagnetic ratio | Gyromagnetic ratio]] | | <math>\gamma_e/2\pi = \frac{g l_p^* m_P^*}{2 k_B^* m_e^*},\; unit = u^{-42}</math> | ''γ_e/2π^*'' = 28024.953 55, unit = u^-42 | ''γ_e/2π'' = 28024.951 64(17) |} Note that ''r, v, Ω, α'' are dimensionless numbers, however when we replace ''u''ⁿ with the SI unit equivalents (''u''¹⁵ → kg, ''u''^-13 → m, ''u''^-30 → s, ...), the ''geometrical objects'' (i.e.: ''c^*'' = 2πΩ²v = 299792458, units = u¹⁷) become '''indistinguishable''' from their respective ''physical constants'' (i.e.: ''c'' = 299792458, units = m/s). If this mathematical relationship can therefore be identified within the SI units themselves, then we have an argument for a Planck scale mathematical universe <ref>[https://codingthecosmos.com/planck-scale.html Planck scale mathematical universe model]</ref>. ===== Electron formula ===== {{main|Electron (mathematical)}} Although the Planck units MLTA are embedded within the electron formula ''f_e'', this formula is both unit-less and non scalable (units = 1, scalars = 1). Furthermore it is the geometry of 2 dimensionless physical constants and so can also be defined as a dimensionless physical constant (if units = scalars = 1, then that constant will be independent of any numerical system and of any system of units, and so would qualify as a "natural unit"). :<math>f_e = 4\pi^2(2^6 3 \pi^2 \alpha \Omega^5)^3 = .23895453...x10^{23}</math> AL as an ampere-meter (ampere-length) are the units for a [[w:magnetic monopole | magnetic monopole]]. :<math>T = \pi \frac{r^9}{v^6},\; u^{-30}</math> :<math>\sigma_{e} = \frac{3 \alpha^2 A L}{2\pi^2} = {2^7 3 \pi^3 \alpha \Omega^5}\frac{r^3}{v^2},\; u^{-10}</math> :<math>f_e = \frac{\sigma_{e}^3}{2 T} = \frac{(2^7 3 \pi^3 \alpha \Omega^5)^3}{2\pi},\; units = \frac{(u^{-10})^3}{u^{-30}} = 1, scalars = (\frac{r^3}{v^2})^3 \frac{v^6}{r^9} = 1</math> The electron has dimensioned parameters, however the dimensions derive from the Planck units, ''f_e'' is a mathematical function that dictates how these Planck objects are applied, it does not have dimension units of its own, consequently there is no physical electron. [[w:electron mass | electron mass]] <math>m_e = \frac{M}{f_e}</math> (M = [[w:Planck mass | Planck mass]]) [[w:Compton wavelength | electron wavelength]] <math>\lambda_e = 2\pi L f_e</math> (L = [[w:Planck length | Planck length]]) [[w:elementary charge | elementary charge]] <math>e = A.T</math> ===== Fine structure constant ===== The Sommerfeld [[w:fine-structure constant | fine structure constant alpha]] is a dimensionless physical constant, the CODATA 2018 inverse alpha = 137.035999084. :<math>\frac{2 (h^*)}{(\mu_0^*) (e^*)^2 (c^*)} = 2({2^3 \pi^4 \Omega^4})/(\frac{\alpha}{2^{11} \pi^5 \Omega^4})(\frac{2^{7} \pi^4 \Omega^3}{\alpha})^2(2 \pi \Omega^2) = \color{blue}\alpha \color{black},\; \frac{u^{19}}{u^{56} (u^{-27})^2 u^{17}} = 1,\; (\frac{r^{13}}{v^5})(\frac{1}{r^7})(\frac{v^6}{r^6})(\frac{1}{v}) = 1</math> ===== Omega ===== The most precise of the experimentally measured constants is the Rydberg ''R = 10973731.568508(65) 1/m''. Here ''c, μ₀, R'' are combined into a unit-less ratio; :<math>\frac{(c^*)^{35}}{(\mu_0^*)^9 (R^*)^7} = (2 \pi \Omega^2)^{35}/(\frac{\alpha}{2^{11} \pi^5 \Omega^4})^9 .(\frac{1} {2^{23} 3^3 \pi^{11} \alpha^5 \Omega^{17}})^7,\;units = \frac{(u^{17})^{35}}{(u^{56})^9 (u^{13})^7} = 1</math> We can now define ''Ω'' using the geometries for (''c^*, μ₀^*, R^*'') and then solve by replacing (''c^*, μ₀^*, R^*'') with the numerical (''c, μ₀, R''). :<math>\Omega^{225}=\frac{(c^*)^{35}}{2^{295} 3^{21} \pi^{157} (\mu_0^*)^9 (R^*)^7 \alpha^{26}}, \;units = 1</math> :<math>\Omega = 2.007\;134\;949\;636...,\; units = 1</math> (CODATA 2014 mean values) :<math>\Omega = 2.007\;134\;949\;687...,\; units = 1</math> (CODATA 2018 mean values) There is a close natural number for ''Ω'' that is a square root implying that ''Ω'' can have a plus or a minus solution and this agrees with theory. This solution would however re-classify Omega as a mathematical constant (as being derivable from other mathematical constants). :<math>\Omega = \sqrt{ \left(\frac{\pi^e}{e^{(e-1)}}\right)} = 2.007\;134\;9543... </math> ===== G, h, e, m_e, k_B ===== As geometrical objects, the physical constants (''G, h, e, m_e, k_B'') can also be defined using the geometrical formulas for (''c^*, μ₀^*, R^*'') and solved using the numerical (mean) values for (''c, μ₀, R, α''), i.e.:. :<math>{(h^*)}^3 = (2^3 \pi^4 \Omega^4 \frac{r^{13} u^{19}}{v^5})^3 = \frac{2\pi^{10} {(\mu_0^*)}^3} {3^6 {(c^*)}^5 \alpha^{13} {(R^*)}^2},\; unit = u^{57}</math> {| class="wikitable" |+Physical constants; calculated vs CODATA 2014 ! Constant ! Geometry ! Calculated from (R, c, μ₀, α) ! CODATA 2014 <ref>[http://www.codata.org/] | CODATA, The Committee on Data for Science and Technology | (2014)</ref> |- | [[w:Planck constant | Planck constant]] | <math>{(h^*)}^3 = \frac{2\pi^{10} {\mu_0}^3} {3^6 {c}^5 \alpha^{13} {R_\infty}^2},\; unit = u^{57}</math> | ''h^*'' = 6.626 069 134 e-34, unit = u¹⁹ | ''h'' = 6.626 070 040(81) e-34 |- | [[w:Gravitational constant | Gravitational constant]] | <math>{(G^*)}^5 = \frac{\pi^3 {\mu_0}}{2^{20} 3^6 \alpha^{11} {R_\infty}^2},\; unit = u^{30}</math> | ''G^*'' = 6.672 497 192 29 e11, unit = u⁶ | ''G'' = 6.674 08(31) e-11 |- | [[w:Elementary charge | Elementary charge]] | <math>{(e^*)}^3 = \frac{4 \pi^5}{3^3 {c}^4 \alpha^8 {R_\infty}},\; unit = u^{-81}</math> | ''e^*'' = 1.602 176 511 30 e-19, unit = u^-19 | ''e'' = 1.602 176 620 8(98) e-19 |- | [[w:Boltzmann constant | Boltzmann constant]] | <math>{(k_B^*)}^3 = \frac{\pi^5 {\mu_0}^3}{3^3 2 {c}^4 \alpha^5 {R_\infty}} ,\; unit = u^{87} </math> | ''k_B^*'' = 1.379 510 147 52 e-23, unit = u²⁹ | ''k_B'' = 1.380 648 52(79) e-23 |- | [[w:Electron mass | Electron mass]] | <math>{(m_e^*)}^3 = \frac{16 \pi^{10} {R_\infty} {\mu_0}^3}{3^6 {c}^8 \alpha^7},\; unit = u^{45} </math> | '' m_e^*'' = 9.109 382 312 56 e-31, unit = u¹⁵ | ''m_e'' = 9.109 383 56(11) e-31 |- | [[w:Planck mass | Planck mass]] | <math>(m_P^*)^{15} = \frac{2^{25}\pi^{13} \mu_0^6}{ 3^6 c^5 \alpha^{16} R_\infty^2},\; unit = (u^{15})^{15}</math> | ''m_P^*'' = .217 672 817 580 e-7, unit = u¹⁵ | ''m_P'' = .217 647 0(51) e-7 |- | [[w:Planck length | Planck length]] | <math>(l_p^*)^{15} = \frac{\pi^{22} \mu_0^9}{2^{35} 3^{24} \alpha^{49} c^{35} R_\infty^8},\; unit = (u^{-13})^{15} </math> | ''l_p^*'' = .161 603 660 096 e-34, unit = u^-13 | ''l_p'' = .161 622 9(38) e-34 |- | [[w:Gyromagnetic ratio | Gyromagnetic ratio]] | <math>(\gamma_e/2\pi)^3 = \frac{g^3 3^3 c^4}{2^8 \pi^8 \alpha \mu_0^3 R_\infty^2},\; unit = u^{-126}</math> | ''γ_e/2π^*'' = 28024.953 55, unit = u^-42 | ''γ_e/2π'' = 28024.951 64(17) |} ==== 2019 SI unit revision ==== Following the 26th General Conference on Weights and Measures ([[w:2019 redefinition of SI base units|2019 redefinition of SI base units]]) are fixed the numerical values of the 4 physical constants (''h, c, e, k_B''). In the context of this model however only 2 base units may be assigned by committee as the rest are then numerically fixed by default and so the revision may lead to unintended consequences. For example, the von Klitzing constant Rk = h/e2, yet to derive h and e using Rk suggests an (inverse) alpha = 137.0359952 <ref>[https://codingthecosmos.com/physical-constants-calc.html Physical constants calculator]</ref>. {| class="wikitable" |+Physical constants ! Constant ! CODATA 2018 <ref>[http://www.codata.org/] | CODATA, The Committee on Data for Science and Technology | (2018)</ref> |- | [[w:Speed of light | Speed of light]] | ''c'' = 299 792 458 (exact) |- | [[w:Planck constant | Planck constant]] | ''h'' = 6.626 070 15 e-34 (exact) |- | [[w:Elementary charge | Elementary charge]] | ''e'' = 1.602 176 634 e-19 (exact) |- | [[w:Boltzmann constant | Boltzmann constant]] | ''k_B'' = 1.380 649 e-23 (exact) |- | [[w:Fine structure constant | Fine structure constant]] | ''α'' = 137.035 999 084(21) |- | [[w:Rydberg constant | Rydberg constant]] | ''R'' = 10973 731.568 160(21) |- | [[w:Electron mass | Electron mass]] | ''m_e'' = 9.109 383 7015(28) e-31 |- | [[w:Vacuum permeability | Vacuum permeability]] | ''μ₀'' = 1.256 637 062 12(19) e-6 |- | [[w:Quantum_Hall_effect#Applications | Von Klitzing constant]] | ''R_K'' = 25812.807 45 (exact) |} For example, if we solve using the above formulas; <math>R^* = \frac{4 \pi^5}{3^3 c^4 \alpha^8 e^3} = 10973\;729.082\;465</math> <math>{(m_e^*)}^3 = \frac{2^4 \pi^{10} R \mu_0^3}{3^6 c^8 \alpha^7},\;m_e^* = 9.109\;382\;3259 \;10^{-31}</math> <math>{(\mu_0^*)}^3 = \frac{3^6 h^3 c^5 \alpha^{13} R^2}{2 \pi^{10}},\;\mu_0^* = 1.256\;637\;251\;88\;10^{-6}</math> <math>{(h^*)}^3 = \frac{2 \pi^{10} \mu_0^3}{3^6 c^5 \alpha^{13} R^2},\;h^* = 6.626\;069\;149\;10^{-34}</math> <math>{(e^*)}^3 = \frac{4 \pi^5}{3^3 c^4 \alpha^8 R},\; e^* = 1.602\;176\;513\;10^{-19}</math> ==== u as sqrt(velocity/mass) ==== We find there is a single base unit '''u''' from which the other units and numerical values can be derived. This base unit incorporates [[v:Sqrt_Planck_momentum |MLT as square roots]]. =====''u = √{L/M.T}''===== :<math>u,\; units = \sqrt{\frac{L}{M T}} = \sqrt{u^{-13-15+30=2}} = u^1</math> Setting: :<math>x,\;units = \sqrt{\frac{M^9 T^{11}}{L^{15}}} = u^0, units = 1, scalars = 1</math> :<math>y,\;units = M^2T = u^0, units = 1, scalars <> 1</math> Gives; :<math>u^3 = \frac{L^{3/2}}{M^{3/2} T^{3/2}} = A,\; (ampere)</math> :<math>u^6 (y) = \frac{L^3}{T^2 M},\; (G)</math> :<math>u^{13} (xy) = \frac{1}{L},\; (1/l_p)</math> :<math>u^{15} (xy^2) = M,\; (m_P)</math> :<math>u^{17} (xy^2) = V,\; (c)</math> :<math>u^{19} (xy^3) = \frac{ML^2}{T},\; (h)</math> :<math>u^{20} (xy^2) = \frac{L^{5/2}}{M^{3/2} T^{5/2}} = AV,\;(T_P)</math> :<math>u^{27} (x^2y^3) = \frac{M^{3/2}\sqrt{T}}{L^{3/2}} = 1/AT,\; (1/e)</math> :<math>u^{29} (x^2y^4) = \frac{M^{5/2}\sqrt{T}}{\sqrt{L}} = ML/AT,\; (k_B)</math> :<math>u^{30} (x^2 y^3) = \frac{1}{T},\; (1/t_p)</math> :<math>u^{56} (x^4 y^7) = \frac{M^4 T}{L^2},\;(\mu_0)</math> :<math>u^{90} (x^6 y^{11}) = \frac{M^4}{T} </math> ===== ''β'' (unit = ''u'') ===== ''i'' (from ''x'') and ''j'' (from ''y''). :<math>R = \sqrt{P} = \sqrt{\Omega} r,\; units = u^8</math> :<math>\beta = \frac{V}{R^2} = \frac{2\pi R^2}{M} = \frac{A^{1/3} \alpha^{1/3}}{2} \;..., \; unit = u</math> :<math>i = \frac{1}{2\pi {(2\pi \Omega)}^{15}},\; units = 1, scalars = 1</math> :<math>j = \frac{r^{17}}{v^8} = k^2t = \frac{k^8}{r^{15}} ...,\; unit = \frac{u^{17*8}}{u^{8*17}} = u^{15*2}u^{-30} ... = 1,\; units = 1, scalars <> 1</math> For example; the constants solved in terms of (''r, v'') :<math>\beta = \frac{V}{R^2} = \frac{2\pi \Omega^2 v}{\Omega r^2} = \frac{2\pi \Omega v}{r^2},\; u^1 = u</math> :<math>A = \beta^3 (\frac{2^4}{\alpha}) = \frac{2^7 \pi^3 \Omega^3}{\alpha}\frac{v^3}{r^6},\; u^3</math> :<math>G = \frac{\beta^6}{2^3 \pi^2} (j) = 2^3 \pi^4 \Omega^6 \frac{r^5}{v^2},\; u^6</math> :<math>L^{-1} = 4\pi \beta^{13} (ij) = \frac{1}{2\pi^2 \Omega^2} \frac{v^5}{r^9},\; u^{13}</math> :<math>M = 2\pi \beta^{15} (ij^2) = \frac{r^4}{v},\; u^{15}</math> :<math>P = \beta^{16} (ij^2) = \Omega r^2,\; u^{16}</math> :<math>V = \beta^{17} (ij^2) = 2\pi \Omega^2 v,\; u^{17}</math> :<math>h = \pi \beta^{19} (ij^3) = 8\pi^4 \Omega^4 \frac{r^{13}}{v^5},\; u^{19}</math> :<math>T_P^* =\frac{2^3 \beta^{20}}{\pi \alpha} (ij^2) = \frac{2^7 \pi^3 \Omega^5}{\alpha} \frac{v^4}{r^6} ,\; u^{20} </math> :<math>e^{-1} = \frac{\alpha \pi \beta^{27} (i^2j^3)}{4} = \frac{\alpha}{128\pi^4 \Omega^3} \frac{v^3}{r^{3}},\; u^{27} </math> :<math>k_B = \frac{\alpha \pi^2 \beta^{29}(i^2j^4)}{4} = \frac{\alpha}{32 \pi \Omega} \frac{r^{10}}{v^3},\; u^{29}</math> :<math>T^{-1} = 4\pi \beta^{30} (i^2 j^3) = \frac{1}{\pi}\frac{v^6}{r^9},\; u^{30}</math> :<math>\mu_0^* = \frac{\pi^3 \alpha \beta^{56}}{2^3} (i^4 j^7) = \frac{\alpha}{2^{11} \pi^5 \Omega^4} r^7,\; u^{56}</math> :<math>\epsilon_0^{*-1} = \mu_0^* (c^*)^2 = \frac{\pi^3 \alpha \beta^{90}}{2^3} (i^6 j^{11}) = \frac{\alpha}{2^9 \pi^3} v^2 r^7,\; u^{90} </math> ===== limit ''j'' ===== The SI values for ''j'' suggest a limit (numerical boundary) to the values the SI constants can have. :<math>j = \frac{r^{17}}{v^8} = k^2 t = \frac{k^{17/4}}{v^{15/4}} = ... </math> gives a range from 0.812997... ''x''10^-59 to 0.123... ''x''10⁶⁰ In SI terms unit ''β'' can be derived via these ratio; :<math>a^{1/3} = \frac{v}{r^2} = \frac{1}{t^{2/15}k^{1/5}} = \frac{\sqrt{v}}{\sqrt{k}} ... = 23326079.1...; unit = u</math> ===== Rydberg formula ===== The [[w:Rydberg_formula |Rydberg formula]] can now be re-written in terms of amperes <math>A^2</math> :<math>\frac{hc}{2\pi \alpha^2} = \frac{j^2 A^2}{2^8 2\pi t_p}</math> ==== External links ==== * [[v:electron_(mathematical) | Mathematical electron]] * [[v:Relativity_(Planck) | Programming relativity at the Planck scale]] * [[v:Quantum_gravity_(Planck) | Programming gravity at the Planck scale]] * [[v:Black-hole_(Planck) | Programming the cosmic microwave background at the Planck scale]] * [[v:Sqrt_Planck_momentum | The sqrt of Planck momentum]] * [[v:God_(programmer) | The Programmer God]] * [[w:Simulation_hypothesis | The Simulation hypothesis]] * [https://codingthecosmos.com/ Programming at the Planck scale using geometrical objects] -Malcolm Macleod's website * [http://www.simulation-argument.com/ Simulation Argument] -Nick Bostrom's website * [https://www.amazon.com/Our-Mathematical-Universe-Ultimate-Reality/dp/0307599809 Our Mathematical Universe: My Quest for the Ultimate Nature of Reality] -Max Tegmark * [https://dx.doi.org/10.2139/ssrn.2531429 The mathematical electron model in a Planck scale universe] -(article) * [https://link.springer.com/article/10.1134/S0202289308020011/ Dirac-Kerr-Newman black-hole electron] -Alexander Burinskii (article) ==== References ==== {{Reflist}} [[Category: Physics]] [[Category: Philosophy of science]] dyvb33lqd7xvuh8n6ob3l5j8oscwemw 2410247 2410238 2022-07-29T16:17:58Z Platos Cave (physics) 2562653 wikitext text/x-wiki {| class="wikitable" |+Dimensioned physical constants ! Constant ! CODATA 2014 <ref>[http://www.codata.org/] | CODATA, The Committee on Data for Science and Technology | (2014)</ref> ! unit |- | <math>\frac{k_B e c}{h} =</math> {{font color|blue|yellow|'''1.000 8254'''}} | <math>\frac{(k_B^*) (e^*) (c^*)}{(h^*)}</math> = {{font color|blue|yellow|'''1.0'''}} | <math>\frac{ (u^{29}) (u^{-27}) (u^{17}) }{ (u^{19}) } = 1</math> |- | <math>\frac{h^3}{e^{13} c^{24}} =</math> {{font color|green|yellow| '''0.228 473 639... 10^-58'''}} | <math>\frac{(h^*)^3}{(e^*)^{13} (c^*)^{24}} =</math> {{font color|green|yellow| '''0.228 473 759... 10^-58'''}} | <math>\frac{(u^{19})^{3}}{(u^{-27})^{13} (u^{17})^{24}} = 1</math> |- | <math>\frac{c^{35}}{\mu_0^9 R^7} =</math> {{font color|red|yellow| '''0.326 103 528 6170... 10³⁰¹'''}} | <math>\frac{(c^*)^{35}}{(\mu_0^*)^9 (R^*)^7} =</math> {{font color|red|yellow| '''0.326 103 528 6170... 10³⁰¹'''}} | <math>\frac{(u^{17})^{35}}{(u^{56})^9 (u^{13})^7} = 1</math> |- | <math>\frac{c^9 e^4}{m_e^3} =</math> {{font color|blue|yellow| '''0.170 514 342... 10⁹²'''}} | <math>\frac{(c^*)^9 (e^*)^4}{(m_e^*)^3} =</math> {{font color|blue|yellow| '''0.170 514 368... 10⁹²'''}} | <math>\frac{ (u^{29}) (u^{-27}) (u^{17}) }{ (u^{19}) } = 1</math> |- | <math>\frac{k_B}{e^2 m_e c^4} =</math> {{font color|green|yellow| '''73 095 507 858.'''}} | <math>\frac{(k_B^*)}{(e^*)^2 (m_e^*) (c^*)^4} =</math> {{font color|green|yellow| '''73 035 235 897.'''}} | <math>\frac{(u^{29})}{(u^{-27})^2 (u^{15}) (u^{17})^4} = 1</math> |- | <math>\frac{h c^2 e m_e}{G^2 k_B} =</math> {{font color|red|yellow| '''0.1413... 10^-21'''}} | <math>\frac{(h^*) (c^*)^2 (e^*) (m_e^*)}{(G^*)^2 (k_B^*)} =</math> {{font color|red|yellow| '''0.1415... 10^-21'''}} | <math>\frac{ (u^{19}) (u^{17})^2 (u^{-27}) (u^{15}) }{ (u^{6})^2 (u^{29}) } = 1</math> |- | <math>\frac{2 h}{\mu_0\; e^2 \;c} = \color{blue}\alpha \color{black}</math> | <math>\frac{2 (h^*)}{(\mu_0^*) (e^*)^2 (c^*)} = \color{blue}\alpha \color{black}</math> | <math> \frac{u^{19}}{u^{56} (u^{-27})^2 u^{17}} = 1</math> |} {| class="wikitable" |+Dimensioned physical constants ! Constant ! CODATA 2014 <ref>[http://www.codata.org/] | CODATA, The Committee on Data for Science and Technology | (2014)</ref> ! unit |- | [[w:Speed of light | Speed of light]] | ''c'' = 299 792 458 (exact) | <math>\frac{m}{s}</math> |- | [[w:Planck constant | Planck constant]] | ''h'' = 6.626 070 040(81) e-34 | <math>\frac{kg \;m^2}{s}</math> |- | [[w:Gravitational constant | Gravitational constant]] | ''G'' = 6.674 08(31) e-11 | <math>\frac{m^3}{kg \;s^2}</math> |- | [[w:Elementary charge | Elementary charge]] | ''e'' = 1.602 176 620 8(98) e-19 | <math>C = A s</math> |- | [[w:Boltzmann constant | Boltzmann constant]] | ''k_B'' = 1.380 648 52(79) e-23 | <math>\frac{kg \;m^2}{s^2 \;K}</math> |- | [[w:Vacuum permeability | Vacuum permeability]] | ''μ₀'' = 4π/10^7 (exact) | <math>\frac{kg \;m}{s^2 \;A^2}</math> |- | [[w:Electron mass | Electron mass]] | ''m_e'' = 9.109 383 56(11) e-31 | <math>kg</math> |} {| class="wikitable" |+Dimensioned physical constants ! SI constant ! CODATA 2014 <ref>[http://www.codata.org/] | CODATA, The Committee on Data for Science and Technology | (2014)</ref> ! unit ! Natural constant* ! unit* |- | [[w:Speed of light | Speed of light]] | ''c'' = 299 792 458 (exact) | <math>\frac{m}{s}</math> | ''c*'' = V | 17 |- | [[w:Planck constant | Planck constant]] | ''h'' = 6.626 070 040(81) e-34 | <math>\frac{kg \;m^2}{s}</math> | <math>h^* = 2 \pi M V L</math> | 19 |- | [[w:Gravitational constant | Gravitational constant]] | ''G'' = 6.674 08(31) e-11 | <math>\frac{m^3}{kg \;s^2}</math> | <math>G^* = \frac{V^2 L}{M}</math> | 6 |- | [[w:Elementary charge | Elementary charge]] | ''e'' = 1.602 176 620 8(98) e-19 | <math>C = A s</math> | <math>e^* = A T</math> | -27 |- | [[w:Boltzmann constant | Boltzmann constant]] | ''k_B'' = 1.380 648 52(79) e-23 | <math>\frac{kg \;m^2}{s^2 \;K}</math> | <math>k_B^* = \frac{2 \pi V M}{A}</math> | 29 |- | [[w:Vacuum permeability | Vacuum permeability]] | ''μ₀'' = 4π/10^7 (exact) | <math>\frac{kg \;m}{s^2 \;A^2}</math> | <math>\mu_0^* = \frac{4 \pi V^2 M}{\alpha L A^2}</math> | 56 |} {| class="wikitable" |+Dimensionless physical constants ! Constant ! value |- | [[w:Fine structure constant | Fine structure constant]] | ''α'' = 137.035 999 139(31) |- | [[v:Planck_units_(geometrical)#Omega | Omega]] | ''Ω'' = 2.007 134 9496 |} {| class="wikitable" |+Dimensioned physical constants ! Constant ! value ! unit |- | [[w:Planck mass | Planck mass]] | ''m_P'' = .217 647 0(51) e-7 | <math>kg</math> |- | [[w:Planck length | Planck length]] | ''l_p'' = .161 622 9(38) e-34 | <math>m</math> |- | [[w:Planck time | Planck time]] | ''t_p'' = 5.391 247(60) e-44 | <math>s</math> |} '''Natural Planck units as geometrical objects''' [[w:Planck units |Planck unit]] theories use basic units for mass, length, time and charge, and operate at the Planck scale. In a geometrical Planck theory, these basic units are assigned geometrical objects (''MLTA'') rather than numerical values, the advantage being that the geometries themselves can encode the function of the unit, for example the object for length (''L'') will encode the function of ''length'', the geometrical ''L'' is 1 unit of (Planck) length, such that, unlike numerical models, a dimensioned descriptive (i.e.: ''kg, m, s, A'', ... ) is not required. The ''MLTA'' geometrical objects are selected whereby they may interact with each other (the mass object for example is not independent of the length and the time objects). This permits a mathematical relationship between them, and so a physical universe can be constructed [[w:Lego |Lego-style]] by combining the base (Planck unit) ''MLTA'' objects to form more complex objects such as electrons (i.e.: by embedding ''L'' and ''A'' into the geometry of the electron, the electron can have wavelength and charge). Furthermore, these objects can overlap and cancel in a particular ratio (according to that mathematical relationship), and this ratio occurs in the electron. And so, although the electron has physical parameters (wavelength, charge ...), '''the electron itself is a mathematical particle (units = 1)''', not a physical particle. Furthermore, due to this unit-less ratio, for any system of units '''if we know the numerical values of any 2 Planck units, then we can solve the dimensioned constants''' (''G, h, c, e, m_e, k_B'') ... for that system of units. === Geometrical objects === Base units for mass <math>M</math>, length <math>L</math>, time <math>T</math>, and ampere <math>A</math> can be constructed from the geometry of 2 [[w:dimensionless physical constant | dimensionless physical constants]], the (inverse) [[w:fine-structure constant | fine structure constant '''α''']] = 137.036 and [[v:Simulation_argument_(coding_Planck_units)#Omega | Omega]] '''Ω''' = 2.007 134 949 <ref>Macleod, M.J. {{Cite journal |title= Programming Planck units from a mathematical electron; a Simulation Hypothesis |journal=Eur. Phys. J. Plus |volume=113 |pages=278 |date=22 March 2018 | doi=10.1140/epjp/i2018-12094-x }}</ref>. Being independent of any numerical system and of any system of units, these MLTA units would qualify as "natural units"; {{bq|''...ihre Bedeutung für alle Zeiten und für alle, auch außerirdische und außermenschliche Kulturen notwendig behalten und welche daher als »natürliche Maßeinheiten« bezeichnet werden können...'' ...These necessarily retain their meaning for all times and for all civilizations, even extraterrestrial and non-human ones, and can therefore be designated as "natural units"... -Max Planck <ref>Planck (1899), p. 479.</ref><ref name="TOM">*Tomilin, K. A., 1999, "[http://www.ihst.ru/personal/tomilin/papers/tomil.pdf Natural Systems of Units: To the Centenary Anniversary of the Planck System]", 287–296.</ref>}} ==== Objects ==== Each object is assigned a geometry and a dimensioned attribute (the object function); :<math>\beta = (2\pi\Omega)</math> [[v:Planck_units_(geometrical)#u_as_sqrt(velocity/mass) |sqrt(velocity/mass)]] :<math>M = (1)</math> mass :<math>T = (\pi)</math> time :<math>P = (\Omega)</math> [[v:Sqrt_Planck_momentum | sqrt of momentum]] :<math>V = (2\pi\Omega^2)</math> velocity :<math>L = (2\pi^2\Omega^2)</math> length :<math>A = (\frac{2^7 \pi^3 \Omega^3}{\alpha})</math> ampere ==== Mathematical relationship ==== A relationship between the objects is defined using '''uⁿ''' whereby; :<math>(\beta)\;u\;</math> :<math>(M)\;u^{15}\;</math> :<math>(T)\;u^{-30}\;</math> :<math>(P)\;u^{16}\;</math> :<math>(V)\;u^{17}\;</math> :<math>(L)\;u^{-13}\;</math> :<math>(A) \;u^{3}\;</math> ==== Scalars ==== To translate from geometrical objects to a numerical system of units requires scalars ('''kltpva''') that can be assigned numerical values. For example, scalars for the SI units; :If we use '''k''' to convert '''M''' to the SI Planck mass <math>m_P</math> (M = 1k = <math>m_P</math>), then '''k''' = 0.2176728e-7kg and '''<math>u^{15}</math>''' will equate to '''kg'''. :To convert '''V''' = 2πΩ² = 25.3123819 to '''c''' requires scalar '''v''' = 11843707.905m/s ('''V''' ''v'' = 2πΩ²''v'' = 299792458m/s) with '''<math>u^{17}</math>''' equating to '''m/s'''. {| class="wikitable" |+Geometrical units ! Attribute ! Geometrical object ! Scalar ! Unit |- | [[v:Planck_units_(geometrical)#u_as_sqrt(velocity/mass) |sqrt(velocity/mass)]] | <math>\beta = (2\pi \Omega)</math> | | <math>unit = u</math> |- | mass | <math>M = (1)</math> | k | <math>unit = u^{15}</math> |- | time | <math>T = (\pi)</math> | t | <math>unit = u^{-30}</math> |- | [[v:Sqrt_Planck_momentum | sqrt(momentum)]] | <math>P = \frac{\beta M}{2\pi} = (\Omega)</math> | p | <math>unit = u^{16}</math> |- | velocity | <math>V = \frac{\beta^2 M}{2\pi} = (2\pi\Omega^2)</math> | v | <math>unit = u^{17}</math> |- | length | <math>L = TV = (2\pi^2\Omega^2)</math> | l | <math>unit = u^{-13}</math> |- | ampere | <math>A = \frac{16 V^3}{\alpha P^3} = (\frac{2^7 \pi^3 \Omega^3}{\alpha})</math> | a | <math>unit = u^3</math> |} ===== Scalar relationships ===== The following ''uⁿ'' groups cancel ('''units = scalars = 1'''), as such '''only 2 numerical scalars are required''', for example, if we know '''a''' and '''l''' then we know '''t''' ('''t = a³l³'''), and from '''l''' and '''t''' we know '''k'''. :<math>\frac{u^{3*3} u^{-13*3}}{u^{-30}}\;(\frac{a^3 l^3}{t}) = \frac{u^{-13*15}}{u^{15*9} u^{-30*11}} \;(\frac{l^{15}}{k^9 t^{11}}) = \;...\; =1</math> This means that if we know any 2 constants, then we can solve the scalars for those constants, and from those 2 scalars we can solve all the Planck units, and from these the dimensioned physical constants. This will apply to any set of units. In this example, to maintain integer exponents, scalar ''p'' is defined in terms of a scalar ''r''. :<math>r = \sqrt{p} = \sqrt{\Omega},\; unit \;u^{16/2=8}</math> The SI Planck units are known with a low precision, conversely 2 of the CODATA 2014 physical constants have been assigned exact numerical values; ''c'' and permeability of vacuum ''μ₀''. Scalars ''r'' and ''v'' were chosen as they can be derived directly from ''V = c'' and ''μ₀'' = 4π/10^7 (see table [[v:Planck_units_(geometrical)#Physical_constants_(as_geometrical_formulas) |geometrical physical constants]] below). Using α = 137.035 999 139 (CODATA 2014), Ω = 2.007 134 949 636... :<math>v = \frac{c}{2 \pi \Omega^2}= 11 843 707.905 ...,\; units = m/s</math> :<math>r^7 = \frac{2^{11} \pi^5 \Omega^4 \mu_0}{\alpha};\; r = 0.712 562 514 304 ...,\; units = (\frac{kg.m}{s})^{1/4}</math> This gives scalars ''klta'' ([[v:Planck_units_(geometrical)#u_as_sqrt(velocity/mass)) |for derivation of units kg, m, s, A from r, v]]); :<math>k = \frac{r^4}{v}</math> = 0.217 672 817 580... ''x'' 10^-7kg, <math>\;\;\;u^{15} = \frac{(u^8)^4}{u^{17}}</math> :<math>l = \frac{r^9}{v^5}</math> = 0.203 220 869 487... ''x'' 10^-36m, <math>\;\;\;u^{-13} = \frac{(u^8)^9}{(u^{17})^5}</math> :<math>t = \frac{r^9}{v^6}</math> = 0.171 585 512 841... ''x'' 10^-43s, <math>\;\;\;u^{-30} = \frac{(u^8)^9}{(u^{17})^6}</math> :<math>a = \frac{v^3}{r^6}</math> = 0.126 918 588 592... ''x'' 10²³A, <math>\;\;\;u^{3} = \frac{(u^{17})^3}{(u^8)^6}</math> ===== Natural units MLTPA ===== Regardless of which system of units we use, alien or terrestrial, any combination of constants where '''scalars = 1''' (i.e.: the scalars overlap and cancel) will give the same numerical result, they will default to the MLTPA objects. This implies that these objects are Planck's 'natural' units, i.e.: that '''all possible systems of units''' are based on these objects, and so, given that these are geometrical objects, they can be construed as evidence of a mathematical universe. The following are examples of '''units = scalars = 1''' ratios using SI units <ref>Macleod, Malcolm J. {{Cite journal |title= Do the fundamental constants embed evidence of a mathematical universe at the Planck scale? |journal=RG | doi=10.13140/RG.2.2.15874.15041/1 }}</ref>. Note: the geometry <math>\color{red}(\Omega^{15})^n\color{black}</math> (integer n ≥ 0) is common to these ratios. ====== m_P, l_p, t_p ====== In this ratio, the MLT units and ''klt'' scalars both cancel; units = scalars = 1, reverting to the base MLT objects. Setting the scalars ''klt'' to convert the MLT objects to the SI Planck units; :k = 0.217 672 817 580... ''x'' 10^-7kg :l = 0.203 220 869 487... ''x'' 10^-36m :t = 0.171 585 512 841... ''x'' 10^-43s :<math>\frac{L^{15}}{M^{9} T^{11}} = \frac{(2\pi^2\Omega^2)^{15}}{(1)^{9} (\pi)^{11}} (\frac{l^{15}}{k^9 t^{11}}) = \frac{l_p^{15}}{m_P^{9} t_p^{11}} </math> (CODATA 2018 mean) The ''klt'' scalars cancel, leaving; :<math>\frac{L^{15}}{M^{9} T^{11}} = \frac{(2\pi^2\Omega^2)^{15}}{(1)^{9} (\pi)^{11}} (\frac{l^{15}}{k^9 t^{11}}) = 2^{15} \pi^{19} \color{red}(\Omega^{15})^2\color{black} = </math>{{font color|blue|yellow|'''0.109 293... 10²⁴ '''}}, <math>(\frac{l^{15}}{k^9 t^{11}}) = 1, \;\frac{u^{-13*15}}{u^{15*9} u^{-30*11}} = 1</math> Solving for the SI units; :<math>\frac{l_p^{15}}{m_P^{9} t_p^{11}} = \frac{(1.616255e-35)^{15}}{(2.176434e-8)^{9} (5.391247e-44)^{11}} = </math> {{font color|blue|yellow| '''0.109 485... 10²⁴'''}} ====== A, l_p, t_p ====== :a = 0.126 918 588 592... ''x'' 10²³A :<math>\frac{A^3 L^3}{T} = (\frac{2^7 \pi^3 \Omega^3}{\alpha})^3 \frac{(2\pi^2\Omega^2)^3}{(\pi)} (\frac{a^3 l^3}{t}) = \frac{2^{24} \pi^{14} \color{red}(\Omega^{15})^1\color{black}}{\alpha^3} = </math> {{font color|green|yellow| '''0.205 571... 10¹³'''}}, <math>(\frac{a^3 l^3}{t}) = 1,\; \frac{u^{3*3} u^{-13*3}}{u^{-30}} = 1</math> :<math>\frac{(e / t_p)^3 l_p^3}{t_p} = \frac{(1.602176634e-19/5.391247e-44)^3 (1.616255e-35)^3}{(5.391247e-44)} = </math> {{font color|green|yellow| '''0.205 543... 10¹³'''}}, <math>units = \frac{(C/s)^3 m^3}{s} </math> The Planck units are known with low precision, and so by defining the 3 most accurately known dimensioned constants in [[v:Planck_units_(geometrical)#Physical_constants_(as_geometrical_formulas) |terms of these objects]] (c, R = Rydberg constant, <math>\mu_0</math>; CODATA 2014 mean values), we can test to greater precision; ====== c, μ₀, R ====== :<math>\frac{(c^*)^{35}}{(\mu_0^*)^9 (R^*)^7} = (2 \pi \Omega^2 v)^{35}/(\frac{\alpha r^7}{2^{11} \pi^5 \Omega^4})^9 .(\frac{v^5} {2^{23} 3^3 \pi^{11} \alpha^5 \Omega^{17} r^9})^7 = 2^{295} \pi^{157} 3^{21} \alpha^{26} \color{red}(\Omega^{15})^{15}\color{black} = </math> {{font color|red|yellow| '''0.326 103 528 6170... 10³⁰¹'''}}, <math>\frac{(u^{17})^{35}}{(u^{56})^9 (u^{13})^7} = 1, \;(v^{35})/(r^7)^9 (\frac{v^5}{r^9})^7 = 1</math> :<math>\frac{c^{35}}{\mu_0^9 R^7} = \frac{(299792458)^{35}}{(4 \pi/10^7)^9 (10973731.568160)^7} = </math> {{font color|red|yellow| '''0.326 103 528 6170... 10³⁰¹'''}}, <math>units = \frac{m^{33}A^{18}}{s^{17}kg^9} == \frac{(u^{-13})^{33} (u^{3})^{18}}{(u^{-30})^{17} (u^{15})^9} = 1</math> The [[w:2019 redefinition of SI base units | 2019 SI unit revision]] assigned exact numerical values to 4 constants (c, e, k_B, h). {{see also |Planck units (geometrical)#2019 SI unit revision}} From the table [[v:Planck_units_(geometrical)#Physical_constants_(as_geometrical_formulas) |geometrical physical constants]], we get geometrical formulas and scalars for; :<math>h^* = 2 \pi MVL = 2^3 \pi^4 \Omega^4 \frac{r^{13}}{v^5},\; u^{15+17-13 = 19}</math> :<math>e^* = AT = \frac{2^7 \pi^4 \Omega^3}{\alpha}\frac{r^3}{v^3},\; u^{3-30 = -27}</math> :<math>k_B^*= 2 \pi MV/A = \frac{\alpha}{2^5 \pi \Omega} \frac{r^{10}}{v^3},\; u^{17+15-3 = 29}</math> ====== c, e, k_B, h ====== :<math>\frac{(k_B^*) (e^*) (c^*)}{(h^*)} = (\frac{\alpha}{2^5 \pi \Omega} \frac{r^{10}}{v^3}) (\frac{2^7 \pi^4 \Omega^3}{\alpha} \frac{r^3}{v^3}) (2 \pi \Omega^2 v) / (2^3 \pi^4 \Omega^4 \frac{r^{13}}{v^5}) </math> = {{font color|blue|yellow|'''1.0'''}}, <math>\frac{ (u^{29}) (u^{-27}) (u^{17}) }{ (u^{19}) } = 1,\; (\frac{r^{10}}{v^3}) (\frac{r^3}{v^3}) (v) / (\frac{r^{13}}{v^5}) = 1</math> :<math>\frac{k_B e c}{h} = </math> {{font color|blue|yellow|'''1.000 8254'''}}, <math>units = \frac{m C}{s^2 K} == \frac{(u^{-13}) (u^{-27})}{(u^{-30})^2 (u^{20})} = 1</math> ====== c, h, e ====== :<math>\frac{(h^*)^3}{(e^*)^{13} (c^*)^{24}} = (2^3 \pi^4 \Omega^4 \frac{r^{13}}{v^5})^3/(\frac{2^7 \pi^4 \Omega^3 r^3}{\alpha v^3})^7.(2\pi\Omega^2 v)^{24} = \frac{\alpha^{13}}{2^{106} \pi^{64} (\color{red}\Omega^{15})^5\color{black}} = </math> {{font color|green|yellow| '''0.228 473 759... 10^-58'''}}, <math>\frac{(u^{19})^{3}}{(u^{-27})^{13} (u^{17})^{24}} = 1, \;(\frac{r^{13}}{v^5})^3 / (\frac{r^3}{v^3})^{13} (v^{24}) = 1</math> :<math>\frac{h^3}{e^{13} c^{24}} = </math> {{font color|green|yellow| '''0.228 473 639... 10^-58'''}}, <math>units = \frac{kg^3 s^{21}}{m^{18} C^{13}} == \frac{(u^{15})^3 (u^{-30})^{21}}{(u^{-13})^{18} (u^{-27})^{13}} = 1</math> ====== m_e, λ_e ====== :<math>\sigma_{e} = \frac{3 \alpha^2 A L}{2\pi^2} = {2^7 3 \pi^3 \alpha \Omega^5}\frac{r^3}{v^2},\; u^{-10}</math> :<math>f_e = \frac{\sigma_{e}^3}{2 T} = 2^{20} 3^3 \pi^8 \alpha^3 (\color{red}\Omega^{15})\color{black},\; \frac{(u^{-10})^3}{u^{-30}} =1,\; (\frac{r^3}{v^2})^3 \frac{v^6}{r^9} = 1</math> :<math>(m_e^*) = \frac{M}{f_e} = \color{blue}9.109\;382\;3227 \;10^{-31}\color{black}\;u^{15}</math> :<math>(m_e^*) = \frac{2^3 \pi^5 (h^*)}{3^3 \alpha^6 (e^*)^3 (c^*)^5} = \frac{1}{2^{20} \pi^8 3^3 \alpha^3 (\color{red}\Omega^{15})\color{black}} \frac{r^4 u^{15}}{v} = \color{blue}9.109\;382\;3227 \;10^{-31}\color{black}\;u^{15}</math> :<math>m_e = \color{blue}9.109\;383\;7015... \;10^{-31}\color{black}\;kg</math> :<math>(\lambda_e^*) = 2 \pi L f_e = \color{purple}2.426\;310\;238\;667 \;10^{-12}\color{black}\;u^{-13}</math> :<math>\lambda_e = \frac{h}{m_e c} = \color{purple}2.426 \;310 \;238 \;67 \;10^{-12}\color{black}\;m</math> ====== c, e, m_e ====== :<math>(m_e^*)= \frac{M}{f_e}, \;f_e = 2^{20} 3^3 \pi^8 \alpha^3 (\color{red}\Omega^{15})^1\color{black} </math>, units = scalars = 1 ([[v:Planck_units_(geometrical)#Electron_formula |m_e formula]]) :<math>\frac{(c^*)^9 (e^*)^4}{(m_e^*)^3} = 2^{97} \pi^{49} 3^9 \alpha^5 (\color{red}\Omega^{15})^5\color{black} = </math> {{font color|red|yellow| '''0.170 514 368... 10⁹²'''}}, <math>\frac{(u^{17})^9 (u^{-27})^4}{(u^{15})^3} = 1,\; (v^9) (\frac{r^3}{v^3})^4 / (\frac{r^4}{v})^3 = 1</math> :<math>\frac{c^9 e^4}{m_e^3} = </math> {{font color|red|yellow| '''0.170 514 342... 10⁹²'''}}, <math>units = \frac{m^9 C^4}{s^9 kg^3} == \frac{(u^{-13})^9 (u^{-27})^4}{(u^{-30})^9 (u^{15})^3} = 1</math> ====== k_B, c, e, m_e ====== :<math>\frac{(k_B^*)}{(e^*)^2 (m_e^*) (c^*)^4} = \frac{3^3 \alpha^6}{2^3 \pi^5} = </math> {{font color|blue|yellow| '''73 035 235 897.'''}}, <math>\frac{(u^{29})}{(u^{-27})^2 (u^{15}) (u^{17})^4} = 1,\; (\frac{r^{10}}{v^3}) / (\frac{r^3}{v^3})^2 (\frac{r^4}{v}) (v)^4 = 1</math> :<math>\frac{k_B}{e^2 m_e c^4} = </math> {{font color|blue|yellow| '''73 095 507 858.'''}}, <math>units = \frac{s^2}{m^2 K C^2} == \frac{(u^{-30})^2}{(u^{-13})^2 (u^{20}) (u^{-27})^2} = 1</math> ====== m_P, t_p, ε₀ ====== These 3 constants, Planck mass, Planck time and the vacuum permittivity have no Omega term. :<math>\frac{M^4 (\epsilon_0^*)}{T} = (1) (\frac{2^9 \pi^3}{\alpha}) / (\pi) = \frac{2^9 \pi^2}{\alpha} = </math> {{font color|green|yellow| '''36.875'''}}, <math>\frac{(u^{15})^4 (u^{-90})}{(u^{-30})} = 1,\; (\frac{r^4}{v})^4 (\frac{1}{r^7 v^2}) / (\frac{r^9}{v^6}) = 1</math> :<math>\frac{m_p^4 (\epsilon_0)}{t_p} = </math> {{font color|green|yellow| '''36.850'''}}, <math>units = \frac{kg^4}{s} \frac{s^4 A^2}{m^3 kg} = \frac{kg^3 A^2 s^3}{m^3} == \frac{(u^{15})^3 (u^{3})^2 (u^{-30})^3}{(u^{-13})^3} = 1</math> ====== G, h, c, e, m_e, K_B ====== :<math>\frac{(h^*) (c^*)^2 (e^*) (m_e^*)}{(G^*)^2 (k_B^*)} = (m_e^*) (\frac{2^{11} \pi^3}{\alpha^2}) = </math> {{font color|red|yellow| '''0.1415... 10^-21'''}}, <math>\frac{ (u^{19}) (u^{17})^2 (u^{-27}) (u^{15}) }{ (u^{6})^2 (u^{29}) } = 1,\; (\frac{r^{13}}{v^5}) v^2 (\frac{r^{3}}{v^3})(\frac{r^{4}}{v^1}) / (\frac{r^5}{v^2})^2 (\frac{r^{10}}{v^3}) = 1</math> :<math>\frac{h c^2 e m_e}{G^2 k_B} = </math> {{font color|red|yellow| '''0.1413... 10^-21'''}}, <math>units = \frac{kg^3 s^3 C K}{m^4} == \frac{(u^{15})^3 (u^{-30})^3 (u^{-27}) (u^{20}) }{(u^{-13})^4} = 1</math> ====== α ====== :<math>\frac{2 (h^*)}{(\mu_0^*) (e^*)^2 (c^*)} = 2({2^3 \pi^4 \Omega^4})/(\frac{\alpha}{2^{11} \pi^5 \Omega^4})(\frac{2^{7} \pi^4 \Omega^3}{\alpha})^2(2 \pi \Omega^2) = \color{blue}\alpha \color{black},\; \frac{u^{19}}{u^{56} (u^{-27})^2 u^{17}} = 1,\; (\frac{r^{13}}{v^5})(\frac{1}{r^7})(\frac{v^6}{r^6})(\frac{1}{v}) = 1</math> Note: This will apply to any combinations of constants where '''units = scalars = 1'''. ===== SI Planck unit scalars ===== :<math>M = m_P = (1)k;\; k = m_P = .217\;672\;817\;58... \;10^{-7},\; u^{15}\; (kg)</math> :<math>T = t_p = {\pi}t;\; t = \frac{t_p}{\pi} = .171\;585\;512\;84... 10^{-43},\; u^{-30}\; (s)</math> :<math>L = l_p = {2\pi^2\Omega^2}l;\; l = \frac{l_p}{2\pi^2\Omega^2} = .203\;220\;869\;48... 10^{-36},\; u^{-13}\; (m)</math> :<math>V = c = {2\pi\Omega^2}v;\; v = \frac{c}{2\pi\Omega^2} = 11\;843\;707.905... ,\; u^{17}\; (m/s)</math> :<math>A = e/t_p = (\frac{2^7 \pi^3 \Omega^3}{\alpha})a = .126\;918\;588\;59... 10^{23},\; u^{3}\; (A)</math> Example MLT (units = scalars = 1); :<math>\frac{l^{15}}{k^9 t^{11}} = \frac{(.203...x10^{-36})^{15}}{(.217...x10^{-7})^9 (.171...x10^{-43})^{11}} \frac{u^{- 13*15}}{u^{15*9} u^{-30*11}} = 1</math> Example ALT (units = scalars = 1); :<math>\frac{a^3 l^3}{t} = \frac{(.126...x10^{23})^3 (.203...x10^{-36})^3}{ (.171...x10^{-43})} \frac{u^{3*3} u^{-13*3}} {u^{-30}} = 1</math> ===== MT to LPVA ===== In this example LPVA are derived from MT. The formulas for MT; :<math>M = (1)k,\; unit = u^{15}</math> :<math>T = (\pi) t,\; unit = u^{-30}</math> Replacing scalars ''pvla'' with ''kt'' :<math>P = (\Omega)\;\frac{k^{12/15}}{t^{2/15}},\; unit = u^{12/15*15-2/15*(-30)=16}</math> :<math>V = \frac{2 \pi P^2}{M} = (2 \pi \Omega^2)\; \frac{k^{9/15}}{t^{4/15}},\; unit = u^{9/15*15-4/15*(-30)=17} </math> :<math>L = T V = (2 \pi^2 \Omega^2) \; k^{9/15} t^{11/15},\; unit = u^{9/15*15+11/15*(-30)=-13}</math> :<math>A = \frac{2^4 V^3}{\alpha P^3} = \left(\frac{2^7 \pi^3 \Omega^3}{\alpha}\right)\; \frac{1}{k^{3/5} t^{2/5}},\; unit = u^{9/15*(-15)+6/15*30=3} </math> ===== PV to MTLA ===== In this example MLTA are derived from PV. The formulas for PV; :<math>P = (\Omega)p,\; unit = u^{16}</math> :<math>V = (2\pi\Omega^2)v,\; unit = u^{17}</math> Replacing scalars ''klta'' with ''pv'' :<math>M = \frac{2\pi P^2}{V} = (1)\frac{p^2}{v},\; unit = u^{16*2-17=15} </math> :<math>T = (\pi) \frac{p^{9/2}}{v^6},\; unit = u^{16*9/2-17*6=-30} </math> :<math>L = T V = (2\pi^2\Omega^2)\frac{p^{9/2}}{v^5},\; unit = u^{16*9/2-17*5=-13}</math> :<math>A = \frac{2^4 V^3}{\alpha P^3} = (\frac{2^7 \pi^3 \Omega^3}{\alpha})\frac{v^3}{p^3},\; unit = u^{17*3-16*3=3}</math> ==== Physical constants (as geometrical formulas) ==== note: <math>\color{red}(u^{15})^n\color{black}</math> constants have no Omega term. {| class="wikitable" |+Dimensioned constants; geometrical vs CODATA 2014 ! Constant ! In Planck units ! Geometrical object ! SI calculated (r, v, Ω, α^*) ! SI CODATA 2014 <ref>[http://www.codata.org/] | CODATA, The Committee on Data for Science and Technology | (2014)</ref> |- | [[w:Speed of light | Speed of light]] | V | <math>c^* = (2\pi\Omega^2)v,\;u^{17} </math> | ''c^*'' = 299 792 458, unit = u¹⁷ | ''c'' = 299 792 458 (exact) |- | [[w:Fine structure constant | Fine structure constant]] | | | ''α^*'' = 137.035 999 139 (mean) | ''α'' = 137.035 999 139(31) |- | [[w:Rydberg constant | Rydberg constant]] | <math>R^* = (\frac{m_e}{4 \pi L \alpha^2 M})</math> | <math>R^* = \frac{1}{2^{23} 3^3 \pi^{11} \alpha^5 \Omega^{17}}\frac{v^5}{r^9},\;u^{13} </math> | ''R^*'' = 10 973 731.568 508, unit = u¹³ | ''R'' = 10 973 731.568 508(65) |- | [[w:Vacuum permeability | Vacuum permeability]] | <math>\mu_0^* = \frac{4 \pi V^2 M}{\alpha L A^2}</math> | <math>\mu_0^* = \frac{\alpha}{2^{11} \pi^5 \Omega^4} r^7,\; u^{17*2+15+13-6=7*8=56}</math> | ''μ₀^*'' = 4π/10^7, unit = u⁵⁶ | ''μ₀'' = 4π/10^7 (exact) |- | [[w:Vacuum permittivity | Vacuum permittivity]] | <math>\epsilon_0^* = \frac{1}{\mu_0^* (c^*)^2}</math> | <math>\epsilon_0^* = \frac{2^9 \pi^3}{\alpha}\frac{1}{r^7 v^2},\; \color{red}1/(u^{15})^6\color{black} = u^{-90}</math> | | |- | [[w:Planck constant | Planck constant]] | <math>h^* = 2 \pi M V L</math> | <math>h^* = 2^3 \pi^4 \Omega^4 \frac{r^{13}}{v^5},\; u^{15+17-13 = 8*13-17*5 = 19}</math> | ''h^*'' = 6.626 069 134 e-34, unit = u¹⁹ | ''h'' = 6.626 070 040(81) e-34 |- | [[w:Gravitational constant | Gravitational constant]] | <math>G^* = \frac{V^2 L}{M}</math> | <math>G^* = 2^3 \pi^4 \Omega^6 \frac{r^5}{v^2},\; u^{34-13-15 = 8*5-17*2 = 6}</math> | ''G^*'' = 6.672 497 192 29 e11, unit = u⁶ | ''G'' = 6.674 08(31) e-11 |- | [[w:Elementary charge | Elementary charge]] | <math>e^* = A T</math> | <math>e^* = \frac{2^7 \pi^4 \Omega^3}{\alpha}\frac{r^3}{v^3},\; u^{3-30=3*8-17*3=-27}</math> | ''e^*'' = 1.602 176 511 30 e-19, unit = u^-19 | ''e'' = 1.602 176 620 8(98) e-19 |- | [[w:Boltzmann constant | Boltzmann constant]] | <math>k_B^* = \frac{2 \pi V M}{A}</math> | <math>k_B^* = \frac{\alpha}{2^5 \pi \Omega} \frac{r^{10}}{v^3},\; u^{17+15-3=10*8-17*3=29}</math> | ''k_B^*'' = 1.379 510 147 52 e-23, unit = u²⁹ | ''k_B'' = 1.380 648 52(79) e-23 |- | [[w:Electron mass | Electron mass]] | | <math>m_e^* = \frac{M}{f_e},\; u^{15}</math> | ''m_e^*'' = 9.109 382 312 56 e-31, unit = u¹⁵ | ''m_e'' = 9.109 383 56(11) e-31 |- | [[w:Classical electron radius | Classical electron radius]] | | <math>\lambda_e^* = 2\pi L f_e,\; u^{-13}</math> | ''λ_e^*'' = 2.426 310 2366 e-12, unit = u^-13 | ''λ_e'' = 2.426 310 236 7(11) e-12 |- | [[w:Planck temperature | Planck temperature]] | <math>T_p^* = \frac{A V}{\pi}</math> | <math>T_p^* = \frac{2^7 \pi^3 \Omega^5}{\alpha} \frac{v^4}{r^6} ,\; u^{3+17=17*4-6*8=20} </math> | ''T_p^*'' = 1.418 145 219 e32, unit = u²⁰ | ''T_p'' = 1.416 784(16) e32 |- | [[w:Planck mass | Planck mass]] | M | <math>m_P^* = (1)\frac{r^4}{v} ,\; \color{red}(u^{15})^1\color{black}</math> | ''m_P^*'' = .217 672 817 580 e-7, unit = u¹⁵ | ''m_P'' = .217 647 0(51) e-7 |- | [[w:Planck length | Planck length]] | L = TV | <math>l_p^* = (2\pi^2\Omega^2)\frac{r^9}{v^5},\;u^{-13} </math> | ''l_p^*'' = .161 603 660 096 e-34, unit = u^-13 | ''l_p'' = .161 622 9(38) e-34 |- | [[w:Planck time | Planck time]] | T | <math>t_p^* = (\pi)\frac{r^9}{v^6} ,\; \color{red}1/(u^{15})^2\color{black} </math> | ''t_p^*'' = 5.390 517 866 e-44, unit = u^-30 | ''t_p'' = 5.391 247(60) e-44 |- | [[w:Ampere | Ampere]] | <math>A = \frac{16 V^3}{\alpha P^3}</math> | <math>A^* = \frac{2^7\pi^3\Omega^3}{\alpha}\frac{v^3}{r^6} ,\; u^3 </math> | A^* = 0.297 221 e25, unit = u³ | ''e/t_p'' = 0.297 181 e25 |- | [[w:Quantum Hall effect | Von Klitzing constant ]] | <math>R_K^* = (\frac{h}{e^2})^*</math> | <math>R_K^* = \frac{\alpha^2}{2^{11} \pi^4 \Omega^2} r^7 v ,\; u^{73}</math> | ''R_K^*'' = 25812.807 455 59, unit = u⁷³ | ''R_K'' = 25812.807 455 5(59) |- | [[w:Gyromagnetic ratio | Gyromagnetic ratio]] | | <math>\gamma_e/2\pi = \frac{g l_p^* m_P^*}{2 k_B^* m_e^*},\; unit = u^{-42}</math> | ''γ_e/2π^*'' = 28024.953 55, unit = u^-42 | ''γ_e/2π'' = 28024.951 64(17) |} Note that ''r, v, Ω, α'' are dimensionless numbers, however when we replace ''u''ⁿ with the SI unit equivalents (''u''¹⁵ → kg, ''u''^-13 → m, ''u''^-30 → s, ...), the ''geometrical objects'' (i.e.: ''c^*'' = 2πΩ²v = 299792458, units = u¹⁷) become '''indistinguishable''' from their respective ''physical constants'' (i.e.: ''c'' = 299792458, units = m/s). If this mathematical relationship can therefore be identified within the SI units themselves, then we have an argument for a Planck scale mathematical universe <ref>[https://codingthecosmos.com/planck-scale.html Planck scale mathematical universe model]</ref>. ===== Electron formula ===== {{main|Electron (mathematical)}} Although the Planck units MLTA are embedded within the electron formula ''f_e'', this formula is both unit-less and non scalable (units = 1, scalars = 1). Furthermore it is the geometry of 2 dimensionless physical constants and so can also be defined as a dimensionless physical constant (if units = scalars = 1, then that constant will be independent of any numerical system and of any system of units, and so would qualify as a "natural unit"). :<math>f_e = 4\pi^2(2^6 3 \pi^2 \alpha \Omega^5)^3 = .23895453...x10^{23}</math> AL as an ampere-meter (ampere-length) are the units for a [[w:magnetic monopole | magnetic monopole]]. :<math>T = \pi \frac{r^9}{v^6},\; u^{-30}</math> :<math>\sigma_{e} = \frac{3 \alpha^2 A L}{2\pi^2} = {2^7 3 \pi^3 \alpha \Omega^5}\frac{r^3}{v^2},\; u^{-10}</math> :<math>f_e = \frac{\sigma_{e}^3}{2 T} = \frac{(2^7 3 \pi^3 \alpha \Omega^5)^3}{2\pi},\; units = \frac{(u^{-10})^3}{u^{-30}} = 1, scalars = (\frac{r^3}{v^2})^3 \frac{v^6}{r^9} = 1</math> The electron has dimensioned parameters, however the dimensions derive from the Planck units, ''f_e'' is a mathematical function that dictates how these Planck objects are applied, it does not have dimension units of its own, consequently there is no physical electron. [[w:electron mass | electron mass]] <math>m_e = \frac{M}{f_e}</math> (M = [[w:Planck mass | Planck mass]]) [[w:Compton wavelength | electron wavelength]] <math>\lambda_e = 2\pi L f_e</math> (L = [[w:Planck length | Planck length]]) [[w:elementary charge | elementary charge]] <math>e = A.T</math> ===== Fine structure constant ===== The Sommerfeld [[w:fine-structure constant | fine structure constant alpha]] is a dimensionless physical constant, the CODATA 2018 inverse alpha = 137.035999084. :<math>\frac{2 (h^*)}{(\mu_0^*) (e^*)^2 (c^*)} = 2({2^3 \pi^4 \Omega^4})/(\frac{\alpha}{2^{11} \pi^5 \Omega^4})(\frac{2^{7} \pi^4 \Omega^3}{\alpha})^2(2 \pi \Omega^2) = \color{blue}\alpha \color{black},\; \frac{u^{19}}{u^{56} (u^{-27})^2 u^{17}} = 1,\; (\frac{r^{13}}{v^5})(\frac{1}{r^7})(\frac{v^6}{r^6})(\frac{1}{v}) = 1</math> ===== Omega ===== The most precise of the experimentally measured constants is the Rydberg ''R = 10973731.568508(65) 1/m''. Here ''c, μ₀, R'' are combined into a unit-less ratio; :<math>\frac{(c^*)^{35}}{(\mu_0^*)^9 (R^*)^7} = (2 \pi \Omega^2)^{35}/(\frac{\alpha}{2^{11} \pi^5 \Omega^4})^9 .(\frac{1} {2^{23} 3^3 \pi^{11} \alpha^5 \Omega^{17}})^7,\;units = \frac{(u^{17})^{35}}{(u^{56})^9 (u^{13})^7} = 1</math> We can now define ''Ω'' using the geometries for (''c^*, μ₀^*, R^*'') and then solve by replacing (''c^*, μ₀^*, R^*'') with the numerical (''c, μ₀, R''). :<math>\Omega^{225}=\frac{(c^*)^{35}}{2^{295} 3^{21} \pi^{157} (\mu_0^*)^9 (R^*)^7 \alpha^{26}}, \;units = 1</math> :<math>\Omega = 2.007\;134\;949\;636...,\; units = 1</math> (CODATA 2014 mean values) :<math>\Omega = 2.007\;134\;949\;687...,\; units = 1</math> (CODATA 2018 mean values) There is a close natural number for ''Ω'' that is a square root implying that ''Ω'' can have a plus or a minus solution and this agrees with theory. This solution would however re-classify Omega as a mathematical constant (as being derivable from other mathematical constants). :<math>\Omega = \sqrt{ \left(\frac{\pi^e}{e^{(e-1)}}\right)} = 2.007\;134\;9543... </math> ===== G, h, e, m_e, k_B ===== As geometrical objects, the physical constants (''G, h, e, m_e, k_B'') can also be defined using the geometrical formulas for (''c^*, μ₀^*, R^*'') and solved using the numerical (mean) values for (''c, μ₀, R, α''), i.e.:. :<math>{(h^*)}^3 = (2^3 \pi^4 \Omega^4 \frac{r^{13} u^{19}}{v^5})^3 = \frac{2\pi^{10} {(\mu_0^*)}^3} {3^6 {(c^*)}^5 \alpha^{13} {(R^*)}^2},\; unit = u^{57}</math> {| class="wikitable" |+Physical constants; calculated vs CODATA 2014 ! Constant ! Geometry ! Calculated from (R, c, μ₀, α) ! CODATA 2014 <ref>[http://www.codata.org/] | CODATA, The Committee on Data for Science and Technology | (2014)</ref> |- | [[w:Planck constant | Planck constant]] | <math>{(h^*)}^3 = \frac{2\pi^{10} {\mu_0}^3} {3^6 {c}^5 \alpha^{13} {R_\infty}^2},\; unit = u^{57}</math> | ''h^*'' = 6.626 069 134 e-34, unit = u¹⁹ | ''h'' = 6.626 070 040(81) e-34 |- | [[w:Gravitational constant | Gravitational constant]] | <math>{(G^*)}^5 = \frac{\pi^3 {\mu_0}}{2^{20} 3^6 \alpha^{11} {R_\infty}^2},\; unit = u^{30}</math> | ''G^*'' = 6.672 497 192 29 e11, unit = u⁶ | ''G'' = 6.674 08(31) e-11 |- | [[w:Elementary charge | Elementary charge]] | <math>{(e^*)}^3 = \frac{4 \pi^5}{3^3 {c}^4 \alpha^8 {R_\infty}},\; unit = u^{-81}</math> | ''e^*'' = 1.602 176 511 30 e-19, unit = u^-19 | ''e'' = 1.602 176 620 8(98) e-19 |- | [[w:Boltzmann constant | Boltzmann constant]] | <math>{(k_B^*)}^3 = \frac{\pi^5 {\mu_0}^3}{3^3 2 {c}^4 \alpha^5 {R_\infty}} ,\; unit = u^{87} </math> | ''k_B^*'' = 1.379 510 147 52 e-23, unit = u²⁹ | ''k_B'' = 1.380 648 52(79) e-23 |- | [[w:Electron mass | Electron mass]] | <math>{(m_e^*)}^3 = \frac{16 \pi^{10} {R_\infty} {\mu_0}^3}{3^6 {c}^8 \alpha^7},\; unit = u^{45} </math> | '' m_e^*'' = 9.109 382 312 56 e-31, unit = u¹⁵ | ''m_e'' = 9.109 383 56(11) e-31 |- | [[w:Planck mass | Planck mass]] | <math>(m_P^*)^{15} = \frac{2^{25}\pi^{13} \mu_0^6}{ 3^6 c^5 \alpha^{16} R_\infty^2},\; unit = (u^{15})^{15}</math> | ''m_P^*'' = .217 672 817 580 e-7, unit = u¹⁵ | ''m_P'' = .217 647 0(51) e-7 |- | [[w:Planck length | Planck length]] | <math>(l_p^*)^{15} = \frac{\pi^{22} \mu_0^9}{2^{35} 3^{24} \alpha^{49} c^{35} R_\infty^8},\; unit = (u^{-13})^{15} </math> | ''l_p^*'' = .161 603 660 096 e-34, unit = u^-13 | ''l_p'' = .161 622 9(38) e-34 |- | [[w:Gyromagnetic ratio | Gyromagnetic ratio]] | <math>(\gamma_e/2\pi)^3 = \frac{g^3 3^3 c^4}{2^8 \pi^8 \alpha \mu_0^3 R_\infty^2},\; unit = u^{-126}</math> | ''γ_e/2π^*'' = 28024.953 55, unit = u^-42 | ''γ_e/2π'' = 28024.951 64(17) |} ==== 2019 SI unit revision ==== Following the 26th General Conference on Weights and Measures ([[w:2019 redefinition of SI base units|2019 redefinition of SI base units]]) are fixed the numerical values of the 4 physical constants (''h, c, e, k_B''). In the context of this model however only 2 base units may be assigned by committee as the rest are then numerically fixed by default and so the revision may lead to unintended consequences. For example, the von Klitzing constant Rk = h/e2, yet to derive h and e using Rk suggests an (inverse) alpha = 137.0359952 <ref>[https://codingthecosmos.com/physical-constants-calc.html Physical constants calculator]</ref>. {| class="wikitable" |+Physical constants ! Constant ! CODATA 2018 <ref>[http://www.codata.org/] | CODATA, The Committee on Data for Science and Technology | (2018)</ref> |- | [[w:Speed of light | Speed of light]] | ''c'' = 299 792 458 (exact) |- | [[w:Planck constant | Planck constant]] | ''h'' = 6.626 070 15 e-34 (exact) |- | [[w:Elementary charge | Elementary charge]] | ''e'' = 1.602 176 634 e-19 (exact) |- | [[w:Boltzmann constant | Boltzmann constant]] | ''k_B'' = 1.380 649 e-23 (exact) |- | [[w:Fine structure constant | Fine structure constant]] | ''α'' = 137.035 999 084(21) |- | [[w:Rydberg constant | Rydberg constant]] | ''R'' = 10973 731.568 160(21) |- | [[w:Electron mass | Electron mass]] | ''m_e'' = 9.109 383 7015(28) e-31 |- | [[w:Vacuum permeability | Vacuum permeability]] | ''μ₀'' = 1.256 637 062 12(19) e-6 |- | [[w:Quantum_Hall_effect#Applications | Von Klitzing constant]] | ''R_K'' = 25812.807 45 (exact) |} For example, if we solve using the above formulas; <math>R^* = \frac{4 \pi^5}{3^3 c^4 \alpha^8 e^3} = 10973\;729.082\;465</math> <math>{(m_e^*)}^3 = \frac{2^4 \pi^{10} R \mu_0^3}{3^6 c^8 \alpha^7},\;m_e^* = 9.109\;382\;3259 \;10^{-31}</math> <math>{(\mu_0^*)}^3 = \frac{3^6 h^3 c^5 \alpha^{13} R^2}{2 \pi^{10}},\;\mu_0^* = 1.256\;637\;251\;88\;10^{-6}</math> <math>{(h^*)}^3 = \frac{2 \pi^{10} \mu_0^3}{3^6 c^5 \alpha^{13} R^2},\;h^* = 6.626\;069\;149\;10^{-34}</math> <math>{(e^*)}^3 = \frac{4 \pi^5}{3^3 c^4 \alpha^8 R},\; e^* = 1.602\;176\;513\;10^{-19}</math> ==== u as sqrt(velocity/mass) ==== We find there is a single base unit '''u''' from which the other units and numerical values can be derived. This base unit incorporates [[v:Sqrt_Planck_momentum |MLT as square roots]]. =====''u = √{L/M.T}''===== :<math>u,\; units = \sqrt{\frac{L}{M T}} = \sqrt{u^{-13-15+30=2}} = u^1</math> Setting: :<math>x,\;units = \sqrt{\frac{M^9 T^{11}}{L^{15}}} = u^0, units = 1, scalars = 1</math> :<math>y,\;units = M^2T = u^0, units = 1, scalars <> 1</math> Gives; :<math>u^3 = \frac{L^{3/2}}{M^{3/2} T^{3/2}} = A,\; (ampere)</math> :<math>u^6 (y) = \frac{L^3}{T^2 M},\; (G)</math> :<math>u^{13} (xy) = \frac{1}{L},\; (1/l_p)</math> :<math>u^{15} (xy^2) = M,\; (m_P)</math> :<math>u^{17} (xy^2) = V,\; (c)</math> :<math>u^{19} (xy^3) = \frac{ML^2}{T},\; (h)</math> :<math>u^{20} (xy^2) = \frac{L^{5/2}}{M^{3/2} T^{5/2}} = AV,\;(T_P)</math> :<math>u^{27} (x^2y^3) = \frac{M^{3/2}\sqrt{T}}{L^{3/2}} = 1/AT,\; (1/e)</math> :<math>u^{29} (x^2y^4) = \frac{M^{5/2}\sqrt{T}}{\sqrt{L}} = ML/AT,\; (k_B)</math> :<math>u^{30} (x^2 y^3) = \frac{1}{T},\; (1/t_p)</math> :<math>u^{56} (x^4 y^7) = \frac{M^4 T}{L^2},\;(\mu_0)</math> :<math>u^{90} (x^6 y^{11}) = \frac{M^4}{T} </math> ===== ''β'' (unit = ''u'') ===== ''i'' (from ''x'') and ''j'' (from ''y''). :<math>R = \sqrt{P} = \sqrt{\Omega} r,\; units = u^8</math> :<math>\beta = \frac{V}{R^2} = \frac{2\pi R^2}{M} = \frac{A^{1/3} \alpha^{1/3}}{2} \;..., \; unit = u</math> :<math>i = \frac{1}{2\pi {(2\pi \Omega)}^{15}},\; units = 1, scalars = 1</math> :<math>j = \frac{r^{17}}{v^8} = k^2t = \frac{k^8}{r^{15}} ...,\; unit = \frac{u^{17*8}}{u^{8*17}} = u^{15*2}u^{-30} ... = 1,\; units = 1, scalars <> 1</math> For example; the constants solved in terms of (''r, v'') :<math>\beta = \frac{V}{R^2} = \frac{2\pi \Omega^2 v}{\Omega r^2} = \frac{2\pi \Omega v}{r^2},\; u^1 = u</math> :<math>A = \beta^3 (\frac{2^4}{\alpha}) = \frac{2^7 \pi^3 \Omega^3}{\alpha}\frac{v^3}{r^6},\; u^3</math> :<math>G = \frac{\beta^6}{2^3 \pi^2} (j) = 2^3 \pi^4 \Omega^6 \frac{r^5}{v^2},\; u^6</math> :<math>L^{-1} = 4\pi \beta^{13} (ij) = \frac{1}{2\pi^2 \Omega^2} \frac{v^5}{r^9},\; u^{13}</math> :<math>M = 2\pi \beta^{15} (ij^2) = \frac{r^4}{v},\; u^{15}</math> :<math>P = \beta^{16} (ij^2) = \Omega r^2,\; u^{16}</math> :<math>V = \beta^{17} (ij^2) = 2\pi \Omega^2 v,\; u^{17}</math> :<math>h = \pi \beta^{19} (ij^3) = 8\pi^4 \Omega^4 \frac{r^{13}}{v^5},\; u^{19}</math> :<math>T_P^* =\frac{2^3 \beta^{20}}{\pi \alpha} (ij^2) = \frac{2^7 \pi^3 \Omega^5}{\alpha} \frac{v^4}{r^6} ,\; u^{20} </math> :<math>e^{-1} = \frac{\alpha \pi \beta^{27} (i^2j^3)}{4} = \frac{\alpha}{128\pi^4 \Omega^3} \frac{v^3}{r^{3}},\; u^{27} </math> :<math>k_B = \frac{\alpha \pi^2 \beta^{29}(i^2j^4)}{4} = \frac{\alpha}{32 \pi \Omega} \frac{r^{10}}{v^3},\; u^{29}</math> :<math>T^{-1} = 4\pi \beta^{30} (i^2 j^3) = \frac{1}{\pi}\frac{v^6}{r^9},\; u^{30}</math> :<math>\mu_0^* = \frac{\pi^3 \alpha \beta^{56}}{2^3} (i^4 j^7) = \frac{\alpha}{2^{11} \pi^5 \Omega^4} r^7,\; u^{56}</math> :<math>\epsilon_0^{*-1} = \mu_0^* (c^*)^2 = \frac{\pi^3 \alpha \beta^{90}}{2^3} (i^6 j^{11}) = \frac{\alpha}{2^9 \pi^3} v^2 r^7,\; u^{90} </math> ===== limit ''j'' ===== The SI values for ''j'' suggest a limit (numerical boundary) to the values the SI constants can have. :<math>j = \frac{r^{17}}{v^8} = k^2 t = \frac{k^{17/4}}{v^{15/4}} = ... </math> gives a range from 0.812997... ''x''10^-59 to 0.123... ''x''10⁶⁰ In SI terms unit ''β'' can be derived via these ratio; :<math>a^{1/3} = \frac{v}{r^2} = \frac{1}{t^{2/15}k^{1/5}} = \frac{\sqrt{v}}{\sqrt{k}} ... = 23326079.1...; unit = u</math> ===== Rydberg formula ===== The [[w:Rydberg_formula |Rydberg formula]] can now be re-written in terms of amperes <math>A^2</math> :<math>\frac{hc}{2\pi \alpha^2} = \frac{j^2 A^2}{2^8 2\pi t_p}</math> ==== External links ==== * [[v:electron_(mathematical) | Mathematical electron]] * [[v:Relativity_(Planck) | Programming relativity at the Planck scale]] * [[v:Quantum_gravity_(Planck) | Programming gravity at the Planck scale]] * [[v:Black-hole_(Planck) | Programming the cosmic microwave background at the Planck scale]] * [[v:Sqrt_Planck_momentum | The sqrt of Planck momentum]] * [[v:God_(programmer) | The Programmer God]] * [[w:Simulation_hypothesis | The Simulation hypothesis]] * [https://codingthecosmos.com/ Programming at the Planck scale using geometrical objects] -Malcolm Macleod's website * [http://www.simulation-argument.com/ Simulation Argument] -Nick Bostrom's website * [https://www.amazon.com/Our-Mathematical-Universe-Ultimate-Reality/dp/0307599809 Our Mathematical Universe: My Quest for the Ultimate Nature of Reality] -Max Tegmark * [https://dx.doi.org/10.2139/ssrn.2531429 The mathematical electron model in a Planck scale universe] -(article) * [https://link.springer.com/article/10.1134/S0202289308020011/ Dirac-Kerr-Newman black-hole electron] -Alexander Burinskii (article) ==== References ==== {{Reflist}} [[Category: Physics]] [[Category: Philosophy of science]] h7e474edzbxwryvco8g0hocsfyktxdi 3 276205 2410362 2410108 2022-07-30T00:35:27Z Dave Braunschweig 426084 /* Subpages */ Reply wikitext text/x-wiki {{Robelbox|theme=9|title=Welcome!|width=100%}} <div style="{{Robelbox/pad}}"> '''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] Jacsaw!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Dave Braunschweig|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOjs UI icon signature-ltr.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity. To [[Wikiversity:Introduction|get started]], you may <!-- The Left column --> <div style="width:50.0%; float:left"> * [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]]. * Visit a (kind of) [[Wikiversity:Random|random project]]. * [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]]. * Find out about [[Wikiversity:Research|research]] activities on Wikiversity. * [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left. </div> <!-- The Right column --> <div style="width:50.0%; float:left"> * Enable VisualEditor under [[Special:Preferences#mw-prefsection-betafeatures|Beta]] settings to make article editing easier. * Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity. * Give [[Wikiversity:Feedback|feedback]] about your initial observations. * Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]]. * [[Wikiversity:Chat|Chat]] with other Wikiversitans on [[:freenode:wikiversity|<kbd>#wikiversity</kbd>]]. </div> <br clear="both"/> You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 22:10, 1 August 2021 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} == Subpages == Please use subpages for course pages. See [https://en.wikiversity.org/w/index.php?title=User%3AJacsaw%2Fsandbox&type=revision&diff=2384897&oldid=2384799] for how to implement this. When you are ready to move this project out of your sandbox and into main space, let me or one of the other custodians know. We can move all of the pages for you at once vs. moving them individually. [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 18:23, 22 March 2022 (UTC) :Hi, I would like to move this content out of sandbox to a main page, and would love to move the subpages at the same time. Are you able to assist? thank you! [[User:Jacsaw|Jacsaw]] ([[User talk:Jacsaw|discuss]] • [[Special:Contributions/Jacsaw|contribs]]) 00:56, 29 July 2022 (UTC) ::Please include my name in the post so I know you've reached out to me. I can do the move. What name do you want for the base page (title for the learning project)? -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 00:35, 30 July 2022 (UTC) If students will be editing this content, it is time to move it out of your user space. It can either be a Draft: space resource, if you prefer, or a regular main space resource. Please let me know which you prefer so it can be moved before additional students work on the content. Thanks! -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 00:12, 28 March 2022 (UTC) :Hi, yes, the students have started adding content to the modules. I was planning on moving it out of sandbox once content was added and reviewed. Does marking it as draft prevent it from being public while it's being built out? thanks! [[Special:Contributions/24.187.47.101|24.187.47.101]] ([[User talk:24.187.47.101|discuss]]) 01:23, 28 March 2022 (UTC) ::The only real difference between Draft: space and main space is whether or not the content appears in local (Wikiversity) search results. Almost all classes with student-developed content are in main space. See [[Solidarity Economy in Latin America]], [[Digital Media Concepts]], and [[Federal Writers' Project – Life Histories]] for current examples. If you tell me what title you want, I can move the pages with one click rather than you having to move each page individually. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 14:10, 28 March 2022 (UTC) qcfexta751zvk90v0vem2qkg39ydfb5 0 277657 2410377 2409813 2022-07-30T02:27:20Z KingMob221 2947327 /* Motivation */ wikitext text/x-wiki {{/Banner}} ==Motivation== # [[/Academic help-seeking/]] - What are the barriers and enablers of AHS and how can AHS be fostered? - [[User:MyUserName|MyUserName]] # [[/Academic self-regulation/]] - What is academic self-regulation, why does it matter, and how can it be fostered? - [[User:MyUserName|MyUserName]] # [[/Actively open-minded thinking/]] - How can AOT be used to improve human performance? - [[User:MyUserName|MyUserName]] # [[/Active transport motivation/]] - What motivates use of active transport and how can people be encouraged to use it? - [[User:MyUserName|MyUserName]] # [[/Antidepressants and motivation/]] - What are the effects of popular antidepressants on motivation? - [[User:MyUserName|MyUserName]] # [[/Approach motivation/]] - What is approach motivation and how does it lead to behaviour? - [[User:U3189370|U3189370]] # [[/Behavioural economics and motivation/]] - What aspects of motivation theory are useful in behavioural economics? - u3141987 # [[/Behavioural model of health services/]] - What is the BMHS and how can it be used? - [[User:MyUserName|MyUserName]] # [[/Beneficence as a psychological need/]] - What is beneficence and what are its implications as a psychological need? - [[User:MyUserName|MyUserName]] # [[/Brief motivational interviewing as a health intervention/]] - How can brief motivational interviewing be used as a health intervention? - [[User:MyUserName|MyUserName]] # [[/Choice overload/]] - How much choose is too much? How much choice is enough? - [[User:MyUserName|MyUserName]] # [[/Chunking and goal pursuit/]] - How does chunking affect goal pursuit? - [[User:MyUserName|MyUserName]] # [[/Cognitive entrenchment/]] - What is cognitive entrenchment and how can it be avoided? - [[User:MyUserName|MyUserName]] # [[/Climate change helplessness/]] - How does learned helpless impact motivation to engage in behaviours to limit climate change? - [[User:MyUserName|MyUserName]] # [[/Closeness communication bias/]] - What is the CCB, why does it occur, and how can it be overcome? - [[User:MyUserName|MyUserName]] # [[/Commitment bias/]] - What motivates escalation of commitment even it does not lead to desirably outcomes? - [[User:MyUserName|MyUserName]] # [[/Conspiracy theory motivation/]] - What motivates people to believe in conspiracy theories? - KingMob221 # [[/Construal level theory/]] - What is construal level theory and how can it be applied? - [[User:MyUserName|MyUserName]] # [[/Courage motivation/]] - What is courage, what motivates courage, and how can courage be enhanced? -[[User:U3213871] # [[/Death drive/]] - What is the death drive and how can it be negotiated? - [[User:U3086459|U3086459]] # [[/Drugs-violence nexus and motivation/]] - What is the role of motivation in the drugs-violence nexus? - [[User:MyUserName|MyUserName]] # [[/Episodic future thinking and delay discounting/]] - What is the relationship between between EFT and DD? - [[User:MyUserName|MyUserName]] # [[/Episodic memory and planning/]] - What role does episodic memory play in planning? - [[User:MyUserName|MyUserName]] # [[/Equity theory/]] - What is equity theory and how can it be applied? - [[U3086459|MyUserName]] # [[/Frame of reference and motivation/]] - How does frame of reference affect motivation? - [[User:MyUserName|MyUserName]] # [[/Freedom and motivation/]] - What is the effect of freedom on motivation? - [[User:MyUserName|MyUserName]] # [[/Fully functioning person/]] - What is a FFP and how can full functioning be developed? - [U3217727] # [[/Functional fixedness/]] - What is functional fixedness and how can it be overcome? - [[User:U3214117]] # [[/Functional imagery training/]] - What is FIT and how can it be applied? - [[User:MyUserName|MyUserName]] # [[/Gamification and work motivation/]] - How can gamification enhance work motivation? - [[User:MyUserName|MyUserName]] # [[/Giving up goals/]] - When should we give up goals and when should we persist? - [[User:MyUserName|MyUserName]] # [[/Green prescription motivation/]] - What motivates green prescription compliance? - [[User:MyUserName|MyUserName]] # [[/Health belief model/]] - What is the HBM and how can it be used to enhance motivation for health-promoting behaviour? - [[User:MyUserName|MyUserName]] # [[/Hijack hypothesis of drug addiction/]] - What is the hijack hypothesis, what is the evidence, and how does it help to understand drug addiction? - [[U3218292]] # [[/Honesty motivation/]] - What motivates honesty? - [[User:MyUserName|MyUserName]] # [[/Humour and work/]] - What is the role of humour in the workplace? - [[User:MyUserName|MyUserName]] # [[/IKEA effect/]] - What is the IKEA effect and how can it be applied? - [[User:MyUserName|MyUserName]] # [[/Intertemporal choice/]] - What are intertemporal choices and how can they be effectively negotiated? - [[User:MyUserName|MyUserName]] # [[/Kindness motivation/]] - What motivates kindness? - [[User:MyUserName|MyUserName]] # [[/Motivational music and exercise/]] - How can music be used to help motivate exercise? - [[User:MyUserName|MyUserName]] # [[/Novelty-variety as a psychological need/]] - What is novelty-variety and what are its implications as a psychological need? - [[User:MyUserName|MyUserName]] # [[/Nucleus accumbens and motivation/]] - What role does the nucleus accumbens play in motivation? - [[User:MyUserName|MyUserName]] # [[/Physiological needs/]] - What are human's physiological needs and how does this influence motivation? - [[U3203655]] # [[/Protection motivation theory and COVID-19/]] - How does PMT apply to managing COVID-19? - [[User:MyUserName|MyUserName]] # [[/Relative deprivation and motivation/]] - What is the effect of relative deprivation on motivation? - [[User:MyUserName|MyUserName]] # [[/Retrospective regret/]] - What is the motivational role of retrospective regret? - [[User:MyUserName|MyUserName]] # [[/Revenge motivation/]] - What motivates revenge and how does it affect us? - [[User:MyUserName|MyUserName]] # [[/Self-efficacy and achievement/]] - What role does self-efficacy play in achievement outcomes? - [[User:U943292|U943292]] # [[/Sexual harassment at work motivation/]] - What motivates sexual harassment at work and what can be done about it? - [[User:MyUserName|MyUserName]] # [[/Signature strengths/]] - What are signature strengths and how can they be applied? - [[User:MyUserName|MyUserName]] # [[/Social cure/]] - What is the social cure and how can it be applied? - [[User:MyUserName|MyUserName]] # [[/System justification theory/]] - What is SJT, how does it affect our lives, and what can be done about it? - [[User:MyUserName|MyUserName]] # [[/Stretch goals/]] - What are stretch goals? Do they work? - [[User:MyUserName|MyUserName]] # [[/Sublimation/]] - What is sublimation and how can it be fostered? - [[User:MyUserName|MyUserName]] # [[/Survival needs and motivation/]] - What are survival needs and how do they influence motivation? - [[User:MyUserName|MyUserName]] # [[/Task initiation/]] - What are the challenges with task initiation and how to get get started? - [[User:MyUserName|MyUserName]] # [[/Theoretical domains framework/]] - What is the TDF and how can be used to guide behaviour change? - [[User:MyUserName|MyUserName]] # [[/Time and motivation/]] - What is the effect of time on motivation? - [[User:MyUserName|MyUserName]] # [[/Time management/]] - How can one's time be managed effectively? - [[User:MyUserName|MyUserName]] # [[/To-do lists/]] - Are to-do lists a good idea? What are their pros and cons? How can they be used effectively? - [[User:MyUserName|MyUserName]] # [[/Uncertainty avoidance/]] - What is uncertainty avoidance, why does it occur, and what are its consequences? - [[User:MyUserName|MyUserName]] # [[/Urgency bias and productivity/]] - What is the impact of urgency bias on productivity and what can be done about it? - [[User:MyUserName|MyUserName]] # [[/Vocational identity/]] - What is vocational identity and how does it develop? - [[User:MyUserName|MyUserName]] # [[/Wanting and liking/]] - What are the similarities and differences between wanting and liking, and what are the implications? - [[User:MyUserName|MyUserName]] # [[/Work breaks, well-being, and productivity/]] - How do work breaks affect well-being and productivity? - [[User:MyUserName|MyUserName]] # [[/Work and flow/]] - What characteristics of work can produce flow and how can flow at work be fostered? - [[User:MyUserName|MyUserName]] ==Emotion== # [[/Animal emotion/]] - What is the emotional experience of animals? - [[User:MyUserName|MyUserName]] # [[/Attributions and emotion/]] - How do attributions affect emotion? - [[User:MyUserName|MyUserName]] # [[/Autonomous sensory meridian response and emotion/]] - What emotions are involved in ASMR experiences and why do they occur? - [[User:MyUserName|MyUserName]] # [[/Benzodiazepines and emotion/]] - What are the effects of benzodiazepines on emotion? - [[User:MyUserName|MyUserName]] # [[/Bewilderment/]] - What is bewilderment and how can it be dealt with? - [[User:MyUserName|MyUserName]] # [[/Burnout/]] - What is burnout and how can be it be managed and prevented? - [[User:MyUserName|MyUserName]] # [[/Cognitive dissonance reduction/]] - What strategies do people use to reduce cognitive dissonance and how effective are they? - [[User:MyUserName|MyUserName]] # [[/Colonisation and emotion in Australia/]] - What are the emotional responses to colonisation in Australia? - [[User:MyUserName|MyUserName]] # [[/Compassion/]] - What is compassion, what are its pros and cons, and how can it be fostered? - [[User:MyUserName|MyUserName]] # [[/Connection to country and well-being/]] - What is the relationship between connection to country and well-being? - [[User:MyUserName|MyUserName]] # [[/Contempt/]] - What is contempt, what causes it, and how can it be managed? - [[User:MyUserName|MyUserName]] # [[/Core emotions/]] - What are the core emotions and what is their function? - [[User:MyUserName|FulaAjeo22]] # [[/Creative arts and trauma/]] - How can creative arts help in dealing with trauma? - [[User:MyUserName|MyUserName]] # [[/Cultural influences on shame, guilt, and pride/]] - How does culture influence shame, guilt, and pride? - [[User:MyUserName|MyUserName]] # [[/Default mode network and the self/]] - What is the relationship between the DMN and the self? - [[User:MyUserName|MyUserName]] # [[/Difficult conversations and emotion/]] - What communication and emotional skills are needed to successfully negotiate difficult conversations? - [[User:MyUserName|MyUserName]] # [[/Disappointment/]] - What is disappointment, what causes it, and how can it be managed? - [[User:MyUserName|MyUserName]] # [[/DMT and spirituality/]] - How can DMT facilitate spiritual experiences? - [[User:MyUserName|MyUserName]] # [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? - [[User:MyUserName|MyUserName]] # [[/Ecological grief/]] - What is ecological grief and what can be done about it? - u3213748 / Brewerjr # [[/Embarrassment/]] - What is embarrassment, what causes it, and how can it be managed? - [[User:MyUserName|MyUserName]] # [[/Emotional intelligence training/]] - How can emotional intelligence be trained? - [[User:MyUserName|MyUserName]] # [[/Emotion knowledge/]] - What is emotion knowledge and how can it be developed? - [[User:MyUserName|MyUserName]] # [[/Emotion across the lifespan/]] - How does emotion develop across the lifespan? - [[User:MyUserName|MyUserName]] # [[/Endocannabinoid system and emotion/]] - What is the role of the endocannabinoid system in emotion? - [[User:MyUserName|MyUserName]] # [[/Environmental grief/]] - What is eco-grief, its causes and consequences, and what can be done? - [[User:MyUserName|MyUserName]] # [[/Exercise and endocannabinoids/]] - What is the relationship between exercise and the endocannabinoid system? - [[User:MyUserName|MyUserName]] # [[/Expressive suppression and emotion regulation/]] - What is the role of expressive suppression in emotion regulation? - [[User:MyUserName|MyUserName]] # [[/Fairness and emotion/]] - What is the relation between fairness and emotion? - [[User:MyUserName|MyUserName]] # [[/Fatigue and emotion/]] - What is the effect of fatigue on emotion and what can be done about it? - [[User:MyUserName|MyUserName]] # [[/Fear/]] - What is fear, what causes it, and how can it be managed? - [[User:MyUserName|MyUserName]] # [[/Fear of working out/]] - What is FOWO and how can it be overcome? - [[User:MyUserName|MyUserName]] # [[/Fundamental attribution error and emotion/]] - What is the relationship between the FAE and emotion? - [[User:MyUserName|MyUserName]] # [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? - [[User:MyUserName|MyUserName]] # [[/Heart rate variability and emotion regulation/]] - What is the relationship between HRV and emotion regulation? - [[User:MyUserName|MyUserName]] # [[/Hedonic adaptation prevention model/]] - What is the HAP model and how can it be applied? - [[User:MyUserName|MyUserName]] # [[/Humility/]] - What is humility, what causes it, and is it desirable? - [[User:MyUserName|MyUserName]] # [[/Hypomania and emotion/]] - What are the emotional characteristics of hypomania? - [[User:MyUserName|MyUserName]] # [[/Impact bias/]] - What is impact bias, what causes it, what are its consequences, and how can it be avoided? - [[User:MyUserName|MyUserName]] # [[Indigenous Australian emotionality]] - In what ways is emotionality experienced by Indigenous Australian people? - [[User:MyUserName|MyUserName]] # [[/Indigenous Australian mindfulness/]] - How has Indigenous Australian culture traditionally conceived of, and practiced, mindfulness? - [[User:MyUserName|MyUserName]] # [[/Inspiration/]] - What is inspiration, what causes it, what are its consequences, and how can it be fostered? - [[User:MyUserName|MyUserName]] # [[/Insular cortex and emotion/]] - What role does the insular cortex play in emotion? - [[User:MyUserName|MyUserName]] # [[/Interoception and emotion/]] - What is the relationship between interoception and emotion? - [[User:MyUserName|MyUserName]] # [[/Kama muta/]] - What is kama muta, what are its effects, and how can it be fostered? - [[User:MyUserName|MyUserName]] # [[/Linguistic relativism and emotion/]] - What is the role of linguistic relativism in emotion? - [[User:MyUserName|MyUserName]] # [[/Menstrual cycle mood disorders/]] - What causes menstrual cycle mood disorders and how can they be managed? - [[User:MyUserName|MyUserName]] # [[/Mindfulness and creativity/]] - How can mindfulness enhance creativity? - [[User:MyUserName|MyUserName]] # [[/Mindfulness and driving/]] - How can mindfulness affect driving? - [[User:MyUserName|MyUserName]] # [[/Mindful self-care/]] - What is mindful self-care, why does it matter, and how can it be developed? - [[User:MyUserName|MyUserName]] # [[/Mixed emotions/]] - What are mixed emotions, what causes them, and how can they be managed? - [[User:MyUserName|MyUserName]] # [[/Mudita/]] - What is mudita and how can it be developed? - [[User:MyUserName|MyUserName]] # [[/Natural disasters and emotion/]] - How do people respond emotionally to natural disasters and how can they be supported? - [[User:MyUserName|MyUserName]] # [[/Nature therapy/]] - What is nature therapy and how can it be applied? - [[User:MyUserName|MyUserName]] # [[/Narcissism and emotion/]] - What is the relationship between narcissism and emotion? - [[User:MyUserName|MyUserName]] # [[/Narrative therapy and emotion/]] - What is the role of emotion in narrative therapy? - [[User:MyUserName|MyUserName]] # [[/Needle fear/]] - How does needle fear develop, what are its consequences, and what can be done about it? - [[User:MyUserName|MyUserName]] # [[/Positivity ratio/]] - What is the positivity ratio and what are its implications? - [[User:MyUserName|MyUserName]] # [[/Post-traumatic stress disorder and emotion/]] - What is the effect of PTSD on emotion? - [[User:JorjaFive|U822459]] # [[/Psychological distress/]] - What is PD, what are the main types, and how can they be managed? - [[User:MyUserName|MyUserName]] # [[/Psychological trauma/]] - What causes psychological trauma, what are the consequences, and how can people recover from psychological trauma? - [[User:MyUserName|MyUserName]] # [[/Psilocybin assisted psychotherapy/]] - How can psilocybin be used to assist psychotherapy? - [[User:MyUserName|MyUserName]] # [[/Rational compassion/]] - What is rational compassion and how can it be cultivated? - [[User:MyUserName|MyUserName]] # [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? - [[User:MyUserName|MyUserName]] # [[/Religiosity and coping/]] - What is the relationship between religiosity and coping? - [[User:MyUserName|MyUserName]] # [[/Resentment/]] - What is resentment, what causes it, and what are its consequences? - [[User:MyUserName|MyUserName]] # [[/Risk-as-feelings/]] - What is the emotional experience of risk and how does it influence decision-making and behaviour? - [[User:MyUserName|MyUserName]] # [[/Self-esteem and culture/]] - What are the cultural influences on self-esteem? - [[User:MyUserName|MyUserName]] # [[/Smiling and emotion/]] - What is the relationship between smiling and emotion? - [[User:MyUserName|MyUserName]] # [[/Social media and suicide prevention/]] - How can social media be used to help prevent suicide? - [[User:MyUserName|MyUserName]] # [[/Sorry business/]] - What is sorry business and what role does it play in Indigenous communities in Australia? - [[User:MyUserName|MyUserName]] # [[/Stress control mindset/]] - What is a SCM, why does it matter, and how can it be cultivated? - [[User:MyUserName|MyUserName]] # [[/Suffering as emotion/]] - What is the emotional experience of suffering and how can people cope with suffering? - [[User:MyUserName|MyUserName]] # [[/Telemental health/]] - What are the pros and cons of TMH and what are the key ingredients for effective TMH practices? - [[User:MyUserName|MyUserName]] # [[/Topophilia/]] - What is topophilia, how does it develop, and what are the psychological impacts? - [[User:MyUserName|MyUserName]] # [[/Triumph/]] - What is triumph, what causes it, and how can it be managed? - [[User:MyUserName|MyUserName]] # [[/Unemployment and mental health/]]: What is the relationship between unemployment and mental health? - [[User:MyUserName|MyUserName]] # [[/Viewing natural scenes and emotion/]] - What is the effect of viewing natural scenes on emotion and how can this be applied? - [[User:MyUserName|MyUserName]] # [[/Volunteer tourism motivation/]] - What motivates volunteer tourism? - [[User:Efost|MyUserName]] # [[/Wave metaphor for emotion/]] - In what respects is an ocean wave a helpful metaphor for understanding human emotions? - [[User:MyUserName|MyUserName]] # [[/Window of tolerance/]] - What is the window of tolerance and how this concept be used? - [[User:MyUserName|MyUserName]] # [[/Workplace mental health training/]] - What is WMHT, what techniques are used, and what are the impacts? - [[User:MyUserName|MyUserName]] # [[/Zoom fatigue/]] - What is Zoom fatigue, what causes it, what are its consequences, and what can be done about it? - [[User:MyUserName|MyUserName]] ==Motivation and emotion== # [[/Financial investing, motivation, and emotion/]] - What role does motivation and emotion play in financial investing? - [[User:MyUserName|MyUserName]] # [[/Hostage negotiation, motivation, and emotion/]] - What role does motivation and emotion play in hostage negotiation? - [[User:U3213549|U3213549]] # [[/Money priming, motivation, and emotion/]] - What is the effect of money priming on motivation and emotion? - [[User:MyUserName|MyUserName]] # [[/Motivational dimensional model of affect/]] - What is the motivational dimensional model of affect and what are its implications? - [[User:MyUserName|MyUserName]] # [[/Napping, motivation, and emotion/]] - What are the motivational and emotional effects of napping? - [[User:MyUserName|MyUserName]] # [[/Overchoice, emotion, and motivation/]] - What are the emotional and motivational effects of overchoice? - [[User:MyUserName|MyUserName]] # [[/Patience and impatience/]] - What are the psychological causes and consequences of patience and impatience? - [[User:MyUserName|MyUserName]] # [[/Reward system, motivation, and emotion/]] - What role does the reward system play in motivation and emotion? - [[User:MyUserName|MyUserName]] [[Category:Motivation and emotion/Book/2022]] rzn2jpgr83v4hkuexmmgoyj8m6tjvqz 0 278551 2410338 2409541 2022-07-30T00:15:35Z Dave Braunschweig 426084 Reverted edits by [[Special:Contributions/122.110.53.53|122.110.53.53]] ([[User_talk:122.110.53.53|talk]]) to last version by [[User:99.232.190.38|99.232.190.38]] using [[Wikiversity:Rollback|rollback]] wikitext text/x-wiki {{DISPLAYTITLE:Digital Media Concepts/Sam Does Arts}} Sam Does Arts, is a digital artist from Toronto. He currently has 594K subscribers on [https://www.youtube.com/channel/UCNNOvB507MRfny7Jcv8MmOw/playlists YouTube], where he uploads a variety of digital art related content. He makes videos about equipment, roasting artworks, challenges, advice, tutorials, and speed paints. He currently has 63 videos uploaded on his channel. Each of his videos have thousands of views and he also has a Patreon and Instagram account. On Patreon, he currently has 2,188 patrons. On [https://www.instagram.com/samdoesarts/ Instagram], he currently has 1.3 million followers with 330 posts. == His Journey == According to his Patreon page, he wrote a short message containing some facts about himself. Sam is a 22 year old man from Toronto, who has a passion for art, specifically in the digital painting area. He states that his passion for art used to only be a hobby but he dropped it after his years in high school. It wasn't until the early start of the year 2020, that he decided to pick up his hobby again and he realized that it was true passion. He was inspired by many great artists and developed his own style through those inspirations. He started his YouTube career on June 20th 2020 and his first Instagram post was on March 1st 2018. == His Art style == His art style has changed over time to what it is now today. This can be seen through his earlier posts on Instagram. Here are some examples of his evolution from his first post to his most recent post. {| class="wikitable" |+Art style Timeline |Post Title |Date published |- |[https://www.instagram.com/p/BfzTwMIAgRd/?utm_source=ig_web_copy_link Portrait study of some little nobody from a galaxy really far] |March 1 2018 |- |[https://www.instagram.com/p/B7gxBzTBpkL/?utm_source=ig_web_copy_link Repainted one of my first ever posts on this account!] |January 19 2020 |- |[https://www.instagram.com/p/CAiXH1uh9ye/?utm_source=ig_web_copy_link 3 hour portrait study!] |May 23 2020 |- |[https://www.instagram.com/p/CBV5aFrBJEk/?utm_source=ig_web_copy_link Colour study, 3hr 30min.] |June 12 2020 |- |[https://www.instagram.com/p/CHnoofAhK35/?utm_source=ig_web_copy_link Can't get enough of golden hour clouds LOL] |November 15 2020 |- |[https://www.instagram.com/p/CKrVBSFhPgt/?utm_source=ig_web_copy_link A vibe (2h)] |January 30 2021 |- |[https://www.instagram.com/p/CPvqc3IhO56/?utm_source=ig_web_copy_link Flash (1h 30m)] |June 5 2021 |- |[https://www.instagram.com/p/CUVQ0hJrbLa/?utm_source=ig_web_copy_link Train station] |September 27 2021 |- |[https://www.instagram.com/p/CU2rP9frCVH/?utm_source=ig_web_copy_link Okay last one i promise] |October 10 2021 |} == His videos (sorted by category) == {| class="wikitable" |+Speed-Paint |Video |Date Published |- |[https://youtu.be/nXBRnts_gfk Spirited Away alley Speed-paint] |Jul 1, 2020 |- |[https://youtu.be/kxB-nOoXLV4 Portrait Study Speed-Paint Process] |Jul 7, 2020 |- |[https://youtu.be/4oTtWa9KR2c Landscape Digital Painting Time-lapse] |Aug 7, 2020 |- |[https://youtu.be/5p8ZL2NLZFY Can you paint without UNDO? | [https://youtu.be/5p8ZL2NLZFY Can you paint without UNDO?] |Nov 10, 2020 |- |[https://youtu.be/F2a9rQEfL_0 Can you paint with ONLY a Basic Round Brush? | [https://youtu.be/F2a9rQEfL_0 Can you paint with ONLY a Basic Round Brush?] |Nov 17, 2020 |- |[https://youtu.be/sPoywuhlqp4 DRAWING MY FAN'S OC!] |Dec 8, 2020 |} {| class="wikitable" |+Tutorials |Video |Date Published |- |[https://youtu.be/0CDd22s3jec Digital Painting Process Explained] |Jul 14, 2020 |- |[https://youtu.be/awasMxRmi50 How I Paint Mood and Atmosphere] |Jul 21, 2020 |- |[https://youtu.be/g9ge4XBNRwA How to Draw Perspective for Beginners] |Jul 28, 2020 |- |[https://youtu.be/GV_iNviuEvg Turning @AmandaRachLee into a Cartoon Character!] |Oct 13, 2020 |- |[https://youtu.be/9oOnK9m9LzQ Painting without OPACITY] |Nov 24, 2020 |- |[https://youtu.be/SIvRcXRaPkg How to Paint Better Backgrounds] |Jan 19, 2021 |- |[https://youtu.be/AkY21_S_IsE Do This Before You Post Your ART] |Mar 16, 2021 |- |[https://youtu.be/TlU-WMJQO9g How to Draw Faces] |Mar 23, 2021 |- |[https://youtu.be/kNQCP3CtHvI The Best Way to Practice DRAWING] |Apr 27, 2021 |- |[https://youtu.be/aDPlWnGcODM Here's how to become Art God] |Jun 8, 2021 |- |[https://youtu.be/2LxJOYXcszI Exposing my art secrets] |Jun 22, 2021 |- |[https://youtu.be/ZoN0dzGjMks What you need to know about ART STYLES] |Jun 29, 2021 |- |[https://youtu.be/pwDs8E3wJ44 What Brushes do you Use??] |Jul 13, 2021 |- |[https://youtu.be/Ed7C-Wddsbc How to become Art God - Guweiz] |Aug 10, 2021 |- |[https://youtu.be/by2W69Ea49U You want to be an INSTAGRAM ARTIST?] |Aug 24, 2021 |} {| class="wikitable" |+Advise |Video |Date Published |- |[https://youtu.be/f87MeKG7Zfg How to improve your art by studying photos] |Aug 4, 2020 |- |[https://youtu.be/hd2il6ND-1E Sam's Digital Art Tips: Why I Paint in 2 sittings] |Aug 18, 2020 |- |[https://youtu.be/RDnFcaJMLjY Sam's Digital Art Tips: How to Paint Faster] |Aug 25, 2020 |- |[https://youtu.be/AwksDpgTmas How to Find Your Art Style] |Sep 8, 2020 |- |[https://youtu.be/DTpYXW_eL5A 3 Ways to Stay Motivated] |Sep 15, 2020 |- |[https://youtu.be/XaVsMOHgFak I Gently Roast My Followers' Art] |Sep 29, 2020 |- |[https://youtu.be/TyKJaZqobcM How to Tell Better Stories with Your Art] |Oct 6, 2020 |- |[https://youtu.be/vzKJ8-vClUo Can You Paint on Just ONE LAYER?] |Oct 20, 2020 |- |[https://youtu.be/BPU-ABkem-k How to Overcome Your Fear of Posting Art Online] |Oct 27, 2020 |- |[https://youtu.be/49hwVIx1OV0 Can You Paint with a MOUSE?] |Dec 1, 2020 |- |[https://youtu.be/bnOWIvg5hG0 Painting with Default Brushes VS. Custom Brushes] |Dec 15, 2020 |- |[https://youtu.be/5qVIhYxlylU Do You REALLY Need Custom Brushes for Digital Art??] |Dec 22, 2020 |- |[https://youtu.be/HZ_c2HktUCE Trying CLIP STUDIO PAINT for the First Time! - First Impressions] |Jan 12, 2021 |- |[https://youtu.be/E6c4tFFbnTY Why You Should Learn Grayscale!] |Feb 2, 2021 |- |[https://youtu.be/_lfPaLpDwXA Can You Paint Without a Sketch?] |Feb 9, 2021 |- |[https://youtu.be/nWRc45xGa40 Repainting My FIRST Instagram Post!] |Feb 16, 2021 |- |[https://youtu.be/mqH4CIiKsgA Can You Paint Without a REFERENCE?] |Mar 2, 2021 |- |[https://youtu.be/ZhCB0a_g9F4 Master Study - Girl with a Pearl Earring] |Mar 9, 2021 |- |[https://youtu.be/YpsBIjmW65M Painting Grayscale to Color - first impressions] |Mar 30, 2021 |- |[https://youtu.be/vKw8oAK48KA REPAINTING MY OC] |Apr 6, 2021 |- |[https://youtu.be/QezAPJYLqcY Trying to Paint in Under 50 Minutes] |Apr 13, 2021 |- |[https://youtu.be/MPVfAIa88lU *GENTLY* ROASTING YOUR ART] |Apr 20, 2021 |- |[https://youtu.be/YUjHG1AYj8M ROASTING samdoesarts' Trash Art] |May 4, 2021 |- |[https://youtu.be/WX61qMzXdLg Why do you only draw girls?] |May 11, 2021 |- |[https://youtu.be/5EjXI-Tz2eU STOP doing this please] |May 25, 2021 |- |[https://youtu.be/7xB74zjyCn0 Instagram we have a Problem] |Jul 20, 2021 |- |[https://youtu.be/5zTVOECaszo The SECRET meaning behind my art] |Aug 3, 2021 |- |[https://youtu.be/yrwotXQeKEE THE MOST underrated artists] |Aug 31, 2021 |- |[https://youtu.be/8tcO_JwMIt4 Some social media advice for Artists] |Sep 23, 2021 |- |[https://youtu.be/9cjo3wiO_j4 THIS COULD BE YOU] |Oct 1, 2021 |- |[https://youtu.be/3O-EwAc7YoU DRAWING MY FAN'S OC!] |Dec 30, 2020 |- |[https://youtu.be/-8a3gTYLfII Drawing Fans with my LEFT HAND + NO UNDO??] |Jan 5, 2021 |- |[https://youtu.be/I98enLsiKdo Goodbye Photoshop?] |Jun 1, 2021 |} {| class="wikitable" |+Roasting |Video |Date Published |- |[https://youtu.be/vM7fvHXi59Q ✨GENTLY✨ ROASTING YOUR ART - 2] |May 18, 2021 |- |[https://youtu.be/T_6PPZAeVds I've had enough] |Jun 15, 2021 |- |[https://youtu.be/hl719pEo_8U Come on man.] |Jul 6, 2021 |- |[https://youtu.be/s_3WHlqToZs GENTLY Roasting your Art 5] |Jul 27, 2021 |- |[https://youtu.be/x-04LcaglzM ✨GENTLY✨ Roasting your Art 6] |Sep 7, 2021 |- |[https://youtu.be/tRTYYISJyAw GENTLY ROASTING your squid game fan art] |Oct 8, 2021 |} == His Equipment and Set Up == {| class="wikitable" |+His Equipment and Set Up Video |Video |Date Published |- |[https://youtu.be/lkaHoWwZ9-I DIGITAL ARTIST Workspace Setup!] |Feb 23, 2021 |} Equipment * [https://www.autonomous.ai/standing-desks/smartdesk-2-home?option1=1&option2=2016&option16=36&option17=1881&purchase_method=1 Smartdesk] * [https://estore.wacom.com/en-US/wacom-cintiq-pro-24-dtk2420k0.html Wacom Cintiq Pro 24] * [https://estore.wacom.com/en-US/wacom-flex-arm-for-cintiq-pro-24-32-ack62803k.html Wacom Flex Arm] * [https://www.samsung.com/semiconductor/minisite/ssd/product/portable/t7/ Samsung SSD] * [https://www.logitech.com/en-ca/products/mice/mx-master-3-mac-wireless-mouse.910-005693.html Mouse] * [https://www.apple.com/us/search/Magic-Keyboard-US-English?tab=accessories Keyboard] * [https://store.huion.com/products/artist-glove?gclid=CjwKCAiAyc2BBhAaEiwA44-wW4IVcXLxXv_AcTv3Zzc6XIII5B3mPQFHwTkOgZur95HaE62ycQ-_zBoCf4gQAvD_BwE Huion Glove] * [https://www.bestbuy.com/site/lg-32-uhd-3840-x-2160-hdr-monitor-with-amd-freesync-white/6419390.p?skuId=6419390 Monitor] * [http://www.rode.com/microphones/videomicpro Microphone] * [https://www.sony.ca/en/electronics/interchangeable-lens-cameras/ilce-6600 Camera] * [https://www.amazon.ca/gp/product/B077BWD2BB/ref=ppx_yo_dt_b_asin_title_o08_s00?ie=UTF8&psc=1 Lens] * [https://www.amazon.ca/gp/product/B075JFF35H/ref=ppx_yo_dt_b_asin_title_o02_s00?ie=UTF8&psc=1 Lights] * [https://www.amazon.ca/gp/product/B07N67D14D/ref=ppx_yo_dt_b_asin_title_o00_s00?ie=UTF8&psc=1 Tripod] * [https://www.amazon.ca/gp/product/B019NY2PKG/ref=ppx_yo_dt_b_asin_title_o01_s00?ie=UTF8&psc=1 Mic stand] == Links == * [https://www.youtube.com/channel/UCNNOvB507MRfny7Jcv8MmOw/featured Sam Does Arts Official YouTube Channel] * [https://www.instagram.com/samdoesarts/ Sam Does Arts Official Instagram] *Sam Does Arts Official Patreon<ref>{{Cite web|url=https://internal-patreonpy-load-2085191141.us-west-1.elb.amazonaws.com//samdoesarts|title=samdoesarts is creating digital paintings, video tutorials, and brushes.|website=Patreon|language=en-US|access-date=2021-10-10}}</ref> [[Category:Digital Media Concepts]] k10frwi886e00k4ntbrame0oq8vegx3 2 280897 2410305 2409918 2022-07-29T20:15:01Z Guy vandegrift 813252 /* Done */ wikitext text/x-wiki Journals: [[User:Guy vandegrift/2021|2021]] [[:Category:Guy vandegrift/Yearbook|Pageviews]]: [[User:Guy_vandegrift/2020#2020 and before|2020]] · [[User:Guy_vandegrift/2021#2021 Pageviews|2021]] · [[User:Guy_vandegrift/2022#Pageviews|2022]]· ----- BIG PROBLEM AT [[User:Guy vandegrift/AAAListen]] ----- *[[Information_theory/Permutations_and_combinations]]. If this works well, send a link to [[User:Guy vandegrift/2022/Proposal]] *[[w:Talk:Permutation#A_nasty_organization_problem_we_need_to_fix]] *[[Sing free]] **[[Sing free/Amazing Grace]] **[[Sing_free/Great_Gate_of_Kiev_(ear_training)]] Go to readme at spreadsheet folder '''audacity file organ''' *[[Special:PrefixIndex/Draft:Sing_free]] **[[Draft:Sing free/Voice training]] *[[The physics of music]] **[[w:User:Guy_vandegrift/sandbox/02#List]] **[[Sing_free/List_of_musical_intervals_(ear_training)]] <big>'''START HERE'''</big> DON'T LOSE [[special:Permalink/2383304#Backup_LilyPond_Great_Gate_Kiev]] DON'T FORGET ABOUT [[Python/Prime factorization]] ----- ===Bi- and Multi-nomial pages to write=== ====[[Information theory/Permutations and combinations]] ... [[Draft:Information theory/Permutations and combinations|(see Draft]] ... [[Template:Information theory/Permutations and combinations|& Template)]]==== *{{Doing}} =====[[Template:Information_theory/Shannon_entropy]]===== {{:Special:PrefixIndex/Information_theory}} [https://en.wikiversity.org/wiki/Special:PrefixIndex?prefix=Draft%3AInformation+theory&namespace=0 PrefixIndex = Draft:Information theory]: *[[Draft:Information theory/Permutations and combinations]] *[[Draft:Information theory/Shannon entropy]] *[[Draft:Information theory/Shannon entropy/a]] ----- ====[[Combinatorics/Multinomial coefficients]]==== Mention two interpretations (only 2!). Provide simplest (related to information theory). Briefly present the conventional using rule of product. Link to images and other proofs all over wikiworld. OR DELETE!!! {{:Special:PrefixIndex/Combinatorics}} ----- #{{See also|w:Combinatorial principles|b:Probability/Combinatorics#What_is_combinatorics?}} #{{See also|w:Rule of product|b:Probability/The Counting Principle|w:Tree diagram (probability theory)}} #{{See also|w:Binomial theorem|w:Multinomial theorem|b:Probability/Combinatorics#Binomial_and_multinomial_coefficients}} :::{{See also|w:Catalan number|w:Lattice path|w:Binomial theorem}} ====[[Combinatorics/Stars & Bars]]==== [https://math.stackexchange.com/questions/3758836/stars-and-bars-but-with-distinct-objects/3758849 colored balls stackexchange] [https://math.stackexchange.com/questions/2593565/stars-and-bars-approach red and blue] *[[Combinatorics/Binomial coefficients]]{{done|Not written by me. Is advanced.}} {{cot|stars bars and beyond (draft)}} :<math>\binom N K =\binom{N\text{ objects}}{K\text{ bars}}</math> Since there is one more bin than bars, <math>k=K-1</math>Example with <math>N=7,\quad K=2</math>:<br>{{spaces|2}} {{Huge|{{spaces|2}}&#124;{{spaces|2}}&bull;{{spaces|2}}&bull;{{spaces|2}}&#124;{{spaces|2}}&bull;{{spaces|2}}&bull;{{spaces|2}}&bull;{{spaces|2}}}} ====Other to do==== *[[Mathematical induction]] needs major work. ====[[Combinatorics/Images]]==== Image collection. Include link to text alternative. ====Allow zero "things" in a bin==== Put <math>n_0</math> into <math>k</math> with bins a minimum of <math>0</math> things per bin. <math>K=k-1</math> and <math>N=K+n_0\Rightarrow N=n_0+k-1</math> <math>\Rightarrow</math><math>\binom N K =\binom{n_0+k-1}{k-1}</math> ====Demand at least 1 "thing" in a bin==== :*Put <math>n_1</math> into <math>k</math> with bins a minimum of <math>1</math> thing per bin. <math>n_1=n_0+k</math> <math>\binom N K = \binom{n_1-1}{k-1}</math> {{Huge|{{spaces|2}}&deg; &#124;{{spaces|2}}&bull;{{spaces|2}}&bull;{{spaces|2}}&deg; &#124;{{spaces|2}}&bull;{{spaces|2}}&bull;{{spaces|2}}&bull; &deg;{{spaces|2}}}} {{cob}} ===Other links=== [[MyOpenMath/Three types of test questions]] *[[:Category:User:Guy vandegrift/Drafts]] *[[:Category:User:Guy vandegrift/Navigation]] ----- A well developed resource I should add to: {{:Special:PrefixIndex/Tensors}} ----- *[https://en.wikiversity.org/w/index.php?target=Guy+vandegrift&namespace=10&tagfilter=&newOnly=1&start=&end=&limit=50&title=Special%3AContributions My templates] ==TO DO== #[[:File:Klaviatur-wIKI.jpg]] needs conversion into two svg files. Use to improve [[w:Piano_key_frequencies]]. From [[w:Musical_keyboard]] and [[w:Keyboard matrix circuit]] it looks like we need a 5 octave version. But which octaves? #[[Free singing lessons]] # Jot ideas for proposal at [[/Proposal]] perhaps link it to Problem at end of [[Information theory/Permutations and combinations]] #*[[:File:5-choose-3 product rule.svg]] would make a good project with an advanced undergraduate student. # Finish [[Information theory/Permutations and combinations]] . Unfortunately, sister link at [[w:Permutations]] is already taken by . # [[Combinatorics]] needs [[Template:BookCat]] enhancement # [[Combinatorics/Rule of product]] needs to be finished # [[Combinatorics/Multinomials]] will have formal proofs and a collection of images # [[Mathematical induction]] needs to have two examples of induction: permutations and combinations #Finish the phasor/poynting sequence at [[:Category:MyOpenMath/Electromagnetism]] #Write essay for and setup transclusion for [[:category:Quizbank/Unfinished]] #After 2022 check images on [[w:Stars and bars (combinatorics)]] #Finish [[w:simple:Radiation pressure]] #[[Draft:Introduction_to_quantum_mechanics]] #Go back to the beginning of E&M and get one unit of TF reading questions completed. ==Done== *3 Jan 2022 [[Combinatorics/Binomial_coefficients]]^{[https://en.wikiversity.org/w/index.php?title=Combinatorics%2FBinomial_coefficients&type=revision&diff=2366084&oldid=1834808 Cleanup]} *8 Jan 2022 [[w:Mathematical induction]]^{[https://simple.wikipedia.org/w/index.php?title=Mathematical_induction&type=revision&diff=7953423&oldid=7120309 Cleanup]}{{Partly done|still needs cleaning up}} *24 Jan 2022 Created table to replace the deprecated image, [[:File:Tabela de Relações Trigonométricas.PNG]] See also: [[w:simple:special:permalink/7984902]], [[w:ca:special:permalink/29331416]], and [[c:special:permalink/624159454]] *2 Feb 2022 Added comment that Cocolmelon's "Ants and the Grasshopper" is to the tune of [https://en.wikipedia.org/w/index.php?title=Red_River_Valley_(song)&diff=prev&oldid=1070383055 w:Red River Valley] *20 Feb 2022 Imported [[Template:Magnify icon]] *26 Feb 2022 Simplified Red River Valley from [[special:permalink/2378796]] *4 Mar 2022 [https://simple.wikipedia.org/w/index.php?title=Hertz&type=revision&diff=8087736&oldid=7897066 Simple English Hz, cps table] *5 Mar 2022 Added [https://en.wikipedia.org/w/index.php?title=Red_River_Valley_(song)&diff=prev&oldid=1075389296 Red River Valley] lilypond and sister link to [[Wikipedia:Red River Valley (song)]]. *6 Mar 2022 [[Sing free/Prelude and Fugue in C major (ear training)]] with sister link to [[w:Just intonation]] *7 Mar 2022 Amazing Grace Lilypond script posted at [https://en.wikipedia.org/w/index.php?title=Amazing_Grace&oldid=1075696156 Amazing_Grace&oldid=1075696156]. Added sister link/ *3 Jun 2022 [https://en.wikipedia.org/w/index.php?title=Stand_by_Me_%28Ben_E._King_song%29&type=revision&diff=1091412161&oldid=1091349828 Minor copyedit] on [[w:Stand by Me (Ben E. King song)]] *7 July 2022 [https://en.wikipedia.org/w/index.php?title=Dyad_%28music%29&type=revision&diff=1096992061&oldid=994729869 Added trivial '''See also:''' to [[w:Dyad (music)]]] *29 July 2022 [[Python/Prime factorization]] with sister link posted on [[w:Trial division]]. ==Pageviews== ===[https://pageviews.toolforge.org/?project=en.wikibooks.org&platform=all-access&agent=user&redirects=0&range=latest-90&pages=Statics/Moment_of_Inertia_(contents)|Probability/Combinatorics|Probability/Conditional_Probability Wikibooks Pageviews:]=== [[b:Statics/Moment of Inertia (contents)]]^{[https://en.wikibooks.org/w/index.php?title=Statics%2FMoment_of_Inertia_%28contents%29&type=revision&diff=3355995&oldid=2674427 added-2-figures]}{{spaces|3}}[[b:Probability/Conditional Probability]]^{[[b:special:permalink/1658724|modified-this-version]]}{{spaces|3}} [[b:Probability/Combinatorics]]^{[[b:special:permalink/3547462|modified-this-version]]} ===[https://pageviews.wmcloud.org/?project=simple.wikipedia.org&platform=all-access&agent=user&redirects=0&range=latest-90&pages=Weighted_average|Radiation_pressure|Pascal%27s_Triangle|Hertz Simple Wikipedia Pageviews:]=== Constructing-[[w:simple:Radiation pressure]]{{spaces|3}} [[w:simple:Weighted average]]{{spaces|3}} [[w:simple:Pascal's Triangle]]{{spaces|3}} [[w:simple:Hertz]] ===[https://pageviews.wmcloud.org/?project=en.wikiversity.org&platform=all-access&agent=user&redirects=0&range=latest-90&pages=Sing_free|Sing_free/Twinkle|Sing_free/Red_River_Valley|Sing_free/Major_sixth_M6|Sing_free/Prelude_and_Fugue_in_C_major_(ear_training)|Sing_free/Four_octave_scale|Sing_free/Amazing_Grace|Python/Prime_factorization Wikiversity Pageviews:]=== [[Sing free]]{{spaces|3}} [[Sing free/Red River Valley]]{{spaces|3}} [[Sing free/Twinkle]]{{spaces|3}} [[Sing free/Major sixth M6]]{{spaces|3}} [[Sing free/Four octave scale]]{{spaces|3}} [[Sing free/Prelude and Fugue in C major (ear training)]]{{spaces|3}} [[Sing free/Amazing Grace]]{{spaces|3}} [[Python/Prime_factorization]]{{spaces|3}} [[]]{{spaces|3}} [[]]{{spaces|3}} [[Category:Guy vandegrift/Yearbook]] ==Subpages== {{Subpages/List}} {{Seealso|Sing free}} 1w6q7bglxlqfb7nwkfuntfdu55oppif 0 284696 2410315 2408439 2022-07-29T22:18:29Z Aherman012 2943941 /* Process phase */ wikitext text/x-wiki <noinclude>{{Helping Give Away Psychological Science Banner}}</noinclude> {{medical disclaimer}} {{:{{ROOTPAGENAME}}/Sidebar}} ==[[Evidence based assessment/Portfolio template/What is a "portfolio"|'''What is a "portfolio"?''']]== * For background information on what assessment portfolios are, click the link in the heading above. Does this page feel like too much information? Click [[Evidence-based assessment/Generalized anxiety disorder (assessment portfolio)|here]] for the condensed version. ==[[Evidence based assessment/Preparation phase|'''Preparation phase''']]== === Diagnostic criteria for generalized anxiety disorder === {{blockquotetop}} <big>'''ICD-11 Diagnostic Criteria<ref>https://icd.who.int/browse11/l-m/en#/http://id.who.int/icd/entity/1712535455</ref>'''</big> *Generalised anxiety disorder is characterized by marked symptoms of anxiety that persist for at least several months, for more days than not, manifested by either general apprehension (i.e. ‘free-floating anxiety’) or excessive worry focused on multiple everyday events, most often concerning family, health, finances, and school or work, together with additional symptoms such as muscular tension or motor restlessness, sympathetic autonomic over-activity, subjective experience of nervousness, difficulty maintaining concentration, irritability, or sleep disturbance. The symptoms result in significant distress or significant impairment in personal, family, social, educational, occupational, or other important areas of functioning. The symptoms are not a manifestation of another health condition and are not due to the effects of a substance or medication on the central nervous system. '''Changes in DSM-5''' * The diagnostic criteria for generalized anxiety disorder changed slightly from DSM-IV-TR to DSM-5. Summaries are available [https://www.ncbi.nlm.nih.gov/books/NBK519712/table/ch3.t9/?report=objectonly here]. {{blockquotebottom}} === Base rates of GAD in different clinical settings === This section describes the demographic setting of the population(s) sampled, base rates of diagnosis, country/region sampled and the diagnostic method that was used. Using this information, clinicians will be able to anchor the rate of GAD that they are likely to see in their clinical practice. * '''''To see prevalence rates across multiple disorders,''''' [[Evidence based assessment/Preparation phase#Base rates for transdiagnostic comparison|'''''click here.''''']] {| class="wikitable sortable" border="1" |- ! Demography ! Setting ! Base Rate ! Diagnostic Method |- | Adults and adolescences in all of U.S.A. | US National Comorbidity Survey Replication (NCS-R; age > = 13) [https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4005415/pdf/nihms-571992.pdf (2012)]<ref name="KesslerEtAl2012" /> | * 0.9% (age 13-17) * 2.9% (age 18-64) * 1.2% (age &gt;= 65) * 2.0% (age &gt;=13) | Fully-structured CIDI Version 3.0 |- | Psychiatric outpatients | Individuals seeking treatment in a Psychiatric Outpatient Clinic (age range not reported) ([https://ajp.psychiatryonline.org/doi/pdf/10.1176/appi.ajp.162.10.1911 2014])<ref name="ZimmermanEtAl2005" /> | * 21% | Structured Clinical Interview for DSM-IV (SCID) |- | Caucasian youth | Children seeking treatment in a Child &amp; Adolescent Anxiety Diagnostic Clinic (age 7 – 18 years old) ([http://journals.sagepub.com/doi/pdf/10.1177/1073191110375792 2011])<ref name="BrownJacobsenEtAl2011" /> | * 0.39% (parent report) * 0.38% (child report) | ADIS-C for DSM-IV Spence Children's Anxiety Scale (SCAS) |- | Caucasian, African American, Asian American, and Hispanic population | Collaborative Psychiatric Epidemiology Studies (CPES; age &gt;= 18, data merged from three representative national database) ([https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3135672/pdf/nihms281333.pdf 2011])<ref>{{Cite journal|last=McLean|first=Carmen P.|last2=Asnaani|first2=Anu|last3=Litz|first3=Brett T.|last4=Hofmann|first4=Stefan G.|date=2011-08-01|title=Gender differences in anxiety disorders: Prevalence, course of illness, comorbidity and burden of illness|url=https://www.sciencedirect.com/science/article/pii/S0022395611000458|journal=Journal of Psychiatric Research|language=en|volume=45|issue=8|pages=1027–1035|doi=10.1016/j.jpsychires.2011.03.006|issn=0022-3956|pmc=PMC3135672|pmid=21439576}}</ref> | * 4.1% (female) * 2.1% (male) | World Mental Health Survey Initiative Version of the World Health Organization Composite International Interview (WMH-CIDI) |- | Pennsylvania | Metropolitan Community Sample, all individuals with eating disorders (ages 13 – 65) ([https://ajp.psychiatryonline.org/doi/pdf/10.1176/appi.ajp.161.12.2215 2014])<ref name="KayeEtAl2004" /> | * 10% | Structured Clinical Interview for DSM-IV (SCID) |- | Adolescents in all of U.S.A. | National Comorbidity Survey Replication Adolescent Supplement (NCS-A; ages 3–18 in the continental U.S) ([https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2946114/pdf/nihms214371.pdf 2011])<ref name="MerikangasEtAl2010" /> | * 2.2% | World Health Organization Composite International Diagnostic Interview (WHO-CIDI) |- |Adolescents in all of U.S.A |National Comorbidity Survey Replication Adolescent Supplement (NCS-A; ages 3–18 in the continental U.S)<ref name=":3">Kessler, R. C., Avenevoli, S., Costello, E. J., Georgiades, K., Green, J. G., Gruber, M. J., . . . Merikangas, K. R. (2012). Prevalence, persistence, and sociodemographic correlates of DSM-IV disorders in the National Comorbidity Survey Replication Adolescent Supplement. Archives of General Psychiatry, 69(4), 372-380. doi:10.1001/archgenpsychiatry.2011.160</ref> | * 5.4% |Composite International Diagnostic Interview (CIDI) |- | North Carolina | Rural community sample African American and White youth (ages 13-16) [https://www.ncbi.nlm.nih.gov/pubmed/12365876 (2002)]<ref>{{Cite journal|last=Angold|first=Adrian|last2=Erkanli|first2=Alaattin|last3=Farmer|first3=Elizabeth M. Z.|last4=Fairbank|first4=John A.|last5=Burns|first5=Barbara J.|last6=Keeler|first6=Gordon|last7=Costello|first7=E. Jane|date=October 2002|title=Psychiatric disorder, impairment, and service use in rural African American and white youth|url=https://www.ncbi.nlm.nih.gov/pubmed/12365876|journal=Archives of General Psychiatry|volume=59|issue=10|pages=893–901|issn=0003-990X|pmid=12365876}}</ref> | * 1.4% | The Child and Adolescent Psychiatric Assessment (CAPA) |- | Texas | Metropolitan Community Sample (ages 11-17) ([https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2736593/pdf/nihms30019.pdf 2007])<ref name="RobertsEtAl2007" /> | * 0.4% | Diagnostic Interview Schedule for Children, Version IV (DISC-IV) |- | Midwestern Urban | Incarcerated adolescents (ages 10-18)<ref>{{Cite journal|last=ABRAM|first=KAREN M.|last2=CHOE|first2=JEANNE Y.|last3=WASHBURN|first3=JASON J.|last4=TEPLIN|first4=LINDA A.|last5=KING|first5=DEVON C.|last6=DULCAN|first6=MINA K.|title=Suicidal Ideation and Behaviors Among Youths in Juvenile Detention|url=http://linkinghub.elsevier.com/retrieve/pii/S0890856709623121|journal=Journal of the American Academy of Child & Adolescent Psychiatry|volume=47|issue=3|pages=291–300|doi=10.1097/chi.0b013e318160b3ce}}</ref> [http://vb3lk7eb4t.search.serialssolutions.com/?ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Suicidal+Ideation+and+Behaviors+Among+Youths+in+Juvenile+Detention&rft.jtitle=Journal+of+the+American+Academy+of+Child+%26+Adolescent+Psychiatry&rft.au=ABRAM%2C+KAREN+M.%2C+Ph.D&rft.au=CHOE%2C+JEANNE+Y.%2C+B.A&rft.au=WASHBURN%2C+JASON+J.%2C+Ph.D.%2C+A.B.P.P&rft.au=TEPLIN%2C+LINDA+A.%2C+Ph.D&rft.date=2008&rft.issn=0890-8567&rft.eissn=1527-5418&rft.volume=47&rft.issue=3&rft.spage=291&rft.epage=300&rft_id=info:doi/10.1097%2FCHI.0b013e318160b3ce&rft.externalDocID=1_s2_0_S0890856709623121 (2002)] | * 1% | Diagnostic Interview Schedule for Children, Version IV (DISC-IV) |- |The southern Appalachian mountain region of North Carolina |Great Smoky Mountain (ages 9-12) ([https://www.ncbi.nlm.nih.gov/pubmed/8956679 1996])<ref name="CostelloEtAl1996" /> | * 1.67% |DSM-III-R, DSM-IV and CAPA |- |New Jersey |Non-referred Adolescent Population (ages 9-17) ([https://www.ncbi.nlm.nih.gov/pubmed/2331210 1990])<ref name="WhitakerEtAl1990" /> | * 3.7% |DSM-III &amp; Beck Depression Inventory (BDI) |- |Non-institutionalized general US population |LGBTQ sample (ages 20-65)<ref>{{Cite journal|last=Bostwick|first=Wendy B.|last2=Boyd|first2=Carol J.|last3=Hughes|first3=Tonda L.|last4=McCabe|first4=Sean Esteban|date=2010-3|title=Dimensions of Sexual Orientation and the Prevalence of Mood and Anxiety Disorders in the United States|url=https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2820045/|journal=American Journal of Public Health|volume=100|issue=3|pages=468–475|doi=10.2105/AJPH.2008.152942|issn=0090-0036|pmc=PMC2820045|pmid=19696380}}</ref> [http://ajph.aphapublications.org/doi/10.2105/AJPH.2008.152942 (2013)] |Women: * 14.8% same-sex * 22.5% bisexual Men: * 16.9% same-sex * 11.5% bisexual |The Alcohol Use Disorder and Associated Disabilities Interview Schedule-IV (AUDADIS-IV) |- |Non-institutionalized general US population |Cross-ethnic American population (ages 18+)<ref>{{Cite journal|last=Asnaani|first=Anu|last2=Richey|first2=J. Anthony|last3=Dimaite|first3=Ruta|last4=Hinton|first4=Devon E.|last5=Hofmann|first5=Stefan G.|date=2010-8|title=A Cross-Ethnic Comparison of Lifetime Prevalence Rates of Anxiety Disorders|url=https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2931265/|journal=The Journal of nervous and mental disease|volume=198|issue=8|pages=551–555|doi=10.1097/NMD.0b013e3181ea169f|issn=0022-3018|pmc=PMC2931265|pmid=20699719}}</ref> [https://journals.lww.com/jonmd/Abstract/2010/08000/A_Cross_Ethnic_Comparison_of_Lifetime_Prevalence.4.aspx (2018)] | * White 8.6% * African Americans 4.9% * Hispanic Americans 5.8% * Asian Americans 2.4% |World Mental Health Survey Initiative Version of the World Health Organization Composite International Interview (WMH-CIDI) |- |Outpatient clinics worldwide |Samples across multiple studies worldwide (all ages)<ref name=":12">{{Cite journal|last=Rettew|first=David C.|last2=Lynch|first2=Alicia Doyle|last3=Achenbach|first3=Thomas M.|last4=Dumenci|first4=Levent|last5=Ivanova|first5=Masha Y.|date=2009-09|title=Meta-analyses of agreement between diagnoses made from clinical evaluations and standardized diagnostic interviews|url=http://dx.doi.org/10.1002/mpr.289|journal=International Journal of Methods in Psychiatric Research|language=en|volume=18|issue=3|pages=169–184|doi=10.1002/mpr.289|issn=1049-8931}}</ref> |5% |Clinical evaluations |- |Outpatient clinic worldwide |Samples across multiple studies worldwide (all ages)<ref name=":12" /> |10% |Standardized Diagnostic Interviews (SDIs) |} '''Search terms:''' [General Anxiety Disorder] AND [youth OR adolescents OR pediatric] AND [prevalence OR incidence] in GoogleScholar and PsycINFO == [[Evidence based assessment/Prediction phase|'''Prediction phase''']] == === Psychometric properties of screening instruments for GAD === The following section contains a list of screening and diagnostic instruments for generalized anxiety disorder. The section includes administration information, psychometric data, and PDFs or links to the screenings. * Screenings are used as part of the [[Evidence based assessment/Prediction phase|prediction phase]] of assessment; for more information on interpretation of this data, or how screenings fit in to the assessment process, click [[Evidence based assessment/Prediction phase|here]]. * '''''For a list of more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&wteswitched=1#Psychometric_properties_of_common_screening_instruments click here.]''''' {| class="wikitable sortable" border="1" ! colspan="10" |Screening measures for GAD |- ! Measure ! Format (Reporter) ! Age Range ! Administration/ Completion Time ! Interrater Reliability ! Test-Retest Reliability ! Construct Validity ! Content Validity ! Highly Recommended !PDF |- | Penn State Worry Questionnaire (PSWQ)<ref name=":0">{{Cite book|url=https://www.worldcat.org/oclc/314222270|title=A guide to assessments that work|date=2008|publisher=Oxford University Press|author=Hunsley, John |author2=Mash, Eric J.|isbn=9780195310641|location=New York|oclc=314222270}}</ref> | Questionnaire (Adult Version, Child Version) | 18+ (Adult Version), 6-18 (Child Version) | 4 minutes | NA<ref name=":0" /> | G<ref name=":0" /> |G<ref name=":0" /> |G<ref name=":0" /> |<ref name=":0" />[[File:Light green check.svg|center|frameless|50x50px]] | * [http://www.midss.org/content/penn-state-worry-questionnaire-pswq PSWQ homepage] '''PDFs of the PSWQ''' *[https://mfr.osf.io/render?url=https://osf.io/s7p38/?action=download%26mode=render Penn State Worry Questionnaire] *[https://mfr.osf.io/render?url=https://osf.io/6q8y9/?action=download%26mode=render PSWQ-C] *[https://mfr.osf.io/render?url=https://osf.io/gx5sr/?action=download%26mode=render PSWC-C Korean] *[https://mfr.osf.io/render?url=https://osf.io/hc6n2/?action=download%26mode=render PSWQ-C Danish] *[https://mfr.osf.io/render?url=https://osf.io/fwbes/?action=download%26mode=render Scoring of the PSWQ-C] |- |[[wikipedia:Screen_for_child_anxiety_related_disorders|Screen for Child Anxiety Related Emotional Disorder (SCARED)]] | Questionnaire (Child, Parent) | 8-19 | 9 or 16 minutes |NA<ref name=":0" /> | G<ref name=":0" /> |G<ref name=":0" /> | G<ref name=":0" /> | | * [http://www.midss.org/content/screen-child-anxiety-related-disorders-scared SCARED] homepage '''PDFs of SCARED''' *[[wikipedia:Screen_for_child_anxiety_related_disorders#PDFs_and_automated_scoring_for_SCARED|SCARED English + Translations & Automatic Scoring]] |- |[[wikipedia:State-Trait_Anxiety_Inventory|State/Trait Anxiety Inventory for Children (STAIC)]] | Questionnaire (Child, Parent) | 6-18 | 5 or 10 minutes |NA<ref name=":0" /> | G<ref name=":0" /> |G<ref name=":0" /> | G<ref name=":0" /> | | * *not free* * [http://www.mindgarden.com/146-state-trait-anxiety-inventory-for-children#horizontalTab2 STAIC Website] |- | Revised Children’s Anxiety and Depression Scale (RCADS) | Questionnaire (Child) | 6-18 | 12 minutes |G<ref name=":2">{{Cite journal|last=Chorpita|first=Bruce F.|last2=Moffitt|first2=Catherine E.|last3=Gray|first3=Jennifer|date=2005-03|title=Psychometric properties of the Revised Child Anxiety and Depression Scale in a clinical sample|url=http://dx.doi.org/10.1016/j.brat.2004.02.004|journal=Behaviour Research and Therapy|volume=43|issue=3|pages=309–322|doi=10.1016/j.brat.2004.02.004|issn=0005-7967}}</ref> |G<ref>{{Cite journal|last=Chorpita|first=Bruce F|last2=Yim|first2=Letitia|last3=Moffitt|first3=Catherine|last4=Umemoto|first4=Lori A|last5=Francis|first5=Sarah E|date=2000-08|title=Assessment of symptoms of DSM-IV anxiety and depression in children: a revised child anxiety and depression scale|url=http://dx.doi.org/10.1016/s0005-7967(99)00130-8|journal=Behaviour Research and Therapy|volume=38|issue=8|pages=835–855|doi=10.1016/s0005-7967(99)00130-8|issn=0005-7967}}</ref> |G<ref name=":2" /> | | | * [https://www.corc.uk.net/outcome-experience-measures/revised-childrens-anxiety-and-depression-scale-rcads/ RCADS homepage] '''PDFs for RCADS''' *[https://mfr.osf.io/render?url=https://osf.io/s3fu2/?action=download%26mode=render RCADS Child Self-reported (8-18 years)] *[https://mfr.osf.io/render?url=https://osf.io/fp9mk/?action=download%26mode=render RCADS Parent-reported] *[https://mfr.osf.io/render?url=https://osf.io/vy7ta/?action=download%26mode=render Child Scoring Aid] *[https://mfr.osf.io/render?url=https://osf.io/t4bz6/?action=download%26mode=render Parent Scoring Aid] '''Subscales''' *[https://mfr.osf.io/render?url=https://osf.io/976xg/?action=download%26mode=render Generalized Anxiety Child Self-reported] *[https://mfr.osf.io/render?url=https://osf.io/y3qp7/?action=download%26mode=render Generalized Anxiety Parent-reported] *[https://mfr.osf.io/render?url=https://osf.io/4gc8d/?action=download%26mode=render Panic Self-Report] *[https://mfr.osf.io/render?url=https://osf.io/nhcsu/?action=download%26mode=render Panic Parent-Report] '''[https://osf.io/2fm4r/download Translations]''' '''[https://mfr.osf.io/render?url=https://osf.io/qsjh9/?action=download%26mode=render User Guide]''' * |- |[[wikipedia:Spence_Children's_Anxiety_Scale|Spence Children’s Anxiety Scale (SCAS)]] | Questionnaire (Child, Parent) | 7-19 | 11 minutes |NA<ref name=":0" /> | A<ref name=":0" /> | E<ref name=":0" /> | E<ref name=":0" /> | |[https://www.scaswebsite.com/ SCAS homepage] [https://www.scaswebsite.com/wp-content/uploads/2021/07/scas.pdf Child Version PDF] [https://www.scaswebsite.com/wp-content/uploads/2021/07/scas-parent-qaire.pdf Parent Version PDF] |- |[[wikipedia:Generalized_Anxiety_Disorder_7|GAD-7 Scale]] |Self report |18+ |5 minutes |G<ref name=":0" /> |Intraclass correlation 0.83<ref>{{Cite journal|last=Spitzer|first=Robert L.|last2=Kroenke|first2=Kurt|last3=Williams|first3=Janet B. W.|last4=Löwe|first4=Bernd|date=2006-05-22|title=A Brief Measure for Assessing Generalized Anxiety Disorder|url=http://archinte.jamanetwork.com/article.aspx?doi=10.1001/archinte.166.10.1092|journal=Archives of Internal Medicine|language=en|volume=166|issue=10|doi=10.1001/archinte.166.10.1092|issn=0003-9926}}</ref> |G<ref name=":0" /> |G<ref name=":0" /> |<ref name=":0" />[[File:Light green check.svg|center|frameless|50x50px]] |GAD-7 homepage [https://www.integration.samhsa.gov/clinical-practice/GAD708.19.08Cartwright.pdf PDF] (english) [http://www.coloradohealthpartnerships.com/provider/integrated/GAD7-Spanish.pdf PDF] (spanish) |- |Kessler Psychological Stress Scale (K10 and K6 Scales) |Self or interview administered | | | | | | | |[https://www.hcp.med.harvard.edu/ncs/k6_scales.php Available in many languages] |- | Worry and Anxiety Questionnaire (WAQ) |Self report | | 10 minutes | NA<ref name=":0" /> | A<ref name=":0" /> | A<ref name=":0" /> | G<ref name=":0" /> |<ref name=":0" />[[File:Light green check.svg|center|frameless|50x50px]] |WAQ homepage [https://uqo.ca/file/19980/download?token=Cmpts0MF PDF] |- |Brown Assessment of Beliefs Scale (BABS) | | | |G<ref name=":0" /> |A<ref name=":0" /> |G<ref name=":0" /> |G<ref name=":0" /> |[[File:Light green check.svg|center|frameless|50x50px]]<ref name=":0" /> | * [https://mfr.osf.io/render?url=https://osf.io/bqr2e/?action=download%26mode=render BABS Adult] * [https://mfr.osf.io/render?url=https://osf.io/z5s8a/?action=download%26mode=render BABS Original Publication] |- |Back Anxiety Inventory (BAI) |Self-report |17-80 |5-10 minutes |G<ref name=":0" /> | |G<ref name=":0" /> |G<ref name=":0" /> | |BAI homepage [https://www.gphealth.org/media/1087/anxiety.pdf PDF] |- |The Clinically Useful Anxiety Outcome Scale (CUXOS) |Self-report |18-85 |Less than 2 minutes |E<ref name=":0" /> | |E<ref name=":0" /> |G<ref name=":0" /> | |CUXOS homepage [https://outcometracker.org/CUXOS.pdf PDF] |- |Achenbach System of Empirically Based Assessments (ASEBA): Child Behavior Checklist (CBCL), Teacher Report Form (TRF), Youth Self-Report (YSR) |CBCL: Parent report, TRF: Teacher report, YSR: Child report |6-18 (CBCL & TRF), 11-18 (YSR)<ref name=":7">{{Cite book|url=https://www.worldcat.org/oclc/1130319849|title=Assessment of disorders in childhood and adolescence|date=2020|others=Eric Arden Youngstrom, Mitchell J. Prinstein, Eric J. Mash, Russell A. Barkley|isbn=978-1-4625-4363-2|edition=Fifth edition|location=New York, NY|oclc=1130319849}}</ref> |10 - 15 minutes<ref name=":7" /> |A<ref name=":0" /> |E<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> | |ASEBA homepage [https://aseba.org/forms/schoolagecbcl.pdf PDF] |} '''Note:''' L = Less than adequate; A = Adequate; G = Good; E = Excellent; U = Unavailable; NA = Not applicable === Likelihood ratios and AUCs of screening instruments for GAD === * '''''For a list of the likelihood ratios for more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&wteswitched=1#Likelihood_ratios_and_AUCs_of_common_screening_instruments click here.]''''' {| class="wikitable sortable" border="1" ! Screening Measure (Primary Reference) ! Format (Reporter) ! Area Under Curve (AUC) ! LR+ (Score) ! LR- (Score) ! Citation ! Clinical Generalizability |- | Penn State Worry Questionnaire (PSWQ''')'''<ref>{{Cite journal|last=Meyer|first=T.J.|last2=Miller|first2=M.L.|last3=Metzger|first3=R.L.|last4=Borkovec|first4=Thomas D.|title=Development and validation of the penn state worry questionnaire|url=https://doi.org/10.1016/0005-7967(90)90135-6|journal=Behaviour Research and Therapy|volume=28|issue=6|pages=487–495|doi=10.1016/0005-7967(90)90135-6}}</ref> | Questionnaire (Child) | 0.74 (N=164) | 1.8 (65+) | 0.5 (< 65) | Fresco, D.M., Mennin, D.S., Heimberg, R.G., Turk, C.L. (2003)<ref>{{Cite journal|last=Fresco|first=David M.|last2=Mennin|first2=Douglas S.|last3=Heimberg|first3=Richard G.|last4=Turk|first4=Cynthia L.|title=Using the Penn State Worry Questionnaire to identify individuals with generalized anxiety disorder: a receiver operating characteristic analysis|url=http://linkinghub.elsevier.com/retrieve/pii/S0005791603000569|journal=Journal of Behavior Therapy and Experimental Psychiatry|volume=34|issue=3-4|pages=283–291|doi=10.1016/j.jbtep.2003.09.001}}</ref> | Generalized Anxiety Disorder vs. social anxiety disorder, adults presenting to specialty anxiety clinic |- | rowspan="2" |[[wikipedia:Screen_for_child_anxiety_related_disorders#PDFs_and_automated_scoring_for_SCARED|Screen for Child Anxiety Related Disorders (SCARED)]]<ref name="BirmaherEtAl1997"/> | rowspan="2" | Questionnaire (Child, Parent) | .70 (N=243) | 5.0 (+32) | .04 | (Birmaher et al., 1997)<ref name="BirmaherEtAl1997"/> | rowspan="2" | High: Pure anxiety disorder versus non-anxiety psychiatric disorder, excluding children with disruptive disorder and depression |- | 0.911 (First screen) (N= 923) | 2.81 (4+; FS) | 0.15 (4-; FS) | Hale III, et al., 2014<ref name="HaleEtAl2014"/> |- | STAIC<ref>{{Cite journal|last=Hodges|first=Kay|title=Depression and anxiety in children: A comparison of self-report questionnaires to clinical interview.|url=http://dx.doi.org/10.1037/1040-3590.2.4.376|journal=Psychological Assessment|language=en|volume=2|issue=4|pages=376–381|doi=10.1037/1040-3590.2.4.376}}</ref> | Questionnaire (Child, Parent) | -- (N=70) | 2 (+69) | .79 | DLR: (Hodges, 1990) | STAIC does well in discriminating between children and adolescents with anxiety disorders and youth without a disorder and moderately well in measuring treatment response and discriminating youth with anxiety disorders from those with externalizing disorders<ref name="SeligmanEtAl2004"/> |- | RCADS<ref name="ChorpitaEtAl2000"/> | Questionnaire (Child) | -- (N=513) | 9.8 | 0.24 | DLR: (Chorpita, Moffitt &amp; Gray, 2005)<ref name="ChorpitaEtAl2005"/> | High: Several studies demonstrate support for the RCADS in non-referred samples of youth |- | SCAS<ref>{{Cite journal|last=Spence|first=Susan H.|title=A measure of anxiety symptoms among children|url=https://doi.org/10.1016/S0005-7967(98)00034-5|journal=Behaviour Research and Therapy|volume=36|issue=5|pages=545–566|doi=10.1016/s0005-7967(98)00034-5}}</ref> | Questionnaire (Child, Parent) | 0.83 (N=654) | -- | -- | (Nauta et al., under review) | |- | Generalized Anxiety Disorder Scale (GADS)<ref name="SpitzerEtAl2006"/> | Questionnaire | 0.88 (N = 438) | 6.3 (5+) | .41 (5-) | Wild et al., 2014 | Elderly persons (ages 58–82) from general population in German |- |Generalized Anxiety Disorder Screener (GAD-7)<ref>{{Cite journal|last=Plummer|first=Faye|last2=Manea|first2=Laura|last3=Trepel|first3=Dominic|last4=McMillan|first4=Dean|date=2016-03-01|title=Screening for anxiety disorders with the GAD-7 and GAD-2: a systematic review and diagnostic metaanalysis|url=https://www.sciencedirect.com/science/article/pii/S0163834315002406|journal=General Hospital Psychiatry|language=en|volume=39|pages=24–31|doi=10.1016/j.genhosppsych.2015.11.005|issn=0163-8343}}</ref> |Questionnaire |0.906<ref>{{Cite journal|last=Spitzer|first=Robert L.|last2=Kroenke|first2=Kurt|last3=Williams|first3=Janet B. W.|last4=Löwe|first4=Bernd|date=2006-05-22|title=A Brief Measure for Assessing Generalized Anxiety Disorder: The GAD-7|url=http://archinte.jamanetwork.com/article.aspx?doi=10.1001/archinte.166.10.1092|journal=Archives of Internal Medicine|language=en|volume=166|issue=10|pages=1092|doi=10.1001/archinte.166.10.1092|issn=0003-9926}}</ref> (N = 2149) |5.17 (8+) |.20 (8-) |Plummer et al., 2016 |Adults aged 16 years and older in any setting (meta-analysis) |- |CBCL Anxious/Depressed Scale T-score<ref>{{Cite journal|last=Eimecke|first=Sylvia D.|last2=Remschmidt|first2=Helmut|last3=Mattejat|first3=Fritz|date=2011-03|title=Utility of the Child Behavior Checklist in screening depressive disorders within clinical samples|url=https://linkinghub.elsevier.com/retrieve/pii/S0165032710005458|journal=Journal of Affective Disorders|language=en|volume=129|issue=1-3|pages=191–197|doi=10.1016/j.jad.2010.08.011}}</ref> |Questionnaire |.75 (N = 1445) |1.49 (9+) |.67(9-) |Eimecke et al., (2011) |Inpatient and outpatient children and adolescents |} '''Note:''' “LR+” refers to the change in likelihood ratio associated with a positive test score, and “LR-” is the likelihood ratio for a low score. Likelihood ratios of 1 indicate that the test result did not change impressions at all. LRs larger than 10 or smaller than .10 are frequently clinically decisive; 5 or .20 are helpful, and between 2.0 and .5 are small enough that they rarely result in clinically meaningful changes of formulation<ref>Sackett, D. L., Straus, S. E., Richardson, W. S., Rosenberg, W., & Haynes, R. B. (2000). Evidence-based medicine: How to practice and teach EBM. Edinburgh: Churchill Livingstone.</ref>. '''Search terms:''' [General Anxiety Disorder] AND [children OR adolescents OR pediatric] AND [sensitivity OR specificity] in GoogleScholar and PsycINFO === Interpreting GAD screening measure scores === * For information on interpreting screening measure scores, click [[Evidence based assessment/Prediction phase#Interpreting screening measure scores|here.]] ==[[Evidence based assessment/Prescription phase|'''Prescription phase''']]== ===Gold standard diagnostic interviews=== * For a list of broad reaching diagnostic interviews sortable by disorder with PDFs (if applicable), [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prescription_phase&wteswitched=1#Common_Diagnostic_Interviews click here.] === Recommended diagnostic instruments for GAD === {| class="wikitable sortable" ! colspan="10" |Diagnostic instruments for GAD |- !Measure !Format (Reporter) !Age Range !Administration/ Completion Time !Interrater Reliability !Test-Retest Reliability !Construct Validity !Content Validity !Highly Recommended !Free and Accessible Measures |- |Anxiety Disorders Interview Schedule for Children (ADIS-C)<ref name=":1">{{Cite journal|date=2001-08-01|title=Test-Retest Reliability of Anxiety Symptoms and Diagnoses With the Anxiety Disorders Interview Schedule for DSM-IV: Child and Parent Versions|url=https://www.sciencedirect.com/science/article/pii/S0890856709603427|journal=Journal of the American Academy of Child & Adolescent Psychiatry|language=en|volume=40|issue=8|pages=937–944|doi=10.1097/00004583-200108000-00016|issn=0890-8567}}</ref> |Child |6-16<ref name=":5">{{Cite journal|last=LYNEHAM|first=HEIDI J.|last2=ABBOTT|first2=MAREE J.|last3=RAPEE|first3=RONALD M.|date=2007-06|title=Interrater Reliability of the Anxiety Disorders Interview Schedule for DSM-IV: Child and Parent Version|url=https://doi.org/10.1097/chi.0b013e3180465a09|journal=Journal of the American Academy of Child &amp; Adolescent Psychiatry|volume=46|issue=6|pages=731–736|doi=10.1097/chi.0b013e3180465a09|issn=0890-8567}}</ref> |Varies |E<ref name=":1" /> |E<ref name=":1" /> |G to E<ref name=":1" /> |N/A | |[https://www.google.com/books/edition/Anxiety_Disorders_Interview_Schedule_ADI/TOtZAAAACAAJ?hl=en Purchase] |- |Anxiety Disorders Interview Schedule for Children (ADIS-P)<ref name=":1" /> |Parent |6-16<ref name=":5" /> |Varies |E<ref name=":1" /> |E<ref name=":1" /> |E<ref name=":1" /> |N/A | |[https://www.google.com/books/edition/Anxiety_Disorders_Interview_Schedule_ADI/TOtZAAAACAAJ?hl=en Purchase] |- |Anxiety Disorders Interview Schedule for DSM-IV (ADIS-IV) <nowiki>*</nowiki>not free |Adult |16+ |Varies |A<ref name=":0" /> |NA<ref name=":0" /> |A<ref name=":0" /> |A<ref name=":0" /> |<ref name=":0" />[[File:Light green check.svg|center|frameless|50x50px]] | |- |Structured Clinical Interview for DSM-IV-TR for Axis I Disorders (SCID-I/P) <nowiki>*</nowiki>not free | | |Varies |A<ref name=":0" /> |NA<ref name=":0" /> |A<ref name=":0" /> |A<ref name=":0" /> | |[http://www.scid4.org/ Website and purchase] |- |Structured Clinical Interview for DSM-IV-TR for Axis II Disorders (SCID-II) <nowiki>*</nowiki>not free | | |Varies |E<ref name=":0" /> |NA<ref name=":0" /> |U<ref name=":0" /> |U<ref name=":0" /> | |[http://www.scid4.org/ Website and purchase] |- |Structured Clinical Interview for DSM-IV (SCID-IV) <nowiki>*</nowiki>not free | | |Varies |A<ref name=":0" /> |A<ref name=":0" /> |E<ref name=":0" /> |E<ref name=":0" /> | |[http://www.scid4.org/ Website and purchase] |- |Anxiety and Related Disorders Interview Schedule for DSM-5 (ADIS-5) |Structured Interview (Adult) |16+ |Varies | | | | | |[https://global.oup.com/academic/product/anxiety-and-related-disorders-interview-schedule-for-dsm-5-adis-5---adult-version-9780199325160?cc=us&lang=en& Purchase] |- |Structured Clinical Interview for DSM-5 Clinician Version (SCID-5- CV)<ref name=":6">{{Cite journal|last=Shabani|first=Amir|last2=Masoumian|first2=Samira|last3=Zamirinejad|first3=Somayeh|last4=Hejri|first4=Maryam|last5=Pirmorad|first5=Tahereh|last6=Yaghmaeezadeh|first6=Hooman|date=2021-05|title=Psychometric properties of Structured Clinical Interview for DSM‐5 Disorders‐Clinician Version (SCID‐5‐CV)|url=https://onlinelibrary.wiley.com/doi/10.1002/brb3.1894|journal=Brain and Behavior|language=en|volume=11|issue=5|doi=10.1002/brb3.1894|issn=2162-3279|pmc=PMC8119811|pmid=33729681}}</ref> |Structured Interview (Adult) |16+ |Varies |E<ref name=":6" /> |A<ref name=":6" /> | | | |[https://www.columbiapsychiatry.org/research/research-labs/diagnostic-and-assessment-lab/structured-clinical-interview-dsm-disorders-11 Purchase] |} '''Note:''' L = Less than adequate; A = Adequate; G = Good; E = Excellent; U = Unavailable; NA = Not applicable === Severity interviews for GAD=== {| class="wikitable sortable" border="1" |- ! Measure ! Format (Reporter) ! Age Range ! Administration/ Completion Time ! Interrater Reliability ! Test-Retest Reliability ! Construct Validity ! Content Validity ! Highly Recommended !Free and Accessible Measures |- | Children's Depression Rating Scale - Revised (CDRS-R) | Structured Interview<ref name=":4">{{Cite journal|last=Mayes|first=Taryn L.|last2=Bernstein|first2=Ira H.|last3=Haley|first3=Charlotte L.|last4=Kennard|first4=Betsy D.|last5=Emslie|first5=Graham J.|date=2010-12|title=Psychometric Properties of the Children's Depression Rating Scale–Revised in Adolescents|url=https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3003451/|journal=Journal of Child and Adolescent Psychopharmacology|volume=20|issue=6|pages=513–516|doi=10.1089/cap.2010.0063|issn=1044-5463|pmc=PMC3003451|pmid=21186970}}</ref> | 6-12 | 15-20 minutes | G | A | G | G | X | * Link to purchase [https://www.wpspublish.com/cdrs-r-childrens-depression-rating-scale-revised] *[http://www.opapc.com/uploads/documents/CDRS-R.pdf PDF] (excerpt) |} '''Note:''' '''L''' = Less than adequate; '''A''' = Adequate; '''G''' = Good; '''E''' = Excellent; '''U''' = Unavailable; '''NA''' = Not applicable ==[[Evidence based assessment/Process phase|'''Process phase''']]== The following section contains a list of process and outcome measures for generalized anxiety disorder. The section includes benchmarks based on published norms for several outcome and severity measures, as well as information about commonly used process measures. Process and outcome measures are used as part of the [[Evidence based assessment/Process phase|process phase]] of assessment. For more information of differences between process and outcome measures, see the page on the [[Evidence based assessment/Process phase|process phase of assessment]]. === Outcome and severity measures === * This table includes clinically significant benchmarks for generalized anxiety disorder specific outcome measures * Information on how to interpret this table can be [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase found here]. * Additionally, these [[Evidence based assessment/Vignettes|vignettes]] might be helpful resources for understanding appropriate adaptation of outcome measures in practice. *''<u>For clinically significant change benchmarks for the CBCL, YSR, and TRF total, externalizing, internalizing, and attention benchmarks,</u>'' [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase&wteswitched=1#Clinically_significant_change_benchmarks_for_widely-used_outcome_measures see here.] {| class="wikitable sortable" border="1" | colspan="7" |'''Clinically significant change benchmarks with common instruments for GAD''' |- | rowspan=1" style="text-align:center;font-size:130%;" | <b> Measure</b> | colspan="3" style="text-align:center;font-size:130%" width="300" | <b> Cut-off scores</b> | colspan="3" style="text-align:center;font-size:120%" | <b> Critical Change <br> (unstandardized scores)</b> |- | colspan="7" span style="font-size:110%; text-align:center;" | <b> Benchmarks Based on Published Norms</b> |- | colspan="1" | | style="text-align:center;font-size:110%" | <b> A</b> | style="text-align:center;font-size:110%" | <b> B</b> | style="text-align:center;font-size:110%" | <b> C</b> | style="text-align:center;font-size:110%" | <b> 95%</b> | style="text-align:center;font-size:110%" | <b> 90%</b> | style="text-align:center;font-size:110%" | <b> SE_difference</b> |- | rowspan="1" style="text-align:center;" | <b> GAD-7</b> | style=“text-align:center;”| -1 | style=“text-align:center;”| 1.3 | style=“text-align:center;”| 0.5 | style=“text-align:center;”| 0.6 | style=“text-align:center;”| 0.5 | style=“text-align:center;”| 0.3 |- | rowspan="1" style="text-align:center;" | <b> PSWQ</b> | style=“text-align:center;”| 51 | style=“text-align:center;”| 73 | style=“text-align:center;”| 59 | style=“text-align:center;”| 9 | style=“text-align:center;”| 8 | style=“text-align:center;”| 4.8 |- | rowspan="1" style="text-align:center;" | <b> SCARED </b> | style=“text-align:center;”| 9.9 | style=“text-align:center;”| 18.1 | style=“text-align:center;”| 15.3 | style=“text-align:center;”| 8.9 | style=“text-align:center;”| 7.5 | style=“text-align:center;”| 4.5 |- | rowspan="1" style="text-align:center;" | <b> STAIC</b> | style=“text-align:center;”| 0.9 | style=“text-align:center;”| 30.1 | style=“text-align:center;”| 18.2 | style=“text-align:center;”| 18.9 | style=“text-align:center;”| 15.9 | style=“text-align:center;”| 9.6 |- | rowspan="1" style="text-align:center;" | <b> RCADS</b> | style=“text-align:center;”| -1.1 | style=“text-align:center;”| 12.7 | style=“text-align:center;”| 6.6 | style=“text-align:center;”| 7.3 | style=“text-align:center;”| 6.1 | style=“text-align:center;”| 3.7 |- | rowspan="1" style="text-align:center;" | <b> SCAS</b> | style=“text-align:center;”| -0.7 | style=“text-align:center;”| 15.1 | style=“text-align:center;”| 5.4 | style=“text-align:center;”| 6.2 | style=“text-align:center;”| 5.2 | style=“text-align:center;”| 3.2 |} '''Note:''' “A” = Away from the clinical range, “B” = Back into the nonclinical range, “C” = Closer to the nonclinical than clinical mean. '''Search terms:''' [General Anxiety Disorder] AND [children OR adolescents OR pediatric] AND [clinical significance OR outcomes] in GoogleScholar and PsycINFO === Treatment === {{collapse top| Click here for treatment information}} Individuals suffering from GAD tend to be high users of outpatient medical care. When treating GAD, physicians should first determine whether pharmacotherapy, psychotherapy, or a combination of the two treatments would be most beneficial to the patient. Literature suggests that treatment of GAD frequently consists of a combination of psychotherapy and pharmacotherapy. Although these therapies have the potential to be effective individually, previous work demonstrates that when combined the degree of clinically significant change increases significantly. Recent studies (e.g., Gorman, 2003<ref name="Gorman2003" />; Walkup et al., 2008<ref name="WalkupEtAl2008" />) have provided evidence to support this claim with the most efficacious medication and behavioral interventions listed below. # '''Medication Interventions''' ## ''Sertraline (Zoloft)'' has been shown to reduce experiences and effects of GAD above and beyond that of placebo conditions. ## ''Pregabalin.'' The mean baseline-to-endpoint decreases in total Hamilton anxiety scale score in the patients given 150 mg/day of pregabalin (–9.2) was significantly greater than the decrease in those given placebo (–6.8)<ref name="PandeEtAl2003" />. ## ''Paroxetine.'' Remission was achieved by 30% of patients in the 20-mg paroxetine groups compared with 20% given placebo. For all three domains of the Sheehan Disability Scale, significantly greater improvement was seen with paroxetine than placebo<ref name="RickelsEtAl2003" />. # '''Behavioral interventions''' ## ''Cognitive behavioral therapy.'' Fourteen 60-minute sessions, which include CBT in anxiety-management skills, followed by behavioral exposure to anxiety-provoking situations have been shown to be effective in treating GAD. A review of studies by Fisher and Durham (1999)<ref name="FisherEtAl1999" /> revealed significant recovery rates at a 6 month follow up after CBT. ## ''Exposure therapy and modeling therapy.'' One meta-analysis found that virtual reality exposure therapy for anxiety disorders had a large effect size (Cohen's d=1.11) compared to controls.<ref>{{Cite journal|last=Powers|first=Mark B.|last2=Emmelkamp|first2=Paul M.G.|title=Virtual reality exposure therapy for anxiety disorders: A meta-analysis|url=https://doi.org/10.1016/j.janxdis.2007.04.006|journal=Journal of Anxiety Disorders|volume=22|issue=3|pages=561–569|doi=10.1016/j.janxdis.2007.04.006}}</ref> ## ''Mindfulness meditation.'' New treatment options such as mindfulness meditation-based stress reduction interventions have also shown to reduce symptoms over the long-term.<ref>{{Cite journal|last=Miller|first=J. J.|last2=Fletcher|first2=K.|last3=Kabat-Zinn|first3=J.|date=May 1995|title=Three-year follow-up and clinical implications of a mindfulness meditation-based stress reduction intervention in the treatment of anxiety disorders|url=https://www.ncbi.nlm.nih.gov/pubmed/7649463|journal=General Hospital Psychiatry|volume=17|issue=3|pages=192–200|issn=0163-8343|pmid=7649463}}</ref> # '''Combination treatment''' ## Previous research suggests that combination therapy that includes components of psychotherapy and pharmacotherapy are the most efficacious in treating GAD. In a study comparing the efficacies GAD treatments, Walkup and colleagues demonstrated a 21-25% improvement of combination therapy over cognitive behavioral therapy or sertraline alone during short-term treatment. These findings suggest that among effective treatments, combination therapy has the potential to provide the best chance for a positive outcome. See Gorman, 2003<ref name="Gorman2003" />; Walkup et al., 2008<ref name="WalkupEtAl2008" />. {{collapse bottom}} * Please refer to the page on [[wikipedia:Generalized_anxiety_disorder|generalized anxiety disorder]] for more information on available treatment or go to [http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/fear-worry-and-anxiety/ Effective Child Therapy] for a curated resource on effective treatments for GAD. *For information on conducting Exposure Therapy for anxiety disordered youth, see [https://www.bravepracticeforkids.com/ www.BravePracticeForKids.com] =='''External Resources'''== # [http://apps.who.int/classifications/icd10/browse/2010/en#/F41.1 ICD-10 diagnostic criteria] # [https://en.wikiversity.org/w/index.php?title=Helping_Give_Away_Psychological_Science/Resources/Annotated_List_of_Where_and_How_to_Find_a_Therapist&wteswitched=1#Other_low-cost_options Find-a-Therapist] #*This is a curated list of find-a-therapist websites where you can find a provider # [https://www.nimh.nih.gov/health/topics/anxiety-disorders/index.shtml NIMH] entry about anxiety disorders # OMIM (Online Mendelian Inheritance in Man) #*[https://www.omim.org/entry/607834 607834] # [https://emedicine.medscape.com/article/286227-overview#a2 eMedicine entry about anxiety disorders] #[https://sccap53.org Society of Clinical Child and Adolescent Psychology] #[http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/fear-worry-and-anxiety/ Effective Child Therapy information on Fear, Worry, & Anxiety] #*Effective Child Therapy is website sponsored by Division 53 of the American Psychological Association (APA), or The [https://sccap53.org Society of Clinical Child and Adolescent Psychology] (SCCAP), in collaboration with the Association for Behavioral and Cognitive Therapies (ABCT). Use for information on symptoms and available treatments. #[http://pediatricbipolar.pitt.edu/resources/instruments Links to SCARED Child, Parent, and Adult + Translations] =='''References'''== {{collapse top|Click here for references}} {{Reflist|3|refs= <ref name="BirmaherEtAl1997">{{cite journal|last1=Birmaher|first1=B|last2=Khetarpal|first2=S|last3=Brent|first3=D|last4=Cully|first4=M|last5=Balach|first5=L|last6=Kaufman|first6=J|last7=Neer|first7=SM|title=The Screen for Child Anxiety Related Emotional Disorders (SCARED): scale construction and psychometric characteristics.|journal=Journal of the American Academy of Child and Adolescent Psychiatry|date=April 1997|volume=36|issue=4|pages=545-53|pmid=9100430}}</ref> <ref name="BrownJacobsenEtAl2011">{{cite journal|last1=Brown-Jacobsen|first1=AM|last2=Wallace|first2=DP|last3=Whiteside|first3=SP|title=Multimethod, multi-informant agreement, and positive predictive value in the identification of child anxiety disorders using the SCAS and ADIS-C.|journal=Assessment|date=September 2011|volume=18|issue=3|pages=382-92|pmid=20644080}}</ref> <ref name="CostelloEtAl1996">{{cite journal|last1=Costello|first1=EJ|last2=Angold|first2=A|last3=Burns|first3=BJ|last4=Stangl|first4=DK|last5=Tweed|first5=DL|last6=Erkanli|first6=A|last7=Worthman|first7=CM|title=The Great Smoky Mountains Study of Youth. 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DSM-IV anxiety disorders.|journal=Depression and anxiety|date=2008|volume=25|issue=12|pages=1046-52|pmid=18833579}}</ref> <ref name="WoodEtAl2002">{{cite journal|last1=Wood|first1=JJ|last2=Piacentini|first2=JC|last3=Bergman|first3=RL|last4=McCracken|first4=J|last5=Barrios|first5=V|title=Concurrent validity of the anxiety disorders section of the Anxiety Disorders Interview Schedule for DSM-IV: child and parent versions.|journal=Journal of clinical child and adolescent psychology : the official journal for the Society of Clinical Child and Adolescent Psychology, American Psychological Association, Division 53|date=September 2002|volume=31|issue=3|pages=335-42|pmid=12149971}}</ref> <ref name="ZimmermanEtAl2005">{{cite journal|last1=Zimmerman|first1=M|last2=Rothschild|first2=L|last3=Chelminski|first3=I|title=The prevalence of DSM-IV personality disorders in psychiatric outpatients.|journal=The American journal of psychiatry|date=October 2005|volume=162|issue=10|pages=1911-8|pmid=16199838}}</ref> }} {{collapse bottom|Click here for references}} [[Category:Psychological disorder portfolios|{{SUBPAGENAME}}]] 26rmemutme0of316i7edtbv98we99cq 2410316 2410315 2022-07-29T22:18:59Z Aherman012 2943941 /* Process phase */ wikitext text/x-wiki <noinclude>{{Helping Give Away Psychological Science Banner}}</noinclude> {{medical disclaimer}} {{:{{ROOTPAGENAME}}/Sidebar}} ==[[Evidence based assessment/Portfolio template/What is a "portfolio"|'''What is a "portfolio"?''']]== * For background information on what assessment portfolios are, click the link in the heading above. Does this page feel like too much information? Click [[Evidence-based assessment/Generalized anxiety disorder (assessment portfolio)|here]] for the condensed version. ==[[Evidence based assessment/Preparation phase|'''Preparation phase''']]== === Diagnostic criteria for generalized anxiety disorder === {{blockquotetop}} <big>'''ICD-11 Diagnostic Criteria<ref>https://icd.who.int/browse11/l-m/en#/http://id.who.int/icd/entity/1712535455</ref>'''</big> *Generalised anxiety disorder is characterized by marked symptoms of anxiety that persist for at least several months, for more days than not, manifested by either general apprehension (i.e. ‘free-floating anxiety’) or excessive worry focused on multiple everyday events, most often concerning family, health, finances, and school or work, together with additional symptoms such as muscular tension or motor restlessness, sympathetic autonomic over-activity, subjective experience of nervousness, difficulty maintaining concentration, irritability, or sleep disturbance. The symptoms result in significant distress or significant impairment in personal, family, social, educational, occupational, or other important areas of functioning. The symptoms are not a manifestation of another health condition and are not due to the effects of a substance or medication on the central nervous system. '''Changes in DSM-5''' * The diagnostic criteria for generalized anxiety disorder changed slightly from DSM-IV-TR to DSM-5. Summaries are available [https://www.ncbi.nlm.nih.gov/books/NBK519712/table/ch3.t9/?report=objectonly here]. {{blockquotebottom}} === Base rates of GAD in different clinical settings === This section describes the demographic setting of the population(s) sampled, base rates of diagnosis, country/region sampled and the diagnostic method that was used. Using this information, clinicians will be able to anchor the rate of GAD that they are likely to see in their clinical practice. * '''''To see prevalence rates across multiple disorders,''''' [[Evidence based assessment/Preparation phase#Base rates for transdiagnostic comparison|'''''click here.''''']] {| class="wikitable sortable" border="1" |- ! Demography ! Setting ! Base Rate ! Diagnostic Method |- | Adults and adolescences in all of U.S.A. | US National Comorbidity Survey Replication (NCS-R; age > = 13) [https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4005415/pdf/nihms-571992.pdf (2012)]<ref name="KesslerEtAl2012" /> | * 0.9% (age 13-17) * 2.9% (age 18-64) * 1.2% (age &gt;= 65) * 2.0% (age &gt;=13) | Fully-structured CIDI Version 3.0 |- | Psychiatric outpatients | Individuals seeking treatment in a Psychiatric Outpatient Clinic (age range not reported) ([https://ajp.psychiatryonline.org/doi/pdf/10.1176/appi.ajp.162.10.1911 2014])<ref name="ZimmermanEtAl2005" /> | * 21% | Structured Clinical Interview for DSM-IV (SCID) |- | Caucasian youth | Children seeking treatment in a Child &amp; Adolescent Anxiety Diagnostic Clinic (age 7 – 18 years old) ([http://journals.sagepub.com/doi/pdf/10.1177/1073191110375792 2011])<ref name="BrownJacobsenEtAl2011" /> | * 0.39% (parent report) * 0.38% (child report) | ADIS-C for DSM-IV Spence Children's Anxiety Scale (SCAS) |- | Caucasian, African American, Asian American, and Hispanic population | Collaborative Psychiatric Epidemiology Studies (CPES; age &gt;= 18, data merged from three representative national database) ([https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3135672/pdf/nihms281333.pdf 2011])<ref>{{Cite journal|last=McLean|first=Carmen P.|last2=Asnaani|first2=Anu|last3=Litz|first3=Brett T.|last4=Hofmann|first4=Stefan G.|date=2011-08-01|title=Gender differences in anxiety disorders: Prevalence, course of illness, comorbidity and burden of illness|url=https://www.sciencedirect.com/science/article/pii/S0022395611000458|journal=Journal of Psychiatric Research|language=en|volume=45|issue=8|pages=1027–1035|doi=10.1016/j.jpsychires.2011.03.006|issn=0022-3956|pmc=PMC3135672|pmid=21439576}}</ref> | * 4.1% (female) * 2.1% (male) | World Mental Health Survey Initiative Version of the World Health Organization Composite International Interview (WMH-CIDI) |- | Pennsylvania | Metropolitan Community Sample, all individuals with eating disorders (ages 13 – 65) ([https://ajp.psychiatryonline.org/doi/pdf/10.1176/appi.ajp.161.12.2215 2014])<ref name="KayeEtAl2004" /> | * 10% | Structured Clinical Interview for DSM-IV (SCID) |- | Adolescents in all of U.S.A. | National Comorbidity Survey Replication Adolescent Supplement (NCS-A; ages 3–18 in the continental U.S) ([https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2946114/pdf/nihms214371.pdf 2011])<ref name="MerikangasEtAl2010" /> | * 2.2% | World Health Organization Composite International Diagnostic Interview (WHO-CIDI) |- |Adolescents in all of U.S.A |National Comorbidity Survey Replication Adolescent Supplement (NCS-A; ages 3–18 in the continental U.S)<ref name=":3">Kessler, R. C., Avenevoli, S., Costello, E. J., Georgiades, K., Green, J. G., Gruber, M. J., . . . Merikangas, K. R. (2012). Prevalence, persistence, and sociodemographic correlates of DSM-IV disorders in the National Comorbidity Survey Replication Adolescent Supplement. Archives of General Psychiatry, 69(4), 372-380. doi:10.1001/archgenpsychiatry.2011.160</ref> | * 5.4% |Composite International Diagnostic Interview (CIDI) |- | North Carolina | Rural community sample African American and White youth (ages 13-16) [https://www.ncbi.nlm.nih.gov/pubmed/12365876 (2002)]<ref>{{Cite journal|last=Angold|first=Adrian|last2=Erkanli|first2=Alaattin|last3=Farmer|first3=Elizabeth M. Z.|last4=Fairbank|first4=John A.|last5=Burns|first5=Barbara J.|last6=Keeler|first6=Gordon|last7=Costello|first7=E. Jane|date=October 2002|title=Psychiatric disorder, impairment, and service use in rural African American and white youth|url=https://www.ncbi.nlm.nih.gov/pubmed/12365876|journal=Archives of General Psychiatry|volume=59|issue=10|pages=893–901|issn=0003-990X|pmid=12365876}}</ref> | * 1.4% | The Child and Adolescent Psychiatric Assessment (CAPA) |- | Texas | Metropolitan Community Sample (ages 11-17) ([https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2736593/pdf/nihms30019.pdf 2007])<ref name="RobertsEtAl2007" /> | * 0.4% | Diagnostic Interview Schedule for Children, Version IV (DISC-IV) |- | Midwestern Urban | Incarcerated adolescents (ages 10-18)<ref>{{Cite journal|last=ABRAM|first=KAREN M.|last2=CHOE|first2=JEANNE Y.|last3=WASHBURN|first3=JASON J.|last4=TEPLIN|first4=LINDA A.|last5=KING|first5=DEVON C.|last6=DULCAN|first6=MINA K.|title=Suicidal Ideation and Behaviors Among Youths in Juvenile Detention|url=http://linkinghub.elsevier.com/retrieve/pii/S0890856709623121|journal=Journal of the American Academy of Child & Adolescent Psychiatry|volume=47|issue=3|pages=291–300|doi=10.1097/chi.0b013e318160b3ce}}</ref> [http://vb3lk7eb4t.search.serialssolutions.com/?ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Suicidal+Ideation+and+Behaviors+Among+Youths+in+Juvenile+Detention&rft.jtitle=Journal+of+the+American+Academy+of+Child+%26+Adolescent+Psychiatry&rft.au=ABRAM%2C+KAREN+M.%2C+Ph.D&rft.au=CHOE%2C+JEANNE+Y.%2C+B.A&rft.au=WASHBURN%2C+JASON+J.%2C+Ph.D.%2C+A.B.P.P&rft.au=TEPLIN%2C+LINDA+A.%2C+Ph.D&rft.date=2008&rft.issn=0890-8567&rft.eissn=1527-5418&rft.volume=47&rft.issue=3&rft.spage=291&rft.epage=300&rft_id=info:doi/10.1097%2FCHI.0b013e318160b3ce&rft.externalDocID=1_s2_0_S0890856709623121 (2002)] | * 1% | Diagnostic Interview Schedule for Children, Version IV (DISC-IV) |- |The southern Appalachian mountain region of North Carolina |Great Smoky Mountain (ages 9-12) ([https://www.ncbi.nlm.nih.gov/pubmed/8956679 1996])<ref name="CostelloEtAl1996" /> | * 1.67% |DSM-III-R, DSM-IV and CAPA |- |New Jersey |Non-referred Adolescent Population (ages 9-17) ([https://www.ncbi.nlm.nih.gov/pubmed/2331210 1990])<ref name="WhitakerEtAl1990" /> | * 3.7% |DSM-III &amp; Beck Depression Inventory (BDI) |- |Non-institutionalized general US population |LGBTQ sample (ages 20-65)<ref>{{Cite journal|last=Bostwick|first=Wendy B.|last2=Boyd|first2=Carol J.|last3=Hughes|first3=Tonda L.|last4=McCabe|first4=Sean Esteban|date=2010-3|title=Dimensions of Sexual Orientation and the Prevalence of Mood and Anxiety Disorders in the United States|url=https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2820045/|journal=American Journal of Public Health|volume=100|issue=3|pages=468–475|doi=10.2105/AJPH.2008.152942|issn=0090-0036|pmc=PMC2820045|pmid=19696380}}</ref> [http://ajph.aphapublications.org/doi/10.2105/AJPH.2008.152942 (2013)] |Women: * 14.8% same-sex * 22.5% bisexual Men: * 16.9% same-sex * 11.5% bisexual |The Alcohol Use Disorder and Associated Disabilities Interview Schedule-IV (AUDADIS-IV) |- |Non-institutionalized general US population |Cross-ethnic American population (ages 18+)<ref>{{Cite journal|last=Asnaani|first=Anu|last2=Richey|first2=J. Anthony|last3=Dimaite|first3=Ruta|last4=Hinton|first4=Devon E.|last5=Hofmann|first5=Stefan G.|date=2010-8|title=A Cross-Ethnic Comparison of Lifetime Prevalence Rates of Anxiety Disorders|url=https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2931265/|journal=The Journal of nervous and mental disease|volume=198|issue=8|pages=551–555|doi=10.1097/NMD.0b013e3181ea169f|issn=0022-3018|pmc=PMC2931265|pmid=20699719}}</ref> [https://journals.lww.com/jonmd/Abstract/2010/08000/A_Cross_Ethnic_Comparison_of_Lifetime_Prevalence.4.aspx (2018)] | * White 8.6% * African Americans 4.9% * Hispanic Americans 5.8% * Asian Americans 2.4% |World Mental Health Survey Initiative Version of the World Health Organization Composite International Interview (WMH-CIDI) |- |Outpatient clinics worldwide |Samples across multiple studies worldwide (all ages)<ref name=":12">{{Cite journal|last=Rettew|first=David C.|last2=Lynch|first2=Alicia Doyle|last3=Achenbach|first3=Thomas M.|last4=Dumenci|first4=Levent|last5=Ivanova|first5=Masha Y.|date=2009-09|title=Meta-analyses of agreement between diagnoses made from clinical evaluations and standardized diagnostic interviews|url=http://dx.doi.org/10.1002/mpr.289|journal=International Journal of Methods in Psychiatric Research|language=en|volume=18|issue=3|pages=169–184|doi=10.1002/mpr.289|issn=1049-8931}}</ref> |5% |Clinical evaluations |- |Outpatient clinic worldwide |Samples across multiple studies worldwide (all ages)<ref name=":12" /> |10% |Standardized Diagnostic Interviews (SDIs) |} '''Search terms:''' [General Anxiety Disorder] AND [youth OR adolescents OR pediatric] AND [prevalence OR incidence] in GoogleScholar and PsycINFO == [[Evidence based assessment/Prediction phase|'''Prediction phase''']] == === Psychometric properties of screening instruments for GAD === The following section contains a list of screening and diagnostic instruments for generalized anxiety disorder. The section includes administration information, psychometric data, and PDFs or links to the screenings. * Screenings are used as part of the [[Evidence based assessment/Prediction phase|prediction phase]] of assessment; for more information on interpretation of this data, or how screenings fit in to the assessment process, click [[Evidence based assessment/Prediction phase|here]]. * '''''For a list of more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&wteswitched=1#Psychometric_properties_of_common_screening_instruments click here.]''''' {| class="wikitable sortable" border="1" ! colspan="10" |Screening measures for GAD |- ! Measure ! Format (Reporter) ! Age Range ! Administration/ Completion Time ! Interrater Reliability ! Test-Retest Reliability ! Construct Validity ! Content Validity ! Highly Recommended !PDF |- | Penn State Worry Questionnaire (PSWQ)<ref name=":0">{{Cite book|url=https://www.worldcat.org/oclc/314222270|title=A guide to assessments that work|date=2008|publisher=Oxford University Press|author=Hunsley, John |author2=Mash, Eric J.|isbn=9780195310641|location=New York|oclc=314222270}}</ref> | Questionnaire (Adult Version, Child Version) | 18+ (Adult Version), 6-18 (Child Version) | 4 minutes | NA<ref name=":0" /> | G<ref name=":0" /> |G<ref name=":0" /> |G<ref name=":0" /> |<ref name=":0" />[[File:Light green check.svg|center|frameless|50x50px]] | * [http://www.midss.org/content/penn-state-worry-questionnaire-pswq PSWQ homepage] '''PDFs of the PSWQ''' *[https://mfr.osf.io/render?url=https://osf.io/s7p38/?action=download%26mode=render Penn State Worry Questionnaire] *[https://mfr.osf.io/render?url=https://osf.io/6q8y9/?action=download%26mode=render PSWQ-C] *[https://mfr.osf.io/render?url=https://osf.io/gx5sr/?action=download%26mode=render PSWC-C Korean] *[https://mfr.osf.io/render?url=https://osf.io/hc6n2/?action=download%26mode=render PSWQ-C Danish] *[https://mfr.osf.io/render?url=https://osf.io/fwbes/?action=download%26mode=render Scoring of the PSWQ-C] |- |[[wikipedia:Screen_for_child_anxiety_related_disorders|Screen for Child Anxiety Related Emotional Disorder (SCARED)]] | Questionnaire (Child, Parent) | 8-19 | 9 or 16 minutes |NA<ref name=":0" /> | G<ref name=":0" /> |G<ref name=":0" /> | G<ref name=":0" /> | | * [http://www.midss.org/content/screen-child-anxiety-related-disorders-scared SCARED] homepage '''PDFs of SCARED''' *[[wikipedia:Screen_for_child_anxiety_related_disorders#PDFs_and_automated_scoring_for_SCARED|SCARED English + Translations & Automatic Scoring]] |- |[[wikipedia:State-Trait_Anxiety_Inventory|State/Trait Anxiety Inventory for Children (STAIC)]] | Questionnaire (Child, Parent) | 6-18 | 5 or 10 minutes |NA<ref name=":0" /> | G<ref name=":0" /> |G<ref name=":0" /> | G<ref name=":0" /> | | * *not free* * [http://www.mindgarden.com/146-state-trait-anxiety-inventory-for-children#horizontalTab2 STAIC Website] |- | Revised Children’s Anxiety and Depression Scale (RCADS) | Questionnaire (Child) | 6-18 | 12 minutes |G<ref name=":2">{{Cite journal|last=Chorpita|first=Bruce F.|last2=Moffitt|first2=Catherine E.|last3=Gray|first3=Jennifer|date=2005-03|title=Psychometric properties of the Revised Child Anxiety and Depression Scale in a clinical sample|url=http://dx.doi.org/10.1016/j.brat.2004.02.004|journal=Behaviour Research and Therapy|volume=43|issue=3|pages=309–322|doi=10.1016/j.brat.2004.02.004|issn=0005-7967}}</ref> |G<ref>{{Cite journal|last=Chorpita|first=Bruce F|last2=Yim|first2=Letitia|last3=Moffitt|first3=Catherine|last4=Umemoto|first4=Lori A|last5=Francis|first5=Sarah E|date=2000-08|title=Assessment of symptoms of DSM-IV anxiety and depression in children: a revised child anxiety and depression scale|url=http://dx.doi.org/10.1016/s0005-7967(99)00130-8|journal=Behaviour Research and Therapy|volume=38|issue=8|pages=835–855|doi=10.1016/s0005-7967(99)00130-8|issn=0005-7967}}</ref> |G<ref name=":2" /> | | | * [https://www.corc.uk.net/outcome-experience-measures/revised-childrens-anxiety-and-depression-scale-rcads/ RCADS homepage] '''PDFs for RCADS''' *[https://mfr.osf.io/render?url=https://osf.io/s3fu2/?action=download%26mode=render RCADS Child Self-reported (8-18 years)] *[https://mfr.osf.io/render?url=https://osf.io/fp9mk/?action=download%26mode=render RCADS Parent-reported] *[https://mfr.osf.io/render?url=https://osf.io/vy7ta/?action=download%26mode=render Child Scoring Aid] *[https://mfr.osf.io/render?url=https://osf.io/t4bz6/?action=download%26mode=render Parent Scoring Aid] '''Subscales''' *[https://mfr.osf.io/render?url=https://osf.io/976xg/?action=download%26mode=render Generalized Anxiety Child Self-reported] *[https://mfr.osf.io/render?url=https://osf.io/y3qp7/?action=download%26mode=render Generalized Anxiety Parent-reported] *[https://mfr.osf.io/render?url=https://osf.io/4gc8d/?action=download%26mode=render Panic Self-Report] *[https://mfr.osf.io/render?url=https://osf.io/nhcsu/?action=download%26mode=render Panic Parent-Report] '''[https://osf.io/2fm4r/download Translations]''' '''[https://mfr.osf.io/render?url=https://osf.io/qsjh9/?action=download%26mode=render User Guide]''' * |- |[[wikipedia:Spence_Children's_Anxiety_Scale|Spence Children’s Anxiety Scale (SCAS)]] | Questionnaire (Child, Parent) | 7-19 | 11 minutes |NA<ref name=":0" /> | A<ref name=":0" /> | E<ref name=":0" /> | E<ref name=":0" /> | |[https://www.scaswebsite.com/ SCAS homepage] [https://www.scaswebsite.com/wp-content/uploads/2021/07/scas.pdf Child Version PDF] [https://www.scaswebsite.com/wp-content/uploads/2021/07/scas-parent-qaire.pdf Parent Version PDF] |- |[[wikipedia:Generalized_Anxiety_Disorder_7|GAD-7 Scale]] |Self report |18+ |5 minutes |G<ref name=":0" /> |Intraclass correlation 0.83<ref>{{Cite journal|last=Spitzer|first=Robert L.|last2=Kroenke|first2=Kurt|last3=Williams|first3=Janet B. W.|last4=Löwe|first4=Bernd|date=2006-05-22|title=A Brief Measure for Assessing Generalized Anxiety Disorder|url=http://archinte.jamanetwork.com/article.aspx?doi=10.1001/archinte.166.10.1092|journal=Archives of Internal Medicine|language=en|volume=166|issue=10|doi=10.1001/archinte.166.10.1092|issn=0003-9926}}</ref> |G<ref name=":0" /> |G<ref name=":0" /> |<ref name=":0" />[[File:Light green check.svg|center|frameless|50x50px]] |GAD-7 homepage [https://www.integration.samhsa.gov/clinical-practice/GAD708.19.08Cartwright.pdf PDF] (english) [http://www.coloradohealthpartnerships.com/provider/integrated/GAD7-Spanish.pdf PDF] (spanish) |- |Kessler Psychological Stress Scale (K10 and K6 Scales) |Self or interview administered | | | | | | | |[https://www.hcp.med.harvard.edu/ncs/k6_scales.php Available in many languages] |- | Worry and Anxiety Questionnaire (WAQ) |Self report | | 10 minutes | NA<ref name=":0" /> | A<ref name=":0" /> | A<ref name=":0" /> | G<ref name=":0" /> |<ref name=":0" />[[File:Light green check.svg|center|frameless|50x50px]] |WAQ homepage [https://uqo.ca/file/19980/download?token=Cmpts0MF PDF] |- |Brown Assessment of Beliefs Scale (BABS) | | | |G<ref name=":0" /> |A<ref name=":0" /> |G<ref name=":0" /> |G<ref name=":0" /> |[[File:Light green check.svg|center|frameless|50x50px]]<ref name=":0" /> | * [https://mfr.osf.io/render?url=https://osf.io/bqr2e/?action=download%26mode=render BABS Adult] * [https://mfr.osf.io/render?url=https://osf.io/z5s8a/?action=download%26mode=render BABS Original Publication] |- |Back Anxiety Inventory (BAI) |Self-report |17-80 |5-10 minutes |G<ref name=":0" /> | |G<ref name=":0" /> |G<ref name=":0" /> | |BAI homepage [https://www.gphealth.org/media/1087/anxiety.pdf PDF] |- |The Clinically Useful Anxiety Outcome Scale (CUXOS) |Self-report |18-85 |Less than 2 minutes |E<ref name=":0" /> | |E<ref name=":0" /> |G<ref name=":0" /> | |CUXOS homepage [https://outcometracker.org/CUXOS.pdf PDF] |- |Achenbach System of Empirically Based Assessments (ASEBA): Child Behavior Checklist (CBCL), Teacher Report Form (TRF), Youth Self-Report (YSR) |CBCL: Parent report, TRF: Teacher report, YSR: Child report |6-18 (CBCL & TRF), 11-18 (YSR)<ref name=":7">{{Cite book|url=https://www.worldcat.org/oclc/1130319849|title=Assessment of disorders in childhood and adolescence|date=2020|others=Eric Arden Youngstrom, Mitchell J. Prinstein, Eric J. Mash, Russell A. Barkley|isbn=978-1-4625-4363-2|edition=Fifth edition|location=New York, NY|oclc=1130319849}}</ref> |10 - 15 minutes<ref name=":7" /> |A<ref name=":0" /> |E<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> | |ASEBA homepage [https://aseba.org/forms/schoolagecbcl.pdf PDF] |} '''Note:''' L = Less than adequate; A = Adequate; G = Good; E = Excellent; U = Unavailable; NA = Not applicable === Likelihood ratios and AUCs of screening instruments for GAD === * '''''For a list of the likelihood ratios for more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&wteswitched=1#Likelihood_ratios_and_AUCs_of_common_screening_instruments click here.]''''' {| class="wikitable sortable" border="1" ! Screening Measure (Primary Reference) ! Format (Reporter) ! Area Under Curve (AUC) ! LR+ (Score) ! LR- (Score) ! Citation ! Clinical Generalizability |- | Penn State Worry Questionnaire (PSWQ''')'''<ref>{{Cite journal|last=Meyer|first=T.J.|last2=Miller|first2=M.L.|last3=Metzger|first3=R.L.|last4=Borkovec|first4=Thomas D.|title=Development and validation of the penn state worry questionnaire|url=https://doi.org/10.1016/0005-7967(90)90135-6|journal=Behaviour Research and Therapy|volume=28|issue=6|pages=487–495|doi=10.1016/0005-7967(90)90135-6}}</ref> | Questionnaire (Child) | 0.74 (N=164) | 1.8 (65+) | 0.5 (< 65) | Fresco, D.M., Mennin, D.S., Heimberg, R.G., Turk, C.L. (2003)<ref>{{Cite journal|last=Fresco|first=David M.|last2=Mennin|first2=Douglas S.|last3=Heimberg|first3=Richard G.|last4=Turk|first4=Cynthia L.|title=Using the Penn State Worry Questionnaire to identify individuals with generalized anxiety disorder: a receiver operating characteristic analysis|url=http://linkinghub.elsevier.com/retrieve/pii/S0005791603000569|journal=Journal of Behavior Therapy and Experimental Psychiatry|volume=34|issue=3-4|pages=283–291|doi=10.1016/j.jbtep.2003.09.001}}</ref> | Generalized Anxiety Disorder vs. social anxiety disorder, adults presenting to specialty anxiety clinic |- | rowspan="2" |[[wikipedia:Screen_for_child_anxiety_related_disorders#PDFs_and_automated_scoring_for_SCARED|Screen for Child Anxiety Related Disorders (SCARED)]]<ref name="BirmaherEtAl1997"/> | rowspan="2" | Questionnaire (Child, Parent) | .70 (N=243) | 5.0 (+32) | .04 | (Birmaher et al., 1997)<ref name="BirmaherEtAl1997"/> | rowspan="2" | High: Pure anxiety disorder versus non-anxiety psychiatric disorder, excluding children with disruptive disorder and depression |- | 0.911 (First screen) (N= 923) | 2.81 (4+; FS) | 0.15 (4-; FS) | Hale III, et al., 2014<ref name="HaleEtAl2014"/> |- | STAIC<ref>{{Cite journal|last=Hodges|first=Kay|title=Depression and anxiety in children: A comparison of self-report questionnaires to clinical interview.|url=http://dx.doi.org/10.1037/1040-3590.2.4.376|journal=Psychological Assessment|language=en|volume=2|issue=4|pages=376–381|doi=10.1037/1040-3590.2.4.376}}</ref> | Questionnaire (Child, Parent) | -- (N=70) | 2 (+69) | .79 | DLR: (Hodges, 1990) | STAIC does well in discriminating between children and adolescents with anxiety disorders and youth without a disorder and moderately well in measuring treatment response and discriminating youth with anxiety disorders from those with externalizing disorders<ref name="SeligmanEtAl2004"/> |- | RCADS<ref name="ChorpitaEtAl2000"/> | Questionnaire (Child) | -- (N=513) | 9.8 | 0.24 | DLR: (Chorpita, Moffitt &amp; Gray, 2005)<ref name="ChorpitaEtAl2005"/> | High: Several studies demonstrate support for the RCADS in non-referred samples of youth |- | SCAS<ref>{{Cite journal|last=Spence|first=Susan H.|title=A measure of anxiety symptoms among children|url=https://doi.org/10.1016/S0005-7967(98)00034-5|journal=Behaviour Research and Therapy|volume=36|issue=5|pages=545–566|doi=10.1016/s0005-7967(98)00034-5}}</ref> | Questionnaire (Child, Parent) | 0.83 (N=654) | -- | -- | (Nauta et al., under review) | |- | Generalized Anxiety Disorder Scale (GADS)<ref name="SpitzerEtAl2006"/> | Questionnaire | 0.88 (N = 438) | 6.3 (5+) | .41 (5-) | Wild et al., 2014 | Elderly persons (ages 58–82) from general population in German |- |Generalized Anxiety Disorder Screener (GAD-7)<ref>{{Cite journal|last=Plummer|first=Faye|last2=Manea|first2=Laura|last3=Trepel|first3=Dominic|last4=McMillan|first4=Dean|date=2016-03-01|title=Screening for anxiety disorders with the GAD-7 and GAD-2: a systematic review and diagnostic metaanalysis|url=https://www.sciencedirect.com/science/article/pii/S0163834315002406|journal=General Hospital Psychiatry|language=en|volume=39|pages=24–31|doi=10.1016/j.genhosppsych.2015.11.005|issn=0163-8343}}</ref> |Questionnaire |0.906<ref>{{Cite journal|last=Spitzer|first=Robert L.|last2=Kroenke|first2=Kurt|last3=Williams|first3=Janet B. W.|last4=Löwe|first4=Bernd|date=2006-05-22|title=A Brief Measure for Assessing Generalized Anxiety Disorder: The GAD-7|url=http://archinte.jamanetwork.com/article.aspx?doi=10.1001/archinte.166.10.1092|journal=Archives of Internal Medicine|language=en|volume=166|issue=10|pages=1092|doi=10.1001/archinte.166.10.1092|issn=0003-9926}}</ref> (N = 2149) |5.17 (8+) |.20 (8-) |Plummer et al., 2016 |Adults aged 16 years and older in any setting (meta-analysis) |- |CBCL Anxious/Depressed Scale T-score<ref>{{Cite journal|last=Eimecke|first=Sylvia D.|last2=Remschmidt|first2=Helmut|last3=Mattejat|first3=Fritz|date=2011-03|title=Utility of the Child Behavior Checklist in screening depressive disorders within clinical samples|url=https://linkinghub.elsevier.com/retrieve/pii/S0165032710005458|journal=Journal of Affective Disorders|language=en|volume=129|issue=1-3|pages=191–197|doi=10.1016/j.jad.2010.08.011}}</ref> |Questionnaire |.75 (N = 1445) |1.49 (9+) |.67(9-) |Eimecke et al., (2011) |Inpatient and outpatient children and adolescents |} '''Note:''' “LR+” refers to the change in likelihood ratio associated with a positive test score, and “LR-” is the likelihood ratio for a low score. Likelihood ratios of 1 indicate that the test result did not change impressions at all. LRs larger than 10 or smaller than .10 are frequently clinically decisive; 5 or .20 are helpful, and between 2.0 and .5 are small enough that they rarely result in clinically meaningful changes of formulation<ref>Sackett, D. L., Straus, S. E., Richardson, W. S., Rosenberg, W., & Haynes, R. B. (2000). Evidence-based medicine: How to practice and teach EBM. Edinburgh: Churchill Livingstone.</ref>. '''Search terms:''' [General Anxiety Disorder] AND [children OR adolescents OR pediatric] AND [sensitivity OR specificity] in GoogleScholar and PsycINFO === Interpreting GAD screening measure scores === * For information on interpreting screening measure scores, click [[Evidence based assessment/Prediction phase#Interpreting screening measure scores|here.]] ==[[Evidence based assessment/Prescription phase|'''Prescription phase''']]== ===Gold standard diagnostic interviews=== * For a list of broad reaching diagnostic interviews sortable by disorder with PDFs (if applicable), [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prescription_phase&wteswitched=1#Common_Diagnostic_Interviews click here.] === Recommended diagnostic instruments for GAD === {| class="wikitable sortable" ! colspan="10" |Diagnostic instruments for GAD |- !Measure !Format (Reporter) !Age Range !Administration/ Completion Time !Interrater Reliability !Test-Retest Reliability !Construct Validity !Content Validity !Highly Recommended !Free and Accessible Measures |- |Anxiety Disorders Interview Schedule for Children (ADIS-C)<ref name=":1">{{Cite journal|date=2001-08-01|title=Test-Retest Reliability of Anxiety Symptoms and Diagnoses With the Anxiety Disorders Interview Schedule for DSM-IV: Child and Parent Versions|url=https://www.sciencedirect.com/science/article/pii/S0890856709603427|journal=Journal of the American Academy of Child & Adolescent Psychiatry|language=en|volume=40|issue=8|pages=937–944|doi=10.1097/00004583-200108000-00016|issn=0890-8567}}</ref> |Child |6-16<ref name=":5">{{Cite journal|last=LYNEHAM|first=HEIDI J.|last2=ABBOTT|first2=MAREE J.|last3=RAPEE|first3=RONALD M.|date=2007-06|title=Interrater Reliability of the Anxiety Disorders Interview Schedule for DSM-IV: Child and Parent Version|url=https://doi.org/10.1097/chi.0b013e3180465a09|journal=Journal of the American Academy of Child &amp; Adolescent Psychiatry|volume=46|issue=6|pages=731–736|doi=10.1097/chi.0b013e3180465a09|issn=0890-8567}}</ref> |Varies |E<ref name=":1" /> |E<ref name=":1" /> |G to E<ref name=":1" /> |N/A | |[https://www.google.com/books/edition/Anxiety_Disorders_Interview_Schedule_ADI/TOtZAAAACAAJ?hl=en Purchase] |- |Anxiety Disorders Interview Schedule for Children (ADIS-P)<ref name=":1" /> |Parent |6-16<ref name=":5" /> |Varies |E<ref name=":1" /> |E<ref name=":1" /> |E<ref name=":1" /> |N/A | |[https://www.google.com/books/edition/Anxiety_Disorders_Interview_Schedule_ADI/TOtZAAAACAAJ?hl=en Purchase] |- |Anxiety Disorders Interview Schedule for DSM-IV (ADIS-IV) <nowiki>*</nowiki>not free |Adult |16+ |Varies |A<ref name=":0" /> |NA<ref name=":0" /> |A<ref name=":0" /> |A<ref name=":0" /> |<ref name=":0" />[[File:Light green check.svg|center|frameless|50x50px]] | |- |Structured Clinical Interview for DSM-IV-TR for Axis I Disorders (SCID-I/P) <nowiki>*</nowiki>not free | | |Varies |A<ref name=":0" /> |NA<ref name=":0" /> |A<ref name=":0" /> |A<ref name=":0" /> | |[http://www.scid4.org/ Website and purchase] |- |Structured Clinical Interview for DSM-IV-TR for Axis II Disorders (SCID-II) <nowiki>*</nowiki>not free | | |Varies |E<ref name=":0" /> |NA<ref name=":0" /> |U<ref name=":0" /> |U<ref name=":0" /> | |[http://www.scid4.org/ Website and purchase] |- |Structured Clinical Interview for DSM-IV (SCID-IV) <nowiki>*</nowiki>not free | | |Varies |A<ref name=":0" /> |A<ref name=":0" /> |E<ref name=":0" /> |E<ref name=":0" /> | |[http://www.scid4.org/ Website and purchase] |- |Anxiety and Related Disorders Interview Schedule for DSM-5 (ADIS-5) |Structured Interview (Adult) |16+ |Varies | | | | | |[https://global.oup.com/academic/product/anxiety-and-related-disorders-interview-schedule-for-dsm-5-adis-5---adult-version-9780199325160?cc=us&lang=en& Purchase] |- |Structured Clinical Interview for DSM-5 Clinician Version (SCID-5- CV)<ref name=":6">{{Cite journal|last=Shabani|first=Amir|last2=Masoumian|first2=Samira|last3=Zamirinejad|first3=Somayeh|last4=Hejri|first4=Maryam|last5=Pirmorad|first5=Tahereh|last6=Yaghmaeezadeh|first6=Hooman|date=2021-05|title=Psychometric properties of Structured Clinical Interview for DSM‐5 Disorders‐Clinician Version (SCID‐5‐CV)|url=https://onlinelibrary.wiley.com/doi/10.1002/brb3.1894|journal=Brain and Behavior|language=en|volume=11|issue=5|doi=10.1002/brb3.1894|issn=2162-3279|pmc=PMC8119811|pmid=33729681}}</ref> |Structured Interview (Adult) |16+ |Varies |E<ref name=":6" /> |A<ref name=":6" /> | | | |[https://www.columbiapsychiatry.org/research/research-labs/diagnostic-and-assessment-lab/structured-clinical-interview-dsm-disorders-11 Purchase] |} '''Note:''' L = Less than adequate; A = Adequate; G = Good; E = Excellent; U = Unavailable; NA = Not applicable === Severity interviews for GAD=== {| class="wikitable sortable" border="1" |- ! Measure ! Format (Reporter) ! Age Range ! Administration/ Completion Time ! Interrater Reliability ! Test-Retest Reliability ! Construct Validity ! Content Validity ! Highly Recommended !Free and Accessible Measures |- | Children's Depression Rating Scale - Revised (CDRS-R) | Structured Interview<ref name=":4">{{Cite journal|last=Mayes|first=Taryn L.|last2=Bernstein|first2=Ira H.|last3=Haley|first3=Charlotte L.|last4=Kennard|first4=Betsy D.|last5=Emslie|first5=Graham J.|date=2010-12|title=Psychometric Properties of the Children's Depression Rating Scale–Revised in Adolescents|url=https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3003451/|journal=Journal of Child and Adolescent Psychopharmacology|volume=20|issue=6|pages=513–516|doi=10.1089/cap.2010.0063|issn=1044-5463|pmc=PMC3003451|pmid=21186970}}</ref> | 6-12 | 15-20 minutes | G | A | G | G | X | * Link to purchase [https://www.wpspublish.com/cdrs-r-childrens-depression-rating-scale-revised] *[http://www.opapc.com/uploads/documents/CDRS-R.pdf PDF] (excerpt) |} '''Note:''' '''L''' = Less than adequate; '''A''' = Adequate; '''G''' = Good; '''E''' = Excellent; '''U''' = Unavailable; '''NA''' = Not applicable ==[[Evidence based assessment/Process phase|'''Process phase''']]== The following section contains a list of process and outcome measures for generalized anxiety disorder. The section includes benchmarks based on published norms for several outcome and severity measures, as well as information about commonly used process measures. Process and outcome measures are used as part of the [[Evidence based assessment/Process phase|process phase]] of assessment. For more information of differences between process and outcome measures, see the page on the [[Evidence based assessment/Process phase|process phase of assessment]]. === Outcome and severity measures === * This table includes clinically significant benchmarks for generalized anxiety disorder specific outcome measures * Information on how to interpret this table can be [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase found here]. * Additionally, these [[Evidence based assessment/Vignettes|vignettes]] might be helpful resources for understanding appropriate adaptation of outcome measures in practice. *''<u>For clinically significant change benchmarks for the CBCL, YSR, and TRF total, externalizing, internalizing, and attention benchmarks,</u>'' [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase&wteswitched=1#Clinically_significant_change_benchmarks_for_widely-used_outcome_measures see here.] {| class="wikitable sortable" border="1" | colspan="7" |'''Clinically significant change benchmarks with common instruments for GAD''' |- | rowspan=1" style="text-align:center;font-size:130%;" | <b> Measure</b> | colspan="3" style="text-align:center;font-size:130%" width="300" | <b> Cut-off scores</b> | colspan="3" style="text-align:center;font-size:120%" | <b> Critical Change <br> (unstandardized scores)</b> |- | colspan="7" span style="font-size:110%; text-align:center;" | <b> Benchmarks Based on Published Norms</b> |- | colspan="1" | | style="text-align:center;font-size:110%" | <b> A</b> | style="text-align:center;font-size:110%" | <b> B</b> | style="text-align:center;font-size:110%" | <b> C</b> | style="text-align:center;font-size:110%" | <b> 95%</b> | style="text-align:center;font-size:110%" | <b> 90%</b> | style="text-align:center;font-size:110%" | <b> SE_difference</b> |- | rowspan="1" style="text-align:center;" | <b> GAD-7</b> | style=“text-align:center;”| -1 | style=“text-align:center;”| 1.3 | style=“text-align:center;”| 0.5 | style=“text-align:center;”| 0.6 | style=“text-align:center;”| 0.5 | style=“text-align:center;”| 0.3 |- | rowspan="1" style="text-align:center;" | <b> PSWQ</b> | style=“text-align:center;”| 51 | style=“text-align:center;”| 73 | style=“text-align:center;”| 59 | style=“text-align:center;”| 9 | style=“text-align:center;”| 8 | style=“text-align:center;”| 4.8 |- | rowspan="1" style="text-align:center;" | <b> SCARED </b> | style=“text-align:center;”| 9.9 | style=“text-align:center;”| 18.1 | style=“text-align:center;”| 15.3 | style=“text-align:center;”| 8.9 | style=“text-align:center;”| 7.5 | style=“text-align:center;”| 4.5 |- | rowspan="1" style="text-align:center;" | <b> STAIC</b> | style=“text-align:center;”| 0.9 | style=“text-align:center;”| 30.1 | style=“text-align:center;”| 18.2 | style=“text-align:center;”| 18.9 | style=“text-align:center;”| 15.9 | style=“text-align:center;”| 9.6 |- | rowspan="1" style="text-align:center;" | <b> RCADS</b> | style=“text-align:center;”| -1.1 | style=“text-align:center;”| 12.7 | style=“text-align:center;”| 6.6 | style=“text-align:center;”| 7.3 | style=“text-align:center;”| 6.1 | style=“text-align:center;”| 3.7 |- | rowspan="1" style="text-align:center;" | <b> SCAS</b> | style=“text-align:center;”| -0.7 | style=“text-align:center;”| 15.1 | style=“text-align:center;”| 5.4 | style=“text-align:center;”| 6.2 | style=“text-align:center;”| 5.2 | style=“text-align:center;”| 3.2 |} '''Note:''' “A” = Away from the clinical range, “B” = Back into the nonclinical range, “C” = Closer to the nonclinical than clinical mean. '''Search terms:''' [General Anxiety Disorder] AND [children OR adolescents OR pediatric] AND [clinical significance OR outcomes] in GoogleScholar and PsycINFO === Treatment === {{collapse top| Click here for treatment information}} Individuals suffering from GAD tend to be high users of outpatient medical care. When treating GAD, physicians should first determine whether pharmacotherapy, psychotherapy, or a combination of the two treatments would be most beneficial to the patient. Literature suggests that treatment of GAD frequently consists of a combination of psychotherapy and pharmacotherapy. Although these therapies have the potential to be effective individually, previous work demonstrates that when combined the degree of clinically significant change increases significantly. Recent studies (e.g., Gorman, 2003<ref name="Gorman2003" />; Walkup et al., 2008<ref name="WalkupEtAl2008" />) have provided evidence to support this claim with the most efficacious medication and behavioral interventions listed below. # '''Medication Interventions''' ## ''Sertraline (Zoloft)'' has been shown to reduce experiences and effects of GAD above and beyond that of placebo conditions. ## ''Pregabalin.'' The mean baseline-to-endpoint decreases in total Hamilton anxiety scale score in the patients given 150 mg/day of pregabalin (–9.2) was significantly greater than the decrease in those given placebo (–6.8)<ref name="PandeEtAl2003" />. ## ''Paroxetine.'' Remission was achieved by 30% of patients in the 20-mg paroxetine groups compared with 20% given placebo. For all three domains of the Sheehan Disability Scale, significantly greater improvement was seen with paroxetine than placebo<ref name="RickelsEtAl2003" />. # '''Behavioral interventions''' ## ''Cognitive behavioral therapy.'' Fourteen 60-minute sessions, which include CBT in anxiety-management skills, followed by behavioral exposure to anxiety-provoking situations have been shown to be effective in treating GAD. A review of studies by Fisher and Durham (1999)<ref name="FisherEtAl1999" /> revealed significant recovery rates at a 6 month follow up after CBT. ## ''Exposure therapy and modeling therapy.'' One meta-analysis found that virtual reality exposure therapy for anxiety disorders had a large effect size (Cohen's d=1.11) compared to controls.<ref>{{Cite journal|last=Powers|first=Mark B.|last2=Emmelkamp|first2=Paul M.G.|title=Virtual reality exposure therapy for anxiety disorders: A meta-analysis|url=https://doi.org/10.1016/j.janxdis.2007.04.006|journal=Journal of Anxiety Disorders|volume=22|issue=3|pages=561–569|doi=10.1016/j.janxdis.2007.04.006}}</ref> ## ''Mindfulness meditation.'' New treatment options such as mindfulness meditation-based stress reduction interventions have also shown to reduce symptoms over the long-term.<ref>{{Cite journal|last=Miller|first=J. J.|last2=Fletcher|first2=K.|last3=Kabat-Zinn|first3=J.|date=May 1995|title=Three-year follow-up and clinical implications of a mindfulness meditation-based stress reduction intervention in the treatment of anxiety disorders|url=https://www.ncbi.nlm.nih.gov/pubmed/7649463|journal=General Hospital Psychiatry|volume=17|issue=3|pages=192–200|issn=0163-8343|pmid=7649463}}</ref> # '''Combination treatment''' ## Previous research suggests that combination therapy that includes components of psychotherapy and pharmacotherapy are the most efficacious in treating GAD. In a study comparing the efficacies GAD treatments, Walkup and colleagues demonstrated a 21-25% improvement of combination therapy over cognitive behavioral therapy or sertraline alone during short-term treatment. These findings suggest that among effective treatments, combination therapy has the potential to provide the best chance for a positive outcome. See Gorman, 2003<ref name="Gorman2003" />; Walkup et al., 2008<ref name="WalkupEtAl2008" />. {{collapse bottom}} * Please refer to the page on [[wikipedia:Generalized_anxiety_disorder|generalized anxiety disorder]] for more information on available treatment or go to [http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/fear-worry-and-anxiety/ Effective Child Therapy] for a curated resource on effective treatments for GAD. *For information on conducting Exposure Therapy for anxiety disordered youth, see [https://www.bravepracticeforkids.com/ www.BravePracticeForKids.com] =='''External Resources'''== # [http://apps.who.int/classifications/icd10/browse/2010/en#/F41.1 ICD-10 diagnostic criteria] # [https://en.wikiversity.org/w/index.php?title=Helping_Give_Away_Psychological_Science/Resources/Annotated_List_of_Where_and_How_to_Find_a_Therapist&wteswitched=1#Other_low-cost_options Find-a-Therapist] #*This is a curated list of find-a-therapist websites where you can find a provider # [https://www.nimh.nih.gov/health/topics/anxiety-disorders/index.shtml NIMH] entry about anxiety disorders # OMIM (Online Mendelian Inheritance in Man) #*[https://www.omim.org/entry/607834 607834] # [https://emedicine.medscape.com/article/286227-overview#a2 eMedicine entry about anxiety disorders] #[https://sccap53.org Society of Clinical Child and Adolescent Psychology] #[http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/fear-worry-and-anxiety/ Effective Child Therapy information on Fear, Worry, & Anxiety] #*Effective Child Therapy is website sponsored by Division 53 of the American Psychological Association (APA), or The [https://sccap53.org Society of Clinical Child and Adolescent Psychology] (SCCAP), in collaboration with the Association for Behavioral and Cognitive Therapies (ABCT). Use for information on symptoms and available treatments. #[http://pediatricbipolar.pitt.edu/resources/instruments Links to SCARED Child, Parent, and Adult + Translations] =='''References'''== {{collapse top|Click here for references}} {{Reflist|3|refs= <ref name="BirmaherEtAl1997">{{cite journal|last1=Birmaher|first1=B|last2=Khetarpal|first2=S|last3=Brent|first3=D|last4=Cully|first4=M|last5=Balach|first5=L|last6=Kaufman|first6=J|last7=Neer|first7=SM|title=The Screen for Child Anxiety Related Emotional Disorders (SCARED): scale construction and psychometric characteristics.|journal=Journal of the American Academy of Child and Adolescent Psychiatry|date=April 1997|volume=36|issue=4|pages=545-53|pmid=9100430}}</ref> <ref name="BrownJacobsenEtAl2011">{{cite journal|last1=Brown-Jacobsen|first1=AM|last2=Wallace|first2=DP|last3=Whiteside|first3=SP|title=Multimethod, multi-informant agreement, and positive predictive value in the identification of child anxiety disorders using the SCAS and ADIS-C.|journal=Assessment|date=September 2011|volume=18|issue=3|pages=382-92|pmid=20644080}}</ref> <ref name="CostelloEtAl1996">{{cite journal|last1=Costello|first1=EJ|last2=Angold|first2=A|last3=Burns|first3=BJ|last4=Stangl|first4=DK|last5=Tweed|first5=DL|last6=Erkanli|first6=A|last7=Worthman|first7=CM|title=The Great Smoky Mountains Study of Youth. 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in generalized anxiety disorder following psychological therapy: an analysis of clinically significant change in the STAI-T across outcome studies since 1990.|journal=Psychological medicine|date=November 1999|volume=29|issue=6|pages=1425-34|pmid=10616949}}</ref> <ref name="Gorman2003">{{cite journal|last1=Gorman|first1=JM|title=Treating generalized anxiety disorder.|journal=The Journal of clinical psychiatry|date=2003|volume=64 Suppl 2|pages=24-9|pmid=12625796}}</ref> <ref name="HaleEtAl2014">{{cite journal|last1=Hale III|first1=WW|last2=Raaijmakers|first2=QA|last3=van Hoof|first3=A|last4=Meeus|first4=WH|title=Improving Screening Cut-Off Scores for DSM-5 Adolescent Anxiety Disorder Symptom Dimensions with the Screen for Child Anxiety Related Emotional Disorders.|journal=Psychiatry journal|date=2014|volume=2014|pages=517527|pmid=24829901}}</ref> <ref name="KayeEtAl2004">{{cite journal|last1=Kaye|first1=WH|last2=Bulik|first2=CM|last3=Thornton|first3=L|last4=Barbarich|first4=N|last5=Masters|first5=K|title=Comorbidity of anxiety disorders with anorexia and bulimia nervosa.|journal=The American journal of psychiatry|date=December 2004|volume=161|issue=12|pages=2215-21|pmid=15569892}}</ref> <ref name="KesslerEtAl2012">{{cite journal|last1=Kessler|first1=RC|last2=Petukhova|first2=M|last3=Sampson|first3=NA|last4=Zaslavsky|first4=AM|last5=Wittchen H|first5=-U|title=Twelve-month and lifetime prevalence and lifetime morbid risk of anxiety and mood disorders in the United States.|journal=International journal of methods in psychiatric research|date=September 2012|volume=21|issue=3|pages=169-84|pmid=22865617}}</ref> <ref name="LynehamEtAl2007">{{cite journal|last1=Lyneham|first1=HJ|last2=Abbott|first2=MJ|last3=Rapee|first3=RM|title=Interrater reliability of the Anxiety Disorders Interview Schedule for DSM-IV: child and parent version.|journal=Journal of the American Academy of Child and Adolescent Psychiatry|date=June 2007|volume=46|issue=6|pages=731-6|pmid=17513985}}</ref> <ref name="MarchEtAl1997">{{cite journal|last1=March|first1=JS|last2=Parker|first2=JD|last3=Sullivan|first3=K|last4=Stallings|first4=P|last5=Conners|first5=CK|title=The Multidimensional Anxiety Scale for Children (MASC): factor structure, reliability, and validity.|journal=Journal of the American Academy of Child and Adolescent Psychiatry|date=April 1997|volume=36|issue=4|pages=554-65|pmid=9100431}}</ref> <ref name="McLeanEtAl2011">{{cite journal|last1=McLean|first1=CP|last2=Asnaani|first2=A|last3=Litz|first3=BT|last4=Hofmann|first4=SG|title=Gender differences in anxiety disorders: prevalence, course of illness, comorbidity and burden of illness.|journal=Journal of psychiatric research|date=August 2011|volume=45|issue=8|pages=1027-35|pmid=21439576}}</ref> <ref name="MerikangasEtAl2010">{{cite journal|last1=Merikangas|first1=KR|last2=He|first2=JP|last3=Burstein|first3=M|last4=Swanson|first4=SA|last5=Avenevoli|first5=S|last6=Cui|first6=L|last7=Benjet|first7=C|last8=Georgiades|first8=K|last9=Swendsen|first9=J|title=Lifetime prevalence of mental disorders in U.S. adolescents: results from the National Comorbidity Survey Replication--Adolescent Supplement (NCS-A).|journal=Journal of the American Academy of Child and Adolescent Psychiatry|date=October 2010|volume=49|issue=10|pages=980-9|pmid=20855043}}</ref> <ref name="PandeEtAl2003">{{cite journal|last1=Pande|first1=AC|last2=Crockatt|first2=JG|last3=Feltner|first3=DE|last4=Janney|first4=CA|last5=Smith|first5=WT|last6=Weisler|first6=R|last7=Londborg|first7=PD|last8=Bielski|first8=RJ|last9=Zimbroff|first9=DL|last10=Davidson|first10=JR|last11=Liu-Dumaw|first11=M|title=Pregabalin in generalized anxiety disorder: a placebo-controlled trial.|journal=The American journal of psychiatry|date=March 2003|volume=160|issue=3|pages=533-40|pmid=12611835}}</ref> <ref name="RickelsEtAl2003">{{cite journal|last1=Rickels|first1=K|last2=Zaninelli|first2=R|last3=McCafferty|first3=J|last4=Bellew|first4=K|last5=Iyengar|first5=M|last6=Sheehan|first6=D|title=Paroxetine treatment of generalized anxiety disorder: a double-blind, placebo-controlled study.|journal=The American journal of psychiatry|date=April 2003|volume=160|issue=4|pages=749-56|pmid=12668365}}</ref> <ref name="RobertsEtAl2007">{{cite journal|last1=Roberts|first1=RE|last2=Roberts|first2=CR|last3=Xing|first3=Y|title=Rates of DSM-IV psychiatric disorders among adolescents in a large metropolitan area.|journal=Journal of psychiatric research|date=December 2007|volume=41|issue=11|pages=959-67|pmid=17107689}}</ref> <ref name="SeligmanEtAl2004">{{cite journal|last1=Seligman|first1=LD|last2=Ollendick|first2=TH|last3=Langley|first3=AK|last4=Baldacci|first4=HB|title=The utility of measures of child and adolescent anxiety: a meta-analytic review of the Revised Children's Manifest Anxiety Scale, the State-Trait Anxiety Inventory for Children, and the Child Behavior Checklist.|journal=Journal of clinical child and adolescent psychology : the official journal for the Society of Clinical Child and Adolescent Psychology, American Psychological Association, Division 53|date=September 2004|volume=33|issue=3|pages=557-65|pmid=15271613}}</ref> <ref name="WalkupEtAl2008">{{cite journal|last1=Walkup|first1=JT|last2=Albano|first2=AM|last3=Piacentini|first3=J|last4=Birmaher|first4=B|last5=Compton|first5=SN|last6=Sherrill|first6=JT|last7=Ginsburg|first7=GS|last8=Rynn|first8=MA|last9=McCracken|first9=J|last10=Waslick|first10=B|last11=Iyengar|first11=S|last12=March|first12=JS|last13=Kendall|first13=PC|title=Cognitive behavioral therapy, sertraline, or a combination in childhood anxiety.|journal=The New England journal of medicine|date=25 December 2008|volume=359|issue=26|pages=2753-66|pmid=18974308}}</ref> <ref name="WhitakerEtAl1990">{{cite journal|last1=Whitaker|first1=A|last2=Johnson|first2=J|last3=Shaffer|first3=D|last4=Rapoport|first4=JL|last5=Kalikow|first5=K|last6=Walsh|first6=BT|last7=Davies|first7=M|last8=Braiman|first8=S|last9=Dolinsky|first9=A|title=Uncommon troubles in young people: prevalence estimates of selected psychiatric disorders in a nonreferred adolescent population.|journal=Archives of general psychiatry|date=May 1990|volume=47|issue=5|pages=487-96|pmid=2331210}}</ref> <ref name="SpitzerEtAl2006">{{cite journal|last1=Spitzer|first1=RL|last2=Kroenke|first2=K|last3=Williams|first3=JB|last4=Löwe|first4=B|title=A brief measure for assessing generalized anxiety disorder: the GAD-7.|journal=Archives of internal medicine|date=22 May 2006|volume=166|issue=10|pages=1092-7|pmid=16717171}}</ref> <ref name="vanGastelEtAl2008">{{cite journal|last1=van Gastel|first1=W|last2=Ferdinand|first2=RF|title=Screening capacity of the Multidimensional Anxiety Scale for Children (MASC) for DSM-IV anxiety disorders.|journal=Depression and anxiety|date=2008|volume=25|issue=12|pages=1046-52|pmid=18833579}}</ref> <ref name="WoodEtAl2002">{{cite journal|last1=Wood|first1=JJ|last2=Piacentini|first2=JC|last3=Bergman|first3=RL|last4=McCracken|first4=J|last5=Barrios|first5=V|title=Concurrent validity of the anxiety disorders section of the Anxiety Disorders Interview Schedule for DSM-IV: child and parent versions.|journal=Journal of clinical child and adolescent psychology : the official journal for the Society of Clinical Child and Adolescent Psychology, American Psychological Association, Division 53|date=September 2002|volume=31|issue=3|pages=335-42|pmid=12149971}}</ref> <ref name="ZimmermanEtAl2005">{{cite journal|last1=Zimmerman|first1=M|last2=Rothschild|first2=L|last3=Chelminski|first3=I|title=The prevalence of DSM-IV personality disorders in psychiatric outpatients.|journal=The American journal of psychiatry|date=October 2005|volume=162|issue=10|pages=1911-8|pmid=16199838}}</ref> }} {{collapse bottom|Click here for references}} [[Category:Psychological disorder portfolios|{{SUBPAGENAME}}]] 2g2cdu8oai0xilm6b8fxp48kvnpj5qn 0 285380 2410227 2409992 2022-07-29T13:26:09Z Young1lim 21186 /* Handling Series of Data */ wikitext text/x-wiki === Introduction === * Overview ([[Media:C01.Intro1.Overview.1.A.20170925.pdf |A.pdf]], [[Media:C01.Intro1.Overview.1.B.20170901.pdf |B.pdf]], [[Media:C01.Intro1.Overview.1.C.20170904.pdf |C.pdf]]) * Number System ([[Media:C01.Intro2.Number.1.A.20171023.pdf |A.pdf]], [[Media:C01.Intro2.Number.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro2.Number.1.C.20170914.pdf |C.pdf]]) * Memory System ([[Media:C01.Intro2.Memory.1.A.20170907.pdf |A.pdf]], [[Media:C01.Intro3.Memory.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro3.Memory.1.C.20170914.pdf |C.pdf]]) === Handling Repetition === * Control ([[Media:C02.Repeat1.Control.1.A.20170925.pdf |A.pdf]], [[Media:C02.Repeat1.Control.1.B.20170918.pdf |B.pdf]], [[Media:C02.Repeat1.Control.1.C.20170926.pdf |C.pdf]]) * Loop ([[Media:C02.Repeat2.Loop.1.A.20170925.pdf |A.pdf]], [[Media:C02.Repeat2.Loop.1.B.20170918.pdf |B.pdf]]) === Handling a Big Work === * Function Overview ([[Media:C03.Func1.Overview.1.A.20171030.pdf |A.pdf]], [[Media:C03.Func1.Oerview.1.B.20161022.pdf |B.pdf]]) * Functions & Variables ([[Media:C03.Func2.Variable.1.A.20161222.pdf |A.pdf]], [[Media:C03.Func2.Variable.1.B.20161222.pdf |B.pdf]]) * Functions & Pointers ([[Media:C03.Func3.Pointer.1.A.20161122.pdf |A.pdf]], [[Media:C03.Func3.Pointer.1.B.20161122.pdf |B.pdf]]) * Functions & Recursions ([[Media:C03.Func4.Recursion.1.A.20161214.pdf |A.pdf]], [[Media:C03.Func4.Recursion.1.B.20161214.pdf |B.pdf]]) === Handling Series of Data === ==== Background ==== * Background ([[Media:C04.Series0.Background.1.A.20180727.pdf |A.pdf]]) ==== Basics ==== * Arrays ([[Media:C04.Series1.Array.1.A.20220728.pdf |A.pdf]], [[Media:C04.Series1.Array.1.B.20161115.pdf |B.pdf]]) * Pointers ([[Media:C04.Series2.Pointer.1.A.20180726.pdf |A.pdf]], [[Media:C04.Series2.Pointer.1.B.20161115.pdf |B.pdf]]) * Array Pointers ([[Media:C04.Series3.ArrayPointer.1.A.20220728.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]]) * Multi-dimensional Arrays ([[Media:C04.Series4.MultiDim.1.A.20220418.pdf |A.pdf]], [[Media:C04.Series4.MultiDim.1.B.11.pdf |B.pdf]]) * Array Access Methods ([[Media:C04.Series4.ArrayAccess.1.A.20190511.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]]) * Structures ([[Media:C04.Series3.Structure.1.A.20171204.pdf |A.pdf]], [[Media:C04.Series2.Structure.1.B.20161130.pdf |B.pdf]]) ==== Applications ==== * Applications of Arrays ([[Media:C04.Series1App.Array.1.A.20220728.pdf |A.pdf]]) * Applications of Pointers ([[Media:C04.Series7.AppPoint.1.A.20200424.pdf |A.pdf]]) * Applications of Array Pointers ([[Media:C04.Series3App.ArrayPointer.1.A.2022024.pdf |A.pdf]]) * Applications of Multi-dimensional Arrays ([[Media:C04.Series4App.MultiDim.1.A.20210719.pdf |A.pdf]]) * Applications of Array Access Methods ([[Media:C04.Series9.AppArrAcess.1.A.20190511.pdf |A.pdf]]) * Applications of Structures ([[Media:C04.Series6.AppStruct.1.A.20190423.pdf |A.pdf]]) ==== Examples ==== * Spreadsheet Example Programs :: Example 1 ([[Media:C04.Series7.Example.1.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.1.C.20171213.pdf |C.pdf]]) :: Example 2 ([[Media:C04.Series7.Example.2.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.2.C.20171213.pdf |C.pdf]]) :: Example 3 ([[Media:C04.Series7.Example.3.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.3.C.20171213.pdf |C.pdf]]) :: Bubble Sort ([[Media:C04.Series7.BubbleSort.1.A.20171211.pdf |A.pdf]]) === Handling Various Kinds of Data === * Types ([[Media:C05.Data1.Type.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data1.Type.1.B.20161212.pdf |B.pdf]]) * Typecasts ([[Media:C05.Data2.TypeCast.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data2.TypeCast.1.B.20161216.pdf |A.pdf]]) * Operators ([[Media:C05.Data3.Operators.1.A.20161219.pdf |A.pdf]], [[Media:C05.Data3.Operators.1.B.20161216.pdf |B.pdf]]) * Files ([[Media:C05.Data4.File.1.A.20161124.pdf |A.pdf]], [[Media:C05.Data4.File.1.B.20161212.pdf |B.pdf]]) === Handling Low Level Operations === * Bitwise Operations ([[Media:BitOp.1.B.20161214.pdf |A.pdf]], [[Media:BitOp.1.B.20161203.pdf |B.pdf]]) * Bit Field ([[Media:BitField.1.A.20161214.pdf |A.pdf]], [[Media:BitField.1.B.20161202.pdf |B.pdf]]) * Union ([[Media:Union.1.A.20161221.pdf |A.pdf]], [[Media:Union.1.B.20161111.pdf |B.pdf]]) * Accessing IO Registers ([[Media:IO.1.A.20141215.pdf |A.pdf]], [[Media:IO.1.B.20161217.pdf |B.pdf]]) === Declarations === * Type Specifiers and Qualifiers ([[Media:C07.Spec1.Type.1.A.20171004.pdf |pdf]]) * Storage Class Specifiers ([[Media:C07.Spec2.Storage.1.A.20171009.pdf |pdf]]) * Scope === Class Notes === * TOC ([[Media:TOC.20171007.pdf |TOC.pdf]]) * Day01 ([[Media:Day01.A.20171007.pdf |A.pdf]], [[Media:Day01.B.20171209.pdf |B.pdf]], [[Media:Day01.C.20171211.pdf |C.pdf]]) ...... Introduction (1) Standard Library * Day02 ([[Media:Day02.A.20171007.pdf |A.pdf]], [[Media:Day02.B.20171209.pdf |B.pdf]], [[Media:Day02.C.20171209.pdf |C.pdf]]) ...... Introduction (2) Basic Elements * Day03 ([[Media:Day03.A.20171007.pdf |A.pdf]], [[Media:Day03.B.20170908.pdf |B.pdf]], [[Media:Day03.C.20171209.pdf |C.pdf]]) ...... Introduction (3) Numbers * Day04 ([[Media:Day04.A.20171007.pdf |A.pdf]], [[Media:Day04.B.20170915.pdf |B.pdf]], [[Media:Day04.C.20171209.pdf |C.pdf]]) ...... Structured Programming (1) Flowcharts * Day05 ([[Media:Day05.A.20171007.pdf |A.pdf]], [[Media:Day05.B.20170915.pdf |B.pdf]], [[Media:Day05.C.20171209.pdf |C.pdf]]) ...... Structured Programming (2) Conditions and Loops * Day06 ([[Media:Day06.A.20171007.pdf |A.pdf]], [[Media:Day06.B.20170923.pdf |B.pdf]], [[Media:Day06.C.20171209.pdf |C.pdf]]) ...... Program Control * Day07 ([[Media:Day07.A.20171007.pdf |A.pdf]], [[Media:Day07.B.20170926.pdf |B.pdf]], [[Media:Day07.C.20171209.pdf |C.pdf]]) ...... Function (1) Definitions * Day08 ([[Media:Day08.A.20171028.pdf |A.pdf]], [[Media:Day08.B.20171016.pdf |B.pdf]], [[Media:Day08.C.20171209.pdf |C.pdf]]) ...... Function (2) Storage Class and Scope * Day09 ([[Media:Day09.A.20171007.pdf |A.pdf]], [[Media:Day09.B.20171017.pdf |B.pdf]], [[Media:Day09.C.20171209.pdf |C.pdf]]) ...... Function (3) Recursion * Day10 ([[Media:Day10.A.20171209.pdf |A.pdf]], [[Media:Day10.B.20171017.pdf |B.pdf]], [[Media:Day10.C.20171209.pdf |C.pdf]]) ...... Arrays (1) Definitions * Day11 ([[Media:Day11.A.20171024.pdf |A.pdf]], [[Media:Day11.B.20171017.pdf |B.pdf]], [[Media:Day11.C.20171212.pdf |C.pdf]]) ...... Arrays (2) Applications * Day12 ([[Media:Day12.A.20171024.pdf |A.pdf]], [[Media:Day12.B.20171020.pdf |B.pdf]], [[Media:Day12.C.20171209.pdf |C.pdf]]) ...... Pointers (1) Definitions * Day13 ([[Media:Day13.A.20171025.pdf |A.pdf]], [[Media:Day13.B.20171024.pdf |B.pdf]], [[Media:Day13.C.20171209.pdf |C.pdf]]) ...... Pointers (2) Applications * Day14 ([[Media:Day14.A.20171226.pdf |A.pdf]], [[Media:Day14.B.20171101.pdf |B.pdf]], [[Media:Day14.C.20171209.pdf |C.pdf]]) ...... C String (1) * Day15 ([[Media:Day15.A.20171209.pdf |A.pdf]], [[Media:Day15.B.20171124.pdf |B.pdf]], [[Media:Day15.C.20171209.pdf |C.pdf]]) ...... C String (2) * Day16 ([[Media:Day16.A.20171208.pdf |A.pdf]], [[Media:Day16.B.20171114.pdf |B.pdf]], [[Media:Day16.C.20171209.pdf |C.pdf]]) ...... C Formatted IO * Day17 ([[Media:Day17.A.20171031.pdf |A.pdf]], [[Media:Day17.B.20171111.pdf |B.pdf]], [[Media:Day17.C.20171209.pdf |C.pdf]]) ...... Structure (1) Definitions * Day18 ([[Media:Day18.A.20171206.pdf |A.pdf]], [[Media:Day18.B.20171128.pdf |B.pdf]], [[Media:Day18.C.20171212.pdf |C.pdf]]) ...... Structure (2) Applications * Day19 ([[Media:Day19.A.20171205.pdf |A.pdf]], [[Media:Day19.B.20171121.pdf |B.pdf]], [[Media:Day19.C.20171209.pdf |C.pdf]]) ...... Union, Bitwise Operators, Enum * Day20 ([[Media:Day20.A.20171205.pdf |A.pdf]], [[Media:Day20.B.20171201.pdf |B.pdf]], [[Media:Day20.C.20171212.pdf |C.pdf]]) ...... Linked List * Day21 ([[Media:Day21.A.20171206.pdf |A.pdf]], [[Media:Day21.B.20171208.pdf |B.pdf]], [[Media:Day21.C.20171212.pdf |C.pdf]]) ...... File Processing * Day22 ([[Media:Day22.A.20171212.pdf |A.pdf]], [[Media:Day22.B.20171213.pdf |B.pdf]], [[Media:Day22.C.20171212.pdf |C.pdf]]) ...... Preprocessing <!----------------------------------------------------------------------> </br> See also https://cprogramex.wordpress.com/ == '''Old Materials '''== until 201201 * Intro.Overview.1.A ([[Media:C.Intro.Overview.1.A.20120107.pdf |pdf]]) * Intro.Memory.1.A ([[Media:C.Intro.Memory.1.A.20120107.pdf |pdf]]) * Intro.Number.1.A ([[Media:C.Intro.Number.1.A.20120107.pdf |pdf]]) * Repeat.Control.1.A ([[Media:C.Repeat.Control.1.A.20120109.pdf |pdf]]) * Repeat.Loop.1.A ([[Media:C.Repeat.Loop.1.A.20120113.pdf |pdf]]) * Work.Function.1.A ([[Media:C.Work.Function.1.A.20120117.pdf |pdf]]) * Work.Scope.1.A ([[Media:C.Work.Scope.1.A.20120117.pdf |pdf]]) * Series.Array.1.A ([[Media:Series.Array.1.A.20110718.pdf |pdf]]) * Series.Pointer.1.A ([[Media:Series.Pointer.1.A.20110719.pdf |pdf]]) * Series.Structure.1.A ([[Media:Series.Structure.1.A.20110805.pdf |pdf]]) * Data.Type.1.A ([[Media:C05.Data2.TypeCast.1.A.20130813.pdf |pdf]]) * Data.TypeCast.1.A ([[Media:Data.TypeCast.1.A.pdf |pdf]]) * Data.Operators.1.A ([[Media:Data.Operators.1.A.20110712.pdf |pdf]]) <br> until 201107 * Intro.1.A ([[Media:Intro.1.A.pdf |pdf]]) * Control.1.A ([[Media:Control.1.A.20110706.pdf |pdf]]) * Iteration.1.A ([[Media:Iteration.1.A.pdf |pdf]]) * Function.1.A ([[Media:Function.1.A.20110705.pdf |pdf]]) * Variable.1.A ([[Media:Variable.1.A.20110708.pdf |pdf]]) * Operators.1.A ([[Media:Operators.1.A.20110712.pdf |pdf]]) * Pointer.1.A ([[Media:Pointer.1.A.pdf |pdf]]) * Pointer.2.A ([[Media:Pointer.2.A.pdf |pdf]]) * Array.1.A ([[Media:Array.1.A.pdf |pdf]]) * Type.1.A ([[Media:Type.1.A.pdf |pdf]]) * Structure.1.A ([[Media:Structure.1.A.pdf |pdf]]) go to [ [[C programming in plain view]] ] [[Category:C programming]] </br> qqlf5xyhylwmojlrzgvssvy4voy1edw 0 285384 2410374 2410122 2022-07-30T02:09:25Z Young1lim 21186 /* Workings of the GNU Compiler */ wikitext text/x-wiki === Workings of the GNU Compiler === * Overview ([[Media:Overview.20200211.pdf |pdf]]) * Access ([[Media:Access.20200409.pdf |pdf]]) * Operators ([[Media:Operator.20200427.pdf |pdf]]) * Conditions ([[Media:Condition.20220730.pdf |pdf]]) * Control ([[Media:Control.20220616.pdf |pdf]]) * Procedure ([[Media:Procedure.20220412.pdf |pdf]]) * Recursion ([[Media:Recursion.20210824-2.pdf |pdf]]) * Arrays ([[Media:Array.20211018.pdf |pdf]]) * Structures ([[Media:Structure.20220101.pdf |pdf]]) * Alignment ([[Media:Alignment.20201117.pdf |pdf]]) * Pointers ([[Media:Pointer.20201106.pdf |pdf]]) </br> === Workings of the GNU Linker === ==== Overview ==== * Static Linking Overview ([[Media:Link.1.StaticOverview.20181120.pdf |pdf]]) * Dynamic Linking Overview ([[Media:Link.2.DynamicOverview.20181120.pdf |pdf]]) ==== Linking Process ==== * Object Files ([[Media:Link.3.A.Object.20190121.pdf |A.pdf]], [[Media:Link.3.B.Object.20190405.pdf |B.pdf]]) * Symbols ([[Media:Link.4.A.Symbol.20190312.pdf |A.pdf]], [[Media:Link.4.B.Symbol.20190312.pdf |B.pdf]]) * Relocation ([[Media:Link.5.A.Relocation.20190320.pdf |A.pdf]], [[Media:Link.5.B.Relocation.20190322.pdf |B.pdf]]) * Loading ([[Media:Link.6.A.Loading.20190501.pdf |A.pdf]], [[Media:Link.6.B.Loading.20190126.pdf |B.pdf]]) * Static Linking ([[Media:Link.7.A.StaticLink.20190122.pdf |A.pdf]], [[Media:Link.7.B.StaticLink.20190128.pdf |B.pdf]]) * Dynamic Linking ([[Media:Link.8.A.DynamicLink.20190207.pdf |A.pdf]], [[Media:Link.8.B.DynamicLink.20190209.pdf |B.pdf]]) * Position Independent Code ([[Media:Link.9.A.PIC.20190304.pdf |A.pdf]], [[Media:Link.9.B.PIC.20190309.pdf |B.pdf]]) ==== Example I ==== * Vector addition ([[Media:Eg1.1A.Vector.20190121.pdf |A.pdf]], [[Media:Eg1.1B.Vector.20190121.pdf |B.pdf]]) * Swapping array elements ([[Media:Eg1.2A.Swap.20190302.pdf |A.pdf]], [[Media:Eg1.2B.Swap.20190121.pdf |B.pdf]]) * Nested functions ([[Media:Eg1.3A.Nest.20190121.pdf |A.pdf]], [[Media:Eg1.3B.Nest.20190121.pdf |B.pdf]]) ==== Examples II ==== * analysis of static linking ([[Media:Ex1.A.StaticLinkEx.20190121.pdf |A.pdf]], [[Media:Ex2.B.StaticLinkEx.20190121.pdf |B.pdf]]) * analysis of dynamic linking ([[Media:Ex2.A.DynamicLinkEx.20190121.pdf |A.pdf]]) * analysis of PIC ([[Media:Ex3.A.PICEx.20190121.pdf |A.pdf]]) </br> go to [ [[C programming in plain view]] ] [[Category:C programming]] 02tlgwomi8mrexfn6spm8a6wtyvu5ak 0 285385 2410252 2409405 2022-07-29T18:58:09Z Young1lim 21186 /* Object Files */ wikitext text/x-wiki === Executable and Linkable Format === ==== Object Files ==== * Introduction * ELF Header ([[Media:ELF1.1B.Header.20220211.pdf |pdf]]) * Group section ([[Media:ELF1.1C.Group.20220426.pdf |pdf]]) * String table section ([[Media:ELF1.1D.StringTbl.20220427.pdf |pdf]]) * Weak and common symbols ([[Media:ELF1.1E.WeakComm.20220727.pdf |pdf]]) * Symbol table section ([[Media:ELF1.1F.SymbolTbl.20220722.pdf |pdf]]) * Special Sections ([[Media:ELF1.7B.Section.20200511.pdf |B.pdf]]) * Relocation ([[Media:ELF1.6A.Relocation.20190413.pdf |A.pdf]]) ==== Program Loading and Dynamic Linking ==== * Introduction * Program Header ([[Media:ELF1.2B.ProgHeader.20220110.pdf |pdf]]) * Program Loading * Dynamic Linking ([[Media:ELF2.4A.DynLinking.20191028.pdf |pdf]]) ==== C Library ==== * C Library === ELF Study === ==== ELF Relocations ==== * Linking ([[Media:ELF1.7A.Linking.20200731.pdf |A.pdf]]) * Loading ([[Media:ELF1.7B.Loading.20201103.pdf |B.pdf]]) * Executing ([[Media:ELF1.7C.Executing.20201221.pdf |C.pdf]]) * Virtual Memory ([[Media:ELF2.1D.VMemory.20211227.pdf |D.pdf]]) * PIC Method ([[Media:ELF1.7B.PICMethod.20200417.pdf |C.pdf]]) * Design Cycles ([[Media:ELF1.7C.DesignCycle.20200317.pdf |D.pdf]]) * Relocs in i386 ([[Media:ELF1.7D.Reloc386.20200413.pdf |E.pdf]]) ==== Relocation Examples ==== * Relocs example introduction ([[Media:ELF1.7Ex.1Intro.20200109.pdf |E1.pdf]]) * Relocs in an object for a library ([[Media:ELF1.7Ex.2ObjectRel.20200319.pdf |E2.pdf]]) * Relocs in an object for an executable ([[Media:ELF1.7Ex.3ObjectMain.20200118.pdf |E3.pdf]]) * Relocs in a library ([[Media:ELF1.7Ex.4Library.20200320.pdf |E4.pdf]]) * Relocs in an executable ([[Media:ELF1.7Ex.5Executable.20200228.pdf |E5.pdf]]) * Result Summary ([[Media:ELF1.7Ex.6Result.20200121.pdf |E6.pdf]]) * Symbol Table Listing ([[Media:ELF1.7Ex.7Symbol.20200120.pdf |E7.pdf]]) * Relocs Listing ([[Media:ELF1.7Ex.8Relocs.20200121.pdf |E8.pdf]]) * Assembly Listing ([[Media:ELF1.7Ex.9Assembly.20200128.pdf |E9.pdf]]) * Reloc Experiments ([[Media:ELF1.7F.Experiments.20191206.pdf |F.pdf]]) </br> go to [ [[C programming in plain view]] ] [[Category:C programming]] ff03brc56uregfiblvquti72iyctlpm 2410254 2410252 2022-07-29T18:59:52Z Young1lim 21186 /* Object Files */ wikitext text/x-wiki === Executable and Linkable Format === ==== Object Files ==== * Introduction * ELF Header ([[Media:ELF1.1B.Header.20220211.pdf |pdf]]) * Group section ([[Media:ELF1.1C.Group.20220426.pdf |pdf]]) * String table section ([[Media:ELF1.1D.StringTbl.20220427.pdf |pdf]]) * Weak and common symbols ([[Media:ELF1.1E.WeakComm.20220728.pdf |pdf]]) * Symbol table section ([[Media:ELF1.1F.SymbolTbl.20220722.pdf |pdf]]) * Special Sections ([[Media:ELF1.7B.Section.20200511.pdf |B.pdf]]) * Relocation ([[Media:ELF1.6A.Relocation.20190413.pdf |A.pdf]]) ==== Program Loading and Dynamic Linking ==== * Introduction * Program Header ([[Media:ELF1.2B.ProgHeader.20220110.pdf |pdf]]) * Program Loading * Dynamic Linking ([[Media:ELF2.4A.DynLinking.20191028.pdf |pdf]]) ==== C Library ==== * C Library === ELF Study === ==== ELF Relocations ==== * Linking ([[Media:ELF1.7A.Linking.20200731.pdf |A.pdf]]) * Loading ([[Media:ELF1.7B.Loading.20201103.pdf |B.pdf]]) * Executing ([[Media:ELF1.7C.Executing.20201221.pdf |C.pdf]]) * Virtual Memory ([[Media:ELF2.1D.VMemory.20211227.pdf |D.pdf]]) * PIC Method ([[Media:ELF1.7B.PICMethod.20200417.pdf |C.pdf]]) * Design Cycles ([[Media:ELF1.7C.DesignCycle.20200317.pdf |D.pdf]]) * Relocs in i386 ([[Media:ELF1.7D.Reloc386.20200413.pdf |E.pdf]]) ==== Relocation Examples ==== * Relocs example introduction ([[Media:ELF1.7Ex.1Intro.20200109.pdf |E1.pdf]]) * Relocs in an object for a library ([[Media:ELF1.7Ex.2ObjectRel.20200319.pdf |E2.pdf]]) * Relocs in an object for an executable ([[Media:ELF1.7Ex.3ObjectMain.20200118.pdf |E3.pdf]]) * Relocs in a library ([[Media:ELF1.7Ex.4Library.20200320.pdf |E4.pdf]]) * Relocs in an executable ([[Media:ELF1.7Ex.5Executable.20200228.pdf |E5.pdf]]) * Result Summary ([[Media:ELF1.7Ex.6Result.20200121.pdf |E6.pdf]]) * Symbol Table Listing ([[Media:ELF1.7Ex.7Symbol.20200120.pdf |E7.pdf]]) * Relocs Listing ([[Media:ELF1.7Ex.8Relocs.20200121.pdf |E8.pdf]]) * Assembly Listing ([[Media:ELF1.7Ex.9Assembly.20200128.pdf |E9.pdf]]) * Reloc Experiments ([[Media:ELF1.7F.Experiments.20191206.pdf |F.pdf]]) </br> go to [ [[C programming in plain view]] ] [[Category:C programming]] sh47ydcg1b5sf3kvprhb5frzd7dh4mc 3 285490 2410357 2409787 2022-07-30T00:28:16Z Dave Braunschweig 426084 /* Sorting Algorithms */ Reply wikitext text/x-wiki {{Robelbox|theme=9|title=Welcome!|width=100%}} <div style="{{Robelbox/pad}}"> '''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] Peter.mlich!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Dave Braunschweig|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOjs UI icon signature-ltr.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity. To [[Wikiversity:Introduction|get started]], you may <!-- The Left column --> <div style="width:50.0%; float:left"> * [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]]. * Visit a (kind of) [[Wikiversity:Random|random project]]. * [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]]. * Find out about [[Wikiversity:Research|research]] activities on Wikiversity. * [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left. </div> <!-- The Right column --> <div style="width:50.0%; float:left"> * Enable VisualEditor under [[Special:Preferences#mw-prefsection-betafeatures|Beta]] settings to make article editing easier. * Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity. * Give [[Wikiversity:Feedback|feedback]] about your initial observations. * Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]]. * [[Wikiversity:Chat|Chat]] with other Wikiversitans on [[:freenode:wikiversity|<kbd>#wikiversity</kbd>]]. </div> <br clear="both"/> You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 00:02, 19 July 2022 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} == Sorting Algorithms == Wikiversity is organized by learning project (more like Wikibooks) rather than by individual resource (such as Wikipedia). Would you like to create a learning project for Sorting Algorithms or would you prefer to have your recent contributions moved under the existing [[Algorithms]] learning project? -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 21:05, 25 July 2022 (UTC) :??? I now try publish 3 pages: :https://en.wikiversity.org/wiki/SortedListMergingSort :https://en.wikiversity.org/wiki/PyramidSelectionSort :https://en.wikiversity.org/wiki/InsertionSortMiddle :concated to page :https://en.wikiversity.org/wiki/Data_Structures_and_Algorithms/Sorting_Data :I try it publish on normal wiki, but moderator deleted it. :If you not need, delete too. :) :I not need publish algorithm code. :But, what is good, when anyone can i run js code on my pages as jsfiddle.net. :But, i know, how works javascript virus. Run is problematic. If virus, can vired user pages. :But, js code is good. Can work on any devices with browsers. Cpp, python or other you cant easy run :) [[User:Peter.mlich|Peter.mlich]] ([[User talk:Peter.mlich|discuss]] • [[Special:Contributions/Peter.mlich|contribs]]) 08:09, 27 July 2022 (UTC) ::Now moved. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 00:28, 30 July 2022 (UTC) pp9ec5qcpih5rv4bf776jftf9pbte79 0 285546 2410243 2409853 2022-07-29T15:15:29Z Ncharamut 2824970 /* Grant funds spent */ updated finances wikitext text/x-wiki == Goals == ''Did you meet your goals? Are you happy with how the project went?''<br>We did meet our goals and are very happy with how the project went. We were able to create a template for conferences to use seen [[Template:Conferences|here]]. And updated all of the previous Future Directions Forum pages seen [[Journal of Clinical Child and Adolescent Psychology Future Directions Forum (JCCAP FDF)|here]]. Additionally, we were able to create Wikipedia pages for some of the previous presenters of the Forum with a focus on women and other minority groups. By starting this, we have decided to try and continue this project moving forward. We have also recruited several new Wiki editors and trained those individuals how to use our template and how to edit Wiki pages. Many of the current editors working on the project have increased their skills on Wiki platforms and have attended workshops on transclusion and creating templates in order to increase our ability to effectively update pages and create the template for conferences to use. == Outcome == ''Please report on your original project targets. Please be sure to '''[[Grants:Metrics#Three_shared_metrics|review and provide metrics required for Rapid Grants]]''''' {| class="wikitable" |+ ! !Target outcome !Achived outcome !Explanation |- |1 |Host 2 edit-a-thons with HGAPS team members from both UNC and UMD |Hosted 2 edit-a-thons |Both edit-a-thons were successful. We had HGAPS members come from both UNC and UMD as well as UCLA. We also had individuals attend the edit-a-thons who are not a part of HGAPS. |- |2 |15+ contributors and attendees per edit-a-thon |10 contributors per edit-a-thon |Although we were not able to get 15+ attendees at our edit-a-thons, we were able to get 10 at both events. |- |3 |5 new editors |10 new editors |We had more than 5 new editors and ended with about 10 new editors who have not edited Wiki before. |- |4 |1 new article created, 20+ articles improved |5 new pages created and 25 pages updated |We were successful on this target and created 5 new pages related to keynote speakers and improved many existing articles with around 25 total. |- |5 |increase views within 6 months of launch |Views have increased |We have had an increase in views on FDF related pages since updating the pages and creating new ones. |} == Learning == ''Projects do not always go according to plan. Sharing what you learned can help you and others plan similar projects in the future. Help the movement learn from your experience by answering the following questions:'' *What worked well? # *What did not work so well? # *What would you do differently next time? # == Finances == ===Grant funds spent=== ''Please describe how much grant money you spent for approved expenses, and tell us what you spent it on.'' * '''Wages for template/tool-kit creators''': $2,040 ($20 per hour x 3 creators x 34 hours) * '''Wages for trainers''': $720 ($20 per hour x 3 trainers x 12 hours) * '''Gift cards to virtually buy lunch for edit-a-thon participants''': $600 ($20 for 15 participants x 2 events) ** We spent $260 on gift cards to buy lunch for edit-a-thon participants. Not everyone that attended wanted one, which led to us being under budget here. We have reallocated these remaining funds ($340) to "wages for template/tool-kit creators". * '''Incentives for sustained editing''': $500 (1st prize $100, 2nd prize $50, and 3rd prize $25 at 1 and 3 month follow-ups after last edit-a-thon = $175 x 4 "contests") ** Upon organizing the edit-a-thons, we decided to change our method for prizes. Since we organized into teams we decided to have participants evaluate each teams work at the end of each edit-a-thon. We used these evaluations to come up with a first place and second place team and these teams were given prizes. We spent $350 in this category and reallocated these remaining funds ($150) to "wages for template/tool-kit creators". * '''HGAPS merch for milestones completed in editor training''': $300 ** We spent $248.27 on HGAPS merch for editors. The remaining funds ($51.73) were reallocated to "wages for template/tool-kit creators". * '''Fiscal sponsor administrative fees (including access to Google Suites for Nonprofits platform and analytics)''': $832 (20% final budget) ** We spent the anticipated $832 for fiscal sponsor administrative fees. '''Total''': USD $4,992 ===Remaining funds=== ''Do you have any remaining grant funds?'' No, we don't have any remaining grant funds left after the completion of the project. <!--Let us know if you would like to use the remaining funds on a similar or new project. Remember, a grants officer must approve this request before you spend the money.--> ==Anything else== ''Anything else you want to share about your project?'' [[Category:Project/Rapid/Report]] 6ggxu14kn60s3xyxq97c3c4pdzewbq2 0 285687 2410348 2409304 2022-07-30T00:26:13Z Dave Braunschweig 426084 Dave Braunschweig moved page [[InsertionSortMiddle]] to [[Data Structures and Algorithms/InsertionSortMiddle]]: Moving under learning project wikitext text/x-wiki {{subpage|Algorithms}} == Description == {{Aligned table |cols=2|class=wikitable |col1header=on |col1align=left | Category | Sorting algorithm | Sub category | Insert sort | Name | '''InsertionSortMiddle''' | Data structure | Array | Comparations | <math>O(n\log n)</math> | Timing | long for log array (base code need to do lot of moves) | Spacing | (original array, input is output) | Stability | Stable algorithm }} How it work? '''InsertionSortMiddle''' work as insert sort. Get next value (i+1) and insert to sorted array. Check possition from middle of sorted array, then middle of part... Properties: Algoritm not need extra memory. Ist slow for large array (moving a large array takes a lot of time). Have minimal comparation operations of all know algorithms (without counting sorts). Stable. == Statistics from real code execution (average) == <pre> n = 1.024 value-min = 0 value-max = n/2 // 50% of array contain some repeating value ------------------ compares ~ 8.825 (Tim-sort ~8.961) cycles ~ 273.362 (Tim-sort ~16.097) moves ~ 263.514 (Tim-sort ~13.659, Select-sort ~3.054) stability = stable </pre> == Schematic of work == <pre style="overflow:auto; width:auto;"> 3 1 2 2 0 3 1 0 // input 3 // is first sorted array, (left = 0), half = 0, mid = 0 + floor(half / 2) (first pos_b), ''' cmp = 0 ''' . 1 // next i = 1, value = array[i] = 1, mid = 0 3-1 // compare 3-1 1 3 // 13 is now output, half = i, mid = 0 + floor(half / 2), ''' cmp + 1 ''' . . 2 // next i, array[i], mid = 0 1---2 // compare(array[i], array[mid]), compare>=0, half = floor(half / 2), left = mid, mid = left + half 3-2 // compare(array[i], array[mid]), compare<0, end (because half = 0, cannot more divide), half = i, mid = 0 + round(half / 2), ''' cmp + 2 ''' 1 2 3 . // ... note: for simplify, i remove lot of repeating text 2 // next i, mid = 1 2---2 // compare>=0, mid = left + half . 3-2 // compare<0, end, ''' cmp + 2 ''' 1 2 2 3 . . 0 // next i, mid = 1 2-----0 // compare<0, mid = left - half 1-------0 // compare<0, end, ''' cmp + 2 ''' 0 1 2 2 3 . 3 // next i, mid = 2 2-----3 // compare>=0, mid = left + half . 2---3 // compare>=0, mid = left + half . 3-3 // compare>=0, end, ''' cmp + 3 ''' 0 1 2 2 3 3 . . 1 // next i, mid = 2 2-------1 // compare<0, mid = left - half 1---------1 // compare>=0, end, ''' cmp + 2 ''' (my alg. code here compute cmp+3, because compare zero) 0 1 1 2 2 3 3 . 0 // next i, mid = 3 2-------0 // compare<0, mid = left - half 1-----------0 // compare<0, mid = left - half 0-------------0 // compare>=0, end, ''' cmp + 3 ''' =============== 0 0 1 1 2 2 3 3 // output, suma(cmp) = 1+2+2+2+3+2+3 = 15 </pre> == Code (javascript) == <syntaxhighlight lang="JavaScript"> <div></div> <script> // Created by Peter Mlich (2017) // insert new item into sorted array, check position from middle of array // note: algorithm code is differend from schema, but simplest function sortInsertMiddle(cmp, start, end, n) { if (o.size<2) {return o.n;} var i, i_start, i_end, left, right, mid, mid_sub; i_start = o.start + 1; i_end = o.end; i = i_start; while (i<i_end) { glob.cycles++; // find position left = o.start - 1; right = i; mid_sub = right - left; while (true) { glob.cycles++; mid = left + (mid_sub>>1); if (o.fn_cmp(arr[o.n][i], arr[o.n][mid])>=0) { left = mid; mid_sub = right - left; if (mid_sub<=1) {mid++; break;} } else { right = mid; mid_sub = right - left; if (mid_sub<=1) {break;} } } // move to position, shift array arrShift(arr[o.n], mid, i); i++; } return o.n; } // ------ // note: code is not optimalized - draft version from my tester function sortCompare (a, b) { glob.cmps++; var c = a - b; return c>0 ? 1 : (c<0 ? -1 : 0); }; function arrShift(list, a, b) // move last (b) on top (a), alternation: splice or copyWithin { if (b<=a || a<0 || b<0) {return;} var tmp = list[b]; glob.cycles += b - a; glob.moves += b - a; while (a<b) { list[b] = list[--b]; } list[a] = tmp; } var arr = [null, [7,7,4,3,4,7,6,7,0,1,0,6,7,2,2,4], [-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1]] var glob = {moves: 0, cycles: 0, cmps: 0}; var o = {start: 0, end: 16, size: 16 - 0, n: 1, moves: 0, cycles: 0, fn_cmp: sortCompare}; var log = [], i=0, n; log[i++] = 'array-before ' + JSON.stringify(arr[1]) o.n = sortInsertMiddle(o.fn_cmp, o.start, o.end, o.n); log[i++] = 'array-after ' + JSON.stringify(arr[o.n]) log[i++] = 'glob ' + JSON.stringify(glob) log[i++] = 'n ' + JSON.stringify(o.end - o.start) document.getElementsByTagName('DIV')[0].innerHTML = log.join('<br>') /* array-before [7,7,4,3,4,7,6,7,0,1,0,6,7,2,2,4] array-after [0,0,1,2,2,3,4,4,4,6,6,7,7,7,7,7] glob {"moves":66,"cycles":125,"cmps":44} n 16 */ </script> </syntaxhighlight> crb7kvtqrak4efs1cw0sb6ag04jzqhb 2410356 2410348 2022-07-30T00:27:45Z Dave Braunschweig 426084 Moving under learning project wikitext text/x-wiki == Description == {{Aligned table |cols=2|class=wikitable |col1header=on |col1align=left | Category | Sorting algorithm | Sub category | Insert sort | Name | '''InsertionSortMiddle''' | Data structure | Array | Comparations | <math>O(n\log n)</math> | Timing | long for log array (base code need to do lot of moves) | Spacing | (original array, input is output) | Stability | Stable algorithm }} How it work? '''InsertionSortMiddle''' work as insert sort. Get next value (i+1) and insert to sorted array. Check possition from middle of sorted array, then middle of part... Properties: Algoritm not need extra memory. Ist slow for large array (moving a large array takes a lot of time). Have minimal comparation operations of all know algorithms (without counting sorts). Stable. == Statistics from real code execution (average) == <pre> n = 1.024 value-min = 0 value-max = n/2 // 50% of array contain some repeating value ------------------ compares ~ 8.825 (Tim-sort ~8.961) cycles ~ 273.362 (Tim-sort ~16.097) moves ~ 263.514 (Tim-sort ~13.659, Select-sort ~3.054) stability = stable </pre> == Schematic of work == <pre style="overflow:auto; width:auto;"> 3 1 2 2 0 3 1 0 // input 3 // is first sorted array, (left = 0), half = 0, mid = 0 + floor(half / 2) (first pos_b), ''' cmp = 0 ''' . 1 // next i = 1, value = array[i] = 1, mid = 0 3-1 // compare 3-1 1 3 // 13 is now output, half = i, mid = 0 + floor(half / 2), ''' cmp + 1 ''' . . 2 // next i, array[i], mid = 0 1---2 // compare(array[i], array[mid]), compare>=0, half = floor(half / 2), left = mid, mid = left + half 3-2 // compare(array[i], array[mid]), compare<0, end (because half = 0, cannot more divide), half = i, mid = 0 + round(half / 2), ''' cmp + 2 ''' 1 2 3 . // ... note: for simplify, i remove lot of repeating text 2 // next i, mid = 1 2---2 // compare>=0, mid = left + half . 3-2 // compare<0, end, ''' cmp + 2 ''' 1 2 2 3 . . 0 // next i, mid = 1 2-----0 // compare<0, mid = left - half 1-------0 // compare<0, end, ''' cmp + 2 ''' 0 1 2 2 3 . 3 // next i, mid = 2 2-----3 // compare>=0, mid = left + half . 2---3 // compare>=0, mid = left + half . 3-3 // compare>=0, end, ''' cmp + 3 ''' 0 1 2 2 3 3 . . 1 // next i, mid = 2 2-------1 // compare<0, mid = left - half 1---------1 // compare>=0, end, ''' cmp + 2 ''' (my alg. code here compute cmp+3, because compare zero) 0 1 1 2 2 3 3 . 0 // next i, mid = 3 2-------0 // compare<0, mid = left - half 1-----------0 // compare<0, mid = left - half 0-------------0 // compare>=0, end, ''' cmp + 3 ''' =============== 0 0 1 1 2 2 3 3 // output, suma(cmp) = 1+2+2+2+3+2+3 = 15 </pre> == Code (javascript) == <syntaxhighlight lang="JavaScript"> <div></div> <script> // Created by Peter Mlich (2017) // insert new item into sorted array, check position from middle of array // note: algorithm code is differend from schema, but simplest function sortInsertMiddle(cmp, start, end, n) { if (o.size<2) {return o.n;} var i, i_start, i_end, left, right, mid, mid_sub; i_start = o.start + 1; i_end = o.end; i = i_start; while (i<i_end) { glob.cycles++; // find position left = o.start - 1; right = i; mid_sub = right - left; while (true) { glob.cycles++; mid = left + (mid_sub>>1); if (o.fn_cmp(arr[o.n][i], arr[o.n][mid])>=0) { left = mid; mid_sub = right - left; if (mid_sub<=1) {mid++; break;} } else { right = mid; mid_sub = right - left; if (mid_sub<=1) {break;} } } // move to position, shift array arrShift(arr[o.n], mid, i); i++; } return o.n; } // ------ // note: code is not optimalized - draft version from my tester function sortCompare (a, b) { glob.cmps++; var c = a - b; return c>0 ? 1 : (c<0 ? -1 : 0); }; function arrShift(list, a, b) // move last (b) on top (a), alternation: splice or copyWithin { if (b<=a || a<0 || b<0) {return;} var tmp = list[b]; glob.cycles += b - a; glob.moves += b - a; while (a<b) { list[b] = list[--b]; } list[a] = tmp; } var arr = [null, [7,7,4,3,4,7,6,7,0,1,0,6,7,2,2,4], [-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1]] var glob = {moves: 0, cycles: 0, cmps: 0}; var o = {start: 0, end: 16, size: 16 - 0, n: 1, moves: 0, cycles: 0, fn_cmp: sortCompare}; var log = [], i=0, n; log[i++] = 'array-before ' + JSON.stringify(arr[1]) o.n = sortInsertMiddle(o.fn_cmp, o.start, o.end, o.n); log[i++] = 'array-after ' + JSON.stringify(arr[o.n]) log[i++] = 'glob ' + JSON.stringify(glob) log[i++] = 'n ' + JSON.stringify(o.end - o.start) document.getElementsByTagName('DIV')[0].innerHTML = log.join('<br>') /* array-before [7,7,4,3,4,7,6,7,0,1,0,6,7,2,2,4] array-after [0,0,1,2,2,3,4,4,4,6,6,7,7,7,7,7] glob {"moves":66,"cycles":125,"cmps":44} n 16 */ </script> </syntaxhighlight> bh3w58pxbq9hv04u29pvzdk1wc1qqse 0 285688 2410350 2409305 2022-07-30T00:26:26Z Dave Braunschweig 426084 Dave Braunschweig moved page [[SortedListMergingSort]] to [[Data Structures and Algorithms/SortedListMergingSort]]: Moving under learning project wikitext text/x-wiki {{subpage|Algorithms}} == Description == {{Aligned table |cols=2|class=wikitable |col1header=on |col1align=left | Category | Sorting algorithm | Sub category | Merge sort | Name | '''SortedListMergingSort''' | Data structure | Array | Comparations | <math>O(n\log n)</math> | Timing | <math>O(n\log n)</math> | Spacing | <math>2*n (+ n)</math> (input + output (+ index table)) | Stability | Stable algorithm }} How it work? '''SortedListMergingSort''' merge sorted list. First, you must detect sorted sub-arrays (by compare values on positions 0-1, 1-2, 2-3...). Or, we can say, always have sorted array size=1. Then merge two arrays (best, arrays with the smallest size) to one. Repeat merging. Trick lies in it, sorted array can compare from top, smaller value move to save, not need more compare with any value in any in this two arrays. Properties: Algoritm need extra memory (copy from array 1 to array 2 and back). Ist fast. Stable. Can be modified to multi-thread. Version with detect sorted sub-arrays can be modified, return ascendecy (asc), descendency (desc) array as <math>O(n)</math>. Can save longer from one of asc or desc sub-arrays. Note: Merging sorted arrays use TimSort, WikiSort. == Statistics (average) == <pre> n = 1.024 value-min = 0 value-max = n/2 // 50% of array contain some repeating value ------------------ compares ~ 8.886 (Tim-sort ~8.961) cycles ~ 11.273 (Tim-sort ~16.097) moves ~ 10.240 (Tim-sort ~13.659, Select-sort ~3.054) stability = stable </pre> == Schematic of work == <pre style="overflow:auto; width:auto;"> 3 1 2 2 0 3 1 0 // input (array_1) 3-1 2-2 0-3 1-0 // compare top of sorted list A (list array-size = 1) with top of sorted list B (list length 1), C-D, E-F, G-H 1 3 2 2 0 3 0 1 // saved output at end in array_2, cmp = 4 (you still copy from array 1 to array 2 and back, need two array or two array with indexes) 1 3 | 2 2 | 0 3 | 0 1 // input (array_2; A-B, C-D, E-F, G-H is now new sorted array with array-size = 2) 1-----2 // compare first from AB with first from CD, smaller save 1 // save 3---2 // compare next from AB with first from CD, smaller save 2 // save 3-----2 // compare from last AB with last from CD, smaller save 2 // save 3 // save (copy) 1 2 2 3 // saved output at end (in array_1), cmp + 3 0-----0 // compare first (EF) with first (GH) 0 // save 3---0 // compare next (EF) with first (GH) 0 // save 3-----1 // compare last (EF) with last (GH) 1 // save 3 // save (copy) 0 0 1 3 // saved output at end (in array_1), cmp + 3 1 2 2 3 | 0 0 1 3 // new sorted lists 1---------0 // compare first (ABCD) with first (EFGH) 0 // save 1-----------0 // compare first (ABCD) with second (EFGH) 0 // save 1-------------1 // compare first (ABCD) with third (EFGH) 1 // save 2------------1 // compare second (ABCD) with third (EFGH) 1 // save 2--------------3 // compare second (ABCD) with last (EFGH) 2 // save 2------------3 // compare third (ABCD) with last (EFGH) 2 // save 3----------3 // compare last (ABCD) with last (EFGH) 3 // save 3 // save (copy), cmp + 7 ================== 0 0 1 1 2 2 3 3 // output in array_2, return handle to array, suma(cmp) = 4+3+3+7 = 17 </pre> == Code (javascript) == <syntaxhighlight lang="JavaScript"> <div></div> <script> // Created by Peter Mlich (2013) // note: code use too Honzik (? Jaroslav) from VUT Brno, but, i create code before seen his document // merging part function listMerging_bounds_part(cmp, i_start, i_end, j_end, m, n) { var cycles = 0; var i,j,k; i = i_start; j = i_end; // i_end = j_start k = i_start; // k_start = i_start while (i<i_end && j<j_end) { cycles++; if (cmp(arr[m][i],arr[m][j])>0) {arr[n][k] = arr[m][j]; j++; k++;} else {arr[n][k] = arr[m][i]; i++; k++;} } while (i<i_end) { cycles++; arr[n][k] = arr[m][i]; i++; k++; } while (j<j_end) { cycles++; arr[n][k] = arr[m][j]; j++; k++; } glob.cycles += cycles; glob.moves += cycles; return n; } // Merge sorted list, first sorted lists have length 1 (or can detect sorted, compare(a,b), b-c, c-d...) function sortedListMergingTop(cmp, start, end, n) { if (o.size<2) {return o.n;} var step, stepmax, tmp, a,b,c, m, n; stepmax = ((o.size + 1) >> 1) << 1; m = o.n; n = o.n==1 ? 2 : 1; for (step=1; step<stepmax; step<<=1) //bounds 1-1, 2-2, 4-4, 8-8... { glob.cycles++; a = o.start; while (a<o.end) { glob.cycles++; b = a + step; c = a + (step<<1); // c = a + step + step; b = b<o.end ? b : o.end; c = c<o.end ? c : o.end; listMerging_bounds_part(o.fn_cmp, a, b, c, m, n); a = c; } tmp = m; m = n; n = tmp; } return m; } // ------ // note: code is not optimalized - draft version from my tester function sortCompare (a, b) { glob.cmps++; var c = a - b; return c>0 ? 1 : (c<0 ? -1 : 0); }; var arr = [null, [7,7,4,3,4,7,6,7,0,1,0,6,7,2,2,4], [-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1]] var glob = {moves: 0, cycles: 0, cmps: 0}; var o = {start: 0, end: 16, size: 16 - 0, n: 1, moves: 0, cycles: 0, fn_cmp: sortCompare}; var log = [], i=0, n; log[i++] = 'array-before ' + JSON.stringify(arr[1]) o.n = sortedListMergingTop(o.fn_cmp, o.start, o.end, o.n); log[i++] = 'array-after ' + JSON.stringify(arr[o.n]) log[i++] = 'glob ' + JSON.stringify(glob) log[i++] = 'n ' + JSON.stringify(o.end - o.start) document.getElementsByTagName('DIV')[0].innerHTML = log.join('<br>') /* array-before [7,7,4,3,4,7,6,7,0,1,0,6,7,2,2,4] array-after [0,0,1,2,2,3,4,4,4,6,6,7,7,7,7,7] glob {"moves":64,"cycles":83,"cmps":47} n 16 */ </script> </syntaxhighlight> 872sly51amwwfdkd9fxsiqted2hi1pf 2410355 2410350 2022-07-30T00:27:29Z Dave Braunschweig 426084 Moving under learning project wikitext text/x-wiki == Description == {{Aligned table |cols=2|class=wikitable |col1header=on |col1align=left | Category | Sorting algorithm | Sub category | Merge sort | Name | '''SortedListMergingSort''' | Data structure | Array | Comparations | <math>O(n\log n)</math> | Timing | <math>O(n\log n)</math> | Spacing | <math>2*n (+ n)</math> (input + output (+ index table)) | Stability | Stable algorithm }} How it work? '''SortedListMergingSort''' merge sorted list. First, you must detect sorted sub-arrays (by compare values on positions 0-1, 1-2, 2-3...). Or, we can say, always have sorted array size=1. Then merge two arrays (best, arrays with the smallest size) to one. Repeat merging. Trick lies in it, sorted array can compare from top, smaller value move to save, not need more compare with any value in any in this two arrays. Properties: Algoritm need extra memory (copy from array 1 to array 2 and back). Ist fast. Stable. Can be modified to multi-thread. Version with detect sorted sub-arrays can be modified, return ascendecy (asc), descendency (desc) array as <math>O(n)</math>. Can save longer from one of asc or desc sub-arrays. Note: Merging sorted arrays use TimSort, WikiSort. == Statistics (average) == <pre> n = 1.024 value-min = 0 value-max = n/2 // 50% of array contain some repeating value ------------------ compares ~ 8.886 (Tim-sort ~8.961) cycles ~ 11.273 (Tim-sort ~16.097) moves ~ 10.240 (Tim-sort ~13.659, Select-sort ~3.054) stability = stable </pre> == Schematic of work == <pre style="overflow:auto; width:auto;"> 3 1 2 2 0 3 1 0 // input (array_1) 3-1 2-2 0-3 1-0 // compare top of sorted list A (list array-size = 1) with top of sorted list B (list length 1), C-D, E-F, G-H 1 3 2 2 0 3 0 1 // saved output at end in array_2, cmp = 4 (you still copy from array 1 to array 2 and back, need two array or two array with indexes) 1 3 | 2 2 | 0 3 | 0 1 // input (array_2; A-B, C-D, E-F, G-H is now new sorted array with array-size = 2) 1-----2 // compare first from AB with first from CD, smaller save 1 // save 3---2 // compare next from AB with first from CD, smaller save 2 // save 3-----2 // compare from last AB with last from CD, smaller save 2 // save 3 // save (copy) 1 2 2 3 // saved output at end (in array_1), cmp + 3 0-----0 // compare first (EF) with first (GH) 0 // save 3---0 // compare next (EF) with first (GH) 0 // save 3-----1 // compare last (EF) with last (GH) 1 // save 3 // save (copy) 0 0 1 3 // saved output at end (in array_1), cmp + 3 1 2 2 3 | 0 0 1 3 // new sorted lists 1---------0 // compare first (ABCD) with first (EFGH) 0 // save 1-----------0 // compare first (ABCD) with second (EFGH) 0 // save 1-------------1 // compare first (ABCD) with third (EFGH) 1 // save 2------------1 // compare second (ABCD) with third (EFGH) 1 // save 2--------------3 // compare second (ABCD) with last (EFGH) 2 // save 2------------3 // compare third (ABCD) with last (EFGH) 2 // save 3----------3 // compare last (ABCD) with last (EFGH) 3 // save 3 // save (copy), cmp + 7 ================== 0 0 1 1 2 2 3 3 // output in array_2, return handle to array, suma(cmp) = 4+3+3+7 = 17 </pre> == Code (javascript) == <syntaxhighlight lang="JavaScript"> <div></div> <script> // Created by Peter Mlich (2013) // note: code use too Honzik (? Jaroslav) from VUT Brno, but, i create code before seen his document // merging part function listMerging_bounds_part(cmp, i_start, i_end, j_end, m, n) { var cycles = 0; var i,j,k; i = i_start; j = i_end; // i_end = j_start k = i_start; // k_start = i_start while (i<i_end && j<j_end) { cycles++; if (cmp(arr[m][i],arr[m][j])>0) {arr[n][k] = arr[m][j]; j++; k++;} else {arr[n][k] = arr[m][i]; i++; k++;} } while (i<i_end) { cycles++; arr[n][k] = arr[m][i]; i++; k++; } while (j<j_end) { cycles++; arr[n][k] = arr[m][j]; j++; k++; } glob.cycles += cycles; glob.moves += cycles; return n; } // Merge sorted list, first sorted lists have length 1 (or can detect sorted, compare(a,b), b-c, c-d...) function sortedListMergingTop(cmp, start, end, n) { if (o.size<2) {return o.n;} var step, stepmax, tmp, a,b,c, m, n; stepmax = ((o.size + 1) >> 1) << 1; m = o.n; n = o.n==1 ? 2 : 1; for (step=1; step<stepmax; step<<=1) //bounds 1-1, 2-2, 4-4, 8-8... { glob.cycles++; a = o.start; while (a<o.end) { glob.cycles++; b = a + step; c = a + (step<<1); // c = a + step + step; b = b<o.end ? b : o.end; c = c<o.end ? c : o.end; listMerging_bounds_part(o.fn_cmp, a, b, c, m, n); a = c; } tmp = m; m = n; n = tmp; } return m; } // ------ // note: code is not optimalized - draft version from my tester function sortCompare (a, b) { glob.cmps++; var c = a - b; return c>0 ? 1 : (c<0 ? -1 : 0); }; var arr = [null, [7,7,4,3,4,7,6,7,0,1,0,6,7,2,2,4], [-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1]] var glob = {moves: 0, cycles: 0, cmps: 0}; var o = {start: 0, end: 16, size: 16 - 0, n: 1, moves: 0, cycles: 0, fn_cmp: sortCompare}; var log = [], i=0, n; log[i++] = 'array-before ' + JSON.stringify(arr[1]) o.n = sortedListMergingTop(o.fn_cmp, o.start, o.end, o.n); log[i++] = 'array-after ' + JSON.stringify(arr[o.n]) log[i++] = 'glob ' + JSON.stringify(glob) log[i++] = 'n ' + JSON.stringify(o.end - o.start) document.getElementsByTagName('DIV')[0].innerHTML = log.join('<br>') /* array-before [7,7,4,3,4,7,6,7,0,1,0,6,7,2,2,4] array-after [0,0,1,2,2,3,4,4,4,6,6,7,7,7,7,7] glob {"moves":64,"cycles":83,"cmps":47} n 16 */ </script> </syntaxhighlight> tasvzgxn1s7nwxvng1j9ey8xvmt1h95 0 285689 2410352 2409306 2022-07-30T00:26:44Z Dave Braunschweig 426084 Dave Braunschweig moved page [[PyramidSelectionSort]] to [[Data Structures and Algorithms/PyramidSelectionSort]]: Moving under learning project wikitext text/x-wiki {{subpage|Algorithms}} == Description == '''PyramidSelectionSort''' get first pair of values, compare it and save minimal value (index) to new array. Repeat for all pair, create row 0. Repeat for row 0, create row 1... Find minimal value. Create tournament table of winners. Then remove minimal and rebuild pyramid branch (where minimal figured) and again find minimal value. {{Aligned table|cols=2|class=wikitable|col1header=on|col1align=left|Category|Sorting algorithm|Sub category|Selection sort|Name|'''PyramidSelectionSort'''|Data structure|Array|Comparations|<math>O(n\log n)</math>|Timing|<math>O(n\log n)</math>|Spacing|<math>2*n + n</math> (input + output + index table)|Stability|Stable algorithm}} == Statistics from real code execution (average) == <pre> n = 1.024 value-min = 0 value-max = n/2 // 50% of array contain some repeating value ------------------ compares ~ 8.886 (Tim-sort ~8.961, Select-sort ~523.776) cycles ~ 11.262 (Tim-sort ~16.097) moves ~ 1.798 (Tim-sort ~13.659, Select-sort ~3.054) stability = stable </pre> == Schematic of work == <pre style="overflow:auto; width:auto;"> pavel vs. tomas zdenek vs. michal | | | | +----+----+ +----+----+ | | tomas zdenek | | +---------+---------+ | zdenek --- out: zdenek pavel vs. tomas - michal --- remove winner and find new winner in this branch | | | | +----+----+ +----+----+ | | tomas michal | | +---------+---------+ | tomas --- out: zdenek, tomas pavel - - michal | | | | +----+----+ +----+----+ | | pavel michal | | +---------+---------+ | pavel --- out: zdenek, tomas, pavel, michal 3 1 2 2 0 3 1 0 // input 3-1 2-2 0-3 1-0 // compare pair from input and create row 0 of minimal 1-2 0-----0 // row 0, pyramid of minimal values / index of position (for scheme i use value, use position in alg. code) 1-----0 . . // row 1 0 . . // row 2, save minimal to out "0", cmp = 7 . . 1 2 3---0 // rebuild branch (row[0][4,5,6,7], row[1][3,4], row[2][1]) and compare new winner in branch 1-----------0 . 0 // save "0", cmp + 2 . x 1 2 3-1 x // rebuild branch 1---------1 1 // save "1", cmp + 2 x 3---2 3 1 // rebuild branch 2-------1 1 // save "1", cmp + 2 x 3 2 3 x // rebuild branch (when not even or odd value from input, use "x" (-1 in alg. code), when "x" copy second index to next level) 2-----3 2 // save "2", cmp + 1 x 3-----2 3 // rebuild branch (when "x", copy index to next level) 2---3 2 // save "2", cmp + 2 x 3 x 3 // rebuild branch (when "x", copy index to next level) 3---------3 3 // save "3", cmp + 1 3 // save last "3" =============== 0 0 1 1 2 2 3 3 // output, suma(cmp) = 7+2+2+2+1+2+1 = 17 </pre> == Code (javascript) == <syntaxhighlight lang="JavaScript"> <div></div> <script> // Created by Peter Mlich (2022) // build first pyramid of minimal values function pyramid_part1_buildPyramid(list, cmp, i_start, i_end, size) { var i,j,k, k_end, lvl, lvlp1; var pyramid = []; i = i_start; j = i_start+1; k = 0; lvl = 0; pyramid[lvl] = []; while (j<i_end) { glob.cycles++; if (cmp(list[i], list[j])>0) {swap(list, i, j);} pyramid[lvl][k] = i; i+=2; j+=2; k++; } if (i<i_end) // pokud je size liche cislo, pak pridej posledni prvek a preswapuj to (toho vyuziji pozdeji v part2) { if (cmp(list[i-2], list[i])>0) { tmp = list[i]; list[i ] = list[i-1]; list[i-1] = list[i-2]; list[i-2] = tmp; glob.moves += 4; pyramid[lvl][k] = i; } else {if (cmp(list[i-1], list[i])>0) { tmp = list[i]; list[i ] = list[i-1]; list[i-1] = tmp; glob.moves += 3; }} } i_end = k; lvlp1 = lvl + 1; while (i_end>1) { glob.cycles++; pyramid[lvlp1] = []; k = 0; i = 0; j = 1; // =i+1 while (j<i_end) { glob.cycles++; if (cmp(list[ pyramid[lvl][i] ], list[ pyramid[lvl][j] ])>0) {pyramid[lvlp1][k] = pyramid[lvl][j]; i+=2; j+=2; k++; continue;} else {pyramid[lvlp1][k] = pyramid[lvl][i]; i+=2; j+=2; k++; continue;} } if (i<i_end) {pyramid[lvlp1][k] = pyramid[lvl][i]; k++;} lvl++; lvlp1++; i_end = k; } return [pyramid, lvl, pyramid[lvl][0], (size>>1)<<1 != size]; // return pyramid, last lvl, last index, boolean for odd-size) } function pyramid_part3_rebuildPyramidEven(pyramid, lvl_end, bool, list, cmp, i_end, pos) { var lvl, val2, empty = -1, a, b; val2 = pyramid[0][pos]; for (lvl=0; lvl<lvl_end; lvl++) { glob.cycles++; if ((pos & 0x01) == 0) { if (pos==pyramid[lvl].length-1) { pos = pos>>1; pyramid[lvl+1][pos] = val2; //val2 = val2; continue; } b = pyramid[lvl][pos+1]; a = pyramid[lvl][pos]; pos = pos>>1; if (b==empty) {pyramid[lvl+1][pos] = a; val2 = a; continue;} if (cmp(list[a], list[b])>0) {pyramid[lvl+1][pos] = b; val2 = b; continue;} pyramid[lvl+1][pos] = a; val2 = a; } else { a = pyramid[lvl][pos-1]; b = pyramid[lvl][pos]; pos = pos>>1; if (a==empty) {pyramid[lvl+1][pos] = b; val2 = b; continue;} if (cmp(list[a], list[b])>0) {pyramid[lvl+1][pos] = b; val2 = b; continue;} pyramid[lvl+1][pos] = a; val2 = a; } } return [pyramid, lvl_end, pyramid[lvl_end][0], bool]; } // rebuild pyramid, rewrite branch by new value function pyramid_part2_rebuildPyramid(pyramid, lvl_end, bool, list, cmp, i_end, i_endm3) { var cycles = 0; var lvl, pos, val, val2, a, b, empty=-1; val = pyramid[lvl_end][0]; pos = val>>1; // pozice zleva if (bool==true && ((pos<<1)==i_endm3) && ((val & 0x01) == 0) ) // kdyz je size liche cislo a dojde k eliminaci n-2, tak posun posledni 2 cisla { bool = false; list[val] = list[val+1]; list[val+1] = list[val+2]; glob.moves += 2; // je sude, pak vymen za liche a prepocitej vsechna nutna porovnani pyramid[0][pos] = val; // pozn.: tento kod je prepsany na funkci, protoze by byl duplicitne return pyramid_part3_rebuildPyramidEven(pyramid, lvl_end, bool, list, cmp, i_end, pos); } else {if ((val & 0x01) == 0) // je sude, pak vymen za liche a prepocitej vsechna nutna porovnani { pyramid[0][pos] = val + 1; return pyramid_part3_rebuildPyramidEven(pyramid, lvl_end, bool, list, cmp, i_end, pos); } else { // je liche, pak odstran a prepocitej vsechna nutna porovnani val2 = empty; pyramid[0][pos] = val2; for (lvl=0; lvl<lvl_end; lvl++) { glob.cycles++; if ((pos & 0x01) == 0) { if (pos==pyramid[lvl].length-1) { pos = pos>>1; pyramid[lvl+1][pos] = val2; //val2 = val2 continue; } a = pyramid[lvl][pos]; b = pyramid[lvl][pos+1]; pos = pos>>1; if (a!==empty && b!==empty) { if (cmp(list[a], list[b])>0) {pyramid[lvl+1][pos] = b; val2 = b; continue;} else {pyramid[lvl+1][pos] = a; val2 = a; continue;} } if (b!==empty) {pyramid[lvl+1][pos] = b; val2 = b; continue;} pyramid[lvl+1][pos] = a; val2 = a; } else { a = pyramid[lvl][pos-1]; b = pyramid[lvl][pos]; pos = pos>>1; if (a!==empty && b!==empty) { if (cmp(list[a], list[b])>0) {pyramid[lvl+1][pos] = b; val2 = b; continue;} else {pyramid[lvl+1][pos] = a; val2 = a; continue;} } if (a!==empty) {pyramid[lvl+1][pos] = a; val2 = a; continue;} pyramid[lvl+1][pos] = b; val2 = b; } } }} return [pyramid, lvl_end, pyramid[lvl_end][0], bool]; } // princip: vyber minimum z kazdeho paru, pak porovnej minima, minima minim ... az ziskas nejmensi cislo // pak vyrad nejmensi cislo z pyramidy a propocitej celou vetev, opet ziskej minimum function PyramidSelectSort(cmp, start, end, n) { if (o.size<2) {return o.n;} var pyramid_data, i, x, y, endm3 = o.end-3; x = o.n; y = o.n==1 ? 2 : 1; pyramid_data = pyramid_part1_buildPyramid(arr[x], o.fn_cmp, o.start, o.end, o.size); // create pyramid of index from minimal values of pair i = o.start; arr[y][i] = arr[x][pyramid_data[2]]; glob.moves++; i++; while (i<o.end) { glob.cycles++; pyramid_data = pyramid_part2_rebuildPyramid(pyramid_data[0], pyramid_data[1], pyramid_data[3], arr[x], o.fn_cmp, o.end, endm3) arr[y][i] = arr[x][pyramid_data[2]]; glob.moves++; i++; } return y; } // note: code is optimalized for my tester function sortCompare (a, b) { glob.cmps++; var c = a - b; return c>0 ? 1 : (c<0 ? -1 : 0); }; function swap (list, a, b) { if (a==b) {return;} var tmp = list[a]; list[a] = list[b]; list[b] = tmp; glob.moves += 3; }; var arr = [null, [7,7,4,3,4,7,6,7,0,1,0,6,7,2,2,4], [-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1]] var glob = {moves: 0, cycles: 0, cmps: 0}; var o = {start: 0, end: 16, size: 16 - 0, n: 1, moves: 0, cycles: 0, fn_cmp: sortCompare}; var log = [], i=0, n; log[i++] = 'array-before ' + JSON.stringify(arr[1]) o.n = PyramidSelectSort(o.fn_cmp, o.start, o.end, o.n); log[i++] = 'array-after ' + JSON.stringify(arr[o.n]) log[i++] = 'glob ' + JSON.stringify(glob) log[i++] = 'n ' + JSON.stringify(o.end - o.start) document.getElementsByTagName('DIV')[0].innerHTML = log.join('<br>') /* array-before [7,7,4,3,4,7,6,7,0,1,0,6,7,2,2,4] array-after [0,0,1,2,2,3,4,4,4,6,6,7,7,7,7,7] glob {"moves":22,"cycles":78,"cmps":47} n 16 */ </script> </syntaxhighlight> ttxq41opla3q1x4m4ab6yijfelmwv7m 2410354 2410352 2022-07-30T00:27:07Z Dave Braunschweig 426084 Moving under learning project wikitext text/x-wiki == Description == '''PyramidSelectionSort''' get first pair of values, compare it and save minimal value (index) to new array. Repeat for all pair, create row 0. Repeat for row 0, create row 1... Find minimal value. Create tournament table of winners. Then remove minimal and rebuild pyramid branch (where minimal figured) and again find minimal value. {{Aligned table|cols=2|class=wikitable|col1header=on|col1align=left|Category|Sorting algorithm|Sub category|Selection sort|Name|'''PyramidSelectionSort'''|Data structure|Array|Comparations|<math>O(n\log n)</math>|Timing|<math>O(n\log n)</math>|Spacing|<math>2*n + n</math> (input + output + index table)|Stability|Stable algorithm}} == Statistics from real code execution (average) == <pre> n = 1.024 value-min = 0 value-max = n/2 // 50% of array contain some repeating value ------------------ compares ~ 8.886 (Tim-sort ~8.961, Select-sort ~523.776) cycles ~ 11.262 (Tim-sort ~16.097) moves ~ 1.798 (Tim-sort ~13.659, Select-sort ~3.054) stability = stable </pre> == Schematic of work == <pre style="overflow:auto; width:auto;"> pavel vs. tomas zdenek vs. michal | | | | +----+----+ +----+----+ | | tomas zdenek | | +---------+---------+ | zdenek --- out: zdenek pavel vs. tomas - michal --- remove winner and find new winner in this branch | | | | +----+----+ +----+----+ | | tomas michal | | +---------+---------+ | tomas --- out: zdenek, tomas pavel - - michal | | | | +----+----+ +----+----+ | | pavel michal | | +---------+---------+ | pavel --- out: zdenek, tomas, pavel, michal 3 1 2 2 0 3 1 0 // input 3-1 2-2 0-3 1-0 // compare pair from input and create row 0 of minimal 1-2 0-----0 // row 0, pyramid of minimal values / index of position (for scheme i use value, use position in alg. code) 1-----0 . . // row 1 0 . . // row 2, save minimal to out "0", cmp = 7 . . 1 2 3---0 // rebuild branch (row[0][4,5,6,7], row[1][3,4], row[2][1]) and compare new winner in branch 1-----------0 . 0 // save "0", cmp + 2 . x 1 2 3-1 x // rebuild branch 1---------1 1 // save "1", cmp + 2 x 3---2 3 1 // rebuild branch 2-------1 1 // save "1", cmp + 2 x 3 2 3 x // rebuild branch (when not even or odd value from input, use "x" (-1 in alg. code), when "x" copy second index to next level) 2-----3 2 // save "2", cmp + 1 x 3-----2 3 // rebuild branch (when "x", copy index to next level) 2---3 2 // save "2", cmp + 2 x 3 x 3 // rebuild branch (when "x", copy index to next level) 3---------3 3 // save "3", cmp + 1 3 // save last "3" =============== 0 0 1 1 2 2 3 3 // output, suma(cmp) = 7+2+2+2+1+2+1 = 17 </pre> == Code (javascript) == <syntaxhighlight lang="JavaScript"> <div></div> <script> // Created by Peter Mlich (2022) // build first pyramid of minimal values function pyramid_part1_buildPyramid(list, cmp, i_start, i_end, size) { var i,j,k, k_end, lvl, lvlp1; var pyramid = []; i = i_start; j = i_start+1; k = 0; lvl = 0; pyramid[lvl] = []; while (j<i_end) { glob.cycles++; if (cmp(list[i], list[j])>0) {swap(list, i, j);} pyramid[lvl][k] = i; i+=2; j+=2; k++; } if (i<i_end) // pokud je size liche cislo, pak pridej posledni prvek a preswapuj to (toho vyuziji pozdeji v part2) { if (cmp(list[i-2], list[i])>0) { tmp = list[i]; list[i ] = list[i-1]; list[i-1] = list[i-2]; list[i-2] = tmp; glob.moves += 4; pyramid[lvl][k] = i; } else {if (cmp(list[i-1], list[i])>0) { tmp = list[i]; list[i ] = list[i-1]; list[i-1] = tmp; glob.moves += 3; }} } i_end = k; lvlp1 = lvl + 1; while (i_end>1) { glob.cycles++; pyramid[lvlp1] = []; k = 0; i = 0; j = 1; // =i+1 while (j<i_end) { glob.cycles++; if (cmp(list[ pyramid[lvl][i] ], list[ pyramid[lvl][j] ])>0) {pyramid[lvlp1][k] = pyramid[lvl][j]; i+=2; j+=2; k++; continue;} else {pyramid[lvlp1][k] = pyramid[lvl][i]; i+=2; j+=2; k++; continue;} } if (i<i_end) {pyramid[lvlp1][k] = pyramid[lvl][i]; k++;} lvl++; lvlp1++; i_end = k; } return [pyramid, lvl, pyramid[lvl][0], (size>>1)<<1 != size]; // return pyramid, last lvl, last index, boolean for odd-size) } function pyramid_part3_rebuildPyramidEven(pyramid, lvl_end, bool, list, cmp, i_end, pos) { var lvl, val2, empty = -1, a, b; val2 = pyramid[0][pos]; for (lvl=0; lvl<lvl_end; lvl++) { glob.cycles++; if ((pos & 0x01) == 0) { if (pos==pyramid[lvl].length-1) { pos = pos>>1; pyramid[lvl+1][pos] = val2; //val2 = val2; continue; } b = pyramid[lvl][pos+1]; a = pyramid[lvl][pos]; pos = pos>>1; if (b==empty) {pyramid[lvl+1][pos] = a; val2 = a; continue;} if (cmp(list[a], list[b])>0) {pyramid[lvl+1][pos] = b; val2 = b; continue;} pyramid[lvl+1][pos] = a; val2 = a; } else { a = pyramid[lvl][pos-1]; b = pyramid[lvl][pos]; pos = pos>>1; if (a==empty) {pyramid[lvl+1][pos] = b; val2 = b; continue;} if (cmp(list[a], list[b])>0) {pyramid[lvl+1][pos] = b; val2 = b; continue;} pyramid[lvl+1][pos] = a; val2 = a; } } return [pyramid, lvl_end, pyramid[lvl_end][0], bool]; } // rebuild pyramid, rewrite branch by new value function pyramid_part2_rebuildPyramid(pyramid, lvl_end, bool, list, cmp, i_end, i_endm3) { var cycles = 0; var lvl, pos, val, val2, a, b, empty=-1; val = pyramid[lvl_end][0]; pos = val>>1; // pozice zleva if (bool==true && ((pos<<1)==i_endm3) && ((val & 0x01) == 0) ) // kdyz je size liche cislo a dojde k eliminaci n-2, tak posun posledni 2 cisla { bool = false; list[val] = list[val+1]; list[val+1] = list[val+2]; glob.moves += 2; // je sude, pak vymen za liche a prepocitej vsechna nutna porovnani pyramid[0][pos] = val; // pozn.: tento kod je prepsany na funkci, protoze by byl duplicitne return pyramid_part3_rebuildPyramidEven(pyramid, lvl_end, bool, list, cmp, i_end, pos); } else {if ((val & 0x01) == 0) // je sude, pak vymen za liche a prepocitej vsechna nutna porovnani { pyramid[0][pos] = val + 1; return pyramid_part3_rebuildPyramidEven(pyramid, lvl_end, bool, list, cmp, i_end, pos); } else { // je liche, pak odstran a prepocitej vsechna nutna porovnani val2 = empty; pyramid[0][pos] = val2; for (lvl=0; lvl<lvl_end; lvl++) { glob.cycles++; if ((pos & 0x01) == 0) { if (pos==pyramid[lvl].length-1) { pos = pos>>1; pyramid[lvl+1][pos] = val2; //val2 = val2 continue; } a = pyramid[lvl][pos]; b = pyramid[lvl][pos+1]; pos = pos>>1; if (a!==empty && b!==empty) { if (cmp(list[a], list[b])>0) {pyramid[lvl+1][pos] = b; val2 = b; continue;} else {pyramid[lvl+1][pos] = a; val2 = a; continue;} } if (b!==empty) {pyramid[lvl+1][pos] = b; val2 = b; continue;} pyramid[lvl+1][pos] = a; val2 = a; } else { a = pyramid[lvl][pos-1]; b = pyramid[lvl][pos]; pos = pos>>1; if (a!==empty && b!==empty) { if (cmp(list[a], list[b])>0) {pyramid[lvl+1][pos] = b; val2 = b; continue;} else {pyramid[lvl+1][pos] = a; val2 = a; continue;} } if (a!==empty) {pyramid[lvl+1][pos] = a; val2 = a; continue;} pyramid[lvl+1][pos] = b; val2 = b; } } }} return [pyramid, lvl_end, pyramid[lvl_end][0], bool]; } // princip: vyber minimum z kazdeho paru, pak porovnej minima, minima minim ... az ziskas nejmensi cislo // pak vyrad nejmensi cislo z pyramidy a propocitej celou vetev, opet ziskej minimum function PyramidSelectSort(cmp, start, end, n) { if (o.size<2) {return o.n;} var pyramid_data, i, x, y, endm3 = o.end-3; x = o.n; y = o.n==1 ? 2 : 1; pyramid_data = pyramid_part1_buildPyramid(arr[x], o.fn_cmp, o.start, o.end, o.size); // create pyramid of index from minimal values of pair i = o.start; arr[y][i] = arr[x][pyramid_data[2]]; glob.moves++; i++; while (i<o.end) { glob.cycles++; pyramid_data = pyramid_part2_rebuildPyramid(pyramid_data[0], pyramid_data[1], pyramid_data[3], arr[x], o.fn_cmp, o.end, endm3) arr[y][i] = arr[x][pyramid_data[2]]; glob.moves++; i++; } return y; } // note: code is optimalized for my tester function sortCompare (a, b) { glob.cmps++; var c = a - b; return c>0 ? 1 : (c<0 ? -1 : 0); }; function swap (list, a, b) { if (a==b) {return;} var tmp = list[a]; list[a] = list[b]; list[b] = tmp; glob.moves += 3; }; var arr = [null, [7,7,4,3,4,7,6,7,0,1,0,6,7,2,2,4], [-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1]] var glob = {moves: 0, cycles: 0, cmps: 0}; var o = {start: 0, end: 16, size: 16 - 0, n: 1, moves: 0, cycles: 0, fn_cmp: sortCompare}; var log = [], i=0, n; log[i++] = 'array-before ' + JSON.stringify(arr[1]) o.n = PyramidSelectSort(o.fn_cmp, o.start, o.end, o.n); log[i++] = 'array-after ' + JSON.stringify(arr[o.n]) log[i++] = 'glob ' + JSON.stringify(glob) log[i++] = 'n ' + JSON.stringify(o.end - o.start) document.getElementsByTagName('DIV')[0].innerHTML = log.join('<br>') /* array-before [7,7,4,3,4,7,6,7,0,1,0,6,7,2,2,4] array-after [0,0,1,2,2,3,4,4,4,6,6,7,7,7,7,7] glob {"moves":22,"cycles":78,"cmps":47} n 16 */ </script> </syntaxhighlight> oo7nwrmvehgw9osasj2pr05kd1lk101 2 285717 2410347 2409410 2022-07-30T00:24:28Z Dave Braunschweig 426084 Dave Braunschweig moved page [[User:Username/Subpage]] to [[User:U3214117]] without leaving a redirect: Rename wikitext text/x-wiki U3214117 omwcwv6bmhic0yqm8x9az00caf0lvxa 0 285721 2410256 2410107 2022-07-29T19:05:29Z Guy vandegrift 813252 wikitext text/x-wiki [[File:Factorization flowchart.svg|thumb|400px|Flowchart for a simple prime factorization code]] This lesson shows how the use of a python dictionary can facilitate the decomposition of a positive integer into a product of prime numbers. There are a few things the reader should know: * Most serious python programmers would probably prefer to download a package that contains a function that reduces an integer to a product of primes. Two promising candidates are [https://pypi.org/project/primefac/ pypi.org/project/primefac] and [https://www.sympy.org/en/index.html www.sympy.org/]. * Wikipedia has a much simpler program [[w:Special:Permalink/1086485306|this permalink]] of [[w:Trial division]] * Lacking the mental wherewithal to seize upon either of these opportunities, I googled "python prime factorization", found many codes, and selected the one that follows for two reasons:<ref>The code that follows is discussed at [https://stackoverflow.com/questions/15347174/python-finding-prime-factors stackoverflow question 15347174] and [https://scientific-python-101.readthedocs.io/_downloads/prime_factorization_solution.py scientific-python-101.readthedocs]</ref> :# The repeated factors were arranged to into exponents, which facilitates conversion into wikitext script. In this sample code, I presented the result in a simple python script.<ref>In the wikitext version, I had to add a few lines so that exponents of 1 would not appear. Example: <math>12=2^2\cdot 3^1</math> should instead read <math>12=2^2\cdot 3.</math> I ommitted the feature here in order to keep the lesson simple.</ref> :# I didn't understand the python code, and wanted master it in order to fully understand how python dictionaries are used. ==Flowchart== ==Spreadsheet== {| class="wikitable" style="text-align: center; vertical-align: bottom;" |+ <math>\text{Establishing } 1550=2\cdot 5^2 \cdot 31</math> |- ! n ! b ! b^2 ! b^2<n? ! b/n ! mod ! replace ! factor ! * |- | 1550 | 2 | 4 | TRUE | 775 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"|n: n/b | style="background:#ffffcc"|2 | |- | 775 | 2 | 4 | TRUE | 387 | 1 | b: b+1 | | |- | 775 | 3 | 9 | TRUE | 258 | 1 | b: b+1 | | |- | 775 | 4 | 16 | TRUE | 193 | 3 | b: b+1 | | |- | 775 | 5 | 25 | TRUE | 155 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/b | style="background:#ffffcc"| 5 | |- | 155 | 5 | 25 | TRUE | 31 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/b | style="background:#ffffcc"| 5 | |- | 31 | 5 | 25 | TRUE | 6 | 1 | b: b+1 | | |- | 31 | 6 | 36 | FALSE | - | - | - | | style="background:#ffffcc"| 31 |} ==Python== ‎<syntaxhighlight lang="python" line="1"> ## In a python code, the function definitions precede the code. We need only 1 function: def prime_factorization(N): ## 12=(2**2)*(2**1)*(5**1), or equivalently, 12 = 2**2 * 3 * 5 ## The dictionary, dict, expresses k**v as a "key-value" pair. In other ## words, if we look up the key, k=5, the dictionary will return the ## value, v=1. In other words: ## k**v <=> key**value <=> base**exponent (for the factored form of N) ## Begin function: n=N #--------------------# n will change in the while loop (getting smaller.) dict = {} #--------------# Start with an empty dictionary if n==1: #---------------# The factorization of 1 is a special case. return {1: 1} # In other words 1 = 1**1 (but recall that 1 is not prime.) k = 2 #------------------# Our first candidate for factorizing any integer. while k**2 <= n: #-------# Do the following unless k is "too big": if n % k: #----------# . Equivalent to: "If k is not a factor of n:" k += 1 #---------# .. increase the value of k by 1 on next attempt. else: #--------------# . Equivalent to "If k is a factor of n:" n /= k #---------# .. Replace n by n/k (for next attempt) try: #-----------# .. If k is in the dictionary, this attempts to: dict[k] += 1 # ... increase the value of k by 1. except KeyError: # .. KeyError occurs if k is not in the dictionary dict[k] = 1 # ... Add k to dictionary and set exponent v=1 # end(while)#------------# (exits while k**2 <= n:) # Here I inserted a diagnostic to be discussed later: # Documented code continues: if n > 1: #--------------# It can be shown that n is either 1 or prime. try: #---------------# . Try to enter n into dictionary dict[n] += 1 #---# .. increase exponent v->v+1 if k in dictinary except KeyError: #---# . If n is not in the dictionary, dict[n] = 1 # .. then enter n into dictionary with v=1 return dict #------# python talk for ending function ################ End def prime_factorization | Begin program ########## #import sys again=True while again == 1: text="\nEnter a positive integer you wish to factor into prime numbers:\n" N=int(input(text)) dict=prime_factorization(N) print("\n"+str(int(N))+"=") text="(" for b in dict.keys(): text+=str(int(b))+"**"+str(int(dict.get(b)))+") * (" print(text.rstrip(" * (")+"\n") print("The dictionary is:\n") print(dict) again=int(input("\nType 1 for another integer, 0 to terminate.")) </syntaxhighlight> ===Line 15: while k**2 <= n:=== This is an important line because it permits the termination of the search before one might think. I leave it as an exercise for the student to learn why any value of k larger than square root of n will never be the next factor of N. It is also left to the student to understand the fact that the time to find the factors can be cut nearly in half if we modify line 17 and increase k by 2 (instead of 1), and why this larger step cannot be taken until after k=2 has been considered. ===Lines in the code that might seem confusing=== The code uses two cleverness tricks that might seem confusing. I lack the formal training in programming to know whether such cleverness would be perceived as a flaw. I know that programmers are allowed to be clever and I know that clear programs are better than confusing programs. The fact that these tricks confused me does not necessarily mean they should be avoided. And, the nice thing about both sources of confusion is that they taught me something about python. ====Line 16: if n%k might seem backwards==== Python follows a common practice of interpreting the number 1 to be True and 0 to be False. But python also accepts any non-zero number as True. Here n%k denotes nmod&nbsp;k, which is the remainder when on divides n by k. If there is no remainder the "if" condition is not satisfied and the code stipulates that the next value of k is attempted. Any number not equal to zero is interpreted as the "if" condition being satisfied, meaning that k is a divisor of n (in that case, we replace n by n/k.) This might be good coding, but it seems to me that it would have been better to construct it along the lines of "if n%k == 0" or even "if n%k !=0". ====Lines 21-23 and 29-31: A failed attempt to change the value of a key==== In this code fragment, we ask if a given value of k has already been identified as a factor. But instead of "asking" if k has been declared a key in the dictionary, an <u>attempt</u> is made to change the value of k. If that attempt fails (because k is not among the dictionary's keys), then k is added to the list of keys, with a value of 1 (a value of 1 indicates that at this point, k, but not k² has been established to be a factor of n.) Again, there might be nothing wrong with using a key error like this. But the code would have been easier for me to follow if the if statement simply called the question of whether k was a key in dict. ==Sample output (with a really big number)== Sometimes you have to wait a few minutes to factor a number this large. But usually it comes out in just a few seconds. I would imagine that a number that was the product of only two roughly equal prime numbers would take forever because k=k+1 must be repeated a virtually infinite number of times. This number came quickly: {{Font color|blue|Enter a positive integer you wish to factor into prime numbers:}} 99283428376482768916467328912876456416375474272263423467896879187289499287374899 &nbsp; <span style="color:blue>99283428376482768916467328912876456416375474272263423467896879187289499287374899=</span> {{Font color|blue|(619**1) * (688**1) * (768**3) * (1024**17) * (64853**1) * (5302548928**1)}} &nbsp; {{Font color|blue|The dictionary is:}} &nbsp; <span style="color:blue>{619: 1, 688: 1, 768: 3, 1024: 17, 64853: 1, 5302548928.0: 1}</span> &nbsp; {{Font color|blue|Type 1 for another integer, 0 to terminate.}} {{center|'''DO NOT TRY TO FACTOR A NUMBER THIS LARGE UNLESS YOU ARE WILLING TO INTERUPT YOUR PYTHON PROGRAM.'''}} {{center|Keep in mind that the universe is only <math>4.4\times10^{26}\text{nanoseconds}</math> old.<ref>http://astronomy.nmsu.edu/geas/lectures/lecture02/slide04.html</ref>}} The number shown rounds up to 10⁸⁰. It is not likely that any computer will ever count that high. If the number to be factored consists of two prime number that are close to 10⁸⁰, the code will iterate all the way up to k coming close to 10⁴⁰. The inability of any computer to perform that many iterations helps explain how cryptocurrency works. &nbsp; 11447jonp5oa21rfm3mglo0uu79j6ht 2410257 2410256 2022-07-29T19:05:55Z Guy vandegrift 813252 wikitext text/x-wiki [[File:Factorization flowchart.svg|thumb|450px|Flowchart for a simple prime factorization code]] This lesson shows how the use of a python dictionary can facilitate the decomposition of a positive integer into a product of prime numbers. There are a few things the reader should know: * Most serious python programmers would probably prefer to download a package that contains a function that reduces an integer to a product of primes. Two promising candidates are [https://pypi.org/project/primefac/ pypi.org/project/primefac] and [https://www.sympy.org/en/index.html www.sympy.org/]. * Wikipedia has a much simpler program [[w:Special:Permalink/1086485306|this permalink]] of [[w:Trial division]] * Lacking the mental wherewithal to seize upon either of these opportunities, I googled "python prime factorization", found many codes, and selected the one that follows for two reasons:<ref>The code that follows is discussed at [https://stackoverflow.com/questions/15347174/python-finding-prime-factors stackoverflow question 15347174] and [https://scientific-python-101.readthedocs.io/_downloads/prime_factorization_solution.py scientific-python-101.readthedocs]</ref> :# The repeated factors were arranged to into exponents, which facilitates conversion into wikitext script. In this sample code, I presented the result in a simple python script.<ref>In the wikitext version, I had to add a few lines so that exponents of 1 would not appear. Example: <math>12=2^2\cdot 3^1</math> should instead read <math>12=2^2\cdot 3.</math> I ommitted the feature here in order to keep the lesson simple.</ref> :# I didn't understand the python code, and wanted master it in order to fully understand how python dictionaries are used. ==Flowchart== ==Spreadsheet== {| class="wikitable" style="text-align: center; vertical-align: bottom;" |+ <math>\text{Establishing } 1550=2\cdot 5^2 \cdot 31</math> |- ! n ! b ! b^2 ! b^2<n? ! b/n ! mod ! replace ! factor ! * |- | 1550 | 2 | 4 | TRUE | 775 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"|n: n/b | style="background:#ffffcc"|2 | |- | 775 | 2 | 4 | TRUE | 387 | 1 | b: b+1 | | |- | 775 | 3 | 9 | TRUE | 258 | 1 | b: b+1 | | |- | 775 | 4 | 16 | TRUE | 193 | 3 | b: b+1 | | |- | 775 | 5 | 25 | TRUE | 155 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/b | style="background:#ffffcc"| 5 | |- | 155 | 5 | 25 | TRUE | 31 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/b | style="background:#ffffcc"| 5 | |- | 31 | 5 | 25 | TRUE | 6 | 1 | b: b+1 | | |- | 31 | 6 | 36 | FALSE | - | - | - | | style="background:#ffffcc"| 31 |} ==Python== ‎<syntaxhighlight lang="python" line="1"> ## In a python code, the function definitions precede the code. We need only 1 function: def prime_factorization(N): ## 12=(2**2)*(2**1)*(5**1), or equivalently, 12 = 2**2 * 3 * 5 ## The dictionary, dict, expresses k**v as a "key-value" pair. In other ## words, if we look up the key, k=5, the dictionary will return the ## value, v=1. In other words: ## k**v <=> key**value <=> base**exponent (for the factored form of N) ## Begin function: n=N #--------------------# n will change in the while loop (getting smaller.) dict = {} #--------------# Start with an empty dictionary if n==1: #---------------# The factorization of 1 is a special case. return {1: 1} # In other words 1 = 1**1 (but recall that 1 is not prime.) k = 2 #------------------# Our first candidate for factorizing any integer. while k**2 <= n: #-------# Do the following unless k is "too big": if n % k: #----------# . Equivalent to: "If k is not a factor of n:" k += 1 #---------# .. increase the value of k by 1 on next attempt. else: #--------------# . Equivalent to "If k is a factor of n:" n /= k #---------# .. Replace n by n/k (for next attempt) try: #-----------# .. If k is in the dictionary, this attempts to: dict[k] += 1 # ... increase the value of k by 1. except KeyError: # .. KeyError occurs if k is not in the dictionary dict[k] = 1 # ... Add k to dictionary and set exponent v=1 # end(while)#------------# (exits while k**2 <= n:) # Here I inserted a diagnostic to be discussed later: # Documented code continues: if n > 1: #--------------# It can be shown that n is either 1 or prime. try: #---------------# . Try to enter n into dictionary dict[n] += 1 #---# .. increase exponent v->v+1 if k in dictinary except KeyError: #---# . If n is not in the dictionary, dict[n] = 1 # .. then enter n into dictionary with v=1 return dict #------# python talk for ending function ################ End def prime_factorization | Begin program ########## #import sys again=True while again == 1: text="\nEnter a positive integer you wish to factor into prime numbers:\n" N=int(input(text)) dict=prime_factorization(N) print("\n"+str(int(N))+"=") text="(" for b in dict.keys(): text+=str(int(b))+"**"+str(int(dict.get(b)))+") * (" print(text.rstrip(" * (")+"\n") print("The dictionary is:\n") print(dict) again=int(input("\nType 1 for another integer, 0 to terminate.")) </syntaxhighlight> ===Line 15: while k**2 <= n:=== This is an important line because it permits the termination of the search before one might think. I leave it as an exercise for the student to learn why any value of k larger than square root of n will never be the next factor of N. It is also left to the student to understand the fact that the time to find the factors can be cut nearly in half if we modify line 17 and increase k by 2 (instead of 1), and why this larger step cannot be taken until after k=2 has been considered. ===Lines in the code that might seem confusing=== The code uses two cleverness tricks that might seem confusing. I lack the formal training in programming to know whether such cleverness would be perceived as a flaw. I know that programmers are allowed to be clever and I know that clear programs are better than confusing programs. The fact that these tricks confused me does not necessarily mean they should be avoided. And, the nice thing about both sources of confusion is that they taught me something about python. ====Line 16: if n%k might seem backwards==== Python follows a common practice of interpreting the number 1 to be True and 0 to be False. But python also accepts any non-zero number as True. Here n%k denotes nmod&nbsp;k, which is the remainder when on divides n by k. If there is no remainder the "if" condition is not satisfied and the code stipulates that the next value of k is attempted. Any number not equal to zero is interpreted as the "if" condition being satisfied, meaning that k is a divisor of n (in that case, we replace n by n/k.) This might be good coding, but it seems to me that it would have been better to construct it along the lines of "if n%k == 0" or even "if n%k !=0". ====Lines 21-23 and 29-31: A failed attempt to change the value of a key==== In this code fragment, we ask if a given value of k has already been identified as a factor. But instead of "asking" if k has been declared a key in the dictionary, an <u>attempt</u> is made to change the value of k. If that attempt fails (because k is not among the dictionary's keys), then k is added to the list of keys, with a value of 1 (a value of 1 indicates that at this point, k, but not k² has been established to be a factor of n.) Again, there might be nothing wrong with using a key error like this. But the code would have been easier for me to follow if the if statement simply called the question of whether k was a key in dict. ==Sample output (with a really big number)== Sometimes you have to wait a few minutes to factor a number this large. But usually it comes out in just a few seconds. I would imagine that a number that was the product of only two roughly equal prime numbers would take forever because k=k+1 must be repeated a virtually infinite number of times. This number came quickly: {{Font color|blue|Enter a positive integer you wish to factor into prime numbers:}} 99283428376482768916467328912876456416375474272263423467896879187289499287374899 &nbsp; <span style="color:blue>99283428376482768916467328912876456416375474272263423467896879187289499287374899=</span> {{Font color|blue|(619**1) * (688**1) * (768**3) * (1024**17) * (64853**1) * (5302548928**1)}} &nbsp; {{Font color|blue|The dictionary is:}} &nbsp; <span style="color:blue>{619: 1, 688: 1, 768: 3, 1024: 17, 64853: 1, 5302548928.0: 1}</span> &nbsp; {{Font color|blue|Type 1 for another integer, 0 to terminate.}} {{center|'''DO NOT TRY TO FACTOR A NUMBER THIS LARGE UNLESS YOU ARE WILLING TO INTERUPT YOUR PYTHON PROGRAM.'''}} {{center|Keep in mind that the universe is only <math>4.4\times10^{26}\text{nanoseconds}</math> old.<ref>http://astronomy.nmsu.edu/geas/lectures/lecture02/slide04.html</ref>}} The number shown rounds up to 10⁸⁰. It is not likely that any computer will ever count that high. If the number to be factored consists of two prime number that are close to 10⁸⁰, the code will iterate all the way up to k coming close to 10⁴⁰. The inability of any computer to perform that many iterations helps explain how cryptocurrency works. &nbsp; hh4sumsawds319o2ftz26590ekdwimu 2410258 2410257 2022-07-29T19:06:08Z Guy vandegrift 813252 /* Flowchart */ wikitext text/x-wiki [[File:Factorization flowchart.svg|thumb|450px|Flowchart for a simple prime factorization code]] This lesson shows how the use of a python dictionary can facilitate the decomposition of a positive integer into a product of prime numbers. There are a few things the reader should know: * Most serious python programmers would probably prefer to download a package that contains a function that reduces an integer to a product of primes. Two promising candidates are [https://pypi.org/project/primefac/ pypi.org/project/primefac] and [https://www.sympy.org/en/index.html www.sympy.org/]. * Wikipedia has a much simpler program [[w:Special:Permalink/1086485306|this permalink]] of [[w:Trial division]] * Lacking the mental wherewithal to seize upon either of these opportunities, I googled "python prime factorization", found many codes, and selected the one that follows for two reasons:<ref>The code that follows is discussed at [https://stackoverflow.com/questions/15347174/python-finding-prime-factors stackoverflow question 15347174] and [https://scientific-python-101.readthedocs.io/_downloads/prime_factorization_solution.py scientific-python-101.readthedocs]</ref> :# The repeated factors were arranged to into exponents, which facilitates conversion into wikitext script. In this sample code, I presented the result in a simple python script.<ref>In the wikitext version, I had to add a few lines so that exponents of 1 would not appear. Example: <math>12=2^2\cdot 3^1</math> should instead read <math>12=2^2\cdot 3.</math> I ommitted the feature here in order to keep the lesson simple.</ref> :# I didn't understand the python code, and wanted master it in order to fully understand how python dictionaries are used. ==Spreadsheet== {| class="wikitable" style="text-align: center; vertical-align: bottom;" |+ <math>\text{Establishing } 1550=2\cdot 5^2 \cdot 31</math> |- ! n ! b ! b^2 ! b^2<n? ! b/n ! mod ! replace ! factor ! * |- | 1550 | 2 | 4 | TRUE | 775 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"|n: n/b | style="background:#ffffcc"|2 | |- | 775 | 2 | 4 | TRUE | 387 | 1 | b: b+1 | | |- | 775 | 3 | 9 | TRUE | 258 | 1 | b: b+1 | | |- | 775 | 4 | 16 | TRUE | 193 | 3 | b: b+1 | | |- | 775 | 5 | 25 | TRUE | 155 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/b | style="background:#ffffcc"| 5 | |- | 155 | 5 | 25 | TRUE | 31 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/b | style="background:#ffffcc"| 5 | |- | 31 | 5 | 25 | TRUE | 6 | 1 | b: b+1 | | |- | 31 | 6 | 36 | FALSE | - | - | - | | style="background:#ffffcc"| 31 |} ==Python== ‎<syntaxhighlight lang="python" line="1"> ## In a python code, the function definitions precede the code. We need only 1 function: def prime_factorization(N): ## 12=(2**2)*(2**1)*(5**1), or equivalently, 12 = 2**2 * 3 * 5 ## The dictionary, dict, expresses k**v as a "key-value" pair. In other ## words, if we look up the key, k=5, the dictionary will return the ## value, v=1. In other words: ## k**v <=> key**value <=> base**exponent (for the factored form of N) ## Begin function: n=N #--------------------# n will change in the while loop (getting smaller.) dict = {} #--------------# Start with an empty dictionary if n==1: #---------------# The factorization of 1 is a special case. return {1: 1} # In other words 1 = 1**1 (but recall that 1 is not prime.) k = 2 #------------------# Our first candidate for factorizing any integer. while k**2 <= n: #-------# Do the following unless k is "too big": if n % k: #----------# . Equivalent to: "If k is not a factor of n:" k += 1 #---------# .. increase the value of k by 1 on next attempt. else: #--------------# . Equivalent to "If k is a factor of n:" n /= k #---------# .. Replace n by n/k (for next attempt) try: #-----------# .. If k is in the dictionary, this attempts to: dict[k] += 1 # ... increase the value of k by 1. except KeyError: # .. KeyError occurs if k is not in the dictionary dict[k] = 1 # ... Add k to dictionary and set exponent v=1 # end(while)#------------# (exits while k**2 <= n:) # Here I inserted a diagnostic to be discussed later: # Documented code continues: if n > 1: #--------------# It can be shown that n is either 1 or prime. try: #---------------# . Try to enter n into dictionary dict[n] += 1 #---# .. increase exponent v->v+1 if k in dictinary except KeyError: #---# . If n is not in the dictionary, dict[n] = 1 # .. then enter n into dictionary with v=1 return dict #------# python talk for ending function ################ End def prime_factorization | Begin program ########## #import sys again=True while again == 1: text="\nEnter a positive integer you wish to factor into prime numbers:\n" N=int(input(text)) dict=prime_factorization(N) print("\n"+str(int(N))+"=") text="(" for b in dict.keys(): text+=str(int(b))+"**"+str(int(dict.get(b)))+") * (" print(text.rstrip(" * (")+"\n") print("The dictionary is:\n") print(dict) again=int(input("\nType 1 for another integer, 0 to terminate.")) </syntaxhighlight> ===Line 15: while k**2 <= n:=== This is an important line because it permits the termination of the search before one might think. I leave it as an exercise for the student to learn why any value of k larger than square root of n will never be the next factor of N. It is also left to the student to understand the fact that the time to find the factors can be cut nearly in half if we modify line 17 and increase k by 2 (instead of 1), and why this larger step cannot be taken until after k=2 has been considered. ===Lines in the code that might seem confusing=== The code uses two cleverness tricks that might seem confusing. I lack the formal training in programming to know whether such cleverness would be perceived as a flaw. I know that programmers are allowed to be clever and I know that clear programs are better than confusing programs. The fact that these tricks confused me does not necessarily mean they should be avoided. And, the nice thing about both sources of confusion is that they taught me something about python. ====Line 16: if n%k might seem backwards==== Python follows a common practice of interpreting the number 1 to be True and 0 to be False. But python also accepts any non-zero number as True. Here n%k denotes nmod&nbsp;k, which is the remainder when on divides n by k. If there is no remainder the "if" condition is not satisfied and the code stipulates that the next value of k is attempted. Any number not equal to zero is interpreted as the "if" condition being satisfied, meaning that k is a divisor of n (in that case, we replace n by n/k.) This might be good coding, but it seems to me that it would have been better to construct it along the lines of "if n%k == 0" or even "if n%k !=0". ====Lines 21-23 and 29-31: A failed attempt to change the value of a key==== In this code fragment, we ask if a given value of k has already been identified as a factor. But instead of "asking" if k has been declared a key in the dictionary, an <u>attempt</u> is made to change the value of k. If that attempt fails (because k is not among the dictionary's keys), then k is added to the list of keys, with a value of 1 (a value of 1 indicates that at this point, k, but not k² has been established to be a factor of n.) Again, there might be nothing wrong with using a key error like this. But the code would have been easier for me to follow if the if statement simply called the question of whether k was a key in dict. ==Sample output (with a really big number)== Sometimes you have to wait a few minutes to factor a number this large. But usually it comes out in just a few seconds. I would imagine that a number that was the product of only two roughly equal prime numbers would take forever because k=k+1 must be repeated a virtually infinite number of times. This number came quickly: {{Font color|blue|Enter a positive integer you wish to factor into prime numbers:}} 99283428376482768916467328912876456416375474272263423467896879187289499287374899 &nbsp; <span style="color:blue>99283428376482768916467328912876456416375474272263423467896879187289499287374899=</span> {{Font color|blue|(619**1) * (688**1) * (768**3) * (1024**17) * (64853**1) * (5302548928**1)}} &nbsp; {{Font color|blue|The dictionary is:}} &nbsp; <span style="color:blue>{619: 1, 688: 1, 768: 3, 1024: 17, 64853: 1, 5302548928.0: 1}</span> &nbsp; {{Font color|blue|Type 1 for another integer, 0 to terminate.}} {{center|'''DO NOT TRY TO FACTOR A NUMBER THIS LARGE UNLESS YOU ARE WILLING TO INTERUPT YOUR PYTHON PROGRAM.'''}} {{center|Keep in mind that the universe is only <math>4.4\times10^{26}\text{nanoseconds}</math> old.<ref>http://astronomy.nmsu.edu/geas/lectures/lecture02/slide04.html</ref>}} The number shown rounds up to 10⁸⁰. It is not likely that any computer will ever count that high. If the number to be factored consists of two prime number that are close to 10⁸⁰, the code will iterate all the way up to k coming close to 10⁴⁰. The inability of any computer to perform that many iterations helps explain how cryptocurrency works. &nbsp; d4pjs277xcbrcb9n2bo03dnp6zkkwgu 2410259 2410258 2022-07-29T19:06:36Z Guy vandegrift 813252 wikitext text/x-wiki [[File:Factorization flowchart.svg|thumb|450px|Flowchart for a simple prime factorization code]] This lesson shows how the use of a python dictionary can facilitate the decomposition of a positive integer into a product of prime numbers. There are a few things the reader should know: * Most serious python programmers would probably prefer to download a package that contains a function that reduces an integer to a product of primes. Two promising candidates are [https://pypi.org/project/primefac/ pypi.org/project/primefac] and [https://www.sympy.org/en/index.html www.sympy.org/]. * Wikipedia has a much simpler program [[w:Special:Permalink/1086485306|this permalink]] of [[w:Trial division]] * Lacking the mental wherewithal to seize upon either of these opportunities, I googled "python prime factorization", found many codes, and selected the one that follows for two reasons:<ref>The code that follows is discussed at [https://stackoverflow.com/questions/15347174/python-finding-prime-factors stackoverflow question 15347174] and [https://scientific-python-101.readthedocs.io/_downloads/prime_factorization_solution.py scientific-python-101.readthedocs]</ref> :# The repeated factors were arranged to into exponents, which facilitates conversion into wikitext script. In this sample code, I presented the result in a simple python script.<ref>In the wikitext version, I had to add a few lines so that exponents of 1 would not appear. Example: <math>12=2^2\cdot 3^1</math> should instead read <math>12=2^2\cdot 3.</math> I ommitted the feature here in order to keep the lesson simple.</ref> :# I didn't understand the python code, and wanted master it in order to fully understand how python dictionaries are used. {| class="wikitable" style="text-align: center; vertical-align: bottom;" |+ <math>\text{Establishing } 1550=2\cdot 5^2 \cdot 31</math> |- ! n ! b ! b^2 ! b^2<n? ! b/n ! mod ! replace ! factor ! * |- | 1550 | 2 | 4 | TRUE | 775 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"|n: n/b | style="background:#ffffcc"|2 | |- | 775 | 2 | 4 | TRUE | 387 | 1 | b: b+1 | | |- | 775 | 3 | 9 | TRUE | 258 | 1 | b: b+1 | | |- | 775 | 4 | 16 | TRUE | 193 | 3 | b: b+1 | | |- | 775 | 5 | 25 | TRUE | 155 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/b | style="background:#ffffcc"| 5 | |- | 155 | 5 | 25 | TRUE | 31 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/b | style="background:#ffffcc"| 5 | |- | 31 | 5 | 25 | TRUE | 6 | 1 | b: b+1 | | |- | 31 | 6 | 36 | FALSE | - | - | - | | style="background:#ffffcc"| 31 |} ==Python== ‎<syntaxhighlight lang="python" line="1"> ## In a python code, the function definitions precede the code. We need only 1 function: def prime_factorization(N): ## 12=(2**2)*(2**1)*(5**1), or equivalently, 12 = 2**2 * 3 * 5 ## The dictionary, dict, expresses k**v as a "key-value" pair. In other ## words, if we look up the key, k=5, the dictionary will return the ## value, v=1. In other words: ## k**v <=> key**value <=> base**exponent (for the factored form of N) ## Begin function: n=N #--------------------# n will change in the while loop (getting smaller.) dict = {} #--------------# Start with an empty dictionary if n==1: #---------------# The factorization of 1 is a special case. return {1: 1} # In other words 1 = 1**1 (but recall that 1 is not prime.) k = 2 #------------------# Our first candidate for factorizing any integer. while k**2 <= n: #-------# Do the following unless k is "too big": if n % k: #----------# . Equivalent to: "If k is not a factor of n:" k += 1 #---------# .. increase the value of k by 1 on next attempt. else: #--------------# . Equivalent to "If k is a factor of n:" n /= k #---------# .. Replace n by n/k (for next attempt) try: #-----------# .. If k is in the dictionary, this attempts to: dict[k] += 1 # ... increase the value of k by 1. except KeyError: # .. KeyError occurs if k is not in the dictionary dict[k] = 1 # ... Add k to dictionary and set exponent v=1 # end(while)#------------# (exits while k**2 <= n:) # Here I inserted a diagnostic to be discussed later: # Documented code continues: if n > 1: #--------------# It can be shown that n is either 1 or prime. try: #---------------# . Try to enter n into dictionary dict[n] += 1 #---# .. increase exponent v->v+1 if k in dictinary except KeyError: #---# . If n is not in the dictionary, dict[n] = 1 # .. then enter n into dictionary with v=1 return dict #------# python talk for ending function ################ End def prime_factorization | Begin program ########## #import sys again=True while again == 1: text="\nEnter a positive integer you wish to factor into prime numbers:\n" N=int(input(text)) dict=prime_factorization(N) print("\n"+str(int(N))+"=") text="(" for b in dict.keys(): text+=str(int(b))+"**"+str(int(dict.get(b)))+") * (" print(text.rstrip(" * (")+"\n") print("The dictionary is:\n") print(dict) again=int(input("\nType 1 for another integer, 0 to terminate.")) </syntaxhighlight> ===Line 15: while k**2 <= n:=== This is an important line because it permits the termination of the search before one might think. I leave it as an exercise for the student to learn why any value of k larger than square root of n will never be the next factor of N. It is also left to the student to understand the fact that the time to find the factors can be cut nearly in half if we modify line 17 and increase k by 2 (instead of 1), and why this larger step cannot be taken until after k=2 has been considered. ===Lines in the code that might seem confusing=== The code uses two cleverness tricks that might seem confusing. I lack the formal training in programming to know whether such cleverness would be perceived as a flaw. I know that programmers are allowed to be clever and I know that clear programs are better than confusing programs. The fact that these tricks confused me does not necessarily mean they should be avoided. And, the nice thing about both sources of confusion is that they taught me something about python. ====Line 16: if n%k might seem backwards==== Python follows a common practice of interpreting the number 1 to be True and 0 to be False. But python also accepts any non-zero number as True. Here n%k denotes nmod&nbsp;k, which is the remainder when on divides n by k. If there is no remainder the "if" condition is not satisfied and the code stipulates that the next value of k is attempted. Any number not equal to zero is interpreted as the "if" condition being satisfied, meaning that k is a divisor of n (in that case, we replace n by n/k.) This might be good coding, but it seems to me that it would have been better to construct it along the lines of "if n%k == 0" or even "if n%k !=0". ====Lines 21-23 and 29-31: A failed attempt to change the value of a key==== In this code fragment, we ask if a given value of k has already been identified as a factor. But instead of "asking" if k has been declared a key in the dictionary, an <u>attempt</u> is made to change the value of k. If that attempt fails (because k is not among the dictionary's keys), then k is added to the list of keys, with a value of 1 (a value of 1 indicates that at this point, k, but not k² has been established to be a factor of n.) Again, there might be nothing wrong with using a key error like this. But the code would have been easier for me to follow if the if statement simply called the question of whether k was a key in dict. ==Sample output (with a really big number)== Sometimes you have to wait a few minutes to factor a number this large. But usually it comes out in just a few seconds. I would imagine that a number that was the product of only two roughly equal prime numbers would take forever because k=k+1 must be repeated a virtually infinite number of times. This number came quickly: {{Font color|blue|Enter a positive integer you wish to factor into prime numbers:}} 99283428376482768916467328912876456416375474272263423467896879187289499287374899 &nbsp; <span style="color:blue>99283428376482768916467328912876456416375474272263423467896879187289499287374899=</span> {{Font color|blue|(619**1) * (688**1) * (768**3) * (1024**17) * (64853**1) * (5302548928**1)}} &nbsp; {{Font color|blue|The dictionary is:}} &nbsp; <span style="color:blue>{619: 1, 688: 1, 768: 3, 1024: 17, 64853: 1, 5302548928.0: 1}</span> &nbsp; {{Font color|blue|Type 1 for another integer, 0 to terminate.}} {{center|'''DO NOT TRY TO FACTOR A NUMBER THIS LARGE UNLESS YOU ARE WILLING TO INTERUPT YOUR PYTHON PROGRAM.'''}} {{center|Keep in mind that the universe is only <math>4.4\times10^{26}\text{nanoseconds}</math> old.<ref>http://astronomy.nmsu.edu/geas/lectures/lecture02/slide04.html</ref>}} The number shown rounds up to 10⁸⁰. It is not likely that any computer will ever count that high. If the number to be factored consists of two prime number that are close to 10⁸⁰, the code will iterate all the way up to k coming close to 10⁴⁰. The inability of any computer to perform that many iterations helps explain how cryptocurrency works. &nbsp; lqbt06ahvzcdh5w29p28dwzk0bz8oiv 2410260 2410259 2022-07-29T19:07:22Z Guy vandegrift 813252 /* Line 15: while k**2 <= n: */ wikitext text/x-wiki [[File:Factorization flowchart.svg|thumb|450px|Flowchart for a simple prime factorization code]] This lesson shows how the use of a python dictionary can facilitate the decomposition of a positive integer into a product of prime numbers. There are a few things the reader should know: * Most serious python programmers would probably prefer to download a package that contains a function that reduces an integer to a product of primes. Two promising candidates are [https://pypi.org/project/primefac/ pypi.org/project/primefac] and [https://www.sympy.org/en/index.html www.sympy.org/]. * Wikipedia has a much simpler program [[w:Special:Permalink/1086485306|this permalink]] of [[w:Trial division]] * Lacking the mental wherewithal to seize upon either of these opportunities, I googled "python prime factorization", found many codes, and selected the one that follows for two reasons:<ref>The code that follows is discussed at [https://stackoverflow.com/questions/15347174/python-finding-prime-factors stackoverflow question 15347174] and [https://scientific-python-101.readthedocs.io/_downloads/prime_factorization_solution.py scientific-python-101.readthedocs]</ref> :# The repeated factors were arranged to into exponents, which facilitates conversion into wikitext script. In this sample code, I presented the result in a simple python script.<ref>In the wikitext version, I had to add a few lines so that exponents of 1 would not appear. Example: <math>12=2^2\cdot 3^1</math> should instead read <math>12=2^2\cdot 3.</math> I ommitted the feature here in order to keep the lesson simple.</ref> :# I didn't understand the python code, and wanted master it in order to fully understand how python dictionaries are used. {| class="wikitable" style="text-align: center; vertical-align: bottom;" |+ <math>\text{Establishing } 1550=2\cdot 5^2 \cdot 31</math> |- ! n ! b ! b^2 ! b^2<n? ! b/n ! mod ! replace ! factor ! * |- | 1550 | 2 | 4 | TRUE | 775 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"|n: n/b | style="background:#ffffcc"|2 | |- | 775 | 2 | 4 | TRUE | 387 | 1 | b: b+1 | | |- | 775 | 3 | 9 | TRUE | 258 | 1 | b: b+1 | | |- | 775 | 4 | 16 | TRUE | 193 | 3 | b: b+1 | | |- | 775 | 5 | 25 | TRUE | 155 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/b | style="background:#ffffcc"| 5 | |- | 155 | 5 | 25 | TRUE | 31 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/b | style="background:#ffffcc"| 5 | |- | 31 | 5 | 25 | TRUE | 6 | 1 | b: b+1 | | |- | 31 | 6 | 36 | FALSE | - | - | - | | style="background:#ffffcc"| 31 |} ==Python== ‎<syntaxhighlight lang="python" line="1"> ## In a python code, the function definitions precede the code. We need only 1 function: def prime_factorization(N): ## 12=(2**2)*(2**1)*(5**1), or equivalently, 12 = 2**2 * 3 * 5 ## The dictionary, dict, expresses k**v as a "key-value" pair. In other ## words, if we look up the key, k=5, the dictionary will return the ## value, v=1. In other words: ## k**v <=> key**value <=> base**exponent (for the factored form of N) ## Begin function: n=N #--------------------# n will change in the while loop (getting smaller.) dict = {} #--------------# Start with an empty dictionary if n==1: #---------------# The factorization of 1 is a special case. return {1: 1} # In other words 1 = 1**1 (but recall that 1 is not prime.) k = 2 #------------------# Our first candidate for factorizing any integer. while k**2 <= n: #-------# Do the following unless k is "too big": if n % k: #----------# . Equivalent to: "If k is not a factor of n:" k += 1 #---------# .. increase the value of k by 1 on next attempt. else: #--------------# . Equivalent to "If k is a factor of n:" n /= k #---------# .. Replace n by n/k (for next attempt) try: #-----------# .. If k is in the dictionary, this attempts to: dict[k] += 1 # ... increase the value of k by 1. except KeyError: # .. KeyError occurs if k is not in the dictionary dict[k] = 1 # ... Add k to dictionary and set exponent v=1 # end(while)#------------# (exits while k**2 <= n:) # Here I inserted a diagnostic to be discussed later: # Documented code continues: if n > 1: #--------------# It can be shown that n is either 1 or prime. try: #---------------# . Try to enter n into dictionary dict[n] += 1 #---# .. increase exponent v->v+1 if k in dictinary except KeyError: #---# . If n is not in the dictionary, dict[n] = 1 # .. then enter n into dictionary with v=1 return dict #------# python talk for ending function ################ End def prime_factorization | Begin program ########## #import sys again=True while again == 1: text="\nEnter a positive integer you wish to factor into prime numbers:\n" N=int(input(text)) dict=prime_factorization(N) print("\n"+str(int(N))+"=") text="(" for b in dict.keys(): text+=str(int(b))+"**"+str(int(dict.get(b)))+") * (" print(text.rstrip(" * (")+"\n") print("The dictionary is:\n") print(dict) again=int(input("\nType 1 for another integer, 0 to terminate.")) </syntaxhighlight> ==Comments on code== ===Line 15: while k**2 <= n:=== This is an important line because it permits the termination of the search before one might think. I leave it as an exercise for the student to learn why any value of k larger than square root of n will never be the next factor of N. It is also left to the student to understand the fact that the time to find the factors can be cut nearly in half if we modify line 17 and increase k by 2 (instead of 1), and why this larger step cannot be taken until after k=2 has been considered. ===Lines in the code that might seem confusing=== The code uses two cleverness tricks that might seem confusing. I lack the formal training in programming to know whether such cleverness would be perceived as a flaw. I know that programmers are allowed to be clever and I know that clear programs are better than confusing programs. The fact that these tricks confused me does not necessarily mean they should be avoided. And, the nice thing about both sources of confusion is that they taught me something about python. ====Line 16: if n%k might seem backwards==== Python follows a common practice of interpreting the number 1 to be True and 0 to be False. But python also accepts any non-zero number as True. Here n%k denotes nmod&nbsp;k, which is the remainder when on divides n by k. If there is no remainder the "if" condition is not satisfied and the code stipulates that the next value of k is attempted. Any number not equal to zero is interpreted as the "if" condition being satisfied, meaning that k is a divisor of n (in that case, we replace n by n/k.) This might be good coding, but it seems to me that it would have been better to construct it along the lines of "if n%k == 0" or even "if n%k !=0". ====Lines 21-23 and 29-31: A failed attempt to change the value of a key==== In this code fragment, we ask if a given value of k has already been identified as a factor. But instead of "asking" if k has been declared a key in the dictionary, an <u>attempt</u> is made to change the value of k. If that attempt fails (because k is not among the dictionary's keys), then k is added to the list of keys, with a value of 1 (a value of 1 indicates that at this point, k, but not k² has been established to be a factor of n.) Again, there might be nothing wrong with using a key error like this. But the code would have been easier for me to follow if the if statement simply called the question of whether k was a key in dict. ==Sample output (with a really big number)== Sometimes you have to wait a few minutes to factor a number this large. But usually it comes out in just a few seconds. I would imagine that a number that was the product of only two roughly equal prime numbers would take forever because k=k+1 must be repeated a virtually infinite number of times. This number came quickly: {{Font color|blue|Enter a positive integer you wish to factor into prime numbers:}} 99283428376482768916467328912876456416375474272263423467896879187289499287374899 &nbsp; <span style="color:blue>99283428376482768916467328912876456416375474272263423467896879187289499287374899=</span> {{Font color|blue|(619**1) * (688**1) * (768**3) * (1024**17) * (64853**1) * (5302548928**1)}} &nbsp; {{Font color|blue|The dictionary is:}} &nbsp; <span style="color:blue>{619: 1, 688: 1, 768: 3, 1024: 17, 64853: 1, 5302548928.0: 1}</span> &nbsp; {{Font color|blue|Type 1 for another integer, 0 to terminate.}} {{center|'''DO NOT TRY TO FACTOR A NUMBER THIS LARGE UNLESS YOU ARE WILLING TO INTERUPT YOUR PYTHON PROGRAM.'''}} {{center|Keep in mind that the universe is only <math>4.4\times10^{26}\text{nanoseconds}</math> old.<ref>http://astronomy.nmsu.edu/geas/lectures/lecture02/slide04.html</ref>}} The number shown rounds up to 10⁸⁰. It is not likely that any computer will ever count that high. If the number to be factored consists of two prime number that are close to 10⁸⁰, the code will iterate all the way up to k coming close to 10⁴⁰. The inability of any computer to perform that many iterations helps explain how cryptocurrency works. &nbsp; rrwcnd7iwt6h0fst2w4eqr2f1a7aag5 2410261 2410260 2022-07-29T19:08:29Z Guy vandegrift 813252 wikitext text/x-wiki [[File:Factorization flowchart.svg|thumb|450px|Flowchart for a simple prime factorization code]] This lesson shows how the use of a python dictionary can facilitate the decomposition of a positive integer into a product of prime numbers. There are a few things the reader should know: * Most serious python programmers would probably prefer to download a package that contains a function that reduces an integer to a product of primes. Two promising candidates are [https://pypi.org/project/primefac/ pypi.org/project/primefac] and [https://www.sympy.org/en/index.html www.sympy.org/]. * Wikipedia has a much simpler program [[w:Special:Permalink/1086485306|this permalink]] of [[w:Trial division]] * Lacking the mental wherewithal to seize upon either of these opportunities, I googled "python prime factorization", found many codes, and selected the one that follows for two reasons:<ref>The code that follows is discussed at [https://stackoverflow.com/questions/15347174/python-finding-prime-factors stackoverflow question 15347174] and [https://scientific-python-101.readthedocs.io/_downloads/prime_factorization_solution.py scientific-python-101.readthedocs]</ref> :# The repeated factors were arranged to into exponents, which facilitates conversion into wikitext script. In this sample code, I presented the result in a simple python script.<ref>In the wikitext version, I had to add a few lines so that exponents of 1 would not appear. Example: <math>12=2^2\cdot 3^1</math> should instead read <math>12=2^2\cdot 3.</math> I ommitted the feature here in order to keep the lesson simple.</ref> :# I didn't understand the python code, and wanted master it in order to fully understand how python dictionaries are used. ===bk1=== {| class="wikitable" style="text-align: center; vertical-align: bottom;" |+ <math>\text{Establishing } 1550=2\cdot 5^2 \cdot 31</math> |- ! n ! b ! b^2 ! b^2<n? ! b/n ! mod ! replace ! factor ! * |- | 1550 | 2 | 4 | TRUE | 775 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"|n: n/b | style="background:#ffffcc"|2 | |- | 775 | 2 | 4 | TRUE | 387 | 1 | b: b+1 | | |- | 775 | 3 | 9 | TRUE | 258 | 1 | b: b+1 | | |- | 775 | 4 | 16 | TRUE | 193 | 3 | b: b+1 | | |- | 775 | 5 | 25 | TRUE | 155 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/b | style="background:#ffffcc"| 5 | |- | 155 | 5 | 25 | TRUE | 31 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/b | style="background:#ffffcc"| 5 | |- | 31 | 5 | 25 | TRUE | 6 | 1 | b: b+1 | | |- | 31 | 6 | 36 | FALSE | - | - | - | | style="background:#ffffcc"| 31 |} ==Python== ‎<syntaxhighlight lang="python" line="1"> ## In a python code, the function definitions precede the code. We need only 1 function: def prime_factorization(N): ## 12=(2**2)*(2**1)*(5**1), or equivalently, 12 = 2**2 * 3 * 5 ## The dictionary, dict, expresses k**v as a "key-value" pair. In other ## words, if we look up the key, k=5, the dictionary will return the ## value, v=1. In other words: ## k**v <=> key**value <=> base**exponent (for the factored form of N) ## Begin function: n=N #--------------------# n will change in the while loop (getting smaller.) dict = {} #--------------# Start with an empty dictionary if n==1: #---------------# The factorization of 1 is a special case. return {1: 1} # In other words 1 = 1**1 (but recall that 1 is not prime.) k = 2 #------------------# Our first candidate for factorizing any integer. while k**2 <= n: #-------# Do the following unless k is "too big": if n % k: #----------# . Equivalent to: "If k is not a factor of n:" k += 1 #---------# .. increase the value of k by 1 on next attempt. else: #--------------# . Equivalent to "If k is a factor of n:" n /= k #---------# .. Replace n by n/k (for next attempt) try: #-----------# .. If k is in the dictionary, this attempts to: dict[k] += 1 # ... increase the value of k by 1. except KeyError: # .. KeyError occurs if k is not in the dictionary dict[k] = 1 # ... Add k to dictionary and set exponent v=1 # end(while)#------------# (exits while k**2 <= n:) # Here I inserted a diagnostic to be discussed later: # Documented code continues: if n > 1: #--------------# It can be shown that n is either 1 or prime. try: #---------------# . Try to enter n into dictionary dict[n] += 1 #---# .. increase exponent v->v+1 if k in dictinary except KeyError: #---# . If n is not in the dictionary, dict[n] = 1 # .. then enter n into dictionary with v=1 return dict #------# python talk for ending function ################ End def prime_factorization | Begin program ########## #import sys again=True while again == 1: text="\nEnter a positive integer you wish to factor into prime numbers:\n" N=int(input(text)) dict=prime_factorization(N) print("\n"+str(int(N))+"=") text="(" for b in dict.keys(): text+=str(int(b))+"**"+str(int(dict.get(b)))+") * (" print(text.rstrip(" * (")+"\n") print("The dictionary is:\n") print(dict) again=int(input("\nType 1 for another integer, 0 to terminate.")) </syntaxhighlight> ==Comments on code== ===Line 15: while k**2 <= n:=== This is an important line because it permits the termination of the search before one might think. I leave it as an exercise for the student to learn why any value of k larger than square root of n will never be the next factor of N. It is also left to the student to understand the fact that the time to find the factors can be cut nearly in half if we modify line 17 and increase k by 2 (instead of 1), and why this larger step cannot be taken until after k=2 has been considered. ===Lines in the code that might seem confusing=== The code uses two cleverness tricks that might seem confusing. I lack the formal training in programming to know whether such cleverness would be perceived as a flaw. I know that programmers are allowed to be clever and I know that clear programs are better than confusing programs. The fact that these tricks confused me does not necessarily mean they should be avoided. And, the nice thing about both sources of confusion is that they taught me something about python. ====Line 16: if n%k might seem backwards==== Python follows a common practice of interpreting the number 1 to be True and 0 to be False. But python also accepts any non-zero number as True. Here n%k denotes nmod&nbsp;k, which is the remainder when on divides n by k. If there is no remainder the "if" condition is not satisfied and the code stipulates that the next value of k is attempted. Any number not equal to zero is interpreted as the "if" condition being satisfied, meaning that k is a divisor of n (in that case, we replace n by n/k.) This might be good coding, but it seems to me that it would have been better to construct it along the lines of "if n%k == 0" or even "if n%k !=0". ====Lines 21-23 and 29-31: A failed attempt to change the value of a key==== In this code fragment, we ask if a given value of k has already been identified as a factor. But instead of "asking" if k has been declared a key in the dictionary, an <u>attempt</u> is made to change the value of k. If that attempt fails (because k is not among the dictionary's keys), then k is added to the list of keys, with a value of 1 (a value of 1 indicates that at this point, k, but not k² has been established to be a factor of n.) Again, there might be nothing wrong with using a key error like this. But the code would have been easier for me to follow if the if statement simply called the question of whether k was a key in dict. ==Sample output (with a really big number)== Sometimes you have to wait a few minutes to factor a number this large. But usually it comes out in just a few seconds. I would imagine that a number that was the product of only two roughly equal prime numbers would take forever because k=k+1 must be repeated a virtually infinite number of times. This number came quickly: {{Font color|blue|Enter a positive integer you wish to factor into prime numbers:}} 99283428376482768916467328912876456416375474272263423467896879187289499287374899 &nbsp; <span style="color:blue>99283428376482768916467328912876456416375474272263423467896879187289499287374899=</span> {{Font color|blue|(619**1) * (688**1) * (768**3) * (1024**17) * (64853**1) * (5302548928**1)}} &nbsp; {{Font color|blue|The dictionary is:}} &nbsp; <span style="color:blue>{619: 1, 688: 1, 768: 3, 1024: 17, 64853: 1, 5302548928.0: 1}</span> &nbsp; {{Font color|blue|Type 1 for another integer, 0 to terminate.}} {{center|'''DO NOT TRY TO FACTOR A NUMBER THIS LARGE UNLESS YOU ARE WILLING TO INTERUPT YOUR PYTHON PROGRAM.'''}} {{center|Keep in mind that the universe is only <math>4.4\times10^{26}\text{nanoseconds}</math> old.<ref>http://astronomy.nmsu.edu/geas/lectures/lecture02/slide04.html</ref>}} The number shown rounds up to 10⁸⁰. It is not likely that any computer will ever count that high. If the number to be factored consists of two prime number that are close to 10⁸⁰, the code will iterate all the way up to k coming close to 10⁴⁰. The inability of any computer to perform that many iterations helps explain how cryptocurrency works. &nbsp; e2knqw4ers25avj6frqgoqqenjwt9ac 2410262 2410261 2022-07-29T19:09:31Z Guy vandegrift 813252 /* bk1 */ wikitext text/x-wiki [[File:Factorization flowchart.svg|thumb|450px|Flowchart for a simple prime factorization code]] This lesson shows how the use of a python dictionary can facilitate the decomposition of a positive integer into a product of prime numbers. There are a few things the reader should know: * Most serious python programmers would probably prefer to download a package that contains a function that reduces an integer to a product of primes. Two promising candidates are [https://pypi.org/project/primefac/ pypi.org/project/primefac] and [https://www.sympy.org/en/index.html www.sympy.org/]. * Wikipedia has a much simpler program [[w:Special:Permalink/1086485306|this permalink]] of [[w:Trial division]] * Lacking the mental wherewithal to seize upon either of these opportunities, I googled "python prime factorization", found many codes, and selected the one that follows for two reasons:<ref>The code that follows is discussed at [https://stackoverflow.com/questions/15347174/python-finding-prime-factors stackoverflow question 15347174] and [https://scientific-python-101.readthedocs.io/_downloads/prime_factorization_solution.py scientific-python-101.readthedocs]</ref> :# The repeated factors were arranged to into exponents, which facilitates conversion into wikitext script. In this sample code, I presented the result in a simple python script.<ref>In the wikitext version, I had to add a few lines so that exponents of 1 would not appear. Example: <math>12=2^2\cdot 3^1</math> should instead read <math>12=2^2\cdot 3.</math> I ommitted the feature here in order to keep the lesson simple.</ref> :# I didn't understand the python code, and wanted master it in order to fully understand how python dictionaries are used. ===bk1=== {| class="wikitable" style="text-align: center; vertical-align: bottom;" |+ <math>\text{Establishing } 1550=2\cdot 5^2 \cdot 31</math> |- ! n ! b ! b^2 ! b^2<n? ! b/k ! mod ! replace ! factor ! * |- | 1550 | 2 | 4 | TRUE | 775 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"|n: n/b | style="background:#ffffcc"|2 | |- | 775 | 2 | 4 | TRUE | 387 | 1 | b: b+1 | | |- | 775 | 3 | 9 | TRUE | 258 | 1 | b: b+1 | | |- | 775 | 4 | 16 | TRUE | 193 | 3 | b: b+1 | | |- | 775 | 5 | 25 | TRUE | 155 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/b | style="background:#ffffcc"| 5 | |- | 155 | 5 | 25 | TRUE | 31 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/b | style="background:#ffffcc"| 5 | |- | 31 | 5 | 25 | TRUE | 6 | 1 | b: b+1 | | |- | 31 | 6 | 36 | FALSE | - | - | - | | style="background:#ffffcc"| 31 |} ==Python== ‎<syntaxhighlight lang="python" line="1"> ## In a python code, the function definitions precede the code. We need only 1 function: def prime_factorization(N): ## 12=(2**2)*(2**1)*(5**1), or equivalently, 12 = 2**2 * 3 * 5 ## The dictionary, dict, expresses k**v as a "key-value" pair. In other ## words, if we look up the key, k=5, the dictionary will return the ## value, v=1. In other words: ## k**v <=> key**value <=> base**exponent (for the factored form of N) ## Begin function: n=N #--------------------# n will change in the while loop (getting smaller.) dict = {} #--------------# Start with an empty dictionary if n==1: #---------------# The factorization of 1 is a special case. return {1: 1} # In other words 1 = 1**1 (but recall that 1 is not prime.) k = 2 #------------------# Our first candidate for factorizing any integer. while k**2 <= n: #-------# Do the following unless k is "too big": if n % k: #----------# . Equivalent to: "If k is not a factor of n:" k += 1 #---------# .. increase the value of k by 1 on next attempt. else: #--------------# . Equivalent to "If k is a factor of n:" n /= k #---------# .. Replace n by n/k (for next attempt) try: #-----------# .. If k is in the dictionary, this attempts to: dict[k] += 1 # ... increase the value of k by 1. except KeyError: # .. KeyError occurs if k is not in the dictionary dict[k] = 1 # ... Add k to dictionary and set exponent v=1 # end(while)#------------# (exits while k**2 <= n:) # Here I inserted a diagnostic to be discussed later: # Documented code continues: if n > 1: #--------------# It can be shown that n is either 1 or prime. try: #---------------# . Try to enter n into dictionary dict[n] += 1 #---# .. increase exponent v->v+1 if k in dictinary except KeyError: #---# . If n is not in the dictionary, dict[n] = 1 # .. then enter n into dictionary with v=1 return dict #------# python talk for ending function ################ End def prime_factorization | Begin program ########## #import sys again=True while again == 1: text="\nEnter a positive integer you wish to factor into prime numbers:\n" N=int(input(text)) dict=prime_factorization(N) print("\n"+str(int(N))+"=") text="(" for b in dict.keys(): text+=str(int(b))+"**"+str(int(dict.get(b)))+") * (" print(text.rstrip(" * (")+"\n") print("The dictionary is:\n") print(dict) again=int(input("\nType 1 for another integer, 0 to terminate.")) </syntaxhighlight> ==Comments on code== ===Line 15: while k**2 <= n:=== This is an important line because it permits the termination of the search before one might think. I leave it as an exercise for the student to learn why any value of k larger than square root of n will never be the next factor of N. It is also left to the student to understand the fact that the time to find the factors can be cut nearly in half if we modify line 17 and increase k by 2 (instead of 1), and why this larger step cannot be taken until after k=2 has been considered. ===Lines in the code that might seem confusing=== The code uses two cleverness tricks that might seem confusing. I lack the formal training in programming to know whether such cleverness would be perceived as a flaw. I know that programmers are allowed to be clever and I know that clear programs are better than confusing programs. The fact that these tricks confused me does not necessarily mean they should be avoided. And, the nice thing about both sources of confusion is that they taught me something about python. ====Line 16: if n%k might seem backwards==== Python follows a common practice of interpreting the number 1 to be True and 0 to be False. But python also accepts any non-zero number as True. Here n%k denotes nmod&nbsp;k, which is the remainder when on divides n by k. If there is no remainder the "if" condition is not satisfied and the code stipulates that the next value of k is attempted. Any number not equal to zero is interpreted as the "if" condition being satisfied, meaning that k is a divisor of n (in that case, we replace n by n/k.) This might be good coding, but it seems to me that it would have been better to construct it along the lines of "if n%k == 0" or even "if n%k !=0". ====Lines 21-23 and 29-31: A failed attempt to change the value of a key==== In this code fragment, we ask if a given value of k has already been identified as a factor. But instead of "asking" if k has been declared a key in the dictionary, an <u>attempt</u> is made to change the value of k. If that attempt fails (because k is not among the dictionary's keys), then k is added to the list of keys, with a value of 1 (a value of 1 indicates that at this point, k, but not k² has been established to be a factor of n.) Again, there might be nothing wrong with using a key error like this. But the code would have been easier for me to follow if the if statement simply called the question of whether k was a key in dict. ==Sample output (with a really big number)== Sometimes you have to wait a few minutes to factor a number this large. But usually it comes out in just a few seconds. I would imagine that a number that was the product of only two roughly equal prime numbers would take forever because k=k+1 must be repeated a virtually infinite number of times. This number came quickly: {{Font color|blue|Enter a positive integer you wish to factor into prime numbers:}} 99283428376482768916467328912876456416375474272263423467896879187289499287374899 &nbsp; <span style="color:blue>99283428376482768916467328912876456416375474272263423467896879187289499287374899=</span> {{Font color|blue|(619**1) * (688**1) * (768**3) * (1024**17) * (64853**1) * (5302548928**1)}} &nbsp; {{Font color|blue|The dictionary is:}} &nbsp; <span style="color:blue>{619: 1, 688: 1, 768: 3, 1024: 17, 64853: 1, 5302548928.0: 1}</span> &nbsp; {{Font color|blue|Type 1 for another integer, 0 to terminate.}} {{center|'''DO NOT TRY TO FACTOR A NUMBER THIS LARGE UNLESS YOU ARE WILLING TO INTERUPT YOUR PYTHON PROGRAM.'''}} {{center|Keep in mind that the universe is only <math>4.4\times10^{26}\text{nanoseconds}</math> old.<ref>http://astronomy.nmsu.edu/geas/lectures/lecture02/slide04.html</ref>}} The number shown rounds up to 10⁸⁰. It is not likely that any computer will ever count that high. If the number to be factored consists of two prime number that are close to 10⁸⁰, the code will iterate all the way up to k coming close to 10⁴⁰. The inability of any computer to perform that many iterations helps explain how cryptocurrency works. &nbsp; k6occhxxamupd36tbey5zl627o5ggha 2410263 2410262 2022-07-29T19:10:28Z Guy vandegrift 813252 /* bk1 */ wikitext text/x-wiki [[File:Factorization flowchart.svg|thumb|450px|Flowchart for a simple prime factorization code]] This lesson shows how the use of a python dictionary can facilitate the decomposition of a positive integer into a product of prime numbers. There are a few things the reader should know: * Most serious python programmers would probably prefer to download a package that contains a function that reduces an integer to a product of primes. Two promising candidates are [https://pypi.org/project/primefac/ pypi.org/project/primefac] and [https://www.sympy.org/en/index.html www.sympy.org/]. * Wikipedia has a much simpler program [[w:Special:Permalink/1086485306|this permalink]] of [[w:Trial division]] * Lacking the mental wherewithal to seize upon either of these opportunities, I googled "python prime factorization", found many codes, and selected the one that follows for two reasons:<ref>The code that follows is discussed at [https://stackoverflow.com/questions/15347174/python-finding-prime-factors stackoverflow question 15347174] and [https://scientific-python-101.readthedocs.io/_downloads/prime_factorization_solution.py scientific-python-101.readthedocs]</ref> :# The repeated factors were arranged to into exponents, which facilitates conversion into wikitext script. In this sample code, I presented the result in a simple python script.<ref>In the wikitext version, I had to add a few lines so that exponents of 1 would not appear. Example: <math>12=2^2\cdot 3^1</math> should instead read <math>12=2^2\cdot 3.</math> I ommitted the feature here in order to keep the lesson simple.</ref> :# I didn't understand the python code, and wanted master it in order to fully understand how python dictionaries are used. ===bk1=== {| class="wikitable" style="text-align: center; vertical-align: bottom;" |+ <math>\text{Establishing } 1550=2\cdot 5^2 \cdot 31</math> |- ! n ! b ! b^2 ! b^2<n? ! b/k ! mod ! replace ! factor ! * |- | 1550 | 2 | 4 | TRUE | 775 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"|n: n/k | style="background:#ffffcc"|2 | |- | 775 | 2 | 4 | TRUE | 387 | 1 | b: b+1 | | |- | 775 | 3 | 9 | TRUE | 258 | 1 | b: b+1 | | |- | 775 | 4 | 16 | TRUE | 193 | 3 | b: b+1 | | |- | 775 | 5 | 25 | TRUE | 155 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 155 | 5 | 25 | TRUE | 31 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 31 | 5 | 25 | TRUE | 6 | 1 | b: b+1 | | |- | 31 | 6 | 36 | FALSE | - | - | - | | style="background:#ffffcc"| 31 |} ==Python== ‎<syntaxhighlight lang="python" line="1"> ## In a python code, the function definitions precede the code. We need only 1 function: def prime_factorization(N): ## 12=(2**2)*(2**1)*(5**1), or equivalently, 12 = 2**2 * 3 * 5 ## The dictionary, dict, expresses k**v as a "key-value" pair. In other ## words, if we look up the key, k=5, the dictionary will return the ## value, v=1. In other words: ## k**v <=> key**value <=> base**exponent (for the factored form of N) ## Begin function: n=N #--------------------# n will change in the while loop (getting smaller.) dict = {} #--------------# Start with an empty dictionary if n==1: #---------------# The factorization of 1 is a special case. return {1: 1} # In other words 1 = 1**1 (but recall that 1 is not prime.) k = 2 #------------------# Our first candidate for factorizing any integer. while k**2 <= n: #-------# Do the following unless k is "too big": if n % k: #----------# . Equivalent to: "If k is not a factor of n:" k += 1 #---------# .. increase the value of k by 1 on next attempt. else: #--------------# . Equivalent to "If k is a factor of n:" n /= k #---------# .. Replace n by n/k (for next attempt) try: #-----------# .. If k is in the dictionary, this attempts to: dict[k] += 1 # ... increase the value of k by 1. except KeyError: # .. KeyError occurs if k is not in the dictionary dict[k] = 1 # ... Add k to dictionary and set exponent v=1 # end(while)#------------# (exits while k**2 <= n:) # Here I inserted a diagnostic to be discussed later: # Documented code continues: if n > 1: #--------------# It can be shown that n is either 1 or prime. try: #---------------# . Try to enter n into dictionary dict[n] += 1 #---# .. increase exponent v->v+1 if k in dictinary except KeyError: #---# . If n is not in the dictionary, dict[n] = 1 # .. then enter n into dictionary with v=1 return dict #------# python talk for ending function ################ End def prime_factorization | Begin program ########## #import sys again=True while again == 1: text="\nEnter a positive integer you wish to factor into prime numbers:\n" N=int(input(text)) dict=prime_factorization(N) print("\n"+str(int(N))+"=") text="(" for b in dict.keys(): text+=str(int(b))+"**"+str(int(dict.get(b)))+") * (" print(text.rstrip(" * (")+"\n") print("The dictionary is:\n") print(dict) again=int(input("\nType 1 for another integer, 0 to terminate.")) </syntaxhighlight> ==Comments on code== ===Line 15: while k**2 <= n:=== This is an important line because it permits the termination of the search before one might think. I leave it as an exercise for the student to learn why any value of k larger than square root of n will never be the next factor of N. It is also left to the student to understand the fact that the time to find the factors can be cut nearly in half if we modify line 17 and increase k by 2 (instead of 1), and why this larger step cannot be taken until after k=2 has been considered. ===Lines in the code that might seem confusing=== The code uses two cleverness tricks that might seem confusing. I lack the formal training in programming to know whether such cleverness would be perceived as a flaw. I know that programmers are allowed to be clever and I know that clear programs are better than confusing programs. The fact that these tricks confused me does not necessarily mean they should be avoided. And, the nice thing about both sources of confusion is that they taught me something about python. ====Line 16: if n%k might seem backwards==== Python follows a common practice of interpreting the number 1 to be True and 0 to be False. But python also accepts any non-zero number as True. Here n%k denotes nmod&nbsp;k, which is the remainder when on divides n by k. If there is no remainder the "if" condition is not satisfied and the code stipulates that the next value of k is attempted. Any number not equal to zero is interpreted as the "if" condition being satisfied, meaning that k is a divisor of n (in that case, we replace n by n/k.) This might be good coding, but it seems to me that it would have been better to construct it along the lines of "if n%k == 0" or even "if n%k !=0". ====Lines 21-23 and 29-31: A failed attempt to change the value of a key==== In this code fragment, we ask if a given value of k has already been identified as a factor. But instead of "asking" if k has been declared a key in the dictionary, an <u>attempt</u> is made to change the value of k. If that attempt fails (because k is not among the dictionary's keys), then k is added to the list of keys, with a value of 1 (a value of 1 indicates that at this point, k, but not k² has been established to be a factor of n.) Again, there might be nothing wrong with using a key error like this. But the code would have been easier for me to follow if the if statement simply called the question of whether k was a key in dict. ==Sample output (with a really big number)== Sometimes you have to wait a few minutes to factor a number this large. But usually it comes out in just a few seconds. I would imagine that a number that was the product of only two roughly equal prime numbers would take forever because k=k+1 must be repeated a virtually infinite number of times. This number came quickly: {{Font color|blue|Enter a positive integer you wish to factor into prime numbers:}} 99283428376482768916467328912876456416375474272263423467896879187289499287374899 &nbsp; <span style="color:blue>99283428376482768916467328912876456416375474272263423467896879187289499287374899=</span> {{Font color|blue|(619**1) * (688**1) * (768**3) * (1024**17) * (64853**1) * (5302548928**1)}} &nbsp; {{Font color|blue|The dictionary is:}} &nbsp; <span style="color:blue>{619: 1, 688: 1, 768: 3, 1024: 17, 64853: 1, 5302548928.0: 1}</span> &nbsp; {{Font color|blue|Type 1 for another integer, 0 to terminate.}} {{center|'''DO NOT TRY TO FACTOR A NUMBER THIS LARGE UNLESS YOU ARE WILLING TO INTERUPT YOUR PYTHON PROGRAM.'''}} {{center|Keep in mind that the universe is only <math>4.4\times10^{26}\text{nanoseconds}</math> old.<ref>http://astronomy.nmsu.edu/geas/lectures/lecture02/slide04.html</ref>}} The number shown rounds up to 10⁸⁰. It is not likely that any computer will ever count that high. If the number to be factored consists of two prime number that are close to 10⁸⁰, the code will iterate all the way up to k coming close to 10⁴⁰. The inability of any computer to perform that many iterations helps explain how cryptocurrency works. &nbsp; httdz2b5bf6xhclaqzps1ogc63n0my9 2410264 2410263 2022-07-29T19:11:09Z Guy vandegrift 813252 /* bk1 */ wikitext text/x-wiki [[File:Factorization flowchart.svg|thumb|450px|Flowchart for a simple prime factorization code]] This lesson shows how the use of a python dictionary can facilitate the decomposition of a positive integer into a product of prime numbers. There are a few things the reader should know: * Most serious python programmers would probably prefer to download a package that contains a function that reduces an integer to a product of primes. Two promising candidates are [https://pypi.org/project/primefac/ pypi.org/project/primefac] and [https://www.sympy.org/en/index.html www.sympy.org/]. * Wikipedia has a much simpler program [[w:Special:Permalink/1086485306|this permalink]] of [[w:Trial division]] * Lacking the mental wherewithal to seize upon either of these opportunities, I googled "python prime factorization", found many codes, and selected the one that follows for two reasons:<ref>The code that follows is discussed at [https://stackoverflow.com/questions/15347174/python-finding-prime-factors stackoverflow question 15347174] and [https://scientific-python-101.readthedocs.io/_downloads/prime_factorization_solution.py scientific-python-101.readthedocs]</ref> :# The repeated factors were arranged to into exponents, which facilitates conversion into wikitext script. In this sample code, I presented the result in a simple python script.<ref>In the wikitext version, I had to add a few lines so that exponents of 1 would not appear. Example: <math>12=2^2\cdot 3^1</math> should instead read <math>12=2^2\cdot 3.</math> I ommitted the feature here in order to keep the lesson simple.</ref> :# I didn't understand the python code, and wanted master it in order to fully understand how python dictionaries are used. ===bk1=== {| class="wikitable" style="text-align: center; vertical-align: bottom;" |+ <math>\text{Establishing } 1550=2\cdot 5^2 \cdot 31</math> |- ! n ! k ! k^2 ! k^2<n? ! b/k ! mod ! replace ! factor ! * |- | 1550 | 2 | 4 | TRUE | 775 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"|n: n/k | style="background:#ffffcc"|2 | |- | 775 | 2 | 4 | TRUE | 387 | 1 | b: b+1 | | |- | 775 | 3 | 9 | TRUE | 258 | 1 | b: b+1 | | |- | 775 | 4 | 16 | TRUE | 193 | 3 | b: b+1 | | |- | 775 | 5 | 25 | TRUE | 155 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 155 | 5 | 25 | TRUE | 31 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 31 | 5 | 25 | TRUE | 6 | 1 | b: b+1 | | |- | 31 | 6 | 36 | FALSE | - | - | - | | style="background:#ffffcc"| 31 |} ==Python== ‎<syntaxhighlight lang="python" line="1"> ## In a python code, the function definitions precede the code. We need only 1 function: def prime_factorization(N): ## 12=(2**2)*(2**1)*(5**1), or equivalently, 12 = 2**2 * 3 * 5 ## The dictionary, dict, expresses k**v as a "key-value" pair. In other ## words, if we look up the key, k=5, the dictionary will return the ## value, v=1. In other words: ## k**v <=> key**value <=> base**exponent (for the factored form of N) ## Begin function: n=N #--------------------# n will change in the while loop (getting smaller.) dict = {} #--------------# Start with an empty dictionary if n==1: #---------------# The factorization of 1 is a special case. return {1: 1} # In other words 1 = 1**1 (but recall that 1 is not prime.) k = 2 #------------------# Our first candidate for factorizing any integer. while k**2 <= n: #-------# Do the following unless k is "too big": if n % k: #----------# . Equivalent to: "If k is not a factor of n:" k += 1 #---------# .. increase the value of k by 1 on next attempt. else: #--------------# . Equivalent to "If k is a factor of n:" n /= k #---------# .. Replace n by n/k (for next attempt) try: #-----------# .. If k is in the dictionary, this attempts to: dict[k] += 1 # ... increase the value of k by 1. except KeyError: # .. KeyError occurs if k is not in the dictionary dict[k] = 1 # ... Add k to dictionary and set exponent v=1 # end(while)#------------# (exits while k**2 <= n:) # Here I inserted a diagnostic to be discussed later: # Documented code continues: if n > 1: #--------------# It can be shown that n is either 1 or prime. try: #---------------# . Try to enter n into dictionary dict[n] += 1 #---# .. increase exponent v->v+1 if k in dictinary except KeyError: #---# . If n is not in the dictionary, dict[n] = 1 # .. then enter n into dictionary with v=1 return dict #------# python talk for ending function ################ End def prime_factorization | Begin program ########## #import sys again=True while again == 1: text="\nEnter a positive integer you wish to factor into prime numbers:\n" N=int(input(text)) dict=prime_factorization(N) print("\n"+str(int(N))+"=") text="(" for b in dict.keys(): text+=str(int(b))+"**"+str(int(dict.get(b)))+") * (" print(text.rstrip(" * (")+"\n") print("The dictionary is:\n") print(dict) again=int(input("\nType 1 for another integer, 0 to terminate.")) </syntaxhighlight> ==Comments on code== ===Line 15: while k**2 <= n:=== This is an important line because it permits the termination of the search before one might think. I leave it as an exercise for the student to learn why any value of k larger than square root of n will never be the next factor of N. It is also left to the student to understand the fact that the time to find the factors can be cut nearly in half if we modify line 17 and increase k by 2 (instead of 1), and why this larger step cannot be taken until after k=2 has been considered. ===Lines in the code that might seem confusing=== The code uses two cleverness tricks that might seem confusing. I lack the formal training in programming to know whether such cleverness would be perceived as a flaw. I know that programmers are allowed to be clever and I know that clear programs are better than confusing programs. The fact that these tricks confused me does not necessarily mean they should be avoided. And, the nice thing about both sources of confusion is that they taught me something about python. ====Line 16: if n%k might seem backwards==== Python follows a common practice of interpreting the number 1 to be True and 0 to be False. But python also accepts any non-zero number as True. Here n%k denotes nmod&nbsp;k, which is the remainder when on divides n by k. If there is no remainder the "if" condition is not satisfied and the code stipulates that the next value of k is attempted. Any number not equal to zero is interpreted as the "if" condition being satisfied, meaning that k is a divisor of n (in that case, we replace n by n/k.) This might be good coding, but it seems to me that it would have been better to construct it along the lines of "if n%k == 0" or even "if n%k !=0". ====Lines 21-23 and 29-31: A failed attempt to change the value of a key==== In this code fragment, we ask if a given value of k has already been identified as a factor. But instead of "asking" if k has been declared a key in the dictionary, an <u>attempt</u> is made to change the value of k. If that attempt fails (because k is not among the dictionary's keys), then k is added to the list of keys, with a value of 1 (a value of 1 indicates that at this point, k, but not k² has been established to be a factor of n.) Again, there might be nothing wrong with using a key error like this. But the code would have been easier for me to follow if the if statement simply called the question of whether k was a key in dict. ==Sample output (with a really big number)== Sometimes you have to wait a few minutes to factor a number this large. But usually it comes out in just a few seconds. I would imagine that a number that was the product of only two roughly equal prime numbers would take forever because k=k+1 must be repeated a virtually infinite number of times. This number came quickly: {{Font color|blue|Enter a positive integer you wish to factor into prime numbers:}} 99283428376482768916467328912876456416375474272263423467896879187289499287374899 &nbsp; <span style="color:blue>99283428376482768916467328912876456416375474272263423467896879187289499287374899=</span> {{Font color|blue|(619**1) * (688**1) * (768**3) * (1024**17) * (64853**1) * (5302548928**1)}} &nbsp; {{Font color|blue|The dictionary is:}} &nbsp; <span style="color:blue>{619: 1, 688: 1, 768: 3, 1024: 17, 64853: 1, 5302548928.0: 1}</span> &nbsp; {{Font color|blue|Type 1 for another integer, 0 to terminate.}} {{center|'''DO NOT TRY TO FACTOR A NUMBER THIS LARGE UNLESS YOU ARE WILLING TO INTERUPT YOUR PYTHON PROGRAM.'''}} {{center|Keep in mind that the universe is only <math>4.4\times10^{26}\text{nanoseconds}</math> old.<ref>http://astronomy.nmsu.edu/geas/lectures/lecture02/slide04.html</ref>}} The number shown rounds up to 10⁸⁰. It is not likely that any computer will ever count that high. If the number to be factored consists of two prime number that are close to 10⁸⁰, the code will iterate all the way up to k coming close to 10⁴⁰. The inability of any computer to perform that many iterations helps explain how cryptocurrency works. &nbsp; 1nolr3meuhjcawhaxz4x4qv110hiune 2410265 2410264 2022-07-29T19:11:23Z Guy vandegrift 813252 /* bk1 */ wikitext text/x-wiki [[File:Factorization flowchart.svg|thumb|450px|Flowchart for a simple prime factorization code]] This lesson shows how the use of a python dictionary can facilitate the decomposition of a positive integer into a product of prime numbers. There are a few things the reader should know: * Most serious python programmers would probably prefer to download a package that contains a function that reduces an integer to a product of primes. Two promising candidates are [https://pypi.org/project/primefac/ pypi.org/project/primefac] and [https://www.sympy.org/en/index.html www.sympy.org/]. * Wikipedia has a much simpler program [[w:Special:Permalink/1086485306|this permalink]] of [[w:Trial division]] * Lacking the mental wherewithal to seize upon either of these opportunities, I googled "python prime factorization", found many codes, and selected the one that follows for two reasons:<ref>The code that follows is discussed at [https://stackoverflow.com/questions/15347174/python-finding-prime-factors stackoverflow question 15347174] and [https://scientific-python-101.readthedocs.io/_downloads/prime_factorization_solution.py scientific-python-101.readthedocs]</ref> :# The repeated factors were arranged to into exponents, which facilitates conversion into wikitext script. In this sample code, I presented the result in a simple python script.<ref>In the wikitext version, I had to add a few lines so that exponents of 1 would not appear. Example: <math>12=2^2\cdot 3^1</math> should instead read <math>12=2^2\cdot 3.</math> I ommitted the feature here in order to keep the lesson simple.</ref> :# I didn't understand the python code, and wanted master it in order to fully understand how python dictionaries are used. ===bk1=== {| class="wikitable" style="text-align: center; vertical-align: bottom;" |+ <math>\text{Establishing } 1550=2\cdot 5^2 \cdot 31</math> |- ! n ! k ! k^2 ! k^2<n? ! n/k ! mod ! replace ! factor ! * |- | 1550 | 2 | 4 | TRUE | 775 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"|n: n/k | style="background:#ffffcc"|2 | |- | 775 | 2 | 4 | TRUE | 387 | 1 | b: b+1 | | |- | 775 | 3 | 9 | TRUE | 258 | 1 | b: b+1 | | |- | 775 | 4 | 16 | TRUE | 193 | 3 | b: b+1 | | |- | 775 | 5 | 25 | TRUE | 155 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 155 | 5 | 25 | TRUE | 31 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 31 | 5 | 25 | TRUE | 6 | 1 | b: b+1 | | |- | 31 | 6 | 36 | FALSE | - | - | - | | style="background:#ffffcc"| 31 |} ==Python== ‎<syntaxhighlight lang="python" line="1"> ## In a python code, the function definitions precede the code. We need only 1 function: def prime_factorization(N): ## 12=(2**2)*(2**1)*(5**1), or equivalently, 12 = 2**2 * 3 * 5 ## The dictionary, dict, expresses k**v as a "key-value" pair. In other ## words, if we look up the key, k=5, the dictionary will return the ## value, v=1. In other words: ## k**v <=> key**value <=> base**exponent (for the factored form of N) ## Begin function: n=N #--------------------# n will change in the while loop (getting smaller.) dict = {} #--------------# Start with an empty dictionary if n==1: #---------------# The factorization of 1 is a special case. return {1: 1} # In other words 1 = 1**1 (but recall that 1 is not prime.) k = 2 #------------------# Our first candidate for factorizing any integer. while k**2 <= n: #-------# Do the following unless k is "too big": if n % k: #----------# . Equivalent to: "If k is not a factor of n:" k += 1 #---------# .. increase the value of k by 1 on next attempt. else: #--------------# . Equivalent to "If k is a factor of n:" n /= k #---------# .. Replace n by n/k (for next attempt) try: #-----------# .. If k is in the dictionary, this attempts to: dict[k] += 1 # ... increase the value of k by 1. except KeyError: # .. KeyError occurs if k is not in the dictionary dict[k] = 1 # ... Add k to dictionary and set exponent v=1 # end(while)#------------# (exits while k**2 <= n:) # Here I inserted a diagnostic to be discussed later: # Documented code continues: if n > 1: #--------------# It can be shown that n is either 1 or prime. try: #---------------# . Try to enter n into dictionary dict[n] += 1 #---# .. increase exponent v->v+1 if k in dictinary except KeyError: #---# . If n is not in the dictionary, dict[n] = 1 # .. then enter n into dictionary with v=1 return dict #------# python talk for ending function ################ End def prime_factorization | Begin program ########## #import sys again=True while again == 1: text="\nEnter a positive integer you wish to factor into prime numbers:\n" N=int(input(text)) dict=prime_factorization(N) print("\n"+str(int(N))+"=") text="(" for b in dict.keys(): text+=str(int(b))+"**"+str(int(dict.get(b)))+") * (" print(text.rstrip(" * (")+"\n") print("The dictionary is:\n") print(dict) again=int(input("\nType 1 for another integer, 0 to terminate.")) </syntaxhighlight> ==Comments on code== ===Line 15: while k**2 <= n:=== This is an important line because it permits the termination of the search before one might think. I leave it as an exercise for the student to learn why any value of k larger than square root of n will never be the next factor of N. It is also left to the student to understand the fact that the time to find the factors can be cut nearly in half if we modify line 17 and increase k by 2 (instead of 1), and why this larger step cannot be taken until after k=2 has been considered. ===Lines in the code that might seem confusing=== The code uses two cleverness tricks that might seem confusing. I lack the formal training in programming to know whether such cleverness would be perceived as a flaw. I know that programmers are allowed to be clever and I know that clear programs are better than confusing programs. The fact that these tricks confused me does not necessarily mean they should be avoided. And, the nice thing about both sources of confusion is that they taught me something about python. ====Line 16: if n%k might seem backwards==== Python follows a common practice of interpreting the number 1 to be True and 0 to be False. But python also accepts any non-zero number as True. Here n%k denotes nmod&nbsp;k, which is the remainder when on divides n by k. If there is no remainder the "if" condition is not satisfied and the code stipulates that the next value of k is attempted. Any number not equal to zero is interpreted as the "if" condition being satisfied, meaning that k is a divisor of n (in that case, we replace n by n/k.) This might be good coding, but it seems to me that it would have been better to construct it along the lines of "if n%k == 0" or even "if n%k !=0". ====Lines 21-23 and 29-31: A failed attempt to change the value of a key==== In this code fragment, we ask if a given value of k has already been identified as a factor. But instead of "asking" if k has been declared a key in the dictionary, an <u>attempt</u> is made to change the value of k. If that attempt fails (because k is not among the dictionary's keys), then k is added to the list of keys, with a value of 1 (a value of 1 indicates that at this point, k, but not k² has been established to be a factor of n.) Again, there might be nothing wrong with using a key error like this. But the code would have been easier for me to follow if the if statement simply called the question of whether k was a key in dict. ==Sample output (with a really big number)== Sometimes you have to wait a few minutes to factor a number this large. But usually it comes out in just a few seconds. I would imagine that a number that was the product of only two roughly equal prime numbers would take forever because k=k+1 must be repeated a virtually infinite number of times. This number came quickly: {{Font color|blue|Enter a positive integer you wish to factor into prime numbers:}} 99283428376482768916467328912876456416375474272263423467896879187289499287374899 &nbsp; <span style="color:blue>99283428376482768916467328912876456416375474272263423467896879187289499287374899=</span> {{Font color|blue|(619**1) * (688**1) * (768**3) * (1024**17) * (64853**1) * (5302548928**1)}} &nbsp; {{Font color|blue|The dictionary is:}} &nbsp; <span style="color:blue>{619: 1, 688: 1, 768: 3, 1024: 17, 64853: 1, 5302548928.0: 1}</span> &nbsp; {{Font color|blue|Type 1 for another integer, 0 to terminate.}} {{center|'''DO NOT TRY TO FACTOR A NUMBER THIS LARGE UNLESS YOU ARE WILLING TO INTERUPT YOUR PYTHON PROGRAM.'''}} {{center|Keep in mind that the universe is only <math>4.4\times10^{26}\text{nanoseconds}</math> old.<ref>http://astronomy.nmsu.edu/geas/lectures/lecture02/slide04.html</ref>}} The number shown rounds up to 10⁸⁰. It is not likely that any computer will ever count that high. If the number to be factored consists of two prime number that are close to 10⁸⁰, the code will iterate all the way up to k coming close to 10⁴⁰. The inability of any computer to perform that many iterations helps explain how cryptocurrency works. &nbsp; 53ul09nwvi1jo1osbpzsl6tqyxlrc0e 2410266 2410265 2022-07-29T19:12:21Z Guy vandegrift 813252 /* bk1 */ wikitext text/x-wiki [[File:Factorization flowchart.svg|thumb|450px|Flowchart for a simple prime factorization code]] This lesson shows how the use of a python dictionary can facilitate the decomposition of a positive integer into a product of prime numbers. There are a few things the reader should know: * Most serious python programmers would probably prefer to download a package that contains a function that reduces an integer to a product of primes. Two promising candidates are [https://pypi.org/project/primefac/ pypi.org/project/primefac] and [https://www.sympy.org/en/index.html www.sympy.org/]. * Wikipedia has a much simpler program [[w:Special:Permalink/1086485306|this permalink]] of [[w:Trial division]] * Lacking the mental wherewithal to seize upon either of these opportunities, I googled "python prime factorization", found many codes, and selected the one that follows for two reasons:<ref>The code that follows is discussed at [https://stackoverflow.com/questions/15347174/python-finding-prime-factors stackoverflow question 15347174] and [https://scientific-python-101.readthedocs.io/_downloads/prime_factorization_solution.py scientific-python-101.readthedocs]</ref> :# The repeated factors were arranged to into exponents, which facilitates conversion into wikitext script. In this sample code, I presented the result in a simple python script.<ref>In the wikitext version, I had to add a few lines so that exponents of 1 would not appear. Example: <math>12=2^2\cdot 3^1</math> should instead read <math>12=2^2\cdot 3.</math> I ommitted the feature here in order to keep the lesson simple.</ref> :# I didn't understand the python code, and wanted master it in order to fully understand how python dictionaries are used. ===bk1=== {| class="wikitable" style="text-align: center; vertical-align: bottom;" |+ <math>\text{Establishing } 1550=2\cdot 5^2 \cdot 31</math> |- ! n ! k ! k^2 ! k^2<n? ! n/k ! mod ! replace ! factor ! * |- | 1550 | 2 | 4 | TRUE | 775 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"|n: n/k | style="background:#ffffcc"|2 | |- | 775 | 2 | 4 | TRUE | 387 | 1 | k: k+1 | | |- | 775 | 3 | 9 | TRUE | 258 | 1 | k: k+1 | | |- | 775 | 4 | 16 | TRUE | 193 | 3 | k: k+1 | | |- | 775 | 5 | 25 | TRUE | 155 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 155 | 5 | 25 | TRUE | 31 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 31 | 5 | 25 | TRUE | 6 | 1 | k: k+1 | | |- | 31 | 6 | 36 | FALSE | - | - | - | | style="background:#ffffcc"| 31 |} ==Python== ‎<syntaxhighlight lang="python" line="1"> ## In a python code, the function definitions precede the code. We need only 1 function: def prime_factorization(N): ## 12=(2**2)*(2**1)*(5**1), or equivalently, 12 = 2**2 * 3 * 5 ## The dictionary, dict, expresses k**v as a "key-value" pair. In other ## words, if we look up the key, k=5, the dictionary will return the ## value, v=1. In other words: ## k**v <=> key**value <=> base**exponent (for the factored form of N) ## Begin function: n=N #--------------------# n will change in the while loop (getting smaller.) dict = {} #--------------# Start with an empty dictionary if n==1: #---------------# The factorization of 1 is a special case. return {1: 1} # In other words 1 = 1**1 (but recall that 1 is not prime.) k = 2 #------------------# Our first candidate for factorizing any integer. while k**2 <= n: #-------# Do the following unless k is "too big": if n % k: #----------# . Equivalent to: "If k is not a factor of n:" k += 1 #---------# .. increase the value of k by 1 on next attempt. else: #--------------# . Equivalent to "If k is a factor of n:" n /= k #---------# .. Replace n by n/k (for next attempt) try: #-----------# .. If k is in the dictionary, this attempts to: dict[k] += 1 # ... increase the value of k by 1. except KeyError: # .. KeyError occurs if k is not in the dictionary dict[k] = 1 # ... Add k to dictionary and set exponent v=1 # end(while)#------------# (exits while k**2 <= n:) # Here I inserted a diagnostic to be discussed later: # Documented code continues: if n > 1: #--------------# It can be shown that n is either 1 or prime. try: #---------------# . Try to enter n into dictionary dict[n] += 1 #---# .. increase exponent v->v+1 if k in dictinary except KeyError: #---# . If n is not in the dictionary, dict[n] = 1 # .. then enter n into dictionary with v=1 return dict #------# python talk for ending function ################ End def prime_factorization | Begin program ########## #import sys again=True while again == 1: text="\nEnter a positive integer you wish to factor into prime numbers:\n" N=int(input(text)) dict=prime_factorization(N) print("\n"+str(int(N))+"=") text="(" for b in dict.keys(): text+=str(int(b))+"**"+str(int(dict.get(b)))+") * (" print(text.rstrip(" * (")+"\n") print("The dictionary is:\n") print(dict) again=int(input("\nType 1 for another integer, 0 to terminate.")) </syntaxhighlight> ==Comments on code== ===Line 15: while k**2 <= n:=== This is an important line because it permits the termination of the search before one might think. I leave it as an exercise for the student to learn why any value of k larger than square root of n will never be the next factor of N. It is also left to the student to understand the fact that the time to find the factors can be cut nearly in half if we modify line 17 and increase k by 2 (instead of 1), and why this larger step cannot be taken until after k=2 has been considered. ===Lines in the code that might seem confusing=== The code uses two cleverness tricks that might seem confusing. I lack the formal training in programming to know whether such cleverness would be perceived as a flaw. I know that programmers are allowed to be clever and I know that clear programs are better than confusing programs. The fact that these tricks confused me does not necessarily mean they should be avoided. And, the nice thing about both sources of confusion is that they taught me something about python. ====Line 16: if n%k might seem backwards==== Python follows a common practice of interpreting the number 1 to be True and 0 to be False. But python also accepts any non-zero number as True. Here n%k denotes nmod&nbsp;k, which is the remainder when on divides n by k. If there is no remainder the "if" condition is not satisfied and the code stipulates that the next value of k is attempted. Any number not equal to zero is interpreted as the "if" condition being satisfied, meaning that k is a divisor of n (in that case, we replace n by n/k.) This might be good coding, but it seems to me that it would have been better to construct it along the lines of "if n%k == 0" or even "if n%k !=0". ====Lines 21-23 and 29-31: A failed attempt to change the value of a key==== In this code fragment, we ask if a given value of k has already been identified as a factor. But instead of "asking" if k has been declared a key in the dictionary, an <u>attempt</u> is made to change the value of k. If that attempt fails (because k is not among the dictionary's keys), then k is added to the list of keys, with a value of 1 (a value of 1 indicates that at this point, k, but not k² has been established to be a factor of n.) Again, there might be nothing wrong with using a key error like this. But the code would have been easier for me to follow if the if statement simply called the question of whether k was a key in dict. ==Sample output (with a really big number)== Sometimes you have to wait a few minutes to factor a number this large. But usually it comes out in just a few seconds. I would imagine that a number that was the product of only two roughly equal prime numbers would take forever because k=k+1 must be repeated a virtually infinite number of times. This number came quickly: {{Font color|blue|Enter a positive integer you wish to factor into prime numbers:}} 99283428376482768916467328912876456416375474272263423467896879187289499287374899 &nbsp; <span style="color:blue>99283428376482768916467328912876456416375474272263423467896879187289499287374899=</span> {{Font color|blue|(619**1) * (688**1) * (768**3) * (1024**17) * (64853**1) * (5302548928**1)}} &nbsp; {{Font color|blue|The dictionary is:}} &nbsp; <span style="color:blue>{619: 1, 688: 1, 768: 3, 1024: 17, 64853: 1, 5302548928.0: 1}</span> &nbsp; {{Font color|blue|Type 1 for another integer, 0 to terminate.}} {{center|'''DO NOT TRY TO FACTOR A NUMBER THIS LARGE UNLESS YOU ARE WILLING TO INTERUPT YOUR PYTHON PROGRAM.'''}} {{center|Keep in mind that the universe is only <math>4.4\times10^{26}\text{nanoseconds}</math> old.<ref>http://astronomy.nmsu.edu/geas/lectures/lecture02/slide04.html</ref>}} The number shown rounds up to 10⁸⁰. It is not likely that any computer will ever count that high. If the number to be factored consists of two prime number that are close to 10⁸⁰, the code will iterate all the way up to k coming close to 10⁴⁰. The inability of any computer to perform that many iterations helps explain how cryptocurrency works. &nbsp; oqeusba7666a86rtcacwaukx73xrw0p 2410267 2410266 2022-07-29T19:13:58Z Guy vandegrift 813252 wikitext text/x-wiki [[File:Factorization flowchart.svg|thumb|420px|Flowchart for a simple prime factorization code]] This lesson shows how the use of a python dictionary can facilitate the decomposition of a positive integer into a product of prime numbers. There are a few things the reader should know: * Most serious python programmers would probably prefer to download a package that contains a function that reduces an integer to a product of primes. Two promising candidates are [https://pypi.org/project/primefac/ pypi.org/project/primefac] and [https://www.sympy.org/en/index.html www.sympy.org/]. * Wikipedia has a much simpler program [[w:Special:Permalink/1086485306|this permalink]] of [[w:Trial division]] * Lacking the mental wherewithal to seize upon either of these opportunities, I googled "python prime factorization", found many codes, and selected the one that follows for two reasons:<ref>The code that follows is discussed at [https://stackoverflow.com/questions/15347174/python-finding-prime-factors stackoverflow question 15347174] and [https://scientific-python-101.readthedocs.io/_downloads/prime_factorization_solution.py scientific-python-101.readthedocs]</ref> :# The repeated factors were arranged to into exponents, which facilitates conversion into wikitext script. In this sample code, I presented the result in a simple python script.<ref>In the wikitext version, I had to add a few lines so that exponents of 1 would not appear. Example: <math>12=2^2\cdot 3^1</math> should instead read <math>12=2^2\cdot 3.</math> I ommitted the feature here in order to keep the lesson simple.</ref> :# I didn't understand the python code, and wanted master it in order to fully understand how python dictionaries are used. ===bk1=== {| class="wikitable" style="text-align: center; vertical-align: bottom;" |+ <math>\text{Establishing } 1550=2\cdot 5^2 \cdot 31</math> |- ! n ! k ! k^2 ! k^2<n? ! n/k ! mod ! replace ! factor ! * |- | 1550 | 2 | 4 | TRUE | 775 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"|n: n/k | style="background:#ffffcc"|2 | |- | 775 | 2 | 4 | TRUE | 387 | 1 | k: k+1 | | |- | 775 | 3 | 9 | TRUE | 258 | 1 | k: k+1 | | |- | 775 | 4 | 16 | TRUE | 193 | 3 | k: k+1 | | |- | 775 | 5 | 25 | TRUE | 155 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 155 | 5 | 25 | TRUE | 31 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 31 | 5 | 25 | TRUE | 6 | 1 | k: k+1 | | |- | 31 | 6 | 36 | FALSE | - | - | - | | style="background:#ffffcc"| 31 |} ==Python== ‎<syntaxhighlight lang="python" line="1"> ## In a python code, the function definitions precede the code. We need only 1 function: def prime_factorization(N): ## 12=(2**2)*(2**1)*(5**1), or equivalently, 12 = 2**2 * 3 * 5 ## The dictionary, dict, expresses k**v as a "key-value" pair. In other ## words, if we look up the key, k=5, the dictionary will return the ## value, v=1. In other words: ## k**v <=> key**value <=> base**exponent (for the factored form of N) ## Begin function: n=N #--------------------# n will change in the while loop (getting smaller.) dict = {} #--------------# Start with an empty dictionary if n==1: #---------------# The factorization of 1 is a special case. return {1: 1} # In other words 1 = 1**1 (but recall that 1 is not prime.) k = 2 #------------------# Our first candidate for factorizing any integer. while k**2 <= n: #-------# Do the following unless k is "too big": if n % k: #----------# . Equivalent to: "If k is not a factor of n:" k += 1 #---------# .. increase the value of k by 1 on next attempt. else: #--------------# . Equivalent to "If k is a factor of n:" n /= k #---------# .. Replace n by n/k (for next attempt) try: #-----------# .. If k is in the dictionary, this attempts to: dict[k] += 1 # ... increase the value of k by 1. except KeyError: # .. KeyError occurs if k is not in the dictionary dict[k] = 1 # ... Add k to dictionary and set exponent v=1 # end(while)#------------# (exits while k**2 <= n:) # Here I inserted a diagnostic to be discussed later: # Documented code continues: if n > 1: #--------------# It can be shown that n is either 1 or prime. try: #---------------# . Try to enter n into dictionary dict[n] += 1 #---# .. increase exponent v->v+1 if k in dictinary except KeyError: #---# . If n is not in the dictionary, dict[n] = 1 # .. then enter n into dictionary with v=1 return dict #------# python talk for ending function ################ End def prime_factorization | Begin program ########## #import sys again=True while again == 1: text="\nEnter a positive integer you wish to factor into prime numbers:\n" N=int(input(text)) dict=prime_factorization(N) print("\n"+str(int(N))+"=") text="(" for b in dict.keys(): text+=str(int(b))+"**"+str(int(dict.get(b)))+") * (" print(text.rstrip(" * (")+"\n") print("The dictionary is:\n") print(dict) again=int(input("\nType 1 for another integer, 0 to terminate.")) </syntaxhighlight> ==Comments on code== ===Line 15: while k**2 <= n:=== This is an important line because it permits the termination of the search before one might think. I leave it as an exercise for the student to learn why any value of k larger than square root of n will never be the next factor of N. It is also left to the student to understand the fact that the time to find the factors can be cut nearly in half if we modify line 17 and increase k by 2 (instead of 1), and why this larger step cannot be taken until after k=2 has been considered. ===Lines in the code that might seem confusing=== The code uses two cleverness tricks that might seem confusing. I lack the formal training in programming to know whether such cleverness would be perceived as a flaw. I know that programmers are allowed to be clever and I know that clear programs are better than confusing programs. The fact that these tricks confused me does not necessarily mean they should be avoided. And, the nice thing about both sources of confusion is that they taught me something about python. ====Line 16: if n%k might seem backwards==== Python follows a common practice of interpreting the number 1 to be True and 0 to be False. But python also accepts any non-zero number as True. Here n%k denotes nmod&nbsp;k, which is the remainder when on divides n by k. If there is no remainder the "if" condition is not satisfied and the code stipulates that the next value of k is attempted. Any number not equal to zero is interpreted as the "if" condition being satisfied, meaning that k is a divisor of n (in that case, we replace n by n/k.) This might be good coding, but it seems to me that it would have been better to construct it along the lines of "if n%k == 0" or even "if n%k !=0". ====Lines 21-23 and 29-31: A failed attempt to change the value of a key==== In this code fragment, we ask if a given value of k has already been identified as a factor. But instead of "asking" if k has been declared a key in the dictionary, an <u>attempt</u> is made to change the value of k. If that attempt fails (because k is not among the dictionary's keys), then k is added to the list of keys, with a value of 1 (a value of 1 indicates that at this point, k, but not k² has been established to be a factor of n.) Again, there might be nothing wrong with using a key error like this. But the code would have been easier for me to follow if the if statement simply called the question of whether k was a key in dict. ==Sample output (with a really big number)== Sometimes you have to wait a few minutes to factor a number this large. But usually it comes out in just a few seconds. I would imagine that a number that was the product of only two roughly equal prime numbers would take forever because k=k+1 must be repeated a virtually infinite number of times. This number came quickly: {{Font color|blue|Enter a positive integer you wish to factor into prime numbers:}} 99283428376482768916467328912876456416375474272263423467896879187289499287374899 &nbsp; <span style="color:blue>99283428376482768916467328912876456416375474272263423467896879187289499287374899=</span> {{Font color|blue|(619**1) * (688**1) * (768**3) * (1024**17) * (64853**1) * (5302548928**1)}} &nbsp; {{Font color|blue|The dictionary is:}} &nbsp; <span style="color:blue>{619: 1, 688: 1, 768: 3, 1024: 17, 64853: 1, 5302548928.0: 1}</span> &nbsp; {{Font color|blue|Type 1 for another integer, 0 to terminate.}} {{center|'''DO NOT TRY TO FACTOR A NUMBER THIS LARGE UNLESS YOU ARE WILLING TO INTERUPT YOUR PYTHON PROGRAM.'''}} {{center|Keep in mind that the universe is only <math>4.4\times10^{26}\text{nanoseconds}</math> old.<ref>http://astronomy.nmsu.edu/geas/lectures/lecture02/slide04.html</ref>}} The number shown rounds up to 10⁸⁰. It is not likely that any computer will ever count that high. If the number to be factored consists of two prime number that are close to 10⁸⁰, the code will iterate all the way up to k coming close to 10⁴⁰. The inability of any computer to perform that many iterations helps explain how cryptocurrency works. &nbsp; 4fclytjdt0cdbzqq4yc7j1lx5gs3vrr 2410268 2410267 2022-07-29T19:16:59Z Guy vandegrift 813252 wikitext text/x-wiki [[File:Factorization flowchart.svg|thumb|420px|Flowchart for a simple prime factorization code]] This lesson shows how the use of a python dictionary can facilitate the decomposition of a positive integer into a product of prime numbers. There are a few things the reader should know: * Most serious python programmers would probably prefer to download a package that contains a function that reduces an integer to a product of primes. Two promising candidates are [https://pypi.org/project/primefac/ pypi.org/project/primefac] and [https://www.sympy.org/en/index.html www.sympy.org/]. * Wikipedia has a much simpler program [[w:Special:Permalink/1086485306|this permalink]] of [[w:Trial division]] * Lacking the mental wherewithal to seize upon either of these opportunities, I googled "python prime factorization", found many codes, and selected the one that follows for two reasons:<ref>The code that follows is discussed at [https://stackoverflow.com/questions/15347174/python-finding-prime-factors stackoverflow question 15347174] and [https://scientific-python-101.readthedocs.io/_downloads/prime_factorization_solution.py scientific-python-101.readthedocs]</ref> :# I wanted a code that counts repeated factors in order to later write the factored form in wikitext with exponents.<ref>In the wikitext version, I had to add a few lines so that exponents of 1 would not appear. Example: <math>12=2^2\cdot 3^1</math> should instead read <math>12=2^2\cdot 3.</math> I ommitted the feature here in order to keep the lesson simple.</ref> :# I didn't understand the python code, and wanted master it in order to fully understand how python dictionaries are used. ===bk1=== {| class="wikitable" style="text-align: center; vertical-align: bottom;" |+ <math>\text{Establishing } 1550=2\cdot 5^2 \cdot 31</math> |- ! n ! k ! k^2 ! k^2<n? ! n/k ! mod ! replace ! factor ! * |- | 1550 | 2 | 4 | TRUE | 775 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"|n: n/k | style="background:#ffffcc"|2 | |- | 775 | 2 | 4 | TRUE | 387 | 1 | k: k+1 | | |- | 775 | 3 | 9 | TRUE | 258 | 1 | k: k+1 | | |- | 775 | 4 | 16 | TRUE | 193 | 3 | k: k+1 | | |- | 775 | 5 | 25 | TRUE | 155 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 155 | 5 | 25 | TRUE | 31 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 31 | 5 | 25 | TRUE | 6 | 1 | k: k+1 | | |- | 31 | 6 | 36 | FALSE | - | - | - | | style="background:#ffffcc"| 31 |} ==Python== ‎<syntaxhighlight lang="python" line="1"> ## In a python code, the function definitions precede the code. We need only 1 function: def prime_factorization(N): ## 12=(2**2)*(2**1)*(5**1), or equivalently, 12 = 2**2 * 3 * 5 ## The dictionary, dict, expresses k**v as a "key-value" pair. In other ## words, if we look up the key, k=5, the dictionary will return the ## value, v=1. In other words: ## k**v <=> key**value <=> base**exponent (for the factored form of N) ## Begin function: n=N #--------------------# n will change in the while loop (getting smaller.) dict = {} #--------------# Start with an empty dictionary if n==1: #---------------# The factorization of 1 is a special case. return {1: 1} # In other words 1 = 1**1 (but recall that 1 is not prime.) k = 2 #------------------# Our first candidate for factorizing any integer. while k**2 <= n: #-------# Do the following unless k is "too big": if n % k: #----------# . Equivalent to: "If k is not a factor of n:" k += 1 #---------# .. increase the value of k by 1 on next attempt. else: #--------------# . Equivalent to "If k is a factor of n:" n /= k #---------# .. Replace n by n/k (for next attempt) try: #-----------# .. If k is in the dictionary, this attempts to: dict[k] += 1 # ... increase the value of k by 1. except KeyError: # .. KeyError occurs if k is not in the dictionary dict[k] = 1 # ... Add k to dictionary and set exponent v=1 # end(while)#------------# (exits while k**2 <= n:) # Here I inserted a diagnostic to be discussed later: # Documented code continues: if n > 1: #--------------# It can be shown that n is either 1 or prime. try: #---------------# . Try to enter n into dictionary dict[n] += 1 #---# .. increase exponent v->v+1 if k in dictinary except KeyError: #---# . If n is not in the dictionary, dict[n] = 1 # .. then enter n into dictionary with v=1 return dict #------# python talk for ending function ################ End def prime_factorization | Begin program ########## #import sys again=True while again == 1: text="\nEnter a positive integer you wish to factor into prime numbers:\n" N=int(input(text)) dict=prime_factorization(N) print("\n"+str(int(N))+"=") text="(" for b in dict.keys(): text+=str(int(b))+"**"+str(int(dict.get(b)))+") * (" print(text.rstrip(" * (")+"\n") print("The dictionary is:\n") print(dict) again=int(input("\nType 1 for another integer, 0 to terminate.")) </syntaxhighlight> ==Comments on code== ===Line 15: while k**2 <= n:=== This is an important line because it permits the termination of the search before one might think. I leave it as an exercise for the student to learn why any value of k larger than square root of n will never be the next factor of N. It is also left to the student to understand the fact that the time to find the factors can be cut nearly in half if we modify line 17 and increase k by 2 (instead of 1), and why this larger step cannot be taken until after k=2 has been considered. ===Lines in the code that might seem confusing=== The code uses two cleverness tricks that might seem confusing. I lack the formal training in programming to know whether such cleverness would be perceived as a flaw. I know that programmers are allowed to be clever and I know that clear programs are better than confusing programs. The fact that these tricks confused me does not necessarily mean they should be avoided. And, the nice thing about both sources of confusion is that they taught me something about python. ====Line 16: if n%k might seem backwards==== Python follows a common practice of interpreting the number 1 to be True and 0 to be False. But python also accepts any non-zero number as True. Here n%k denotes nmod&nbsp;k, which is the remainder when on divides n by k. If there is no remainder the "if" condition is not satisfied and the code stipulates that the next value of k is attempted. Any number not equal to zero is interpreted as the "if" condition being satisfied, meaning that k is a divisor of n (in that case, we replace n by n/k.) This might be good coding, but it seems to me that it would have been better to construct it along the lines of "if n%k == 0" or even "if n%k !=0". ====Lines 21-23 and 29-31: A failed attempt to change the value of a key==== In this code fragment, we ask if a given value of k has already been identified as a factor. But instead of "asking" if k has been declared a key in the dictionary, an <u>attempt</u> is made to change the value of k. If that attempt fails (because k is not among the dictionary's keys), then k is added to the list of keys, with a value of 1 (a value of 1 indicates that at this point, k, but not k² has been established to be a factor of n.) Again, there might be nothing wrong with using a key error like this. But the code would have been easier for me to follow if the if statement simply called the question of whether k was a key in dict. ==Sample output (with a really big number)== Sometimes you have to wait a few minutes to factor a number this large. But usually it comes out in just a few seconds. I would imagine that a number that was the product of only two roughly equal prime numbers would take forever because k=k+1 must be repeated a virtually infinite number of times. This number came quickly: {{Font color|blue|Enter a positive integer you wish to factor into prime numbers:}} 99283428376482768916467328912876456416375474272263423467896879187289499287374899 &nbsp; <span style="color:blue>99283428376482768916467328912876456416375474272263423467896879187289499287374899=</span> {{Font color|blue|(619**1) * (688**1) * (768**3) * (1024**17) * (64853**1) * (5302548928**1)}} &nbsp; {{Font color|blue|The dictionary is:}} &nbsp; <span style="color:blue>{619: 1, 688: 1, 768: 3, 1024: 17, 64853: 1, 5302548928.0: 1}</span> &nbsp; {{Font color|blue|Type 1 for another integer, 0 to terminate.}} {{center|'''DO NOT TRY TO FACTOR A NUMBER THIS LARGE UNLESS YOU ARE WILLING TO INTERUPT YOUR PYTHON PROGRAM.'''}} {{center|Keep in mind that the universe is only <math>4.4\times10^{26}\text{nanoseconds}</math> old.<ref>http://astronomy.nmsu.edu/geas/lectures/lecture02/slide04.html</ref>}} The number shown rounds up to 10⁸⁰. It is not likely that any computer will ever count that high. If the number to be factored consists of two prime number that are close to 10⁸⁰, the code will iterate all the way up to k coming close to 10⁴⁰. The inability of any computer to perform that many iterations helps explain how cryptocurrency works. &nbsp; o6vgvsmidir5bjuulveocez2lpqzvqc 2410269 2410268 2022-07-29T19:17:38Z Guy vandegrift 813252 /* bk1 */ wikitext text/x-wiki [[File:Factorization flowchart.svg|thumb|420px|Flowchart for a simple prime factorization code]] This lesson shows how the use of a python dictionary can facilitate the decomposition of a positive integer into a product of prime numbers. There are a few things the reader should know: * Most serious python programmers would probably prefer to download a package that contains a function that reduces an integer to a product of primes. Two promising candidates are [https://pypi.org/project/primefac/ pypi.org/project/primefac] and [https://www.sympy.org/en/index.html www.sympy.org/]. * Wikipedia has a much simpler program [[w:Special:Permalink/1086485306|this permalink]] of [[w:Trial division]] * Lacking the mental wherewithal to seize upon either of these opportunities, I googled "python prime factorization", found many codes, and selected the one that follows for two reasons:<ref>The code that follows is discussed at [https://stackoverflow.com/questions/15347174/python-finding-prime-factors stackoverflow question 15347174] and [https://scientific-python-101.readthedocs.io/_downloads/prime_factorization_solution.py scientific-python-101.readthedocs]</ref> :# I wanted a code that counts repeated factors in order to later write the factored form in wikitext with exponents.<ref>In the wikitext version, I had to add a few lines so that exponents of 1 would not appear. Example: <math>12=2^2\cdot 3^1</math> should instead read <math>12=2^2\cdot 3.</math> I ommitted the feature here in order to keep the lesson simple.</ref> :# I didn't understand the python code, and wanted master it in order to fully understand how python dictionaries are used. {| class="wikitable" style="text-align: center; vertical-align: bottom;" |+ <math>\text{Establishing } 1550=2\cdot 5^2 \cdot 31</math> |- ! n ! k ! k^2 ! k^2<n? ! n/k ! mod ! replace ! factor ! * |- | 1550 | 2 | 4 | TRUE | 775 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"|n: n/k | style="background:#ffffcc"|2 | |- | 775 | 2 | 4 | TRUE | 387 | 1 | k: k+1 | | |- | 775 | 3 | 9 | TRUE | 258 | 1 | k: k+1 | | |- | 775 | 4 | 16 | TRUE | 193 | 3 | k: k+1 | | |- | 775 | 5 | 25 | TRUE | 155 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 155 | 5 | 25 | TRUE | 31 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 31 | 5 | 25 | TRUE | 6 | 1 | k: k+1 | | |- | 31 | 6 | 36 | FALSE | - | - | - | | style="background:#ffffcc"| 31 |} ==Python== ‎<syntaxhighlight lang="python" line="1"> ## In a python code, the function definitions precede the code. We need only 1 function: def prime_factorization(N): ## 12=(2**2)*(2**1)*(5**1), or equivalently, 12 = 2**2 * 3 * 5 ## The dictionary, dict, expresses k**v as a "key-value" pair. In other ## words, if we look up the key, k=5, the dictionary will return the ## value, v=1. In other words: ## k**v <=> key**value <=> base**exponent (for the factored form of N) ## Begin function: n=N #--------------------# n will change in the while loop (getting smaller.) dict = {} #--------------# Start with an empty dictionary if n==1: #---------------# The factorization of 1 is a special case. return {1: 1} # In other words 1 = 1**1 (but recall that 1 is not prime.) k = 2 #------------------# Our first candidate for factorizing any integer. while k**2 <= n: #-------# Do the following unless k is "too big": if n % k: #----------# . Equivalent to: "If k is not a factor of n:" k += 1 #---------# .. increase the value of k by 1 on next attempt. else: #--------------# . Equivalent to "If k is a factor of n:" n /= k #---------# .. Replace n by n/k (for next attempt) try: #-----------# .. If k is in the dictionary, this attempts to: dict[k] += 1 # ... increase the value of k by 1. except KeyError: # .. KeyError occurs if k is not in the dictionary dict[k] = 1 # ... Add k to dictionary and set exponent v=1 # end(while)#------------# (exits while k**2 <= n:) # Here I inserted a diagnostic to be discussed later: # Documented code continues: if n > 1: #--------------# It can be shown that n is either 1 or prime. try: #---------------# . Try to enter n into dictionary dict[n] += 1 #---# .. increase exponent v->v+1 if k in dictinary except KeyError: #---# . If n is not in the dictionary, dict[n] = 1 # .. then enter n into dictionary with v=1 return dict #------# python talk for ending function ################ End def prime_factorization | Begin program ########## #import sys again=True while again == 1: text="\nEnter a positive integer you wish to factor into prime numbers:\n" N=int(input(text)) dict=prime_factorization(N) print("\n"+str(int(N))+"=") text="(" for b in dict.keys(): text+=str(int(b))+"**"+str(int(dict.get(b)))+") * (" print(text.rstrip(" * (")+"\n") print("The dictionary is:\n") print(dict) again=int(input("\nType 1 for another integer, 0 to terminate.")) </syntaxhighlight> ==Comments on code== ===Line 15: while k**2 <= n:=== This is an important line because it permits the termination of the search before one might think. I leave it as an exercise for the student to learn why any value of k larger than square root of n will never be the next factor of N. It is also left to the student to understand the fact that the time to find the factors can be cut nearly in half if we modify line 17 and increase k by 2 (instead of 1), and why this larger step cannot be taken until after k=2 has been considered. ===Lines in the code that might seem confusing=== The code uses two cleverness tricks that might seem confusing. I lack the formal training in programming to know whether such cleverness would be perceived as a flaw. I know that programmers are allowed to be clever and I know that clear programs are better than confusing programs. The fact that these tricks confused me does not necessarily mean they should be avoided. And, the nice thing about both sources of confusion is that they taught me something about python. ====Line 16: if n%k might seem backwards==== Python follows a common practice of interpreting the number 1 to be True and 0 to be False. But python also accepts any non-zero number as True. Here n%k denotes nmod&nbsp;k, which is the remainder when on divides n by k. If there is no remainder the "if" condition is not satisfied and the code stipulates that the next value of k is attempted. Any number not equal to zero is interpreted as the "if" condition being satisfied, meaning that k is a divisor of n (in that case, we replace n by n/k.) This might be good coding, but it seems to me that it would have been better to construct it along the lines of "if n%k == 0" or even "if n%k !=0". ====Lines 21-23 and 29-31: A failed attempt to change the value of a key==== In this code fragment, we ask if a given value of k has already been identified as a factor. But instead of "asking" if k has been declared a key in the dictionary, an <u>attempt</u> is made to change the value of k. If that attempt fails (because k is not among the dictionary's keys), then k is added to the list of keys, with a value of 1 (a value of 1 indicates that at this point, k, but not k² has been established to be a factor of n.) Again, there might be nothing wrong with using a key error like this. But the code would have been easier for me to follow if the if statement simply called the question of whether k was a key in dict. ==Sample output (with a really big number)== Sometimes you have to wait a few minutes to factor a number this large. But usually it comes out in just a few seconds. I would imagine that a number that was the product of only two roughly equal prime numbers would take forever because k=k+1 must be repeated a virtually infinite number of times. This number came quickly: {{Font color|blue|Enter a positive integer you wish to factor into prime numbers:}} 99283428376482768916467328912876456416375474272263423467896879187289499287374899 &nbsp; <span style="color:blue>99283428376482768916467328912876456416375474272263423467896879187289499287374899=</span> {{Font color|blue|(619**1) * (688**1) * (768**3) * (1024**17) * (64853**1) * (5302548928**1)}} &nbsp; {{Font color|blue|The dictionary is:}} &nbsp; <span style="color:blue>{619: 1, 688: 1, 768: 3, 1024: 17, 64853: 1, 5302548928.0: 1}</span> &nbsp; {{Font color|blue|Type 1 for another integer, 0 to terminate.}} {{center|'''DO NOT TRY TO FACTOR A NUMBER THIS LARGE UNLESS YOU ARE WILLING TO INTERUPT YOUR PYTHON PROGRAM.'''}} {{center|Keep in mind that the universe is only <math>4.4\times10^{26}\text{nanoseconds}</math> old.<ref>http://astronomy.nmsu.edu/geas/lectures/lecture02/slide04.html</ref>}} The number shown rounds up to 10⁸⁰. It is not likely that any computer will ever count that high. If the number to be factored consists of two prime number that are close to 10⁸⁰, the code will iterate all the way up to k coming close to 10⁴⁰. The inability of any computer to perform that many iterations helps explain how cryptocurrency works. &nbsp; 5euycac6qhi37ckbpxyatorppb1702p 2410270 2410269 2022-07-29T19:19:15Z Guy vandegrift 813252 wikitext text/x-wiki [[File:Factorization flowchart.svg|thumb|420px|Flowchart for a simple prime factorization code]] This lesson shows how the use of a python dictionary can facilitate the decomposition of a positive integer into a product of prime numbers. There are a few things the reader should know: * Most serious python programmers would probably prefer to download a package that contains a function that reduces an integer to a product of primes. Two promising candidates are [https://pypi.org/project/primefac/ pypi.org/project/primefac] and [https://www.sympy.org/en/index.html www.sympy.org/]. * Wikipedia has a much simpler program [[w:Special:Permalink/1086485306|this permalink]] of [[w:Trial division]] * Lacking the mental wherewithal to seize upon either of these opportunities, I googled "python prime factorization", found many codes, and selected the one that follows for two reasons:<ref>The code that follows is discussed at [https://stackoverflow.com/questions/15347174/python-finding-prime-factors stackoverflow question 15347174] and [https://scientific-python-101.readthedocs.io/_downloads/prime_factorization_solution.py scientific-python-101.readthedocs]</ref> :# I wanted a code that counts repeated factors in order to later write the factored form in wikitext with exponents.<ref>In the wikitext version, I had to add a few lines so that exponents of 1 would not appear. Example: <math>12=2^2\cdot 3^1</math> should instead read <math>12=2^2\cdot 3.</math> I ommitted the feature here in order to keep the lesson simple.</ref> :# I didn't understand the python code, and wanted master it in order to fully understand how python dictionaries are used. A good student exercise is to modify the code so that the output can be shown in a table. This one creates a wikitable: {| class="wikitable" style="text-align: center; vertical-align: bottom;" |+ <math>\text{Establishing } 1550=2\cdot 5^2 \cdot 31</math> |- ! n ! k ! k^2 ! k^2<n? ! n/k ! mod ! replace ! factor ! * |- | 1550 | 2 | 4 | TRUE | 775 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"|n: n/k | style="background:#ffffcc"|2 | |- | 775 | 2 | 4 | TRUE | 387 | 1 | k: k+1 | | |- | 775 | 3 | 9 | TRUE | 258 | 1 | k: k+1 | | |- | 775 | 4 | 16 | TRUE | 193 | 3 | k: k+1 | | |- | 775 | 5 | 25 | TRUE | 155 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 155 | 5 | 25 | TRUE | 31 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 31 | 5 | 25 | TRUE | 6 | 1 | k: k+1 | | |- | 31 | 6 | 36 | FALSE | - | - | - | | style="background:#ffffcc"| 31 |} ==Python== ‎<syntaxhighlight lang="python" line="1"> ## In a python code, the function definitions precede the code. We need only 1 function: def prime_factorization(N): ## 12=(2**2)*(2**1)*(5**1), or equivalently, 12 = 2**2 * 3 * 5 ## The dictionary, dict, expresses k**v as a "key-value" pair. In other ## words, if we look up the key, k=5, the dictionary will return the ## value, v=1. In other words: ## k**v <=> key**value <=> base**exponent (for the factored form of N) ## Begin function: n=N #--------------------# n will change in the while loop (getting smaller.) dict = {} #--------------# Start with an empty dictionary if n==1: #---------------# The factorization of 1 is a special case. return {1: 1} # In other words 1 = 1**1 (but recall that 1 is not prime.) k = 2 #------------------# Our first candidate for factorizing any integer. while k**2 <= n: #-------# Do the following unless k is "too big": if n % k: #----------# . Equivalent to: "If k is not a factor of n:" k += 1 #---------# .. increase the value of k by 1 on next attempt. else: #--------------# . Equivalent to "If k is a factor of n:" n /= k #---------# .. Replace n by n/k (for next attempt) try: #-----------# .. If k is in the dictionary, this attempts to: dict[k] += 1 # ... increase the value of k by 1. except KeyError: # .. KeyError occurs if k is not in the dictionary dict[k] = 1 # ... Add k to dictionary and set exponent v=1 # end(while)#------------# (exits while k**2 <= n:) # Here I inserted a diagnostic to be discussed later: # Documented code continues: if n > 1: #--------------# It can be shown that n is either 1 or prime. try: #---------------# . Try to enter n into dictionary dict[n] += 1 #---# .. increase exponent v->v+1 if k in dictinary except KeyError: #---# . If n is not in the dictionary, dict[n] = 1 # .. then enter n into dictionary with v=1 return dict #------# python talk for ending function ################ End def prime_factorization | Begin program ########## #import sys again=True while again == 1: text="\nEnter a positive integer you wish to factor into prime numbers:\n" N=int(input(text)) dict=prime_factorization(N) print("\n"+str(int(N))+"=") text="(" for b in dict.keys(): text+=str(int(b))+"**"+str(int(dict.get(b)))+") * (" print(text.rstrip(" * (")+"\n") print("The dictionary is:\n") print(dict) again=int(input("\nType 1 for another integer, 0 to terminate.")) </syntaxhighlight> ==Comments on code== ===Line 15: while k**2 <= n:=== This is an important line because it permits the termination of the search before one might think. I leave it as an exercise for the student to learn why any value of k larger than square root of n will never be the next factor of N. It is also left to the student to understand the fact that the time to find the factors can be cut nearly in half if we modify line 17 and increase k by 2 (instead of 1), and why this larger step cannot be taken until after k=2 has been considered. ===Lines in the code that might seem confusing=== The code uses two cleverness tricks that might seem confusing. I lack the formal training in programming to know whether such cleverness would be perceived as a flaw. I know that programmers are allowed to be clever and I know that clear programs are better than confusing programs. The fact that these tricks confused me does not necessarily mean they should be avoided. And, the nice thing about both sources of confusion is that they taught me something about python. ====Line 16: if n%k might seem backwards==== Python follows a common practice of interpreting the number 1 to be True and 0 to be False. But python also accepts any non-zero number as True. Here n%k denotes nmod&nbsp;k, which is the remainder when on divides n by k. If there is no remainder the "if" condition is not satisfied and the code stipulates that the next value of k is attempted. Any number not equal to zero is interpreted as the "if" condition being satisfied, meaning that k is a divisor of n (in that case, we replace n by n/k.) This might be good coding, but it seems to me that it would have been better to construct it along the lines of "if n%k == 0" or even "if n%k !=0". ====Lines 21-23 and 29-31: A failed attempt to change the value of a key==== In this code fragment, we ask if a given value of k has already been identified as a factor. But instead of "asking" if k has been declared a key in the dictionary, an <u>attempt</u> is made to change the value of k. If that attempt fails (because k is not among the dictionary's keys), then k is added to the list of keys, with a value of 1 (a value of 1 indicates that at this point, k, but not k² has been established to be a factor of n.) Again, there might be nothing wrong with using a key error like this. But the code would have been easier for me to follow if the if statement simply called the question of whether k was a key in dict. ==Sample output (with a really big number)== Sometimes you have to wait a few minutes to factor a number this large. But usually it comes out in just a few seconds. I would imagine that a number that was the product of only two roughly equal prime numbers would take forever because k=k+1 must be repeated a virtually infinite number of times. This number came quickly: {{Font color|blue|Enter a positive integer you wish to factor into prime numbers:}} 99283428376482768916467328912876456416375474272263423467896879187289499287374899 &nbsp; <span style="color:blue>99283428376482768916467328912876456416375474272263423467896879187289499287374899=</span> {{Font color|blue|(619**1) * (688**1) * (768**3) * (1024**17) * (64853**1) * (5302548928**1)}} &nbsp; {{Font color|blue|The dictionary is:}} &nbsp; <span style="color:blue>{619: 1, 688: 1, 768: 3, 1024: 17, 64853: 1, 5302548928.0: 1}</span> &nbsp; {{Font color|blue|Type 1 for another integer, 0 to terminate.}} {{center|'''DO NOT TRY TO FACTOR A NUMBER THIS LARGE UNLESS YOU ARE WILLING TO INTERUPT YOUR PYTHON PROGRAM.'''}} {{center|Keep in mind that the universe is only <math>4.4\times10^{26}\text{nanoseconds}</math> old.<ref>http://astronomy.nmsu.edu/geas/lectures/lecture02/slide04.html</ref>}} The number shown rounds up to 10⁸⁰. It is not likely that any computer will ever count that high. If the number to be factored consists of two prime number that are close to 10⁸⁰, the code will iterate all the way up to k coming close to 10⁴⁰. The inability of any computer to perform that many iterations helps explain how cryptocurrency works. &nbsp; 1pm59sv9dj8gyptxatb1ejqox4krjxn 2410271 2410270 2022-07-29T19:20:06Z Guy vandegrift 813252 wikitext text/x-wiki [[File:Factorization flowchart.svg|thumb|420px|Flowchart for a simple prime factorization code]] This lesson shows how the use of a python dictionary can facilitate the decomposition of a positive integer into a product of prime numbers. There are a few things the reader should know: * Most serious python programmers would probably prefer to download a package that contains a function that reduces an integer to a product of primes. Two promising candidates are [https://pypi.org/project/primefac/ pypi.org/project/primefac] and [https://www.sympy.org/en/index.html www.sympy.org/]. * Wikipedia has a much simpler program [[w:Special:Permalink/1086485306|this permalink]] of [[w:Trial division]] * Lacking the mental wherewithal to seize upon either of these opportunities, I googled "python prime factorization", found many codes, and selected the one that follows for two reasons:<ref>The code that follows is discussed at [https://stackoverflow.com/questions/15347174/python-finding-prime-factors stackoverflow question 15347174] and [https://scientific-python-101.readthedocs.io/_downloads/prime_factorization_solution.py scientific-python-101.readthedocs]</ref> :# I wanted a code that counts repeated factors in order to later write the factored form in wikitext with exponents.<ref>In the wikitext version, I had to add a few lines so that exponents of 1 would not appear. Example: <math>12=2^2\cdot 3^1</math> should instead read <math>12=2^2\cdot 3.</math> I ommitted the feature here in order to keep the lesson simple.</ref> :# I didn't understand the python code, and wanted master it in order to fully understand how python dictionaries are used. Instructors can task students with the exercise of modifying this code so that the output can be shown in a table. This one creates a wikitable: {| class="wikitable" style="text-align: center; vertical-align: bottom;" |+ <math>\text{Establishing } 1550=2\cdot 5^2 \cdot 31</math> |- ! n ! k ! k^2 ! k^2<n? ! n/k ! mod ! replace ! factor ! * |- | 1550 | 2 | 4 | TRUE | 775 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"|n: n/k | style="background:#ffffcc"|2 | |- | 775 | 2 | 4 | TRUE | 387 | 1 | k: k+1 | | |- | 775 | 3 | 9 | TRUE | 258 | 1 | k: k+1 | | |- | 775 | 4 | 16 | TRUE | 193 | 3 | k: k+1 | | |- | 775 | 5 | 25 | TRUE | 155 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 155 | 5 | 25 | TRUE | 31 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 31 | 5 | 25 | TRUE | 6 | 1 | k: k+1 | | |- | 31 | 6 | 36 | FALSE | - | - | - | | style="background:#ffffcc"| 31 |} ==Python== ‎<syntaxhighlight lang="python" line="1"> ## In a python code, the function definitions precede the code. We need only 1 function: def prime_factorization(N): ## 12=(2**2)*(2**1)*(5**1), or equivalently, 12 = 2**2 * 3 * 5 ## The dictionary, dict, expresses k**v as a "key-value" pair. In other ## words, if we look up the key, k=5, the dictionary will return the ## value, v=1. In other words: ## k**v <=> key**value <=> base**exponent (for the factored form of N) ## Begin function: n=N #--------------------# n will change in the while loop (getting smaller.) dict = {} #--------------# Start with an empty dictionary if n==1: #---------------# The factorization of 1 is a special case. return {1: 1} # In other words 1 = 1**1 (but recall that 1 is not prime.) k = 2 #------------------# Our first candidate for factorizing any integer. while k**2 <= n: #-------# Do the following unless k is "too big": if n % k: #----------# . Equivalent to: "If k is not a factor of n:" k += 1 #---------# .. increase the value of k by 1 on next attempt. else: #--------------# . Equivalent to "If k is a factor of n:" n /= k #---------# .. Replace n by n/k (for next attempt) try: #-----------# .. If k is in the dictionary, this attempts to: dict[k] += 1 # ... increase the value of k by 1. except KeyError: # .. KeyError occurs if k is not in the dictionary dict[k] = 1 # ... Add k to dictionary and set exponent v=1 # end(while)#------------# (exits while k**2 <= n:) # Here I inserted a diagnostic to be discussed later: # Documented code continues: if n > 1: #--------------# It can be shown that n is either 1 or prime. try: #---------------# . Try to enter n into dictionary dict[n] += 1 #---# .. increase exponent v->v+1 if k in dictinary except KeyError: #---# . If n is not in the dictionary, dict[n] = 1 # .. then enter n into dictionary with v=1 return dict #------# python talk for ending function ################ End def prime_factorization | Begin program ########## #import sys again=True while again == 1: text="\nEnter a positive integer you wish to factor into prime numbers:\n" N=int(input(text)) dict=prime_factorization(N) print("\n"+str(int(N))+"=") text="(" for b in dict.keys(): text+=str(int(b))+"**"+str(int(dict.get(b)))+") * (" print(text.rstrip(" * (")+"\n") print("The dictionary is:\n") print(dict) again=int(input("\nType 1 for another integer, 0 to terminate.")) </syntaxhighlight> ==Comments on code== ===Line 15: while k**2 <= n:=== This is an important line because it permits the termination of the search before one might think. I leave it as an exercise for the student to learn why any value of k larger than square root of n will never be the next factor of N. It is also left to the student to understand the fact that the time to find the factors can be cut nearly in half if we modify line 17 and increase k by 2 (instead of 1), and why this larger step cannot be taken until after k=2 has been considered. ===Lines in the code that might seem confusing=== The code uses two cleverness tricks that might seem confusing. I lack the formal training in programming to know whether such cleverness would be perceived as a flaw. I know that programmers are allowed to be clever and I know that clear programs are better than confusing programs. The fact that these tricks confused me does not necessarily mean they should be avoided. And, the nice thing about both sources of confusion is that they taught me something about python. ====Line 16: if n%k might seem backwards==== Python follows a common practice of interpreting the number 1 to be True and 0 to be False. But python also accepts any non-zero number as True. Here n%k denotes nmod&nbsp;k, which is the remainder when on divides n by k. If there is no remainder the "if" condition is not satisfied and the code stipulates that the next value of k is attempted. Any number not equal to zero is interpreted as the "if" condition being satisfied, meaning that k is a divisor of n (in that case, we replace n by n/k.) This might be good coding, but it seems to me that it would have been better to construct it along the lines of "if n%k == 0" or even "if n%k !=0". ====Lines 21-23 and 29-31: A failed attempt to change the value of a key==== In this code fragment, we ask if a given value of k has already been identified as a factor. But instead of "asking" if k has been declared a key in the dictionary, an <u>attempt</u> is made to change the value of k. If that attempt fails (because k is not among the dictionary's keys), then k is added to the list of keys, with a value of 1 (a value of 1 indicates that at this point, k, but not k² has been established to be a factor of n.) Again, there might be nothing wrong with using a key error like this. But the code would have been easier for me to follow if the if statement simply called the question of whether k was a key in dict. ==Sample output (with a really big number)== Sometimes you have to wait a few minutes to factor a number this large. But usually it comes out in just a few seconds. I would imagine that a number that was the product of only two roughly equal prime numbers would take forever because k=k+1 must be repeated a virtually infinite number of times. This number came quickly: {{Font color|blue|Enter a positive integer you wish to factor into prime numbers:}} 99283428376482768916467328912876456416375474272263423467896879187289499287374899 &nbsp; <span style="color:blue>99283428376482768916467328912876456416375474272263423467896879187289499287374899=</span> {{Font color|blue|(619**1) * (688**1) * (768**3) * (1024**17) * (64853**1) * (5302548928**1)}} &nbsp; {{Font color|blue|The dictionary is:}} &nbsp; <span style="color:blue>{619: 1, 688: 1, 768: 3, 1024: 17, 64853: 1, 5302548928.0: 1}</span> &nbsp; {{Font color|blue|Type 1 for another integer, 0 to terminate.}} {{center|'''DO NOT TRY TO FACTOR A NUMBER THIS LARGE UNLESS YOU ARE WILLING TO INTERUPT YOUR PYTHON PROGRAM.'''}} {{center|Keep in mind that the universe is only <math>4.4\times10^{26}\text{nanoseconds}</math> old.<ref>http://astronomy.nmsu.edu/geas/lectures/lecture02/slide04.html</ref>}} The number shown rounds up to 10⁸⁰. It is not likely that any computer will ever count that high. If the number to be factored consists of two prime number that are close to 10⁸⁰, the code will iterate all the way up to k coming close to 10⁴⁰. The inability of any computer to perform that many iterations helps explain how cryptocurrency works. &nbsp; ekvq47k1wvc3f5oey4fp4b09ze2qflk 2410272 2410271 2022-07-29T19:20:42Z Guy vandegrift 813252 /* Python */ wikitext text/x-wiki [[File:Factorization flowchart.svg|thumb|420px|Flowchart for a simple prime factorization code]] This lesson shows how the use of a python dictionary can facilitate the decomposition of a positive integer into a product of prime numbers. There are a few things the reader should know: * Most serious python programmers would probably prefer to download a package that contains a function that reduces an integer to a product of primes. Two promising candidates are [https://pypi.org/project/primefac/ pypi.org/project/primefac] and [https://www.sympy.org/en/index.html www.sympy.org/]. * Wikipedia has a much simpler program [[w:Special:Permalink/1086485306|this permalink]] of [[w:Trial division]] * Lacking the mental wherewithal to seize upon either of these opportunities, I googled "python prime factorization", found many codes, and selected the one that follows for two reasons:<ref>The code that follows is discussed at [https://stackoverflow.com/questions/15347174/python-finding-prime-factors stackoverflow question 15347174] and [https://scientific-python-101.readthedocs.io/_downloads/prime_factorization_solution.py scientific-python-101.readthedocs]</ref> :# I wanted a code that counts repeated factors in order to later write the factored form in wikitext with exponents.<ref>In the wikitext version, I had to add a few lines so that exponents of 1 would not appear. Example: <math>12=2^2\cdot 3^1</math> should instead read <math>12=2^2\cdot 3.</math> I ommitted the feature here in order to keep the lesson simple.</ref> :# I didn't understand the python code, and wanted master it in order to fully understand how python dictionaries are used. Instructors can task students with the exercise of modifying this code so that the output can be shown in a table. This one creates a wikitable: {| class="wikitable" style="text-align: center; vertical-align: bottom;" |+ <math>\text{Establishing } 1550=2\cdot 5^2 \cdot 31</math> |- ! n ! k ! k^2 ! k^2<n? ! n/k ! mod ! replace ! factor ! * |- | 1550 | 2 | 4 | TRUE | 775 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"|n: n/k | style="background:#ffffcc"|2 | |- | 775 | 2 | 4 | TRUE | 387 | 1 | k: k+1 | | |- | 775 | 3 | 9 | TRUE | 258 | 1 | k: k+1 | | |- | 775 | 4 | 16 | TRUE | 193 | 3 | k: k+1 | | |- | 775 | 5 | 25 | TRUE | 155 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 155 | 5 | 25 | TRUE | 31 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 31 | 5 | 25 | TRUE | 6 | 1 | k: k+1 | | |- | 31 | 6 | 36 | FALSE | - | - | - | | style="background:#ffffcc"| 31 |} ==Python code== ‎<syntaxhighlight lang="python" line="1"> ## In a python code, the function definitions precede the code. We need only 1 function: def prime_factorization(N): ## 12=(2**2)*(2**1)*(5**1), or equivalently, 12 = 2**2 * 3 * 5 ## The dictionary, dict, expresses k**v as a "key-value" pair. In other ## words, if we look up the key, k=5, the dictionary will return the ## value, v=1. In other words: ## k**v <=> key**value <=> base**exponent (for the factored form of N) ## Begin function: n=N #--------------------# n will change in the while loop (getting smaller.) dict = {} #--------------# Start with an empty dictionary if n==1: #---------------# The factorization of 1 is a special case. return {1: 1} # In other words 1 = 1**1 (but recall that 1 is not prime.) k = 2 #------------------# Our first candidate for factorizing any integer. while k**2 <= n: #-------# Do the following unless k is "too big": if n % k: #----------# . Equivalent to: "If k is not a factor of n:" k += 1 #---------# .. increase the value of k by 1 on next attempt. else: #--------------# . Equivalent to "If k is a factor of n:" n /= k #---------# .. Replace n by n/k (for next attempt) try: #-----------# .. If k is in the dictionary, this attempts to: dict[k] += 1 # ... increase the value of k by 1. except KeyError: # .. KeyError occurs if k is not in the dictionary dict[k] = 1 # ... Add k to dictionary and set exponent v=1 # end(while)#------------# (exits while k**2 <= n:) # Here I inserted a diagnostic to be discussed later: # Documented code continues: if n > 1: #--------------# It can be shown that n is either 1 or prime. try: #---------------# . Try to enter n into dictionary dict[n] += 1 #---# .. increase exponent v->v+1 if k in dictinary except KeyError: #---# . If n is not in the dictionary, dict[n] = 1 # .. then enter n into dictionary with v=1 return dict #------# python talk for ending function ################ End def prime_factorization | Begin program ########## #import sys again=True while again == 1: text="\nEnter a positive integer you wish to factor into prime numbers:\n" N=int(input(text)) dict=prime_factorization(N) print("\n"+str(int(N))+"=") text="(" for b in dict.keys(): text+=str(int(b))+"**"+str(int(dict.get(b)))+") * (" print(text.rstrip(" * (")+"\n") print("The dictionary is:\n") print(dict) again=int(input("\nType 1 for another integer, 0 to terminate.")) </syntaxhighlight> ==Comments on code== ===Line 15: while k**2 <= n:=== This is an important line because it permits the termination of the search before one might think. I leave it as an exercise for the student to learn why any value of k larger than square root of n will never be the next factor of N. It is also left to the student to understand the fact that the time to find the factors can be cut nearly in half if we modify line 17 and increase k by 2 (instead of 1), and why this larger step cannot be taken until after k=2 has been considered. ===Lines in the code that might seem confusing=== The code uses two cleverness tricks that might seem confusing. I lack the formal training in programming to know whether such cleverness would be perceived as a flaw. I know that programmers are allowed to be clever and I know that clear programs are better than confusing programs. The fact that these tricks confused me does not necessarily mean they should be avoided. And, the nice thing about both sources of confusion is that they taught me something about python. ====Line 16: if n%k might seem backwards==== Python follows a common practice of interpreting the number 1 to be True and 0 to be False. But python also accepts any non-zero number as True. Here n%k denotes nmod&nbsp;k, which is the remainder when on divides n by k. If there is no remainder the "if" condition is not satisfied and the code stipulates that the next value of k is attempted. Any number not equal to zero is interpreted as the "if" condition being satisfied, meaning that k is a divisor of n (in that case, we replace n by n/k.) This might be good coding, but it seems to me that it would have been better to construct it along the lines of "if n%k == 0" or even "if n%k !=0". ====Lines 21-23 and 29-31: A failed attempt to change the value of a key==== In this code fragment, we ask if a given value of k has already been identified as a factor. But instead of "asking" if k has been declared a key in the dictionary, an <u>attempt</u> is made to change the value of k. If that attempt fails (because k is not among the dictionary's keys), then k is added to the list of keys, with a value of 1 (a value of 1 indicates that at this point, k, but not k² has been established to be a factor of n.) Again, there might be nothing wrong with using a key error like this. But the code would have been easier for me to follow if the if statement simply called the question of whether k was a key in dict. ==Sample output (with a really big number)== Sometimes you have to wait a few minutes to factor a number this large. But usually it comes out in just a few seconds. I would imagine that a number that was the product of only two roughly equal prime numbers would take forever because k=k+1 must be repeated a virtually infinite number of times. This number came quickly: {{Font color|blue|Enter a positive integer you wish to factor into prime numbers:}} 99283428376482768916467328912876456416375474272263423467896879187289499287374899 &nbsp; <span style="color:blue>99283428376482768916467328912876456416375474272263423467896879187289499287374899=</span> {{Font color|blue|(619**1) * (688**1) * (768**3) * (1024**17) * (64853**1) * (5302548928**1)}} &nbsp; {{Font color|blue|The dictionary is:}} &nbsp; <span style="color:blue>{619: 1, 688: 1, 768: 3, 1024: 17, 64853: 1, 5302548928.0: 1}</span> &nbsp; {{Font color|blue|Type 1 for another integer, 0 to terminate.}} {{center|'''DO NOT TRY TO FACTOR A NUMBER THIS LARGE UNLESS YOU ARE WILLING TO INTERUPT YOUR PYTHON PROGRAM.'''}} {{center|Keep in mind that the universe is only <math>4.4\times10^{26}\text{nanoseconds}</math> old.<ref>http://astronomy.nmsu.edu/geas/lectures/lecture02/slide04.html</ref>}} The number shown rounds up to 10⁸⁰. It is not likely that any computer will ever count that high. If the number to be factored consists of two prime number that are close to 10⁸⁰, the code will iterate all the way up to k coming close to 10⁴⁰. The inability of any computer to perform that many iterations helps explain how cryptocurrency works. &nbsp; cn18y2vz2ceo6my2sgcf84elaea7rtp 2410273 2410272 2022-07-29T19:23:28Z Guy vandegrift 813252 wikitext text/x-wiki [[File:Factorization flowchart.svg|thumb|420px|Flowchart for a simple prime factorization code]] This lesson shows how the use of a python dictionary can facilitate the decomposition of a positive integer into a product of prime numbers. There are a few things the reader should know: * Most serious python programmers would probably prefer to download a package that contains a function that reduces an integer to a product of primes. Two promising candidates are [https://pypi.org/project/primefac/ pypi.org/project/primefac] and [https://www.sympy.org/en/index.html www.sympy.org/]. * Wikipedia has a much simpler program [[w:Special:Permalink/1086485306|this permalink]] of [[w:Trial division]] * Lacking the mental wherewithal to seize upon either of these opportunities, I googled "python prime factorization", found many codes, and selected the one that follows for two reasons:<ref>The code that follows is discussed at [https://stackoverflow.com/questions/15347174/python-finding-prime-factors stackoverflow question 15347174] and [https://scientific-python-101.readthedocs.io/_downloads/prime_factorization_solution.py scientific-python-101.readthedocs]</ref> :# I wanted a code that counts repeated factors in order to later write the factored form in wikitext with exponents.<ref>In the wikitext version, I had to add a few lines so that exponents of 1 would not appear. Example: <math>12=2^2\cdot 3^1</math> should instead read <math>12=2^2\cdot 3.</math> I ommitted the feature here in order to keep the lesson simple.</ref> :# I didn't understand the python code, and wanted master it in order to fully understand how python dictionaries are used. A good student exercise would be to modify this code so that it runs approximately twice as fast.Instructors can task students with modifying this code so that the output can be shown in a table. This one creates a wikitable that illustrates the "wasted" efforts to see if even numbers can be added to the list of primes (see: [[Wikipedia:Trial division#Method]].): {| class="wikitable" style="text-align: center; vertical-align: bottom;" |+ <math>\text{Establishing } 1550=2\cdot 5^2 \cdot 31</math> |- ! n ! k ! k^2 ! k^2<n? ! n/k ! mod ! replace ! factor ! * |- | 1550 | 2 | 4 | TRUE | 775 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"|n: n/k | style="background:#ffffcc"|2 | |- | 775 | 2 | 4 | TRUE | 387 | 1 | k: k+1 | | |- | 775 | 3 | 9 | TRUE | 258 | 1 | k: k+1 | | |- | 775 | 4 | 16 | TRUE | 193 | 3 | k: k+1 | | |- | 775 | 5 | 25 | TRUE | 155 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 155 | 5 | 25 | TRUE | 31 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 31 | 5 | 25 | TRUE | 6 | 1 | k: k+1 | | |- | 31 | 6 | 36 | FALSE | - | - | - | | style="background:#ffffcc"| 31 |} ==Python code== ‎<syntaxhighlight lang="python" line="1"> ## In a python code, the function definitions precede the code. We need only 1 function: def prime_factorization(N): ## 12=(2**2)*(2**1)*(5**1), or equivalently, 12 = 2**2 * 3 * 5 ## The dictionary, dict, expresses k**v as a "key-value" pair. In other ## words, if we look up the key, k=5, the dictionary will return the ## value, v=1. In other words: ## k**v <=> key**value <=> base**exponent (for the factored form of N) ## Begin function: n=N #--------------------# n will change in the while loop (getting smaller.) dict = {} #--------------# Start with an empty dictionary if n==1: #---------------# The factorization of 1 is a special case. return {1: 1} # In other words 1 = 1**1 (but recall that 1 is not prime.) k = 2 #------------------# Our first candidate for factorizing any integer. while k**2 <= n: #-------# Do the following unless k is "too big": if n % k: #----------# . Equivalent to: "If k is not a factor of n:" k += 1 #---------# .. increase the value of k by 1 on next attempt. else: #--------------# . Equivalent to "If k is a factor of n:" n /= k #---------# .. Replace n by n/k (for next attempt) try: #-----------# .. If k is in the dictionary, this attempts to: dict[k] += 1 # ... increase the value of k by 1. except KeyError: # .. KeyError occurs if k is not in the dictionary dict[k] = 1 # ... Add k to dictionary and set exponent v=1 # end(while)#------------# (exits while k**2 <= n:) # Here I inserted a diagnostic to be discussed later: # Documented code continues: if n > 1: #--------------# It can be shown that n is either 1 or prime. try: #---------------# . Try to enter n into dictionary dict[n] += 1 #---# .. increase exponent v->v+1 if k in dictinary except KeyError: #---# . If n is not in the dictionary, dict[n] = 1 # .. then enter n into dictionary with v=1 return dict #------# python talk for ending function ################ End def prime_factorization | Begin program ########## #import sys again=True while again == 1: text="\nEnter a positive integer you wish to factor into prime numbers:\n" N=int(input(text)) dict=prime_factorization(N) print("\n"+str(int(N))+"=") text="(" for b in dict.keys(): text+=str(int(b))+"**"+str(int(dict.get(b)))+") * (" print(text.rstrip(" * (")+"\n") print("The dictionary is:\n") print(dict) again=int(input("\nType 1 for another integer, 0 to terminate.")) </syntaxhighlight> ==Comments on code== ===Line 15: while k**2 <= n:=== This is an important line because it permits the termination of the search before one might think. I leave it as an exercise for the student to learn why any value of k larger than square root of n will never be the next factor of N. It is also left to the student to understand the fact that the time to find the factors can be cut nearly in half if we modify line 17 and increase k by 2 (instead of 1), and why this larger step cannot be taken until after k=2 has been considered. ===Lines in the code that might seem confusing=== The code uses two cleverness tricks that might seem confusing. I lack the formal training in programming to know whether such cleverness would be perceived as a flaw. I know that programmers are allowed to be clever and I know that clear programs are better than confusing programs. The fact that these tricks confused me does not necessarily mean they should be avoided. And, the nice thing about both sources of confusion is that they taught me something about python. ====Line 16: if n%k might seem backwards==== Python follows a common practice of interpreting the number 1 to be True and 0 to be False. But python also accepts any non-zero number as True. Here n%k denotes nmod&nbsp;k, which is the remainder when on divides n by k. If there is no remainder the "if" condition is not satisfied and the code stipulates that the next value of k is attempted. Any number not equal to zero is interpreted as the "if" condition being satisfied, meaning that k is a divisor of n (in that case, we replace n by n/k.) This might be good coding, but it seems to me that it would have been better to construct it along the lines of "if n%k == 0" or even "if n%k !=0". ====Lines 21-23 and 29-31: A failed attempt to change the value of a key==== In this code fragment, we ask if a given value of k has already been identified as a factor. But instead of "asking" if k has been declared a key in the dictionary, an <u>attempt</u> is made to change the value of k. If that attempt fails (because k is not among the dictionary's keys), then k is added to the list of keys, with a value of 1 (a value of 1 indicates that at this point, k, but not k² has been established to be a factor of n.) Again, there might be nothing wrong with using a key error like this. But the code would have been easier for me to follow if the if statement simply called the question of whether k was a key in dict. ==Sample output (with a really big number)== Sometimes you have to wait a few minutes to factor a number this large. But usually it comes out in just a few seconds. I would imagine that a number that was the product of only two roughly equal prime numbers would take forever because k=k+1 must be repeated a virtually infinite number of times. This number came quickly: {{Font color|blue|Enter a positive integer you wish to factor into prime numbers:}} 99283428376482768916467328912876456416375474272263423467896879187289499287374899 &nbsp; <span style="color:blue>99283428376482768916467328912876456416375474272263423467896879187289499287374899=</span> {{Font color|blue|(619**1) * (688**1) * (768**3) * (1024**17) * (64853**1) * (5302548928**1)}} &nbsp; {{Font color|blue|The dictionary is:}} &nbsp; <span style="color:blue>{619: 1, 688: 1, 768: 3, 1024: 17, 64853: 1, 5302548928.0: 1}</span> &nbsp; {{Font color|blue|Type 1 for another integer, 0 to terminate.}} {{center|'''DO NOT TRY TO FACTOR A NUMBER THIS LARGE UNLESS YOU ARE WILLING TO INTERUPT YOUR PYTHON PROGRAM.'''}} {{center|Keep in mind that the universe is only <math>4.4\times10^{26}\text{nanoseconds}</math> old.<ref>http://astronomy.nmsu.edu/geas/lectures/lecture02/slide04.html</ref>}} The number shown rounds up to 10⁸⁰. It is not likely that any computer will ever count that high. If the number to be factored consists of two prime number that are close to 10⁸⁰, the code will iterate all the way up to k coming close to 10⁴⁰. The inability of any computer to perform that many iterations helps explain how cryptocurrency works. &nbsp; ald6st9n4qbegdi66ptfj7lm38hvtob 2410274 2410273 2022-07-29T19:25:01Z Guy vandegrift 813252 wikitext text/x-wiki [[File:Factorization flowchart.svg|thumb|420px|Flowchart for a simple prime factorization code]] This lesson shows how the use of a python dictionary can facilitate the decomposition of a positive integer into a product of prime numbers. There are a few things the reader should know: * Most serious python programmers would probably prefer to download a package that contains a function that reduces an integer to a product of primes. Two promising candidates are [https://pypi.org/project/primefac/ pypi.org/project/primefac] and [https://www.sympy.org/en/index.html www.sympy.org/]. * Wikipedia has a much simpler program [[w:Special:Permalink/1086485306|this permalink]] of [[w:Trial division]] * Lacking the mental wherewithal to seize upon either of these opportunities, I googled "python prime factorization", found many codes, and selected the one that follows for two reasons:<ref>The code that follows is discussed at [https://stackoverflow.com/questions/15347174/python-finding-prime-factors stackoverflow question 15347174] and [https://scientific-python-101.readthedocs.io/_downloads/prime_factorization_solution.py scientific-python-101.readthedocs]</ref> :# I wanted a code that counts repeated factors in order to later write the factored form in wikitext with exponents.<ref>In the wikitext version, I had to add a few lines so that exponents of 1 would not appear. Example: <math>12=2^2\cdot 3^1</math> should instead read <math>12=2^2\cdot 3.</math> I ommitted the feature here in order to keep the lesson simple.</ref> :# I didn't understand the python code, and wanted master it in order to fully understand how python dictionaries are used. A good student exercise would be to modify this code so that it runs approximately twice as fast.Instructors can task students with modifying this code so that the output can be shown in a table. This one creates a wikitable that illustrates the "wasted" efforts to see if 4 and 6 are prime factors (see: [[Wikipedia:Trial division#Method]].): {| class="wikitable" style="text-align: center; vertical-align: bottom;" |+ <math>\text{Establishing } 1550=2\cdot 5^2 \cdot 31</math> |- ! n ! k ! k^2 ! k^2<n? ! n/k ! mod ! replace ! factor ! * |- | 1550 | 2 | 4 | TRUE | 775 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"|n: n/k | style="background:#ffffcc"|2 | |- | 775 | 2 | 4 | TRUE | 387 | 1 | k: k+1 | | |- | 775 | 3 | 9 | TRUE | 258 | 1 | k: k+1 | | |- | 775 | 4 | 16 | TRUE | 193 | 3 | k: k+1 | | |- | 775 | 5 | 25 | TRUE | 155 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 155 | 5 | 25 | TRUE | 31 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 31 | 5 | 25 | TRUE | 6 | 1 | k: k+1 | | |- | 31 | 6 | 36 | FALSE | - | - | - | | style="background:#ffffcc"| 31 |} ==Python code== ‎<syntaxhighlight lang="python" line="1"> ## In a python code, the function definitions precede the code. We need only 1 function: def prime_factorization(N): ## 12=(2**2)*(2**1)*(5**1), or equivalently, 12 = 2**2 * 3 * 5 ## The dictionary, dict, expresses k**v as a "key-value" pair. In other ## words, if we look up the key, k=5, the dictionary will return the ## value, v=1. In other words: ## k**v <=> key**value <=> base**exponent (for the factored form of N) ## Begin function: n=N #--------------------# n will change in the while loop (getting smaller.) dict = {} #--------------# Start with an empty dictionary if n==1: #---------------# The factorization of 1 is a special case. return {1: 1} # In other words 1 = 1**1 (but recall that 1 is not prime.) k = 2 #------------------# Our first candidate for factorizing any integer. while k**2 <= n: #-------# Do the following unless k is "too big": if n % k: #----------# . Equivalent to: "If k is not a factor of n:" k += 1 #---------# .. increase the value of k by 1 on next attempt. else: #--------------# . Equivalent to "If k is a factor of n:" n /= k #---------# .. Replace n by n/k (for next attempt) try: #-----------# .. If k is in the dictionary, this attempts to: dict[k] += 1 # ... increase the value of k by 1. except KeyError: # .. KeyError occurs if k is not in the dictionary dict[k] = 1 # ... Add k to dictionary and set exponent v=1 # end(while)#------------# (exits while k**2 <= n:) # Here I inserted a diagnostic to be discussed later: # Documented code continues: if n > 1: #--------------# It can be shown that n is either 1 or prime. try: #---------------# . Try to enter n into dictionary dict[n] += 1 #---# .. increase exponent v->v+1 if k in dictinary except KeyError: #---# . If n is not in the dictionary, dict[n] = 1 # .. then enter n into dictionary with v=1 return dict #------# python talk for ending function ################ End def prime_factorization | Begin program ########## #import sys again=True while again == 1: text="\nEnter a positive integer you wish to factor into prime numbers:\n" N=int(input(text)) dict=prime_factorization(N) print("\n"+str(int(N))+"=") text="(" for b in dict.keys(): text+=str(int(b))+"**"+str(int(dict.get(b)))+") * (" print(text.rstrip(" * (")+"\n") print("The dictionary is:\n") print(dict) again=int(input("\nType 1 for another integer, 0 to terminate.")) </syntaxhighlight> ==Comments on code== ===Line 15: while k**2 <= n:=== This is an important line because it permits the termination of the search before one might think. I leave it as an exercise for the student to learn why any value of k larger than square root of n will never be the next factor of N. It is also left to the student to understand the fact that the time to find the factors can be cut nearly in half if we modify line 17 and increase k by 2 (instead of 1), and why this larger step cannot be taken until after k=2 has been considered. ===Lines in the code that might seem confusing=== The code uses two cleverness tricks that might seem confusing. I lack the formal training in programming to know whether such cleverness would be perceived as a flaw. I know that programmers are allowed to be clever and I know that clear programs are better than confusing programs. The fact that these tricks confused me does not necessarily mean they should be avoided. And, the nice thing about both sources of confusion is that they taught me something about python. ====Line 16: if n%k might seem backwards==== Python follows a common practice of interpreting the number 1 to be True and 0 to be False. But python also accepts any non-zero number as True. Here n%k denotes nmod&nbsp;k, which is the remainder when on divides n by k. If there is no remainder the "if" condition is not satisfied and the code stipulates that the next value of k is attempted. Any number not equal to zero is interpreted as the "if" condition being satisfied, meaning that k is a divisor of n (in that case, we replace n by n/k.) This might be good coding, but it seems to me that it would have been better to construct it along the lines of "if n%k == 0" or even "if n%k !=0". ====Lines 21-23 and 29-31: A failed attempt to change the value of a key==== In this code fragment, we ask if a given value of k has already been identified as a factor. But instead of "asking" if k has been declared a key in the dictionary, an <u>attempt</u> is made to change the value of k. If that attempt fails (because k is not among the dictionary's keys), then k is added to the list of keys, with a value of 1 (a value of 1 indicates that at this point, k, but not k² has been established to be a factor of n.) Again, there might be nothing wrong with using a key error like this. But the code would have been easier for me to follow if the if statement simply called the question of whether k was a key in dict. ==Sample output (with a really big number)== Sometimes you have to wait a few minutes to factor a number this large. But usually it comes out in just a few seconds. I would imagine that a number that was the product of only two roughly equal prime numbers would take forever because k=k+1 must be repeated a virtually infinite number of times. This number came quickly: {{Font color|blue|Enter a positive integer you wish to factor into prime numbers:}} 99283428376482768916467328912876456416375474272263423467896879187289499287374899 &nbsp; <span style="color:blue>99283428376482768916467328912876456416375474272263423467896879187289499287374899=</span> {{Font color|blue|(619**1) * (688**1) * (768**3) * (1024**17) * (64853**1) * (5302548928**1)}} &nbsp; {{Font color|blue|The dictionary is:}} &nbsp; <span style="color:blue>{619: 1, 688: 1, 768: 3, 1024: 17, 64853: 1, 5302548928.0: 1}</span> &nbsp; {{Font color|blue|Type 1 for another integer, 0 to terminate.}} {{center|'''DO NOT TRY TO FACTOR A NUMBER THIS LARGE UNLESS YOU ARE WILLING TO INTERUPT YOUR PYTHON PROGRAM.'''}} {{center|Keep in mind that the universe is only <math>4.4\times10^{26}\text{nanoseconds}</math> old.<ref>http://astronomy.nmsu.edu/geas/lectures/lecture02/slide04.html</ref>}} The number shown rounds up to 10⁸⁰. It is not likely that any computer will ever count that high. If the number to be factored consists of two prime number that are close to 10⁸⁰, the code will iterate all the way up to k coming close to 10⁴⁰. The inability of any computer to perform that many iterations helps explain how cryptocurrency works. &nbsp; 7oabdy4jeq60z3pf6b0pcrl8j9sgtur 2410275 2410274 2022-07-29T19:34:36Z Guy vandegrift 813252 wikitext text/x-wiki [[File:Factorization flowchart.svg|thumb|420px|Flowchart for a simple prime factorization code]] '''Topics covered:''' ''prime factorization, flowcharts, python dictionaries'' This lesson shows how the use of a python dictionary can facilitate the decomposition of a positive integer into a product of prime numbers. There are a few things the reader should know: * Most serious python programmers would probably prefer to download a package that contains a function that reduces an integer to a product of primes. Two promising candidates are [https://pypi.org/project/primefac/ pypi.org/project/primefac] and [https://www.sympy.org/en/index.html www.sympy.org/]. * Wikipedia has a much simpler program [[w:Special:Permalink/1086485306|this permalink]] of [[w:Trial division]] * Lacking the mental wherewithal to seize upon either of these opportunities, I googled "python prime factorization", found many codes, and selected the one that follows for two reasons:<ref>The code that follows is discussed at [https://stackoverflow.com/questions/15347174/python-finding-prime-factors stackoverflow question 15347174] and [https://scientific-python-101.readthedocs.io/_downloads/prime_factorization_solution.py scientific-python-101.readthedocs]</ref> :# I wanted a code that counts repeated factors in order to later write the factored form in wikitext with exponents.<ref>In the wikitext version, I had to add a few lines so that exponents of 1 would not appear. Example: <math>12=2^2\cdot 3^1</math> should instead read <math>12=2^2\cdot 3.</math> I ommitted the feature here in order to keep the lesson simple.</ref> :# I didn't understand the python code, and wanted master it in order to fully understand how python dictionaries are used. A good student exercise would be to modify this code so that it runs approximately twice as fast.Instructors can task students with modifying this code so that the output can be shown in a table. This one creates a wikitable that illustrates the "wasted" efforts to see if 4 and 6 are prime factors (see: [[Wikipedia:Trial division#Method]].): {| class="wikitable" style="text-align: center; vertical-align: bottom;" |+ <math>\text{Establishing } 1550=2\cdot 5^2 \cdot 31</math> |- ! n ! k ! k^2 ! k^2<n? ! n/k ! mod ! replace ! factor ! * |- | 1550 | 2 | 4 | TRUE | 775 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"|n: n/k | style="background:#ffffcc"|2 | |- | 775 | 2 | 4 | TRUE | 387 | 1 | k: k+1 | | |- | 775 | 3 | 9 | TRUE | 258 | 1 | k: k+1 | | |- | 775 | 4 | 16 | TRUE | 193 | 3 | k: k+1 | | |- | 775 | 5 | 25 | TRUE | 155 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 155 | 5 | 25 | TRUE | 31 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 31 | 5 | 25 | TRUE | 6 | 1 | k: k+1 | | |- | 31 | 6 | 36 | FALSE | - | - | - | | style="background:#ffffcc"| 31 |} ==Python code== ‎<syntaxhighlight lang="python" line="1"> ## In a python code, the function definitions precede the code. We need only 1 function: def prime_factorization(N): ## 12=(2**2)*(2**1)*(5**1), or equivalently, 12 = 2**2 * 3 * 5 ## The dictionary, dict, expresses k**v as a "key-value" pair. In other ## words, if we look up the key, k=5, the dictionary will return the ## value, v=1. In other words: ## k**v <=> key**value <=> base**exponent (for the factored form of N) ## Begin function: n=N #--------------------# n will change in the while loop (getting smaller.) dict = {} #--------------# Start with an empty dictionary if n==1: #---------------# The factorization of 1 is a special case. return {1: 1} # In other words 1 = 1**1 (but recall that 1 is not prime.) k = 2 #------------------# Our first candidate for factorizing any integer. while k**2 <= n: #-------# Do the following unless k is "too big": if n % k: #----------# . Equivalent to: "If k is not a factor of n:" k += 1 #---------# .. increase the value of k by 1 on next attempt. else: #--------------# . Equivalent to "If k is a factor of n:" n /= k #---------# .. Replace n by n/k (for next attempt) try: #-----------# .. If k is in the dictionary, this attempts to: dict[k] += 1 # ... increase the value of k by 1. except KeyError: # .. KeyError occurs if k is not in the dictionary dict[k] = 1 # ... Add k to dictionary and set exponent v=1 # end(while)#------------# (exits while k**2 <= n:) # Here I inserted a diagnostic to be discussed later: # Documented code continues: if n > 1: #--------------# It can be shown that n is either 1 or prime. try: #---------------# . Try to enter n into dictionary dict[n] += 1 #---# .. increase exponent v->v+1 if k in dictinary except KeyError: #---# . If n is not in the dictionary, dict[n] = 1 # .. then enter n into dictionary with v=1 return dict #------# python talk for ending function ################ End def prime_factorization | Begin program ########## #import sys again=True while again == 1: text="\nEnter a positive integer you wish to factor into prime numbers:\n" N=int(input(text)) dict=prime_factorization(N) print("\n"+str(int(N))+"=") text="(" for b in dict.keys(): text+=str(int(b))+"**"+str(int(dict.get(b)))+") * (" print(text.rstrip(" * (")+"\n") print("The dictionary is:\n") print(dict) again=int(input("\nType 1 for another integer, 0 to terminate.")) </syntaxhighlight> ==Comments on code== ===Line 15: while k**2 <= n:=== This is an important line because it permits the termination of the search before one might think. I leave it as an exercise for the student to learn why any value of k larger than square root of n will never be the next factor of N. It is also left to the student to understand the fact that the time to find the factors can be cut nearly in half if we modify line 17 and increase k by 2 (instead of 1), and why this larger step cannot be taken until after k=2 has been considered. ===Lines in the code that might seem confusing=== The code uses two cleverness tricks that might seem confusing. I lack the formal training in programming to know whether such cleverness would be perceived as a flaw. I know that programmers are allowed to be clever and I know that clear programs are better than confusing programs. The fact that these tricks confused me does not necessarily mean they should be avoided. And, the nice thing about both sources of confusion is that they taught me something about python. ====Line 16: if n%k might seem backwards==== Python follows a common practice of interpreting the number 1 to be True and 0 to be False. But python also accepts any non-zero number as True. Here n%k denotes nmod&nbsp;k, which is the remainder when on divides n by k. If there is no remainder the "if" condition is not satisfied and the code stipulates that the next value of k is attempted. Any number not equal to zero is interpreted as the "if" condition being satisfied, meaning that k is a divisor of n (in that case, we replace n by n/k.) This might be good coding, but it seems to me that it would have been better to construct it along the lines of "if n%k == 0" or even "if n%k !=0". ====Lines 21-23 and 29-31: A failed attempt to change the value of a key==== In this code fragment, we ask if a given value of k has already been identified as a factor. But instead of "asking" if k has been declared a key in the dictionary, an <u>attempt</u> is made to change the value of k. If that attempt fails (because k is not among the dictionary's keys), then k is added to the list of keys, with a value of 1 (a value of 1 indicates that at this point, k, but not k² has been established to be a factor of n.) Again, there might be nothing wrong with using a key error like this. But the code would have been easier for me to follow if the if statement simply called the question of whether k was a key in dict. ==Sample output (with a really big number)== Sometimes you have to wait a few minutes to factor a number this large. But usually it comes out in just a few seconds. I would imagine that a number that was the product of only two roughly equal prime numbers would take forever because k=k+1 must be repeated a virtually infinite number of times. This number came quickly: {{Font color|blue|Enter a positive integer you wish to factor into prime numbers:}} 99283428376482768916467328912876456416375474272263423467896879187289499287374899 &nbsp; <span style="color:blue>99283428376482768916467328912876456416375474272263423467896879187289499287374899=</span> {{Font color|blue|(619**1) * (688**1) * (768**3) * (1024**17) * (64853**1) * (5302548928**1)}} &nbsp; {{Font color|blue|The dictionary is:}} &nbsp; <span style="color:blue>{619: 1, 688: 1, 768: 3, 1024: 17, 64853: 1, 5302548928.0: 1}</span> &nbsp; {{Font color|blue|Type 1 for another integer, 0 to terminate.}} {{center|'''DO NOT TRY TO FACTOR A NUMBER THIS LARGE UNLESS YOU ARE WILLING TO INTERUPT YOUR PYTHON PROGRAM.'''}} {{center|Keep in mind that the universe is only <math>4.4\times10^{26}\text{nanoseconds}</math> old.<ref>http://astronomy.nmsu.edu/geas/lectures/lecture02/slide04.html</ref>}} The number shown rounds up to 10⁸⁰. It is not likely that any computer will ever count that high. If the number to be factored consists of two prime number that are close to 10⁸⁰, the code will iterate all the way up to k coming close to 10⁴⁰. The inability of any computer to perform that many iterations helps explain how cryptocurrency works. &nbsp; j2441029ohg0j8j86poebq48antcpxf 2410276 2410275 2022-07-29T19:37:51Z Guy vandegrift 813252 wikitext text/x-wiki [[File:Factorization flowchart.svg|thumb|420px|Flowchart for a simple prime factorization code]] '''Topics covered:''' ''prime factorization, flowcharts, python dictionaries'' This lesson shows how the use of a python dictionary can product of prime numbers. There are a few things the reader should know: * A serious python programmer would probably prefer to download a package that gives you a function that performs prime factorization. Two promising candidates are [https://pypi.org/project/primefac/ pypi.org/project/primefac] and [https://www.sympy.org/en/index.html www.sympy.org/]. * A simpler version of this program can be found at simpler program [[w:Special:Permalink/1086485306|this permalink]] of [[w:Trial division]] * Lacking the mental wherewithal to seize upon either of these opportunities, I googled "python prime factorization", found many codes, and selected the one that follows for two reasons:<ref>The code that follows is discussed at [https://stackoverflow.com/questions/15347174/python-finding-prime-factors stackoverflow question 15347174] and [https://scientific-python-101.readthedocs.io/_downloads/prime_factorization_solution.py scientific-python-101.readthedocs]</ref> :# I wanted a code that counts repeated factors in order to later write the factored form in wikitext with exponents.<ref>In the wikitext version, I had to add a few lines so that exponents of 1 would not appear. Example: <math>12=2^2\cdot 3^1</math> should instead read <math>12=2^2\cdot 3.</math> I ommitted the feature here in order to keep the lesson simple.</ref> :# I didn't understand the python code, and wanted master it in order to fully understand how python dictionaries are used. A good student exercise would be to modify this code so that it runs approximately twice as fast.Instructors can task students with modifying this code so that the output can be shown in a table. This one creates a wikitable that illustrates the "wasted" efforts to see if 4 and 6 are prime factors (see: [[Wikipedia:Trial division#Method]].): {| class="wikitable" style="text-align: center; vertical-align: bottom;" |+ <math>\text{Establishing } 1550=2\cdot 5^2 \cdot 31</math> |- ! n ! k ! k^2 ! k^2<n? ! n/k ! mod ! replace ! factor ! * |- | 1550 | 2 | 4 | TRUE | 775 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"|n: n/k | style="background:#ffffcc"|2 | |- | 775 | 2 | 4 | TRUE | 387 | 1 | k: k+1 | | |- | 775 | 3 | 9 | TRUE | 258 | 1 | k: k+1 | | |- | 775 | 4 | 16 | TRUE | 193 | 3 | k: k+1 | | |- | 775 | 5 | 25 | TRUE | 155 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 155 | 5 | 25 | TRUE | 31 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 31 | 5 | 25 | TRUE | 6 | 1 | k: k+1 | | |- | 31 | 6 | 36 | FALSE | - | - | - | | style="background:#ffffcc"| 31 |} ==Python code== ‎<syntaxhighlight lang="python" line="1"> ## In a python code, the function definitions precede the code. We need only 1 function: def prime_factorization(N): ## 12=(2**2)*(2**1)*(5**1), or equivalently, 12 = 2**2 * 3 * 5 ## The dictionary, dict, expresses k**v as a "key-value" pair. In other ## words, if we look up the key, k=5, the dictionary will return the ## value, v=1. In other words: ## k**v <=> key**value <=> base**exponent (for the factored form of N) ## Begin function: n=N #--------------------# n will change in the while loop (getting smaller.) dict = {} #--------------# Start with an empty dictionary if n==1: #---------------# The factorization of 1 is a special case. return {1: 1} # In other words 1 = 1**1 (but recall that 1 is not prime.) k = 2 #------------------# Our first candidate for factorizing any integer. while k**2 <= n: #-------# Do the following unless k is "too big": if n % k: #----------# . Equivalent to: "If k is not a factor of n:" k += 1 #---------# .. increase the value of k by 1 on next attempt. else: #--------------# . Equivalent to "If k is a factor of n:" n /= k #---------# .. Replace n by n/k (for next attempt) try: #-----------# .. If k is in the dictionary, this attempts to: dict[k] += 1 # ... increase the value of k by 1. except KeyError: # .. KeyError occurs if k is not in the dictionary dict[k] = 1 # ... Add k to dictionary and set exponent v=1 # end(while)#------------# (exits while k**2 <= n:) # Here I inserted a diagnostic to be discussed later: # Documented code continues: if n > 1: #--------------# It can be shown that n is either 1 or prime. try: #---------------# . Try to enter n into dictionary dict[n] += 1 #---# .. increase exponent v->v+1 if k in dictinary except KeyError: #---# . If n is not in the dictionary, dict[n] = 1 # .. then enter n into dictionary with v=1 return dict #------# python talk for ending function ################ End def prime_factorization | Begin program ########## #import sys again=True while again == 1: text="\nEnter a positive integer you wish to factor into prime numbers:\n" N=int(input(text)) dict=prime_factorization(N) print("\n"+str(int(N))+"=") text="(" for b in dict.keys(): text+=str(int(b))+"**"+str(int(dict.get(b)))+") * (" print(text.rstrip(" * (")+"\n") print("The dictionary is:\n") print(dict) again=int(input("\nType 1 for another integer, 0 to terminate.")) </syntaxhighlight> ==Comments on code== ===Line 15: while k**2 <= n:=== This is an important line because it permits the termination of the search before one might think. I leave it as an exercise for the student to learn why any value of k larger than square root of n will never be the next factor of N. It is also left to the student to understand the fact that the time to find the factors can be cut nearly in half if we modify line 17 and increase k by 2 (instead of 1), and why this larger step cannot be taken until after k=2 has been considered. ===Lines in the code that might seem confusing=== The code uses two cleverness tricks that might seem confusing. I lack the formal training in programming to know whether such cleverness would be perceived as a flaw. I know that programmers are allowed to be clever and I know that clear programs are better than confusing programs. The fact that these tricks confused me does not necessarily mean they should be avoided. And, the nice thing about both sources of confusion is that they taught me something about python. ====Line 16: if n%k might seem backwards==== Python follows a common practice of interpreting the number 1 to be True and 0 to be False. But python also accepts any non-zero number as True. Here n%k denotes nmod&nbsp;k, which is the remainder when on divides n by k. If there is no remainder the "if" condition is not satisfied and the code stipulates that the next value of k is attempted. Any number not equal to zero is interpreted as the "if" condition being satisfied, meaning that k is a divisor of n (in that case, we replace n by n/k.) This might be good coding, but it seems to me that it would have been better to construct it along the lines of "if n%k == 0" or even "if n%k !=0". ====Lines 21-23 and 29-31: A failed attempt to change the value of a key==== In this code fragment, we ask if a given value of k has already been identified as a factor. But instead of "asking" if k has been declared a key in the dictionary, an <u>attempt</u> is made to change the value of k. If that attempt fails (because k is not among the dictionary's keys), then k is added to the list of keys, with a value of 1 (a value of 1 indicates that at this point, k, but not k² has been established to be a factor of n.) Again, there might be nothing wrong with using a key error like this. But the code would have been easier for me to follow if the if statement simply called the question of whether k was a key in dict. ==Sample output (with a really big number)== Sometimes you have to wait a few minutes to factor a number this large. But usually it comes out in just a few seconds. I would imagine that a number that was the product of only two roughly equal prime numbers would take forever because k=k+1 must be repeated a virtually infinite number of times. This number came quickly: {{Font color|blue|Enter a positive integer you wish to factor into prime numbers:}} 99283428376482768916467328912876456416375474272263423467896879187289499287374899 &nbsp; <span style="color:blue>99283428376482768916467328912876456416375474272263423467896879187289499287374899=</span> {{Font color|blue|(619**1) * (688**1) * (768**3) * (1024**17) * (64853**1) * (5302548928**1)}} &nbsp; {{Font color|blue|The dictionary is:}} &nbsp; <span style="color:blue>{619: 1, 688: 1, 768: 3, 1024: 17, 64853: 1, 5302548928.0: 1}</span> &nbsp; {{Font color|blue|Type 1 for another integer, 0 to terminate.}} {{center|'''DO NOT TRY TO FACTOR A NUMBER THIS LARGE UNLESS YOU ARE WILLING TO INTERUPT YOUR PYTHON PROGRAM.'''}} {{center|Keep in mind that the universe is only <math>4.4\times10^{26}\text{nanoseconds}</math> old.<ref>http://astronomy.nmsu.edu/geas/lectures/lecture02/slide04.html</ref>}} The number shown rounds up to 10⁸⁰. It is not likely that any computer will ever count that high. If the number to be factored consists of two prime number that are close to 10⁸⁰, the code will iterate all the way up to k coming close to 10⁴⁰. The inability of any computer to perform that many iterations helps explain how cryptocurrency works. &nbsp; j44wzo3tviz63xtp6aylrq0im4e4x08 2410277 2410276 2022-07-29T19:39:22Z Guy vandegrift 813252 wikitext text/x-wiki [[File:Factorization flowchart.svg|thumb|420px|Flowchart for a simple prime factorization code]] '''Topics covered:''' ''prime factorization, flowcharts, python dictionaries'' This lesson shows how the use of a python dictionary can product of prime numbers. There are a few things the reader should know: * A serious python programmer would probably prefer to download a package that gives you a function that performs prime factorization. Two promising candidates are [https://pypi.org/project/primefac/ pypi.org/project/primefac] and [https://www.sympy.org/en/index.html www.sympy.org/]. * A simpler version of this program can be found at [[Wikipedia:Trial division]] (a working python script is at [[w:Special:Permalink/1086485306]].) * Lacking the mental wherewithal to seize upon either of these opportunities, I googled "python prime factorization", found many codes, and selected the one that follows for two reasons:<ref>The code that follows is discussed at [https://stackoverflow.com/questions/15347174/python-finding-prime-factors stackoverflow question 15347174] and [https://scientific-python-101.readthedocs.io/_downloads/prime_factorization_solution.py scientific-python-101.readthedocs]</ref> :# I wanted a code that counts repeated factors in order to later write the factored form in wikitext with exponents.<ref>In the wikitext version, I had to add a few lines so that exponents of 1 would not appear. Example: <math>12=2^2\cdot 3^1</math> should instead read <math>12=2^2\cdot 3.</math> I ommitted the feature here in order to keep the lesson simple.</ref> :# I didn't understand the python code, and wanted master it in order to fully understand how python dictionaries are used. A good student exercise would be to modify this code so that it runs approximately twice as fast.Instructors can task students with modifying this code so that the output can be shown in a table. This one creates a wikitable that illustrates the "wasted" efforts to see if 4 and 6 are prime factors (see: [[Wikipedia:Trial division#Method]].): {| class="wikitable" style="text-align: center; vertical-align: bottom;" |+ <math>\text{Establishing } 1550=2\cdot 5^2 \cdot 31</math> |- ! n ! k ! k^2 ! k^2<n? ! n/k ! mod ! replace ! factor ! * |- | 1550 | 2 | 4 | TRUE | 775 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"|n: n/k | style="background:#ffffcc"|2 | |- | 775 | 2 | 4 | TRUE | 387 | 1 | k: k+1 | | |- | 775 | 3 | 9 | TRUE | 258 | 1 | k: k+1 | | |- | 775 | 4 | 16 | TRUE | 193 | 3 | k: k+1 | | |- | 775 | 5 | 25 | TRUE | 155 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 155 | 5 | 25 | TRUE | 31 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 31 | 5 | 25 | TRUE | 6 | 1 | k: k+1 | | |- | 31 | 6 | 36 | FALSE | - | - | - | | style="background:#ffffcc"| 31 |} ==Python code== ‎<syntaxhighlight lang="python" line="1"> ## In a python code, the function definitions precede the code. We need only 1 function: def prime_factorization(N): ## 12=(2**2)*(2**1)*(5**1), or equivalently, 12 = 2**2 * 3 * 5 ## The dictionary, dict, expresses k**v as a "key-value" pair. In other ## words, if we look up the key, k=5, the dictionary will return the ## value, v=1. In other words: ## k**v <=> key**value <=> base**exponent (for the factored form of N) ## Begin function: n=N #--------------------# n will change in the while loop (getting smaller.) dict = {} #--------------# Start with an empty dictionary if n==1: #---------------# The factorization of 1 is a special case. return {1: 1} # In other words 1 = 1**1 (but recall that 1 is not prime.) k = 2 #------------------# Our first candidate for factorizing any integer. while k**2 <= n: #-------# Do the following unless k is "too big": if n % k: #----------# . Equivalent to: "If k is not a factor of n:" k += 1 #---------# .. increase the value of k by 1 on next attempt. else: #--------------# . Equivalent to "If k is a factor of n:" n /= k #---------# .. Replace n by n/k (for next attempt) try: #-----------# .. If k is in the dictionary, this attempts to: dict[k] += 1 # ... increase the value of k by 1. except KeyError: # .. KeyError occurs if k is not in the dictionary dict[k] = 1 # ... Add k to dictionary and set exponent v=1 # end(while)#------------# (exits while k**2 <= n:) # Here I inserted a diagnostic to be discussed later: # Documented code continues: if n > 1: #--------------# It can be shown that n is either 1 or prime. try: #---------------# . Try to enter n into dictionary dict[n] += 1 #---# .. increase exponent v->v+1 if k in dictinary except KeyError: #---# . If n is not in the dictionary, dict[n] = 1 # .. then enter n into dictionary with v=1 return dict #------# python talk for ending function ################ End def prime_factorization | Begin program ########## #import sys again=True while again == 1: text="\nEnter a positive integer you wish to factor into prime numbers:\n" N=int(input(text)) dict=prime_factorization(N) print("\n"+str(int(N))+"=") text="(" for b in dict.keys(): text+=str(int(b))+"**"+str(int(dict.get(b)))+") * (" print(text.rstrip(" * (")+"\n") print("The dictionary is:\n") print(dict) again=int(input("\nType 1 for another integer, 0 to terminate.")) </syntaxhighlight> ==Comments on code== ===Line 15: while k**2 <= n:=== This is an important line because it permits the termination of the search before one might think. I leave it as an exercise for the student to learn why any value of k larger than square root of n will never be the next factor of N. It is also left to the student to understand the fact that the time to find the factors can be cut nearly in half if we modify line 17 and increase k by 2 (instead of 1), and why this larger step cannot be taken until after k=2 has been considered. ===Lines in the code that might seem confusing=== The code uses two cleverness tricks that might seem confusing. I lack the formal training in programming to know whether such cleverness would be perceived as a flaw. I know that programmers are allowed to be clever and I know that clear programs are better than confusing programs. The fact that these tricks confused me does not necessarily mean they should be avoided. And, the nice thing about both sources of confusion is that they taught me something about python. ====Line 16: if n%k might seem backwards==== Python follows a common practice of interpreting the number 1 to be True and 0 to be False. But python also accepts any non-zero number as True. Here n%k denotes nmod&nbsp;k, which is the remainder when on divides n by k. If there is no remainder the "if" condition is not satisfied and the code stipulates that the next value of k is attempted. Any number not equal to zero is interpreted as the "if" condition being satisfied, meaning that k is a divisor of n (in that case, we replace n by n/k.) This might be good coding, but it seems to me that it would have been better to construct it along the lines of "if n%k == 0" or even "if n%k !=0". ====Lines 21-23 and 29-31: A failed attempt to change the value of a key==== In this code fragment, we ask if a given value of k has already been identified as a factor. But instead of "asking" if k has been declared a key in the dictionary, an <u>attempt</u> is made to change the value of k. If that attempt fails (because k is not among the dictionary's keys), then k is added to the list of keys, with a value of 1 (a value of 1 indicates that at this point, k, but not k² has been established to be a factor of n.) Again, there might be nothing wrong with using a key error like this. But the code would have been easier for me to follow if the if statement simply called the question of whether k was a key in dict. ==Sample output (with a really big number)== Sometimes you have to wait a few minutes to factor a number this large. But usually it comes out in just a few seconds. I would imagine that a number that was the product of only two roughly equal prime numbers would take forever because k=k+1 must be repeated a virtually infinite number of times. This number came quickly: {{Font color|blue|Enter a positive integer you wish to factor into prime numbers:}} 99283428376482768916467328912876456416375474272263423467896879187289499287374899 &nbsp; <span style="color:blue>99283428376482768916467328912876456416375474272263423467896879187289499287374899=</span> {{Font color|blue|(619**1) * (688**1) * (768**3) * (1024**17) * (64853**1) * (5302548928**1)}} &nbsp; {{Font color|blue|The dictionary is:}} &nbsp; <span style="color:blue>{619: 1, 688: 1, 768: 3, 1024: 17, 64853: 1, 5302548928.0: 1}</span> &nbsp; {{Font color|blue|Type 1 for another integer, 0 to terminate.}} {{center|'''DO NOT TRY TO FACTOR A NUMBER THIS LARGE UNLESS YOU ARE WILLING TO INTERUPT YOUR PYTHON PROGRAM.'''}} {{center|Keep in mind that the universe is only <math>4.4\times10^{26}\text{nanoseconds}</math> old.<ref>http://astronomy.nmsu.edu/geas/lectures/lecture02/slide04.html</ref>}} The number shown rounds up to 10⁸⁰. It is not likely that any computer will ever count that high. If the number to be factored consists of two prime number that are close to 10⁸⁰, the code will iterate all the way up to k coming close to 10⁴⁰. The inability of any computer to perform that many iterations helps explain how cryptocurrency works. &nbsp; bci77y5u9oa84jpmffnhl1didfo0nay 2410278 2410277 2022-07-29T19:43:51Z Guy vandegrift 813252 wikitext text/x-wiki [[File:Factorization flowchart.svg|thumb|420px|Flowchart for a simple prime factorization code]] '''Topics covered:''' ''prime factorization, flowcharts, python dictionaries'' This lesson shows how the use of a python dictionary can product of prime numbers. There are a few things the reader should know: * A serious python programmer would probably prefer to download a package that gives you a function that performs prime factorization. Two promising candidates are [https://pypi.org/project/primefac/ pypi.org/project/primefac] and [https://www.sympy.org/en/index.html www.sympy.org/]. * A simpler version of this program can be found at [[Wikipedia:Trial division#Method]].<ref>A working python script is at [[w:Special:Permalink/1086485306]]</ref. * Lacking the mental wherewithal to seize upon either of these opportunities, I googled "python prime factorization", found many codes, and selected the one that follows for two reasons:<ref>The code that follows is discussed at [https://stackoverflow.com/questions/15347174/python-finding-prime-factors stackoverflow question 15347174] and [https://scientific-python-101.readthedocs.io/_downloads/prime_factorization_solution.py scientific-python-101.readthedocs]</ref> :# I wanted a code that counts repeated factors in order to later write the factored form in wikitext with exponents.<ref>In the wikitext version, I had to add a few lines so that exponents of 1 would not appear. Example: <math>12=2^2\cdot 3^1</math> should instead read <math>12=2^2\cdot 3.</math> I ommitted the feature here in order to keep the lesson simple.</ref> :# I didn't understand the python code, and wanted master it in order to fully understand how python dictionaries are used. A good student exercise would be to modify this code so that it runs approximately twice as fast. Instructors can task students with modifying this code so that the output can be shown in a table. This table creates a wikitable that illustrates the "wasted" efforts associated with investigating the even numbers 4 and 6 as candidates to be prime factors of 1550 (see: [[Wikipedia:Trial division#Method]].): {| class="wikitable" style="text-align: center; vertical-align: bottom;" |+ <math>\text{Establishing } 1550=2\cdot 5^2 \cdot 31</math> |- ! n ! k ! k^2 ! k^2<n? ! n/k ! mod ! replace ! factor ! * |- | 1550 | 2 | 4 | TRUE | 775 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"|n: n/k | style="background:#ffffcc"|2 | |- | 775 | 2 | 4 | TRUE | 387 | 1 | k: k+1 | | |- | 775 | 3 | 9 | TRUE | 258 | 1 | k: k+1 | | |- | 775 | 4 | 16 | TRUE | 193 | 3 | k: k+1 | | |- | 775 | 5 | 25 | TRUE | 155 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 155 | 5 | 25 | TRUE | 31 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 31 | 5 | 25 | TRUE | 6 | 1 | k: k+1 | | |- | 31 | 6 | 36 | FALSE | - | - | - | | style="background:#ffffcc"| 31 |} ==Python code== ‎<syntaxhighlight lang="python" line="1"> ## In a python code, the function definitions precede the code. We need only 1 function: def prime_factorization(N): ## 12=(2**2)*(2**1)*(5**1), or equivalently, 12 = 2**2 * 3 * 5 ## The dictionary, dict, expresses k**v as a "key-value" pair. In other ## words, if we look up the key, k=5, the dictionary will return the ## value, v=1. In other words: ## k**v <=> key**value <=> base**exponent (for the factored form of N) ## Begin function: n=N #--------------------# n will change in the while loop (getting smaller.) dict = {} #--------------# Start with an empty dictionary if n==1: #---------------# The factorization of 1 is a special case. return {1: 1} # In other words 1 = 1**1 (but recall that 1 is not prime.) k = 2 #------------------# Our first candidate for factorizing any integer. while k**2 <= n: #-------# Do the following unless k is "too big": if n % k: #----------# . Equivalent to: "If k is not a factor of n:" k += 1 #---------# .. increase the value of k by 1 on next attempt. else: #--------------# . Equivalent to "If k is a factor of n:" n /= k #---------# .. Replace n by n/k (for next attempt) try: #-----------# .. If k is in the dictionary, this attempts to: dict[k] += 1 # ... increase the value of k by 1. except KeyError: # .. KeyError occurs if k is not in the dictionary dict[k] = 1 # ... Add k to dictionary and set exponent v=1 # end(while)#------------# (exits while k**2 <= n:) # Here I inserted a diagnostic to be discussed later: # Documented code continues: if n > 1: #--------------# It can be shown that n is either 1 or prime. try: #---------------# . Try to enter n into dictionary dict[n] += 1 #---# .. increase exponent v->v+1 if k in dictinary except KeyError: #---# . If n is not in the dictionary, dict[n] = 1 # .. then enter n into dictionary with v=1 return dict #------# python talk for ending function ################ End def prime_factorization | Begin program ########## #import sys again=True while again == 1: text="\nEnter a positive integer you wish to factor into prime numbers:\n" N=int(input(text)) dict=prime_factorization(N) print("\n"+str(int(N))+"=") text="(" for b in dict.keys(): text+=str(int(b))+"**"+str(int(dict.get(b)))+") * (" print(text.rstrip(" * (")+"\n") print("The dictionary is:\n") print(dict) again=int(input("\nType 1 for another integer, 0 to terminate.")) </syntaxhighlight> ==Comments on code== ===Line 15: while k**2 <= n:=== This is an important line because it permits the termination of the search before one might think. I leave it as an exercise for the student to learn why any value of k larger than square root of n will never be the next factor of N. It is also left to the student to understand the fact that the time to find the factors can be cut nearly in half if we modify line 17 and increase k by 2 (instead of 1), and why this larger step cannot be taken until after k=2 has been considered. ===Lines in the code that might seem confusing=== The code uses two cleverness tricks that might seem confusing. I lack the formal training in programming to know whether such cleverness would be perceived as a flaw. I know that programmers are allowed to be clever and I know that clear programs are better than confusing programs. The fact that these tricks confused me does not necessarily mean they should be avoided. And, the nice thing about both sources of confusion is that they taught me something about python. ====Line 16: if n%k might seem backwards==== Python follows a common practice of interpreting the number 1 to be True and 0 to be False. But python also accepts any non-zero number as True. Here n%k denotes nmod&nbsp;k, which is the remainder when on divides n by k. If there is no remainder the "if" condition is not satisfied and the code stipulates that the next value of k is attempted. Any number not equal to zero is interpreted as the "if" condition being satisfied, meaning that k is a divisor of n (in that case, we replace n by n/k.) This might be good coding, but it seems to me that it would have been better to construct it along the lines of "if n%k == 0" or even "if n%k !=0". ====Lines 21-23 and 29-31: A failed attempt to change the value of a key==== In this code fragment, we ask if a given value of k has already been identified as a factor. But instead of "asking" if k has been declared a key in the dictionary, an <u>attempt</u> is made to change the value of k. If that attempt fails (because k is not among the dictionary's keys), then k is added to the list of keys, with a value of 1 (a value of 1 indicates that at this point, k, but not k² has been established to be a factor of n.) Again, there might be nothing wrong with using a key error like this. But the code would have been easier for me to follow if the if statement simply called the question of whether k was a key in dict. ==Sample output (with a really big number)== Sometimes you have to wait a few minutes to factor a number this large. But usually it comes out in just a few seconds. I would imagine that a number that was the product of only two roughly equal prime numbers would take forever because k=k+1 must be repeated a virtually infinite number of times. This number came quickly: {{Font color|blue|Enter a positive integer you wish to factor into prime numbers:}} 99283428376482768916467328912876456416375474272263423467896879187289499287374899 &nbsp; <span style="color:blue>99283428376482768916467328912876456416375474272263423467896879187289499287374899=</span> {{Font color|blue|(619**1) * (688**1) * (768**3) * (1024**17) * (64853**1) * (5302548928**1)}} &nbsp; {{Font color|blue|The dictionary is:}} &nbsp; <span style="color:blue>{619: 1, 688: 1, 768: 3, 1024: 17, 64853: 1, 5302548928.0: 1}</span> &nbsp; {{Font color|blue|Type 1 for another integer, 0 to terminate.}} {{center|'''DO NOT TRY TO FACTOR A NUMBER THIS LARGE UNLESS YOU ARE WILLING TO INTERUPT YOUR PYTHON PROGRAM.'''}} {{center|Keep in mind that the universe is only <math>4.4\times10^{26}\text{nanoseconds}</math> old.<ref>http://astronomy.nmsu.edu/geas/lectures/lecture02/slide04.html</ref>}} The number shown rounds up to 10⁸⁰. It is not likely that any computer will ever count that high. If the number to be factored consists of two prime number that are close to 10⁸⁰, the code will iterate all the way up to k coming close to 10⁴⁰. The inability of any computer to perform that many iterations helps explain how cryptocurrency works. &nbsp; 7swncrmbtoe2x9xkd8utmkfx4hbzkbt 2410279 2410278 2022-07-29T19:44:06Z Guy vandegrift 813252 wikitext text/x-wiki [[File:Factorization flowchart.svg|thumb|420px|Flowchart for a simple prime factorization code]] '''Topics covered:''' ''prime factorization, flowcharts, python dictionaries'' This lesson shows how the use of a python dictionary can product of prime numbers. There are a few things the reader should know: * A serious python programmer would probably prefer to download a package that gives you a function that performs prime factorization. Two promising candidates are [https://pypi.org/project/primefac/ pypi.org/project/primefac] and [https://www.sympy.org/en/index.html www.sympy.org/]. * A simpler version of this program can be found at [[Wikipedia:Trial division#Method]].<ref>A working python script is at [[w:Special:Permalink/1086485306]]</ref>. * Lacking the mental wherewithal to seize upon either of these opportunities, I googled "python prime factorization", found many codes, and selected the one that follows for two reasons:<ref>The code that follows is discussed at [https://stackoverflow.com/questions/15347174/python-finding-prime-factors stackoverflow question 15347174] and [https://scientific-python-101.readthedocs.io/_downloads/prime_factorization_solution.py scientific-python-101.readthedocs]</ref> :# I wanted a code that counts repeated factors in order to later write the factored form in wikitext with exponents.<ref>In the wikitext version, I had to add a few lines so that exponents of 1 would not appear. Example: <math>12=2^2\cdot 3^1</math> should instead read <math>12=2^2\cdot 3.</math> I ommitted the feature here in order to keep the lesson simple.</ref> :# I didn't understand the python code, and wanted master it in order to fully understand how python dictionaries are used. A good student exercise would be to modify this code so that it runs approximately twice as fast. Instructors can task students with modifying this code so that the output can be shown in a table. This table creates a wikitable that illustrates the "wasted" efforts associated with investigating the even numbers 4 and 6 as candidates to be prime factors of 1550 (see: [[Wikipedia:Trial division#Method]].): {| class="wikitable" style="text-align: center; vertical-align: bottom;" |+ <math>\text{Establishing } 1550=2\cdot 5^2 \cdot 31</math> |- ! n ! k ! k^2 ! k^2<n? ! n/k ! mod ! replace ! factor ! * |- | 1550 | 2 | 4 | TRUE | 775 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"|n: n/k | style="background:#ffffcc"|2 | |- | 775 | 2 | 4 | TRUE | 387 | 1 | k: k+1 | | |- | 775 | 3 | 9 | TRUE | 258 | 1 | k: k+1 | | |- | 775 | 4 | 16 | TRUE | 193 | 3 | k: k+1 | | |- | 775 | 5 | 25 | TRUE | 155 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 155 | 5 | 25 | TRUE | 31 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 31 | 5 | 25 | TRUE | 6 | 1 | k: k+1 | | |- | 31 | 6 | 36 | FALSE | - | - | - | | style="background:#ffffcc"| 31 |} ==Python code== ‎<syntaxhighlight lang="python" line="1"> ## In a python code, the function definitions precede the code. We need only 1 function: def prime_factorization(N): ## 12=(2**2)*(2**1)*(5**1), or equivalently, 12 = 2**2 * 3 * 5 ## The dictionary, dict, expresses k**v as a "key-value" pair. In other ## words, if we look up the key, k=5, the dictionary will return the ## value, v=1. In other words: ## k**v <=> key**value <=> base**exponent (for the factored form of N) ## Begin function: n=N #--------------------# n will change in the while loop (getting smaller.) dict = {} #--------------# Start with an empty dictionary if n==1: #---------------# The factorization of 1 is a special case. return {1: 1} # In other words 1 = 1**1 (but recall that 1 is not prime.) k = 2 #------------------# Our first candidate for factorizing any integer. while k**2 <= n: #-------# Do the following unless k is "too big": if n % k: #----------# . Equivalent to: "If k is not a factor of n:" k += 1 #---------# .. increase the value of k by 1 on next attempt. else: #--------------# . Equivalent to "If k is a factor of n:" n /= k #---------# .. Replace n by n/k (for next attempt) try: #-----------# .. If k is in the dictionary, this attempts to: dict[k] += 1 # ... increase the value of k by 1. except KeyError: # .. KeyError occurs if k is not in the dictionary dict[k] = 1 # ... Add k to dictionary and set exponent v=1 # end(while)#------------# (exits while k**2 <= n:) # Here I inserted a diagnostic to be discussed later: # Documented code continues: if n > 1: #--------------# It can be shown that n is either 1 or prime. try: #---------------# . Try to enter n into dictionary dict[n] += 1 #---# .. increase exponent v->v+1 if k in dictinary except KeyError: #---# . If n is not in the dictionary, dict[n] = 1 # .. then enter n into dictionary with v=1 return dict #------# python talk for ending function ################ End def prime_factorization | Begin program ########## #import sys again=True while again == 1: text="\nEnter a positive integer you wish to factor into prime numbers:\n" N=int(input(text)) dict=prime_factorization(N) print("\n"+str(int(N))+"=") text="(" for b in dict.keys(): text+=str(int(b))+"**"+str(int(dict.get(b)))+") * (" print(text.rstrip(" * (")+"\n") print("The dictionary is:\n") print(dict) again=int(input("\nType 1 for another integer, 0 to terminate.")) </syntaxhighlight> ==Comments on code== ===Line 15: while k**2 <= n:=== This is an important line because it permits the termination of the search before one might think. I leave it as an exercise for the student to learn why any value of k larger than square root of n will never be the next factor of N. It is also left to the student to understand the fact that the time to find the factors can be cut nearly in half if we modify line 17 and increase k by 2 (instead of 1), and why this larger step cannot be taken until after k=2 has been considered. ===Lines in the code that might seem confusing=== The code uses two cleverness tricks that might seem confusing. I lack the formal training in programming to know whether such cleverness would be perceived as a flaw. I know that programmers are allowed to be clever and I know that clear programs are better than confusing programs. The fact that these tricks confused me does not necessarily mean they should be avoided. And, the nice thing about both sources of confusion is that they taught me something about python. ====Line 16: if n%k might seem backwards==== Python follows a common practice of interpreting the number 1 to be True and 0 to be False. But python also accepts any non-zero number as True. Here n%k denotes nmod&nbsp;k, which is the remainder when on divides n by k. If there is no remainder the "if" condition is not satisfied and the code stipulates that the next value of k is attempted. Any number not equal to zero is interpreted as the "if" condition being satisfied, meaning that k is a divisor of n (in that case, we replace n by n/k.) This might be good coding, but it seems to me that it would have been better to construct it along the lines of "if n%k == 0" or even "if n%k !=0". ====Lines 21-23 and 29-31: A failed attempt to change the value of a key==== In this code fragment, we ask if a given value of k has already been identified as a factor. But instead of "asking" if k has been declared a key in the dictionary, an <u>attempt</u> is made to change the value of k. If that attempt fails (because k is not among the dictionary's keys), then k is added to the list of keys, with a value of 1 (a value of 1 indicates that at this point, k, but not k² has been established to be a factor of n.) Again, there might be nothing wrong with using a key error like this. But the code would have been easier for me to follow if the if statement simply called the question of whether k was a key in dict. ==Sample output (with a really big number)== Sometimes you have to wait a few minutes to factor a number this large. But usually it comes out in just a few seconds. I would imagine that a number that was the product of only two roughly equal prime numbers would take forever because k=k+1 must be repeated a virtually infinite number of times. This number came quickly: {{Font color|blue|Enter a positive integer you wish to factor into prime numbers:}} 99283428376482768916467328912876456416375474272263423467896879187289499287374899 &nbsp; <span style="color:blue>99283428376482768916467328912876456416375474272263423467896879187289499287374899=</span> {{Font color|blue|(619**1) * (688**1) * (768**3) * (1024**17) * (64853**1) * (5302548928**1)}} &nbsp; {{Font color|blue|The dictionary is:}} &nbsp; <span style="color:blue>{619: 1, 688: 1, 768: 3, 1024: 17, 64853: 1, 5302548928.0: 1}</span> &nbsp; {{Font color|blue|Type 1 for another integer, 0 to terminate.}} {{center|'''DO NOT TRY TO FACTOR A NUMBER THIS LARGE UNLESS YOU ARE WILLING TO INTERUPT YOUR PYTHON PROGRAM.'''}} {{center|Keep in mind that the universe is only <math>4.4\times10^{26}\text{nanoseconds}</math> old.<ref>http://astronomy.nmsu.edu/geas/lectures/lecture02/slide04.html</ref>}} The number shown rounds up to 10⁸⁰. It is not likely that any computer will ever count that high. If the number to be factored consists of two prime number that are close to 10⁸⁰, the code will iterate all the way up to k coming close to 10⁴⁰. The inability of any computer to perform that many iterations helps explain how cryptocurrency works. &nbsp; sv1cs6potaefl0sxrb4qkb5eqc7lrd8 2410280 2410279 2022-07-29T19:45:10Z Guy vandegrift 813252 wikitext text/x-wiki [[File:Factorization flowchart.svg|thumb|420px|Flowchart for a simple prime factorization code]] '''Topics covered:''' ''prime factorization, flowcharts, python dictionaries'' This lesson shows how the use of a python dictionary can product of prime numbers. There are a few things the reader should know: * A serious python programmer would probably prefer to download a package that gives you a function that performs prime factorization. Two promising candidates are [https://pypi.org/project/primefac/ pypi.org/project/primefac] and [https://www.sympy.org/en/index.html www.sympy.org/]. * A simpler version of this program can be found at [[Wikipedia:Trial division#Method]].<ref>A working python script is at [[w:Special:Permalink/1086485306]]</ref>. * Lacking the mental wherewithal to seize upon either of these opportunities, I googled "python prime factorization", found many codes, and selected the one [[#Python code|shown below]] for two reasons:<ref>The code that follows is discussed at [https://stackoverflow.com/questions/15347174/python-finding-prime-factors stackoverflow question 15347174] and [https://scientific-python-101.readthedocs.io/_downloads/prime_factorization_solution.py scientific-python-101.readthedocs]</ref> :# I wanted a code that counts repeated factors in order to later write the factored form in wikitext with exponents.<ref>In the wikitext version, I had to add a few lines so that exponents of 1 would not appear. Example: <math>12=2^2\cdot 3^1</math> should instead read <math>12=2^2\cdot 3.</math> I ommitted the feature here in order to keep the lesson simple.</ref> :# I didn't understand the python code, and wanted master it in order to fully understand how python dictionaries are used. A good student exercise would be to modify this code so that it runs approximately twice as fast. Instructors can task students with modifying this code so that the output can be shown in a table. This table creates a wikitable that illustrates the "wasted" efforts associated with investigating the even numbers 4 and 6 as candidates to be prime factors of 1550 (see: [[Wikipedia:Trial division#Method]].): {| class="wikitable" style="text-align: center; vertical-align: bottom;" |+ <math>\text{Establishing } 1550=2\cdot 5^2 \cdot 31</math> |- ! n ! k ! k^2 ! k^2<n? ! n/k ! mod ! replace ! factor ! * |- | 1550 | 2 | 4 | TRUE | 775 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"|n: n/k | style="background:#ffffcc"|2 | |- | 775 | 2 | 4 | TRUE | 387 | 1 | k: k+1 | | |- | 775 | 3 | 9 | TRUE | 258 | 1 | k: k+1 | | |- | 775 | 4 | 16 | TRUE | 193 | 3 | k: k+1 | | |- | 775 | 5 | 25 | TRUE | 155 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 155 | 5 | 25 | TRUE | 31 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 31 | 5 | 25 | TRUE | 6 | 1 | k: k+1 | | |- | 31 | 6 | 36 | FALSE | - | - | - | | style="background:#ffffcc"| 31 |} ==Python code== ‎<syntaxhighlight lang="python" line="1"> ## In a python code, the function definitions precede the code. We need only 1 function: def prime_factorization(N): ## 12=(2**2)*(2**1)*(5**1), or equivalently, 12 = 2**2 * 3 * 5 ## The dictionary, dict, expresses k**v as a "key-value" pair. In other ## words, if we look up the key, k=5, the dictionary will return the ## value, v=1. In other words: ## k**v <=> key**value <=> base**exponent (for the factored form of N) ## Begin function: n=N #--------------------# n will change in the while loop (getting smaller.) dict = {} #--------------# Start with an empty dictionary if n==1: #---------------# The factorization of 1 is a special case. return {1: 1} # In other words 1 = 1**1 (but recall that 1 is not prime.) k = 2 #------------------# Our first candidate for factorizing any integer. while k**2 <= n: #-------# Do the following unless k is "too big": if n % k: #----------# . Equivalent to: "If k is not a factor of n:" k += 1 #---------# .. increase the value of k by 1 on next attempt. else: #--------------# . Equivalent to "If k is a factor of n:" n /= k #---------# .. Replace n by n/k (for next attempt) try: #-----------# .. If k is in the dictionary, this attempts to: dict[k] += 1 # ... increase the value of k by 1. except KeyError: # .. KeyError occurs if k is not in the dictionary dict[k] = 1 # ... Add k to dictionary and set exponent v=1 # end(while)#------------# (exits while k**2 <= n:) # Here I inserted a diagnostic to be discussed later: # Documented code continues: if n > 1: #--------------# It can be shown that n is either 1 or prime. try: #---------------# . Try to enter n into dictionary dict[n] += 1 #---# .. increase exponent v->v+1 if k in dictinary except KeyError: #---# . If n is not in the dictionary, dict[n] = 1 # .. then enter n into dictionary with v=1 return dict #------# python talk for ending function ################ End def prime_factorization | Begin program ########## #import sys again=True while again == 1: text="\nEnter a positive integer you wish to factor into prime numbers:\n" N=int(input(text)) dict=prime_factorization(N) print("\n"+str(int(N))+"=") text="(" for b in dict.keys(): text+=str(int(b))+"**"+str(int(dict.get(b)))+") * (" print(text.rstrip(" * (")+"\n") print("The dictionary is:\n") print(dict) again=int(input("\nType 1 for another integer, 0 to terminate.")) </syntaxhighlight> ==Comments on code== ===Line 15: while k**2 <= n:=== This is an important line because it permits the termination of the search before one might think. I leave it as an exercise for the student to learn why any value of k larger than square root of n will never be the next factor of N. It is also left to the student to understand the fact that the time to find the factors can be cut nearly in half if we modify line 17 and increase k by 2 (instead of 1), and why this larger step cannot be taken until after k=2 has been considered. ===Lines in the code that might seem confusing=== The code uses two cleverness tricks that might seem confusing. I lack the formal training in programming to know whether such cleverness would be perceived as a flaw. I know that programmers are allowed to be clever and I know that clear programs are better than confusing programs. The fact that these tricks confused me does not necessarily mean they should be avoided. And, the nice thing about both sources of confusion is that they taught me something about python. ====Line 16: if n%k might seem backwards==== Python follows a common practice of interpreting the number 1 to be True and 0 to be False. But python also accepts any non-zero number as True. Here n%k denotes nmod&nbsp;k, which is the remainder when on divides n by k. If there is no remainder the "if" condition is not satisfied and the code stipulates that the next value of k is attempted. Any number not equal to zero is interpreted as the "if" condition being satisfied, meaning that k is a divisor of n (in that case, we replace n by n/k.) This might be good coding, but it seems to me that it would have been better to construct it along the lines of "if n%k == 0" or even "if n%k !=0". ====Lines 21-23 and 29-31: A failed attempt to change the value of a key==== In this code fragment, we ask if a given value of k has already been identified as a factor. But instead of "asking" if k has been declared a key in the dictionary, an <u>attempt</u> is made to change the value of k. If that attempt fails (because k is not among the dictionary's keys), then k is added to the list of keys, with a value of 1 (a value of 1 indicates that at this point, k, but not k² has been established to be a factor of n.) Again, there might be nothing wrong with using a key error like this. But the code would have been easier for me to follow if the if statement simply called the question of whether k was a key in dict. ==Sample output (with a really big number)== Sometimes you have to wait a few minutes to factor a number this large. But usually it comes out in just a few seconds. I would imagine that a number that was the product of only two roughly equal prime numbers would take forever because k=k+1 must be repeated a virtually infinite number of times. This number came quickly: {{Font color|blue|Enter a positive integer you wish to factor into prime numbers:}} 99283428376482768916467328912876456416375474272263423467896879187289499287374899 &nbsp; <span style="color:blue>99283428376482768916467328912876456416375474272263423467896879187289499287374899=</span> {{Font color|blue|(619**1) * (688**1) * (768**3) * (1024**17) * (64853**1) * (5302548928**1)}} &nbsp; {{Font color|blue|The dictionary is:}} &nbsp; <span style="color:blue>{619: 1, 688: 1, 768: 3, 1024: 17, 64853: 1, 5302548928.0: 1}</span> &nbsp; {{Font color|blue|Type 1 for another integer, 0 to terminate.}} {{center|'''DO NOT TRY TO FACTOR A NUMBER THIS LARGE UNLESS YOU ARE WILLING TO INTERUPT YOUR PYTHON PROGRAM.'''}} {{center|Keep in mind that the universe is only <math>4.4\times10^{26}\text{nanoseconds}</math> old.<ref>http://astronomy.nmsu.edu/geas/lectures/lecture02/slide04.html</ref>}} The number shown rounds up to 10⁸⁰. It is not likely that any computer will ever count that high. If the number to be factored consists of two prime number that are close to 10⁸⁰, the code will iterate all the way up to k coming close to 10⁴⁰. The inability of any computer to perform that many iterations helps explain how cryptocurrency works. &nbsp; 276omv4pdcl3hnwrjz1oo0jut8we3l7 2410281 2410280 2022-07-29T19:45:52Z Guy vandegrift 813252 wikitext text/x-wiki [[File:Factorization flowchart.svg|thumb|420px|Flowchart for a simple prime factorization code]] '''Topics covered:''' ''prime factorization, flowcharts, python dictionaries'' This lesson shows how the use of a python dictionary can product of prime numbers. There are a few things the reader should know: * A serious python programmer would probably prefer to download a package that gives you a function that performs prime factorization. Two promising candidates are [https://pypi.org/project/primefac/ pypi.org/project/primefac] and [https://www.sympy.org/en/index.html www.sympy.org/]. * A simpler version of this program can be found at [[Wikipedia:Trial division#Method]].<ref>A working python script is at [[w:Special:Permalink/1086485306]]</ref>. * Lacking the mental wherewithal to seize upon either of these opportunities, I googled "python prime factorization", found many codes, and selected the one [[#Python code|shown below]] for two reasons:<ref>The code that follows is discussed at [https://stackoverflow.com/questions/15347174/python-finding-prime-factors stackoverflow question 15347174] and [https://scientific-python-101.readthedocs.io/_downloads/prime_factorization_solution.py scientific-python-101.readthedocs]</ref> :# I wanted a code that counts repeated factors in order to later write the factored form in wikitext with exponents.<ref>In the wikitext version, I had to add a few lines so that exponents of 1 would not appear. Example: <math>12=2^2\cdot 3^1</math> should instead read <math>12=2^2\cdot 3.</math> I ommitted the feature here in order to keep the lesson simple.</ref> :# I didn't understand the python code, and wanted master it in order to fully understand how python dictionaries are used. A good student exercise would be to modify this code so that it runs approximately twice as fast. Instructors could also task students with modifying this code so that the output can be shown in a table. This table creates a wikitable that illustrates the "wasted" efforts associated with investigating the even numbers 4 and 6 as candidates to be prime factors of 1550 (see: [[Wikipedia:Trial division#Method]].): {| class="wikitable" style="text-align: center; vertical-align: bottom;" |+ <math>\text{Establishing } 1550=2\cdot 5^2 \cdot 31</math> |- ! n ! k ! k^2 ! k^2<n? ! n/k ! mod ! replace ! factor ! * |- | 1550 | 2 | 4 | TRUE | 775 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"|n: n/k | style="background:#ffffcc"|2 | |- | 775 | 2 | 4 | TRUE | 387 | 1 | k: k+1 | | |- | 775 | 3 | 9 | TRUE | 258 | 1 | k: k+1 | | |- | 775 | 4 | 16 | TRUE | 193 | 3 | k: k+1 | | |- | 775 | 5 | 25 | TRUE | 155 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 155 | 5 | 25 | TRUE | 31 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 31 | 5 | 25 | TRUE | 6 | 1 | k: k+1 | | |- | 31 | 6 | 36 | FALSE | - | - | - | | style="background:#ffffcc"| 31 |} ==Python code== ‎<syntaxhighlight lang="python" line="1"> ## In a python code, the function definitions precede the code. We need only 1 function: def prime_factorization(N): ## 12=(2**2)*(2**1)*(5**1), or equivalently, 12 = 2**2 * 3 * 5 ## The dictionary, dict, expresses k**v as a "key-value" pair. In other ## words, if we look up the key, k=5, the dictionary will return the ## value, v=1. In other words: ## k**v <=> key**value <=> base**exponent (for the factored form of N) ## Begin function: n=N #--------------------# n will change in the while loop (getting smaller.) dict = {} #--------------# Start with an empty dictionary if n==1: #---------------# The factorization of 1 is a special case. return {1: 1} # In other words 1 = 1**1 (but recall that 1 is not prime.) k = 2 #------------------# Our first candidate for factorizing any integer. while k**2 <= n: #-------# Do the following unless k is "too big": if n % k: #----------# . Equivalent to: "If k is not a factor of n:" k += 1 #---------# .. increase the value of k by 1 on next attempt. else: #--------------# . Equivalent to "If k is a factor of n:" n /= k #---------# .. Replace n by n/k (for next attempt) try: #-----------# .. If k is in the dictionary, this attempts to: dict[k] += 1 # ... increase the value of k by 1. except KeyError: # .. KeyError occurs if k is not in the dictionary dict[k] = 1 # ... Add k to dictionary and set exponent v=1 # end(while)#------------# (exits while k**2 <= n:) # Here I inserted a diagnostic to be discussed later: # Documented code continues: if n > 1: #--------------# It can be shown that n is either 1 or prime. try: #---------------# . Try to enter n into dictionary dict[n] += 1 #---# .. increase exponent v->v+1 if k in dictinary except KeyError: #---# . If n is not in the dictionary, dict[n] = 1 # .. then enter n into dictionary with v=1 return dict #------# python talk for ending function ################ End def prime_factorization | Begin program ########## #import sys again=True while again == 1: text="\nEnter a positive integer you wish to factor into prime numbers:\n" N=int(input(text)) dict=prime_factorization(N) print("\n"+str(int(N))+"=") text="(" for b in dict.keys(): text+=str(int(b))+"**"+str(int(dict.get(b)))+") * (" print(text.rstrip(" * (")+"\n") print("The dictionary is:\n") print(dict) again=int(input("\nType 1 for another integer, 0 to terminate.")) </syntaxhighlight> ==Comments on code== ===Line 15: while k**2 <= n:=== This is an important line because it permits the termination of the search before one might think. I leave it as an exercise for the student to learn why any value of k larger than square root of n will never be the next factor of N. It is also left to the student to understand the fact that the time to find the factors can be cut nearly in half if we modify line 17 and increase k by 2 (instead of 1), and why this larger step cannot be taken until after k=2 has been considered. ===Lines in the code that might seem confusing=== The code uses two cleverness tricks that might seem confusing. I lack the formal training in programming to know whether such cleverness would be perceived as a flaw. I know that programmers are allowed to be clever and I know that clear programs are better than confusing programs. The fact that these tricks confused me does not necessarily mean they should be avoided. And, the nice thing about both sources of confusion is that they taught me something about python. ====Line 16: if n%k might seem backwards==== Python follows a common practice of interpreting the number 1 to be True and 0 to be False. But python also accepts any non-zero number as True. Here n%k denotes nmod&nbsp;k, which is the remainder when on divides n by k. If there is no remainder the "if" condition is not satisfied and the code stipulates that the next value of k is attempted. Any number not equal to zero is interpreted as the "if" condition being satisfied, meaning that k is a divisor of n (in that case, we replace n by n/k.) This might be good coding, but it seems to me that it would have been better to construct it along the lines of "if n%k == 0" or even "if n%k !=0". ====Lines 21-23 and 29-31: A failed attempt to change the value of a key==== In this code fragment, we ask if a given value of k has already been identified as a factor. But instead of "asking" if k has been declared a key in the dictionary, an <u>attempt</u> is made to change the value of k. If that attempt fails (because k is not among the dictionary's keys), then k is added to the list of keys, with a value of 1 (a value of 1 indicates that at this point, k, but not k² has been established to be a factor of n.) Again, there might be nothing wrong with using a key error like this. But the code would have been easier for me to follow if the if statement simply called the question of whether k was a key in dict. ==Sample output (with a really big number)== Sometimes you have to wait a few minutes to factor a number this large. But usually it comes out in just a few seconds. I would imagine that a number that was the product of only two roughly equal prime numbers would take forever because k=k+1 must be repeated a virtually infinite number of times. This number came quickly: {{Font color|blue|Enter a positive integer you wish to factor into prime numbers:}} 99283428376482768916467328912876456416375474272263423467896879187289499287374899 &nbsp; <span style="color:blue>99283428376482768916467328912876456416375474272263423467896879187289499287374899=</span> {{Font color|blue|(619**1) * (688**1) * (768**3) * (1024**17) * (64853**1) * (5302548928**1)}} &nbsp; {{Font color|blue|The dictionary is:}} &nbsp; <span style="color:blue>{619: 1, 688: 1, 768: 3, 1024: 17, 64853: 1, 5302548928.0: 1}</span> &nbsp; {{Font color|blue|Type 1 for another integer, 0 to terminate.}} {{center|'''DO NOT TRY TO FACTOR A NUMBER THIS LARGE UNLESS YOU ARE WILLING TO INTERUPT YOUR PYTHON PROGRAM.'''}} {{center|Keep in mind that the universe is only <math>4.4\times10^{26}\text{nanoseconds}</math> old.<ref>http://astronomy.nmsu.edu/geas/lectures/lecture02/slide04.html</ref>}} The number shown rounds up to 10⁸⁰. It is not likely that any computer will ever count that high. If the number to be factored consists of two prime number that are close to 10⁸⁰, the code will iterate all the way up to k coming close to 10⁴⁰. The inability of any computer to perform that many iterations helps explain how cryptocurrency works. &nbsp; hsqn41wowf5fb4afl5w8t9qbkff9vj5 2410282 2410281 2022-07-29T19:48:13Z Guy vandegrift 813252 wikitext text/x-wiki [[File:Factorization flowchart.svg|thumb|420px|Flowchart for a simple prime factorization code]] '''Topics covered:''' ''prime factorization, flowcharts, python dictionaries'' This lesson shows how the use of a python dictionary can product of prime numbers. There are a few things the reader should know: * A serious python programmer would probably prefer to download a package that gives you a function that performs prime factorization. Two promising candidates are [https://pypi.org/project/primefac/ pypi.org/project/primefac] and [https://www.sympy.org/en/index.html www.sympy.org/]. * A simpler version of this program can be found at [[Wikipedia:Trial division#Method]].<ref>A working python script is at [[w:Special:Permalink/1086485306]]</ref>. * Lacking the mental wherewithal to seize upon either of these opportunities, I googled "python prime factorization", found many codes, and selected the one [[#Python code|shown below]] for two reasons:<ref>The code that follows is discussed at [https://stackoverflow.com/questions/15347174/python-finding-prime-factors stackoverflow question 15347174] and [https://scientific-python-101.readthedocs.io/_downloads/prime_factorization_solution.py scientific-python-101.readthedocs]</ref> :# I wanted a code that counts repeated factors in order to later write the factored form in wikitext with exponents.<ref>In the wikitext version, I had to add a few lines so that exponents of 1 would not appear. Example: <math>12=2^2\cdot 3^1</math> should instead read <math>12=2^2\cdot 3.</math> I ommitted the feature here in order to keep the lesson simple.</ref> :# I didn't understand the python code, and wanted master it in order to fully understand how python dictionaries are used. A good student exercise would be to modify this code so that it runs approximately twice as fast. Instructors could also task students with modifying this code so that the output can be shown in a table that displays each effort to see if any given integer <math>k</math> is a prime factor of <math>N</math> This table creates a wikitable that illustrates the "wasted" efforts associated with investigating the even numbers 4 and 6 as candidates to be prime factors of 1550: {| class="wikitable" style="text-align: center; vertical-align: bottom;" |+ <math>\text{Establishing } 1550=2\cdot 5^2 \cdot 31</math> |- ! n ! k ! k^2 ! k^2<n? ! n/k ! mod ! replace ! factor ! * |- | 1550 | 2 | 4 | TRUE | 775 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"|n: n/k | style="background:#ffffcc"|2 | |- | 775 | 2 | 4 | TRUE | 387 | 1 | k: k+1 | | |- | 775 | 3 | 9 | TRUE | 258 | 1 | k: k+1 | | |- | 775 | 4 | 16 | TRUE | 193 | 3 | k: k+1 | | |- | 775 | 5 | 25 | TRUE | 155 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 155 | 5 | 25 | TRUE | 31 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 31 | 5 | 25 | TRUE | 6 | 1 | k: k+1 | | |- | 31 | 6 | 36 | FALSE | - | - | - | | style="background:#ffffcc"| 31 |} ==Python code== ‎<syntaxhighlight lang="python" line="1"> ## In a python code, the function definitions precede the code. We need only 1 function: def prime_factorization(N): ## 12=(2**2)*(2**1)*(5**1), or equivalently, 12 = 2**2 * 3 * 5 ## The dictionary, dict, expresses k**v as a "key-value" pair. In other ## words, if we look up the key, k=5, the dictionary will return the ## value, v=1. In other words: ## k**v <=> key**value <=> base**exponent (for the factored form of N) ## Begin function: n=N #--------------------# n will change in the while loop (getting smaller.) dict = {} #--------------# Start with an empty dictionary if n==1: #---------------# The factorization of 1 is a special case. return {1: 1} # In other words 1 = 1**1 (but recall that 1 is not prime.) k = 2 #------------------# Our first candidate for factorizing any integer. while k**2 <= n: #-------# Do the following unless k is "too big": if n % k: #----------# . Equivalent to: "If k is not a factor of n:" k += 1 #---------# .. increase the value of k by 1 on next attempt. else: #--------------# . Equivalent to "If k is a factor of n:" n /= k #---------# .. Replace n by n/k (for next attempt) try: #-----------# .. If k is in the dictionary, this attempts to: dict[k] += 1 # ... increase the value of k by 1. except KeyError: # .. KeyError occurs if k is not in the dictionary dict[k] = 1 # ... Add k to dictionary and set exponent v=1 # end(while)#------------# (exits while k**2 <= n:) # Here I inserted a diagnostic to be discussed later: # Documented code continues: if n > 1: #--------------# It can be shown that n is either 1 or prime. try: #---------------# . Try to enter n into dictionary dict[n] += 1 #---# .. increase exponent v->v+1 if k in dictinary except KeyError: #---# . If n is not in the dictionary, dict[n] = 1 # .. then enter n into dictionary with v=1 return dict #------# python talk for ending function ################ End def prime_factorization | Begin program ########## #import sys again=True while again == 1: text="\nEnter a positive integer you wish to factor into prime numbers:\n" N=int(input(text)) dict=prime_factorization(N) print("\n"+str(int(N))+"=") text="(" for b in dict.keys(): text+=str(int(b))+"**"+str(int(dict.get(b)))+") * (" print(text.rstrip(" * (")+"\n") print("The dictionary is:\n") print(dict) again=int(input("\nType 1 for another integer, 0 to terminate.")) </syntaxhighlight> ==Comments on code== ===Line 15: while k**2 <= n:=== This is an important line because it permits the termination of the search before one might think. I leave it as an exercise for the student to learn why any value of k larger than square root of n will never be the next factor of N. It is also left to the student to understand the fact that the time to find the factors can be cut nearly in half if we modify line 17 and increase k by 2 (instead of 1), and why this larger step cannot be taken until after k=2 has been considered. ===Lines in the code that might seem confusing=== The code uses two cleverness tricks that might seem confusing. I lack the formal training in programming to know whether such cleverness would be perceived as a flaw. I know that programmers are allowed to be clever and I know that clear programs are better than confusing programs. The fact that these tricks confused me does not necessarily mean they should be avoided. And, the nice thing about both sources of confusion is that they taught me something about python. ====Line 16: if n%k might seem backwards==== Python follows a common practice of interpreting the number 1 to be True and 0 to be False. But python also accepts any non-zero number as True. Here n%k denotes nmod&nbsp;k, which is the remainder when on divides n by k. If there is no remainder the "if" condition is not satisfied and the code stipulates that the next value of k is attempted. Any number not equal to zero is interpreted as the "if" condition being satisfied, meaning that k is a divisor of n (in that case, we replace n by n/k.) This might be good coding, but it seems to me that it would have been better to construct it along the lines of "if n%k == 0" or even "if n%k !=0". ====Lines 21-23 and 29-31: A failed attempt to change the value of a key==== In this code fragment, we ask if a given value of k has already been identified as a factor. But instead of "asking" if k has been declared a key in the dictionary, an <u>attempt</u> is made to change the value of k. If that attempt fails (because k is not among the dictionary's keys), then k is added to the list of keys, with a value of 1 (a value of 1 indicates that at this point, k, but not k² has been established to be a factor of n.) Again, there might be nothing wrong with using a key error like this. But the code would have been easier for me to follow if the if statement simply called the question of whether k was a key in dict. ==Sample output (with a really big number)== Sometimes you have to wait a few minutes to factor a number this large. But usually it comes out in just a few seconds. I would imagine that a number that was the product of only two roughly equal prime numbers would take forever because k=k+1 must be repeated a virtually infinite number of times. This number came quickly: {{Font color|blue|Enter a positive integer you wish to factor into prime numbers:}} 99283428376482768916467328912876456416375474272263423467896879187289499287374899 &nbsp; <span style="color:blue>99283428376482768916467328912876456416375474272263423467896879187289499287374899=</span> {{Font color|blue|(619**1) * (688**1) * (768**3) * (1024**17) * (64853**1) * (5302548928**1)}} &nbsp; {{Font color|blue|The dictionary is:}} &nbsp; <span style="color:blue>{619: 1, 688: 1, 768: 3, 1024: 17, 64853: 1, 5302548928.0: 1}</span> &nbsp; {{Font color|blue|Type 1 for another integer, 0 to terminate.}} {{center|'''DO NOT TRY TO FACTOR A NUMBER THIS LARGE UNLESS YOU ARE WILLING TO INTERUPT YOUR PYTHON PROGRAM.'''}} {{center|Keep in mind that the universe is only <math>4.4\times10^{26}\text{nanoseconds}</math> old.<ref>http://astronomy.nmsu.edu/geas/lectures/lecture02/slide04.html</ref>}} The number shown rounds up to 10⁸⁰. It is not likely that any computer will ever count that high. If the number to be factored consists of two prime number that are close to 10⁸⁰, the code will iterate all the way up to k coming close to 10⁴⁰. The inability of any computer to perform that many iterations helps explain how cryptocurrency works. &nbsp; mc5gjxhoe51eoxvwk7ebygz067a1oe9 2410283 2410282 2022-07-29T19:48:27Z Guy vandegrift 813252 wikitext text/x-wiki [[File:Factorization flowchart.svg|thumb|420px|Flowchart for a simple prime factorization code]] '''Topics covered:''' ''prime factorization, flowcharts, python dictionaries'' This lesson shows how the use of a python dictionary can product of prime numbers. There are a few things the reader should know: * A serious python programmer would probably prefer to download a package that gives you a function that performs prime factorization. Two promising candidates are [https://pypi.org/project/primefac/ pypi.org/project/primefac] and [https://www.sympy.org/en/index.html www.sympy.org/]. * A simpler version of this program can be found at [[Wikipedia:Trial division#Method]].<ref>A working python script is at [[w:Special:Permalink/1086485306]]</ref>. * Lacking the mental wherewithal to seize upon either of these opportunities, I googled "python prime factorization", found many codes, and selected the one [[#Python code|shown below]] for two reasons:<ref>The code that follows is discussed at [https://stackoverflow.com/questions/15347174/python-finding-prime-factors stackoverflow question 15347174] and [https://scientific-python-101.readthedocs.io/_downloads/prime_factorization_solution.py scientific-python-101.readthedocs]</ref> :# I wanted a code that counts repeated factors in order to later write the factored form in wikitext with exponents.<ref>In the wikitext version, I had to add a few lines so that exponents of 1 would not appear. Example: <math>12=2^2\cdot 3^1</math> should instead read <math>12=2^2\cdot 3.</math> I ommitted the feature here in order to keep the lesson simple.</ref> :# I didn't understand the python code, and wanted master it in order to fully understand how python dictionaries are used. A good student exercise would be to modify this code so that it runs approximately twice as fast. Instructors could also task students with modifying this code so that the output can be shown in a table that displays each effort to see if any given integer <math>k</math> is a prime factor of <math>N.</math> This table creates a wikitable that illustrates the "wasted" efforts associated with investigating the even numbers 4 and 6 as candidates to be prime factors of 1550: {| class="wikitable" style="text-align: center; vertical-align: bottom;" |+ <math>\text{Establishing } 1550=2\cdot 5^2 \cdot 31</math> |- ! n ! k ! k^2 ! k^2<n? ! n/k ! mod ! replace ! factor ! * |- | 1550 | 2 | 4 | TRUE | 775 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"|n: n/k | style="background:#ffffcc"|2 | |- | 775 | 2 | 4 | TRUE | 387 | 1 | k: k+1 | | |- | 775 | 3 | 9 | TRUE | 258 | 1 | k: k+1 | | |- | 775 | 4 | 16 | TRUE | 193 | 3 | k: k+1 | | |- | 775 | 5 | 25 | TRUE | 155 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 155 | 5 | 25 | TRUE | 31 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 31 | 5 | 25 | TRUE | 6 | 1 | k: k+1 | | |- | 31 | 6 | 36 | FALSE | - | - | - | | style="background:#ffffcc"| 31 |} ==Python code== ‎<syntaxhighlight lang="python" line="1"> ## In a python code, the function definitions precede the code. We need only 1 function: def prime_factorization(N): ## 12=(2**2)*(2**1)*(5**1), or equivalently, 12 = 2**2 * 3 * 5 ## The dictionary, dict, expresses k**v as a "key-value" pair. In other ## words, if we look up the key, k=5, the dictionary will return the ## value, v=1. In other words: ## k**v <=> key**value <=> base**exponent (for the factored form of N) ## Begin function: n=N #--------------------# n will change in the while loop (getting smaller.) dict = {} #--------------# Start with an empty dictionary if n==1: #---------------# The factorization of 1 is a special case. return {1: 1} # In other words 1 = 1**1 (but recall that 1 is not prime.) k = 2 #------------------# Our first candidate for factorizing any integer. while k**2 <= n: #-------# Do the following unless k is "too big": if n % k: #----------# . Equivalent to: "If k is not a factor of n:" k += 1 #---------# .. increase the value of k by 1 on next attempt. else: #--------------# . Equivalent to "If k is a factor of n:" n /= k #---------# .. Replace n by n/k (for next attempt) try: #-----------# .. If k is in the dictionary, this attempts to: dict[k] += 1 # ... increase the value of k by 1. except KeyError: # .. KeyError occurs if k is not in the dictionary dict[k] = 1 # ... Add k to dictionary and set exponent v=1 # end(while)#------------# (exits while k**2 <= n:) # Here I inserted a diagnostic to be discussed later: # Documented code continues: if n > 1: #--------------# It can be shown that n is either 1 or prime. try: #---------------# . Try to enter n into dictionary dict[n] += 1 #---# .. increase exponent v->v+1 if k in dictinary except KeyError: #---# . If n is not in the dictionary, dict[n] = 1 # .. then enter n into dictionary with v=1 return dict #------# python talk for ending function ################ End def prime_factorization | Begin program ########## #import sys again=True while again == 1: text="\nEnter a positive integer you wish to factor into prime numbers:\n" N=int(input(text)) dict=prime_factorization(N) print("\n"+str(int(N))+"=") text="(" for b in dict.keys(): text+=str(int(b))+"**"+str(int(dict.get(b)))+") * (" print(text.rstrip(" * (")+"\n") print("The dictionary is:\n") print(dict) again=int(input("\nType 1 for another integer, 0 to terminate.")) </syntaxhighlight> ==Comments on code== ===Line 15: while k**2 <= n:=== This is an important line because it permits the termination of the search before one might think. I leave it as an exercise for the student to learn why any value of k larger than square root of n will never be the next factor of N. It is also left to the student to understand the fact that the time to find the factors can be cut nearly in half if we modify line 17 and increase k by 2 (instead of 1), and why this larger step cannot be taken until after k=2 has been considered. ===Lines in the code that might seem confusing=== The code uses two cleverness tricks that might seem confusing. I lack the formal training in programming to know whether such cleverness would be perceived as a flaw. I know that programmers are allowed to be clever and I know that clear programs are better than confusing programs. The fact that these tricks confused me does not necessarily mean they should be avoided. And, the nice thing about both sources of confusion is that they taught me something about python. ====Line 16: if n%k might seem backwards==== Python follows a common practice of interpreting the number 1 to be True and 0 to be False. But python also accepts any non-zero number as True. Here n%k denotes nmod&nbsp;k, which is the remainder when on divides n by k. If there is no remainder the "if" condition is not satisfied and the code stipulates that the next value of k is attempted. Any number not equal to zero is interpreted as the "if" condition being satisfied, meaning that k is a divisor of n (in that case, we replace n by n/k.) This might be good coding, but it seems to me that it would have been better to construct it along the lines of "if n%k == 0" or even "if n%k !=0". ====Lines 21-23 and 29-31: A failed attempt to change the value of a key==== In this code fragment, we ask if a given value of k has already been identified as a factor. But instead of "asking" if k has been declared a key in the dictionary, an <u>attempt</u> is made to change the value of k. If that attempt fails (because k is not among the dictionary's keys), then k is added to the list of keys, with a value of 1 (a value of 1 indicates that at this point, k, but not k² has been established to be a factor of n.) Again, there might be nothing wrong with using a key error like this. But the code would have been easier for me to follow if the if statement simply called the question of whether k was a key in dict. ==Sample output (with a really big number)== Sometimes you have to wait a few minutes to factor a number this large. But usually it comes out in just a few seconds. I would imagine that a number that was the product of only two roughly equal prime numbers would take forever because k=k+1 must be repeated a virtually infinite number of times. This number came quickly: {{Font color|blue|Enter a positive integer you wish to factor into prime numbers:}} 99283428376482768916467328912876456416375474272263423467896879187289499287374899 &nbsp; <span style="color:blue>99283428376482768916467328912876456416375474272263423467896879187289499287374899=</span> {{Font color|blue|(619**1) * (688**1) * (768**3) * (1024**17) * (64853**1) * (5302548928**1)}} &nbsp; {{Font color|blue|The dictionary is:}} &nbsp; <span style="color:blue>{619: 1, 688: 1, 768: 3, 1024: 17, 64853: 1, 5302548928.0: 1}</span> &nbsp; {{Font color|blue|Type 1 for another integer, 0 to terminate.}} {{center|'''DO NOT TRY TO FACTOR A NUMBER THIS LARGE UNLESS YOU ARE WILLING TO INTERUPT YOUR PYTHON PROGRAM.'''}} {{center|Keep in mind that the universe is only <math>4.4\times10^{26}\text{nanoseconds}</math> old.<ref>http://astronomy.nmsu.edu/geas/lectures/lecture02/slide04.html</ref>}} The number shown rounds up to 10⁸⁰. It is not likely that any computer will ever count that high. If the number to be factored consists of two prime number that are close to 10⁸⁰, the code will iterate all the way up to k coming close to 10⁴⁰. The inability of any computer to perform that many iterations helps explain how cryptocurrency works. &nbsp; pwsgc2ejz4xvdqg7rj563y5ghddb2vl 2410284 2410283 2022-07-29T19:49:30Z Guy vandegrift 813252 wikitext text/x-wiki [[File:Factorization flowchart.svg|thumb|420px|Flowchart for a simple prime factorization code]] '''Topics covered:''' ''prime factorization, flowcharts, python dictionaries'' This lesson shows how the use of a python dictionary can product of prime numbers. There are a few things the reader should know: * A serious python programmer would probably prefer to download a package that gives you a function that performs prime factorization. Two promising candidates are [https://pypi.org/project/primefac/ pypi.org/project/primefac] and [https://www.sympy.org/en/index.html www.sympy.org/]. * A simpler version of this program can be found at [[Wikipedia:Trial division#Method]].<ref>A working python script is at [[w:Special:Permalink/1086485306]]</ref>. * Lacking the mental wherewithal to seize upon either of these opportunities, I googled "python prime factorization", found many codes, and selected the one [[#Python code|shown below]] for two reasons:<ref>The code that follows is discussed at [https://stackoverflow.com/questions/15347174/python-finding-prime-factors stackoverflow question 15347174] and [https://scientific-python-101.readthedocs.io/_downloads/prime_factorization_solution.py scientific-python-101.readthedocs]</ref> :# I wanted a code that counts repeated factors in order to later write the factored form in wikitext with exponents.<ref>In the wikitext version, I had to add a few lines so that exponents of 1 would not appear. Example: <math>12=2^2\cdot 3^1</math> should instead read <math>12=2^2\cdot 3.</math> I ommitted the feature here in order to keep the lesson simple.</ref> :# I didn't understand the python code, and wanted master it in order to fully understand how python dictionaries are used. A good student exercise would be to modify this code so that it runs approximately twice as fast. Instructors could also task students with modifying this code so that the output can be shown in a table that displays each effort to see if any given integer <math>k</math> is a prime factor of <math>N.</math> A wikitable version of this code was used to create the following table. It illustrates the "wasted" efforts associated with investigating the even numbers 4 and 6 as candidates to be prime factors of 1550: {| class="wikitable" style="text-align: center; vertical-align: bottom;" |+ <math>\text{Establishing } 1550=2\cdot 5^2 \cdot 31</math> |- ! n ! k ! k^2 ! k^2<n? ! n/k ! mod ! replace ! factor ! * |- | 1550 | 2 | 4 | TRUE | 775 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"|n: n/k | style="background:#ffffcc"|2 | |- | 775 | 2 | 4 | TRUE | 387 | 1 | k: k+1 | | |- | 775 | 3 | 9 | TRUE | 258 | 1 | k: k+1 | | |- | 775 | 4 | 16 | TRUE | 193 | 3 | k: k+1 | | |- | 775 | 5 | 25 | TRUE | 155 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 155 | 5 | 25 | TRUE | 31 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 31 | 5 | 25 | TRUE | 6 | 1 | k: k+1 | | |- | 31 | 6 | 36 | FALSE | - | - | - | | style="background:#ffffcc"| 31 |} ==Python code== ‎<syntaxhighlight lang="python" line="1"> ## In a python code, the function definitions precede the code. We need only 1 function: def prime_factorization(N): ## 12=(2**2)*(2**1)*(5**1), or equivalently, 12 = 2**2 * 3 * 5 ## The dictionary, dict, expresses k**v as a "key-value" pair. In other ## words, if we look up the key, k=5, the dictionary will return the ## value, v=1. In other words: ## k**v <=> key**value <=> base**exponent (for the factored form of N) ## Begin function: n=N #--------------------# n will change in the while loop (getting smaller.) dict = {} #--------------# Start with an empty dictionary if n==1: #---------------# The factorization of 1 is a special case. return {1: 1} # In other words 1 = 1**1 (but recall that 1 is not prime.) k = 2 #------------------# Our first candidate for factorizing any integer. while k**2 <= n: #-------# Do the following unless k is "too big": if n % k: #----------# . Equivalent to: "If k is not a factor of n:" k += 1 #---------# .. increase the value of k by 1 on next attempt. else: #--------------# . Equivalent to "If k is a factor of n:" n /= k #---------# .. Replace n by n/k (for next attempt) try: #-----------# .. If k is in the dictionary, this attempts to: dict[k] += 1 # ... increase the value of k by 1. except KeyError: # .. KeyError occurs if k is not in the dictionary dict[k] = 1 # ... Add k to dictionary and set exponent v=1 # end(while)#------------# (exits while k**2 <= n:) # Here I inserted a diagnostic to be discussed later: # Documented code continues: if n > 1: #--------------# It can be shown that n is either 1 or prime. try: #---------------# . Try to enter n into dictionary dict[n] += 1 #---# .. increase exponent v->v+1 if k in dictinary except KeyError: #---# . If n is not in the dictionary, dict[n] = 1 # .. then enter n into dictionary with v=1 return dict #------# python talk for ending function ################ End def prime_factorization | Begin program ########## #import sys again=True while again == 1: text="\nEnter a positive integer you wish to factor into prime numbers:\n" N=int(input(text)) dict=prime_factorization(N) print("\n"+str(int(N))+"=") text="(" for b in dict.keys(): text+=str(int(b))+"**"+str(int(dict.get(b)))+") * (" print(text.rstrip(" * (")+"\n") print("The dictionary is:\n") print(dict) again=int(input("\nType 1 for another integer, 0 to terminate.")) </syntaxhighlight> ==Comments on code== ===Line 15: while k**2 <= n:=== This is an important line because it permits the termination of the search before one might think. I leave it as an exercise for the student to learn why any value of k larger than square root of n will never be the next factor of N. It is also left to the student to understand the fact that the time to find the factors can be cut nearly in half if we modify line 17 and increase k by 2 (instead of 1), and why this larger step cannot be taken until after k=2 has been considered. ===Lines in the code that might seem confusing=== The code uses two cleverness tricks that might seem confusing. I lack the formal training in programming to know whether such cleverness would be perceived as a flaw. I know that programmers are allowed to be clever and I know that clear programs are better than confusing programs. The fact that these tricks confused me does not necessarily mean they should be avoided. And, the nice thing about both sources of confusion is that they taught me something about python. ====Line 16: if n%k might seem backwards==== Python follows a common practice of interpreting the number 1 to be True and 0 to be False. But python also accepts any non-zero number as True. Here n%k denotes nmod&nbsp;k, which is the remainder when on divides n by k. If there is no remainder the "if" condition is not satisfied and the code stipulates that the next value of k is attempted. Any number not equal to zero is interpreted as the "if" condition being satisfied, meaning that k is a divisor of n (in that case, we replace n by n/k.) This might be good coding, but it seems to me that it would have been better to construct it along the lines of "if n%k == 0" or even "if n%k !=0". ====Lines 21-23 and 29-31: A failed attempt to change the value of a key==== In this code fragment, we ask if a given value of k has already been identified as a factor. But instead of "asking" if k has been declared a key in the dictionary, an <u>attempt</u> is made to change the value of k. If that attempt fails (because k is not among the dictionary's keys), then k is added to the list of keys, with a value of 1 (a value of 1 indicates that at this point, k, but not k² has been established to be a factor of n.) Again, there might be nothing wrong with using a key error like this. But the code would have been easier for me to follow if the if statement simply called the question of whether k was a key in dict. ==Sample output (with a really big number)== Sometimes you have to wait a few minutes to factor a number this large. But usually it comes out in just a few seconds. I would imagine that a number that was the product of only two roughly equal prime numbers would take forever because k=k+1 must be repeated a virtually infinite number of times. This number came quickly: {{Font color|blue|Enter a positive integer you wish to factor into prime numbers:}} 99283428376482768916467328912876456416375474272263423467896879187289499287374899 &nbsp; <span style="color:blue>99283428376482768916467328912876456416375474272263423467896879187289499287374899=</span> {{Font color|blue|(619**1) * (688**1) * (768**3) * (1024**17) * (64853**1) * (5302548928**1)}} &nbsp; {{Font color|blue|The dictionary is:}} &nbsp; <span style="color:blue>{619: 1, 688: 1, 768: 3, 1024: 17, 64853: 1, 5302548928.0: 1}</span> &nbsp; {{Font color|blue|Type 1 for another integer, 0 to terminate.}} {{center|'''DO NOT TRY TO FACTOR A NUMBER THIS LARGE UNLESS YOU ARE WILLING TO INTERUPT YOUR PYTHON PROGRAM.'''}} {{center|Keep in mind that the universe is only <math>4.4\times10^{26}\text{nanoseconds}</math> old.<ref>http://astronomy.nmsu.edu/geas/lectures/lecture02/slide04.html</ref>}} The number shown rounds up to 10⁸⁰. It is not likely that any computer will ever count that high. If the number to be factored consists of two prime number that are close to 10⁸⁰, the code will iterate all the way up to k coming close to 10⁴⁰. The inability of any computer to perform that many iterations helps explain how cryptocurrency works. &nbsp; df7ylphg8kwmz58q2i7rj8m7zgbhux6 2410285 2410284 2022-07-29T19:53:15Z Guy vandegrift 813252 wikitext text/x-wiki [[File:Factorization flowchart.svg|thumb|420px|Flowchart for a simple prime factorization code]] '''Topics covered:''' ''prime factorization, flowcharts, python dictionaries'' This lesson shows how the use of a python dictionary can product of prime numbers. There are a few things the reader should know: * A serious python programmer would probably prefer to download a package that gives you a function that performs prime factorization. Two promising candidates are [https://pypi.org/project/primefac/ pypi.org/project/primefac] and [https://www.sympy.org/en/index.html www.sympy.org/]. * A simpler version of this program can be found at [[Wikipedia:Trial division#Method]].<ref>A working python script is at [[w:Special:Permalink/1086485306]]</ref>. * Lacking the mental wherewithal to seize upon either of these opportunities, I googled "python prime factorization", found many codes, and selected the one [[#Python code|shown below]] for two reasons:<ref>The code that follows is discussed at [https://stackoverflow.com/questions/15347174/python-finding-prime-factors stackoverflow question 15347174] and [https://scientific-python-101.readthedocs.io/_downloads/prime_factorization_solution.py scientific-python-101.readthedocs]</ref> :# I wanted a code that counts repeated factors in order to later write the factored form in wikitext with exponents.<ref>In the wikitext version, I had to add a few lines so that exponents of 1 would not appear. Example: <math>12=2^2\cdot 3^1</math> should instead read <math>12=2^2\cdot 3.</math> I ommitted the feature here in order to keep the lesson simple.</ref> :# I didn't understand the python code, and wanted master it in order to fully understand how python dictionaries are used. A good student exercise would be to modify this code so that it runs approximately twice as fast. Instructors could also task students with modifying this code so that the output can be shown in a table that displays each effort to see if any given integer <math>k</math> is a prime factor of <math>N.</math> A version of such a modification created the wikitable shown below. It illustrates the "wasted" efforts to see if 4 and 6 might be prime factors of 1550. And it illustrates the efficiency of dividing a candidate integer into the number <math>n</math>, which soon becomes much smaller than <math>N.</math> {| class="wikitable" style="text-align: center; vertical-align: bottom;" |+ <math>\text{Establishing } 1550=2\cdot 5^2 \cdot 31</math> |- ! n ! k ! k^2 ! k^2<n? ! n/k ! mod ! replace ! factor ! * |- | 1550 | 2 | 4 | TRUE | 775 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"|n: n/k | style="background:#ffffcc"|2 | |- | 775 | 2 | 4 | TRUE | 387 | 1 | k: k+1 | | |- | 775 | 3 | 9 | TRUE | 258 | 1 | k: k+1 | | |- | 775 | 4 | 16 | TRUE | 193 | 3 | k: k+1 | | |- | 775 | 5 | 25 | TRUE | 155 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 155 | 5 | 25 | TRUE | 31 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 31 | 5 | 25 | TRUE | 6 | 1 | k: k+1 | | |- | 31 | 6 | 36 | FALSE | - | - | - | | style="background:#ffffcc"| 31 |} ==Python code== ‎<syntaxhighlight lang="python" line="1"> ## In a python code, the function definitions precede the code. We need only 1 function: def prime_factorization(N): ## 12=(2**2)*(2**1)*(5**1), or equivalently, 12 = 2**2 * 3 * 5 ## The dictionary, dict, expresses k**v as a "key-value" pair. In other ## words, if we look up the key, k=5, the dictionary will return the ## value, v=1. In other words: ## k**v <=> key**value <=> base**exponent (for the factored form of N) ## Begin function: n=N #--------------------# n will change in the while loop (getting smaller.) dict = {} #--------------# Start with an empty dictionary if n==1: #---------------# The factorization of 1 is a special case. return {1: 1} # In other words 1 = 1**1 (but recall that 1 is not prime.) k = 2 #------------------# Our first candidate for factorizing any integer. while k**2 <= n: #-------# Do the following unless k is "too big": if n % k: #----------# . Equivalent to: "If k is not a factor of n:" k += 1 #---------# .. increase the value of k by 1 on next attempt. else: #--------------# . Equivalent to "If k is a factor of n:" n /= k #---------# .. Replace n by n/k (for next attempt) try: #-----------# .. If k is in the dictionary, this attempts to: dict[k] += 1 # ... increase the value of k by 1. except KeyError: # .. KeyError occurs if k is not in the dictionary dict[k] = 1 # ... Add k to dictionary and set exponent v=1 # end(while)#------------# (exits while k**2 <= n:) # Here I inserted a diagnostic to be discussed later: # Documented code continues: if n > 1: #--------------# It can be shown that n is either 1 or prime. try: #---------------# . Try to enter n into dictionary dict[n] += 1 #---# .. increase exponent v->v+1 if k in dictinary except KeyError: #---# . If n is not in the dictionary, dict[n] = 1 # .. then enter n into dictionary with v=1 return dict #------# python talk for ending function ################ End def prime_factorization | Begin program ########## #import sys again=True while again == 1: text="\nEnter a positive integer you wish to factor into prime numbers:\n" N=int(input(text)) dict=prime_factorization(N) print("\n"+str(int(N))+"=") text="(" for b in dict.keys(): text+=str(int(b))+"**"+str(int(dict.get(b)))+") * (" print(text.rstrip(" * (")+"\n") print("The dictionary is:\n") print(dict) again=int(input("\nType 1 for another integer, 0 to terminate.")) </syntaxhighlight> ==Comments on code== ===Line 15: while k**2 <= n:=== This is an important line because it permits the termination of the search before one might think. I leave it as an exercise for the student to learn why any value of k larger than square root of n will never be the next factor of N. It is also left to the student to understand the fact that the time to find the factors can be cut nearly in half if we modify line 17 and increase k by 2 (instead of 1), and why this larger step cannot be taken until after k=2 has been considered. ===Lines in the code that might seem confusing=== The code uses two cleverness tricks that might seem confusing. I lack the formal training in programming to know whether such cleverness would be perceived as a flaw. I know that programmers are allowed to be clever and I know that clear programs are better than confusing programs. The fact that these tricks confused me does not necessarily mean they should be avoided. And, the nice thing about both sources of confusion is that they taught me something about python. ====Line 16: if n%k might seem backwards==== Python follows a common practice of interpreting the number 1 to be True and 0 to be False. But python also accepts any non-zero number as True. Here n%k denotes nmod&nbsp;k, which is the remainder when on divides n by k. If there is no remainder the "if" condition is not satisfied and the code stipulates that the next value of k is attempted. Any number not equal to zero is interpreted as the "if" condition being satisfied, meaning that k is a divisor of n (in that case, we replace n by n/k.) This might be good coding, but it seems to me that it would have been better to construct it along the lines of "if n%k == 0" or even "if n%k !=0". ====Lines 21-23 and 29-31: A failed attempt to change the value of a key==== In this code fragment, we ask if a given value of k has already been identified as a factor. But instead of "asking" if k has been declared a key in the dictionary, an <u>attempt</u> is made to change the value of k. If that attempt fails (because k is not among the dictionary's keys), then k is added to the list of keys, with a value of 1 (a value of 1 indicates that at this point, k, but not k² has been established to be a factor of n.) Again, there might be nothing wrong with using a key error like this. But the code would have been easier for me to follow if the if statement simply called the question of whether k was a key in dict. ==Sample output (with a really big number)== Sometimes you have to wait a few minutes to factor a number this large. But usually it comes out in just a few seconds. I would imagine that a number that was the product of only two roughly equal prime numbers would take forever because k=k+1 must be repeated a virtually infinite number of times. This number came quickly: {{Font color|blue|Enter a positive integer you wish to factor into prime numbers:}} 99283428376482768916467328912876456416375474272263423467896879187289499287374899 &nbsp; <span style="color:blue>99283428376482768916467328912876456416375474272263423467896879187289499287374899=</span> {{Font color|blue|(619**1) * (688**1) * (768**3) * (1024**17) * (64853**1) * (5302548928**1)}} &nbsp; {{Font color|blue|The dictionary is:}} &nbsp; <span style="color:blue>{619: 1, 688: 1, 768: 3, 1024: 17, 64853: 1, 5302548928.0: 1}</span> &nbsp; {{Font color|blue|Type 1 for another integer, 0 to terminate.}} {{center|'''DO NOT TRY TO FACTOR A NUMBER THIS LARGE UNLESS YOU ARE WILLING TO INTERUPT YOUR PYTHON PROGRAM.'''}} {{center|Keep in mind that the universe is only <math>4.4\times10^{26}\text{nanoseconds}</math> old.<ref>http://astronomy.nmsu.edu/geas/lectures/lecture02/slide04.html</ref>}} The number shown rounds up to 10⁸⁰. It is not likely that any computer will ever count that high. If the number to be factored consists of two prime number that are close to 10⁸⁰, the code will iterate all the way up to k coming close to 10⁴⁰. The inability of any computer to perform that many iterations helps explain how cryptocurrency works. &nbsp; sadwyt041lyw2d2rznd6zz8369akx2b 2410286 2410285 2022-07-29T19:55:14Z Guy vandegrift 813252 wikitext text/x-wiki [[File:Factorization flowchart.svg|thumb|420px|Flowchart for a simple prime factorization code]] '''Topics covered:''' ''prime factorization, flowcharts, python dictionaries'' Here we learn how a python dictionary can reduce an integer into a product of prime numbers. There are a few things the student should know about the code you are about to study: * A serious python programmer would probably prefer to download a package that gives you a function that performs prime factorization. Two promising candidates are [https://pypi.org/project/primefac/ pypi.org/project/primefac] and [https://www.sympy.org/en/index.html www.sympy.org/]. * A simpler version of this program can be found at [[Wikipedia:Trial division#Method]].<ref>A working python script is at [[w:Special:Permalink/1086485306]]</ref>. * Lacking the mental wherewithal to seize upon either of these opportunities, I googled "python prime factorization", found many codes, and selected the one [[#Python code|shown below]] for two reasons:<ref>The code that follows is discussed at [https://stackoverflow.com/questions/15347174/python-finding-prime-factors stackoverflow question 15347174] and [https://scientific-python-101.readthedocs.io/_downloads/prime_factorization_solution.py scientific-python-101.readthedocs]</ref> :# I wanted a code that counts repeated factors in order to later write the factored form in wikitext with exponents.<ref>In the wikitext version, I had to add a few lines so that exponents of 1 would not appear. Example: <math>12=2^2\cdot 3^1</math> should instead read <math>12=2^2\cdot 3.</math> I ommitted the feature here in order to keep the lesson simple.</ref> :# I didn't understand the python code, and wanted master it in order to fully understand how python dictionaries are used. A good student exercise would be to modify this code so that it runs approximately twice as fast. Instructors could also task students with modifying this code so that the output can be shown in a table that displays each effort to see if any given integer <math>k</math> is a prime factor of <math>N.</math> A version of such a modification created the wikitable shown below. It illustrates the "wasted" efforts to see if 4 and 6 might be prime factors of 1550. And it illustrates the efficiency of dividing a candidate integer into the number <math>n</math>, which soon becomes much smaller than <math>N.</math> {| class="wikitable" style="text-align: center; vertical-align: bottom;" |+ <math>\text{Establishing } 1550=2\cdot 5^2 \cdot 31</math> |- ! n ! k ! k^2 ! k^2<n? ! n/k ! mod ! replace ! factor ! * |- | 1550 | 2 | 4 | TRUE | 775 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"|n: n/k | style="background:#ffffcc"|2 | |- | 775 | 2 | 4 | TRUE | 387 | 1 | k: k+1 | | |- | 775 | 3 | 9 | TRUE | 258 | 1 | k: k+1 | | |- | 775 | 4 | 16 | TRUE | 193 | 3 | k: k+1 | | |- | 775 | 5 | 25 | TRUE | 155 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 155 | 5 | 25 | TRUE | 31 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 31 | 5 | 25 | TRUE | 6 | 1 | k: k+1 | | |- | 31 | 6 | 36 | FALSE | - | - | - | | style="background:#ffffcc"| 31 |} ==Python code== ‎<syntaxhighlight lang="python" line="1"> ## In a python code, the function definitions precede the code. We need only 1 function: def prime_factorization(N): ## 12=(2**2)*(2**1)*(5**1), or equivalently, 12 = 2**2 * 3 * 5 ## The dictionary, dict, expresses k**v as a "key-value" pair. In other ## words, if we look up the key, k=5, the dictionary will return the ## value, v=1. In other words: ## k**v <=> key**value <=> base**exponent (for the factored form of N) ## Begin function: n=N #--------------------# n will change in the while loop (getting smaller.) dict = {} #--------------# Start with an empty dictionary if n==1: #---------------# The factorization of 1 is a special case. return {1: 1} # In other words 1 = 1**1 (but recall that 1 is not prime.) k = 2 #------------------# Our first candidate for factorizing any integer. while k**2 <= n: #-------# Do the following unless k is "too big": if n % k: #----------# . Equivalent to: "If k is not a factor of n:" k += 1 #---------# .. increase the value of k by 1 on next attempt. else: #--------------# . Equivalent to "If k is a factor of n:" n /= k #---------# .. Replace n by n/k (for next attempt) try: #-----------# .. If k is in the dictionary, this attempts to: dict[k] += 1 # ... increase the value of k by 1. except KeyError: # .. KeyError occurs if k is not in the dictionary dict[k] = 1 # ... Add k to dictionary and set exponent v=1 # end(while)#------------# (exits while k**2 <= n:) # Here I inserted a diagnostic to be discussed later: # Documented code continues: if n > 1: #--------------# It can be shown that n is either 1 or prime. try: #---------------# . Try to enter n into dictionary dict[n] += 1 #---# .. increase exponent v->v+1 if k in dictinary except KeyError: #---# . If n is not in the dictionary, dict[n] = 1 # .. then enter n into dictionary with v=1 return dict #------# python talk for ending function ################ End def prime_factorization | Begin program ########## #import sys again=True while again == 1: text="\nEnter a positive integer you wish to factor into prime numbers:\n" N=int(input(text)) dict=prime_factorization(N) print("\n"+str(int(N))+"=") text="(" for b in dict.keys(): text+=str(int(b))+"**"+str(int(dict.get(b)))+") * (" print(text.rstrip(" * (")+"\n") print("The dictionary is:\n") print(dict) again=int(input("\nType 1 for another integer, 0 to terminate.")) </syntaxhighlight> ==Comments on code== ===Line 15: while k**2 <= n:=== This is an important line because it permits the termination of the search before one might think. I leave it as an exercise for the student to learn why any value of k larger than square root of n will never be the next factor of N. It is also left to the student to understand the fact that the time to find the factors can be cut nearly in half if we modify line 17 and increase k by 2 (instead of 1), and why this larger step cannot be taken until after k=2 has been considered. ===Lines in the code that might seem confusing=== The code uses two cleverness tricks that might seem confusing. I lack the formal training in programming to know whether such cleverness would be perceived as a flaw. I know that programmers are allowed to be clever and I know that clear programs are better than confusing programs. The fact that these tricks confused me does not necessarily mean they should be avoided. And, the nice thing about both sources of confusion is that they taught me something about python. ====Line 16: if n%k might seem backwards==== Python follows a common practice of interpreting the number 1 to be True and 0 to be False. But python also accepts any non-zero number as True. Here n%k denotes nmod&nbsp;k, which is the remainder when on divides n by k. If there is no remainder the "if" condition is not satisfied and the code stipulates that the next value of k is attempted. Any number not equal to zero is interpreted as the "if" condition being satisfied, meaning that k is a divisor of n (in that case, we replace n by n/k.) This might be good coding, but it seems to me that it would have been better to construct it along the lines of "if n%k == 0" or even "if n%k !=0". ====Lines 21-23 and 29-31: A failed attempt to change the value of a key==== In this code fragment, we ask if a given value of k has already been identified as a factor. But instead of "asking" if k has been declared a key in the dictionary, an <u>attempt</u> is made to change the value of k. If that attempt fails (because k is not among the dictionary's keys), then k is added to the list of keys, with a value of 1 (a value of 1 indicates that at this point, k, but not k² has been established to be a factor of n.) Again, there might be nothing wrong with using a key error like this. But the code would have been easier for me to follow if the if statement simply called the question of whether k was a key in dict. ==Sample output (with a really big number)== Sometimes you have to wait a few minutes to factor a number this large. But usually it comes out in just a few seconds. I would imagine that a number that was the product of only two roughly equal prime numbers would take forever because k=k+1 must be repeated a virtually infinite number of times. This number came quickly: {{Font color|blue|Enter a positive integer you wish to factor into prime numbers:}} 99283428376482768916467328912876456416375474272263423467896879187289499287374899 &nbsp; <span style="color:blue>99283428376482768916467328912876456416375474272263423467896879187289499287374899=</span> {{Font color|blue|(619**1) * (688**1) * (768**3) * (1024**17) * (64853**1) * (5302548928**1)}} &nbsp; {{Font color|blue|The dictionary is:}} &nbsp; <span style="color:blue>{619: 1, 688: 1, 768: 3, 1024: 17, 64853: 1, 5302548928.0: 1}</span> &nbsp; {{Font color|blue|Type 1 for another integer, 0 to terminate.}} {{center|'''DO NOT TRY TO FACTOR A NUMBER THIS LARGE UNLESS YOU ARE WILLING TO INTERUPT YOUR PYTHON PROGRAM.'''}} {{center|Keep in mind that the universe is only <math>4.4\times10^{26}\text{nanoseconds}</math> old.<ref>http://astronomy.nmsu.edu/geas/lectures/lecture02/slide04.html</ref>}} The number shown rounds up to 10⁸⁰. It is not likely that any computer will ever count that high. If the number to be factored consists of two prime number that are close to 10⁸⁰, the code will iterate all the way up to k coming close to 10⁴⁰. The inability of any computer to perform that many iterations helps explain how cryptocurrency works. &nbsp; d7hme790mcym0oqra36jclczhfnsu8x 2410287 2410286 2022-07-29T19:57:28Z Guy vandegrift 813252 wikitext text/x-wiki [[File:Factorization flowchart.svg|thumb|420px|Flowchart for a simple prime factorization code]] '''Topics covered:''' ''prime factorization, flowcharts, python dictionaries'' Here we learn how a [[Python Concepts/Dictionaries|python dictionary]] can reduce an integer into a product of prime numbers. There are a few things the student should know about the code you are about to study: * A serious python programmer would probably prefer to download a package that gives you a function that performs prime factorization. Two promising candidates are [https://pypi.org/project/primefac/ pypi.org/project/primefac] and [https://www.sympy.org/en/index.html www.sympy.org/]. * A simpler version of this program can be found at [[Wikipedia:Trial division#Method]].<ref>A working python script is at [[w:Special:Permalink/1086485306]]</ref>. * Lacking the mental wherewithal to seize upon either of these opportunities, I googled "python prime factorization", found many codes, and selected the one [[#Python code|shown below]] for two reasons:<ref>The code that follows is discussed at [https://stackoverflow.com/questions/15347174/python-finding-prime-factors stackoverflow question 15347174] and [https://scientific-python-101.readthedocs.io/_downloads/prime_factorization_solution.py scientific-python-101.readthedocs]</ref> :# I wanted a code that counts repeated factors in order to later write the factored form in wikitext with exponents.<ref>In the wikitext version, I had to add a few lines so that exponents of 1 would not appear. Example: <math>12=2^2\cdot 3^1</math> should instead read <math>12=2^2\cdot 3.</math> I ommitted the feature here in order to keep the lesson simple.</ref> :# I didn't understand the python code, and wanted master it in order to fully understand how python dictionaries are used. A good student exercise would be to modify this code so that it runs approximately twice as fast. Instructors could also task students with modifying this code so that the output can be shown in a table that displays each effort to see if any given integer <math>k</math> is a prime factor of <math>N.</math> A version of such a modification created the wikitable shown below. It illustrates the "wasted" efforts to see if 4 and 6 might be prime factors of 1550. And it illustrates the efficiency of dividing a candidate integer into the number <math>n</math>, which soon becomes much smaller than <math>N.</math> {| class="wikitable" style="text-align: center; vertical-align: bottom;" |+ <math>\text{Establishing } 1550=2\cdot 5^2 \cdot 31</math> |- ! n ! k ! k^2 ! k^2<n? ! n/k ! mod ! replace ! factor ! * |- | 1550 | 2 | 4 | TRUE | 775 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"|n: n/k | style="background:#ffffcc"|2 | |- | 775 | 2 | 4 | TRUE | 387 | 1 | k: k+1 | | |- | 775 | 3 | 9 | TRUE | 258 | 1 | k: k+1 | | |- | 775 | 4 | 16 | TRUE | 193 | 3 | k: k+1 | | |- | 775 | 5 | 25 | TRUE | 155 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 155 | 5 | 25 | TRUE | 31 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 31 | 5 | 25 | TRUE | 6 | 1 | k: k+1 | | |- | 31 | 6 | 36 | FALSE | - | - | - | | style="background:#ffffcc"| 31 |} ==Python code== ‎<syntaxhighlight lang="python" line="1"> ## In a python code, the function definitions precede the code. We need only 1 function: def prime_factorization(N): ## 12=(2**2)*(2**1)*(5**1), or equivalently, 12 = 2**2 * 3 * 5 ## The dictionary, dict, expresses k**v as a "key-value" pair. In other ## words, if we look up the key, k=5, the dictionary will return the ## value, v=1. In other words: ## k**v <=> key**value <=> base**exponent (for the factored form of N) ## Begin function: n=N #--------------------# n will change in the while loop (getting smaller.) dict = {} #--------------# Start with an empty dictionary if n==1: #---------------# The factorization of 1 is a special case. return {1: 1} # In other words 1 = 1**1 (but recall that 1 is not prime.) k = 2 #------------------# Our first candidate for factorizing any integer. while k**2 <= n: #-------# Do the following unless k is "too big": if n % k: #----------# . Equivalent to: "If k is not a factor of n:" k += 1 #---------# .. increase the value of k by 1 on next attempt. else: #--------------# . Equivalent to "If k is a factor of n:" n /= k #---------# .. Replace n by n/k (for next attempt) try: #-----------# .. If k is in the dictionary, this attempts to: dict[k] += 1 # ... increase the value of k by 1. except KeyError: # .. KeyError occurs if k is not in the dictionary dict[k] = 1 # ... Add k to dictionary and set exponent v=1 # end(while)#------------# (exits while k**2 <= n:) # Here I inserted a diagnostic to be discussed later: # Documented code continues: if n > 1: #--------------# It can be shown that n is either 1 or prime. try: #---------------# . Try to enter n into dictionary dict[n] += 1 #---# .. increase exponent v->v+1 if k in dictinary except KeyError: #---# . If n is not in the dictionary, dict[n] = 1 # .. then enter n into dictionary with v=1 return dict #------# python talk for ending function ################ End def prime_factorization | Begin program ########## #import sys again=True while again == 1: text="\nEnter a positive integer you wish to factor into prime numbers:\n" N=int(input(text)) dict=prime_factorization(N) print("\n"+str(int(N))+"=") text="(" for b in dict.keys(): text+=str(int(b))+"**"+str(int(dict.get(b)))+") * (" print(text.rstrip(" * (")+"\n") print("The dictionary is:\n") print(dict) again=int(input("\nType 1 for another integer, 0 to terminate.")) </syntaxhighlight> ==Comments on code== ===Line 15: while k**2 <= n:=== This is an important line because it permits the termination of the search before one might think. I leave it as an exercise for the student to learn why any value of k larger than square root of n will never be the next factor of N. It is also left to the student to understand the fact that the time to find the factors can be cut nearly in half if we modify line 17 and increase k by 2 (instead of 1), and why this larger step cannot be taken until after k=2 has been considered. ===Lines in the code that might seem confusing=== The code uses two cleverness tricks that might seem confusing. I lack the formal training in programming to know whether such cleverness would be perceived as a flaw. I know that programmers are allowed to be clever and I know that clear programs are better than confusing programs. The fact that these tricks confused me does not necessarily mean they should be avoided. And, the nice thing about both sources of confusion is that they taught me something about python. ====Line 16: if n%k might seem backwards==== Python follows a common practice of interpreting the number 1 to be True and 0 to be False. But python also accepts any non-zero number as True. Here n%k denotes nmod&nbsp;k, which is the remainder when on divides n by k. If there is no remainder the "if" condition is not satisfied and the code stipulates that the next value of k is attempted. Any number not equal to zero is interpreted as the "if" condition being satisfied, meaning that k is a divisor of n (in that case, we replace n by n/k.) This might be good coding, but it seems to me that it would have been better to construct it along the lines of "if n%k == 0" or even "if n%k !=0". ====Lines 21-23 and 29-31: A failed attempt to change the value of a key==== In this code fragment, we ask if a given value of k has already been identified as a factor. But instead of "asking" if k has been declared a key in the dictionary, an <u>attempt</u> is made to change the value of k. If that attempt fails (because k is not among the dictionary's keys), then k is added to the list of keys, with a value of 1 (a value of 1 indicates that at this point, k, but not k² has been established to be a factor of n.) Again, there might be nothing wrong with using a key error like this. But the code would have been easier for me to follow if the if statement simply called the question of whether k was a key in dict. ==Sample output (with a really big number)== Sometimes you have to wait a few minutes to factor a number this large. But usually it comes out in just a few seconds. I would imagine that a number that was the product of only two roughly equal prime numbers would take forever because k=k+1 must be repeated a virtually infinite number of times. This number came quickly: {{Font color|blue|Enter a positive integer you wish to factor into prime numbers:}} 99283428376482768916467328912876456416375474272263423467896879187289499287374899 &nbsp; <span style="color:blue>99283428376482768916467328912876456416375474272263423467896879187289499287374899=</span> {{Font color|blue|(619**1) * (688**1) * (768**3) * (1024**17) * (64853**1) * (5302548928**1)}} &nbsp; {{Font color|blue|The dictionary is:}} &nbsp; <span style="color:blue>{619: 1, 688: 1, 768: 3, 1024: 17, 64853: 1, 5302548928.0: 1}</span> &nbsp; {{Font color|blue|Type 1 for another integer, 0 to terminate.}} {{center|'''DO NOT TRY TO FACTOR A NUMBER THIS LARGE UNLESS YOU ARE WILLING TO INTERUPT YOUR PYTHON PROGRAM.'''}} {{center|Keep in mind that the universe is only <math>4.4\times10^{26}\text{nanoseconds}</math> old.<ref>http://astronomy.nmsu.edu/geas/lectures/lecture02/slide04.html</ref>}} The number shown rounds up to 10⁸⁰. It is not likely that any computer will ever count that high. If the number to be factored consists of two prime number that are close to 10⁸⁰, the code will iterate all the way up to k coming close to 10⁴⁰. The inability of any computer to perform that many iterations helps explain how cryptocurrency works. &nbsp; tj1yz8tb9vtqeoh4vn6oz6jzy73nc6t 2410288 2410287 2022-07-29T19:58:59Z Guy vandegrift 813252 wikitext text/x-wiki [[File:Factorization flowchart.svg|thumb|420px|Flowchart for a simple prime factorization code]] '''Topics covered:''' ''prime factorization, flowcharts, python dictionaries'' Here we learn how a [[Python Concepts/Dictionaries|python dictionary]] can reduce an integer into a product of prime numbers. There are a few things the student should know about the code you are about to study: * A serious python programmer would probably prefer to download a package that gives you a function that performs prime factorization. Two promising candidates are [https://pypi.org/project/primefac/ pypi.org/project/primefac] and [https://www.sympy.org/en/index.html www.sympy.org/]. * A simpler version of this program can be found at [[Wikipedia:Trial division#Method]].<ref>A working python script is at [[w:Special:Permalink/1086485306]]</ref>. * Lacking the mental wherewithal to seize upon either of these opportunities, I googled "python prime factorization", found many codes, and selected the [[#Python code|one we shall study]] for two reasons:<ref>The code that follows is discussed at [https://stackoverflow.com/questions/15347174/python-finding-prime-factors stackoverflow question 15347174] and [https://scientific-python-101.readthedocs.io/_downloads/prime_factorization_solution.py scientific-python-101.readthedocs]</ref> :# I wanted a code that counts repeated factors in order to later write the factored form in wikitext with exponents.<ref>In the wikitext version, I had to add a few lines so that exponents of 1 would not appear. Example: <math>12=2^2\cdot 3^1</math> should instead read <math>12=2^2\cdot 3.</math> I ommitted the feature here in order to keep the lesson simple.</ref> :# I didn't understand the python code, and wanted master it in order to fully understand how python dictionaries are used. A good student exercise would be to modify this code so that it runs approximately twice as fast. Instructors could also task students with modifying this code so that the output can be shown in a table that displays each effort to see if any given integer <math>k</math> is a prime factor of <math>N.</math> A version of such a modification created the wikitable shown below. It illustrates the "wasted" efforts to see if 4 and 6 might be prime factors of 1550. And it illustrates the efficiency of dividing a candidate integer into the number <math>n</math>, which soon becomes much smaller than <math>N.</math> {| class="wikitable" style="text-align: center; vertical-align: bottom;" |+ <math>\text{Establishing } 1550=2\cdot 5^2 \cdot 31</math> |- ! n ! k ! k^2 ! k^2<n? ! n/k ! mod ! replace ! factor ! * |- | 1550 | 2 | 4 | TRUE | 775 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"|n: n/k | style="background:#ffffcc"|2 | |- | 775 | 2 | 4 | TRUE | 387 | 1 | k: k+1 | | |- | 775 | 3 | 9 | TRUE | 258 | 1 | k: k+1 | | |- | 775 | 4 | 16 | TRUE | 193 | 3 | k: k+1 | | |- | 775 | 5 | 25 | TRUE | 155 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 155 | 5 | 25 | TRUE | 31 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 31 | 5 | 25 | TRUE | 6 | 1 | k: k+1 | | |- | 31 | 6 | 36 | FALSE | - | - | - | | style="background:#ffffcc"| 31 |} ==Python code== ‎<syntaxhighlight lang="python" line="1"> ## In a python code, the function definitions precede the code. We need only 1 function: def prime_factorization(N): ## 12=(2**2)*(2**1)*(5**1), or equivalently, 12 = 2**2 * 3 * 5 ## The dictionary, dict, expresses k**v as a "key-value" pair. In other ## words, if we look up the key, k=5, the dictionary will return the ## value, v=1. In other words: ## k**v <=> key**value <=> base**exponent (for the factored form of N) ## Begin function: n=N #--------------------# n will change in the while loop (getting smaller.) dict = {} #--------------# Start with an empty dictionary if n==1: #---------------# The factorization of 1 is a special case. return {1: 1} # In other words 1 = 1**1 (but recall that 1 is not prime.) k = 2 #------------------# Our first candidate for factorizing any integer. while k**2 <= n: #-------# Do the following unless k is "too big": if n % k: #----------# . Equivalent to: "If k is not a factor of n:" k += 1 #---------# .. increase the value of k by 1 on next attempt. else: #--------------# . Equivalent to "If k is a factor of n:" n /= k #---------# .. Replace n by n/k (for next attempt) try: #-----------# .. If k is in the dictionary, this attempts to: dict[k] += 1 # ... increase the value of k by 1. except KeyError: # .. KeyError occurs if k is not in the dictionary dict[k] = 1 # ... Add k to dictionary and set exponent v=1 # end(while)#------------# (exits while k**2 <= n:) # Here I inserted a diagnostic to be discussed later: # Documented code continues: if n > 1: #--------------# It can be shown that n is either 1 or prime. try: #---------------# . Try to enter n into dictionary dict[n] += 1 #---# .. increase exponent v->v+1 if k in dictinary except KeyError: #---# . If n is not in the dictionary, dict[n] = 1 # .. then enter n into dictionary with v=1 return dict #------# python talk for ending function ################ End def prime_factorization | Begin program ########## #import sys again=True while again == 1: text="\nEnter a positive integer you wish to factor into prime numbers:\n" N=int(input(text)) dict=prime_factorization(N) print("\n"+str(int(N))+"=") text="(" for b in dict.keys(): text+=str(int(b))+"**"+str(int(dict.get(b)))+") * (" print(text.rstrip(" * (")+"\n") print("The dictionary is:\n") print(dict) again=int(input("\nType 1 for another integer, 0 to terminate.")) </syntaxhighlight> ==Comments on code== ===Line 15: while k**2 <= n:=== This is an important line because it permits the termination of the search before one might think. I leave it as an exercise for the student to learn why any value of k larger than square root of n will never be the next factor of N. It is also left to the student to understand the fact that the time to find the factors can be cut nearly in half if we modify line 17 and increase k by 2 (instead of 1), and why this larger step cannot be taken until after k=2 has been considered. ===Lines in the code that might seem confusing=== The code uses two cleverness tricks that might seem confusing. I lack the formal training in programming to know whether such cleverness would be perceived as a flaw. I know that programmers are allowed to be clever and I know that clear programs are better than confusing programs. The fact that these tricks confused me does not necessarily mean they should be avoided. And, the nice thing about both sources of confusion is that they taught me something about python. ====Line 16: if n%k might seem backwards==== Python follows a common practice of interpreting the number 1 to be True and 0 to be False. But python also accepts any non-zero number as True. Here n%k denotes nmod&nbsp;k, which is the remainder when on divides n by k. If there is no remainder the "if" condition is not satisfied and the code stipulates that the next value of k is attempted. Any number not equal to zero is interpreted as the "if" condition being satisfied, meaning that k is a divisor of n (in that case, we replace n by n/k.) This might be good coding, but it seems to me that it would have been better to construct it along the lines of "if n%k == 0" or even "if n%k !=0". ====Lines 21-23 and 29-31: A failed attempt to change the value of a key==== In this code fragment, we ask if a given value of k has already been identified as a factor. But instead of "asking" if k has been declared a key in the dictionary, an <u>attempt</u> is made to change the value of k. If that attempt fails (because k is not among the dictionary's keys), then k is added to the list of keys, with a value of 1 (a value of 1 indicates that at this point, k, but not k² has been established to be a factor of n.) Again, there might be nothing wrong with using a key error like this. But the code would have been easier for me to follow if the if statement simply called the question of whether k was a key in dict. ==Sample output (with a really big number)== Sometimes you have to wait a few minutes to factor a number this large. But usually it comes out in just a few seconds. I would imagine that a number that was the product of only two roughly equal prime numbers would take forever because k=k+1 must be repeated a virtually infinite number of times. This number came quickly: {{Font color|blue|Enter a positive integer you wish to factor into prime numbers:}} 99283428376482768916467328912876456416375474272263423467896879187289499287374899 &nbsp; <span style="color:blue>99283428376482768916467328912876456416375474272263423467896879187289499287374899=</span> {{Font color|blue|(619**1) * (688**1) * (768**3) * (1024**17) * (64853**1) * (5302548928**1)}} &nbsp; {{Font color|blue|The dictionary is:}} &nbsp; <span style="color:blue>{619: 1, 688: 1, 768: 3, 1024: 17, 64853: 1, 5302548928.0: 1}</span> &nbsp; {{Font color|blue|Type 1 for another integer, 0 to terminate.}} {{center|'''DO NOT TRY TO FACTOR A NUMBER THIS LARGE UNLESS YOU ARE WILLING TO INTERUPT YOUR PYTHON PROGRAM.'''}} {{center|Keep in mind that the universe is only <math>4.4\times10^{26}\text{nanoseconds}</math> old.<ref>http://astronomy.nmsu.edu/geas/lectures/lecture02/slide04.html</ref>}} The number shown rounds up to 10⁸⁰. It is not likely that any computer will ever count that high. If the number to be factored consists of two prime number that are close to 10⁸⁰, the code will iterate all the way up to k coming close to 10⁴⁰. The inability of any computer to perform that many iterations helps explain how cryptocurrency works. &nbsp; p11i3ns40tfmxfpmpuwf28iynr1tat3 2410289 2410288 2022-07-29T19:59:40Z Guy vandegrift 813252 wikitext text/x-wiki [[File:Factorization flowchart.svg|thumb|420px|Flowchart for a simple prime factorization code]] '''Topics covered:''' ''prime factorization, flowcharts, python dictionaries'' Here we learn how a [[Python Concepts/Dictionaries|python dictionary]] can reduce an integer into a product of prime numbers. There are a few things the student should know about the code you are about to study: * A serious python programmer would probably prefer to download a package that gives you a function that performs prime factorization. Two promising candidates are [https://pypi.org/project/primefac/ pypi.org/project/primefac] and [https://www.sympy.org/en/index.html www.sympy.org/]. * A simpler version of this program can be found at [[Wikipedia:Trial division#Method]].<ref>A working python script is at [[w:Special:Permalink/1086485306]]</ref>. * Lacking the mental wherewithal to seize upon either of these opportunities, I googled "python prime factorization", found many codes, and selected the [[#Python code|one we shall examine]] for two reasons:<ref>The code that follows is discussed at [https://stackoverflow.com/questions/15347174/python-finding-prime-factors stackoverflow question 15347174] and [https://scientific-python-101.readthedocs.io/_downloads/prime_factorization_solution.py scientific-python-101.readthedocs]</ref> :# I wanted a code that counts repeated factors in order to later write the factored form in wikitext with exponents.<ref>In the wikitext version, I had to add a few lines so that exponents of 1 would not appear. Example: <math>12=2^2\cdot 3^1</math> should instead read <math>12=2^2\cdot 3.</math> I ommitted the feature here in order to keep the lesson simple.</ref> :# I didn't understand the python code, and wanted master it in order to fully understand how python dictionaries are used. A good student exercise would be to modify this code so that it runs approximately twice as fast. Instructors could also task students with modifying this code so that the output can be shown in a table that displays each effort to see if any given integer <math>k</math> is a prime factor of <math>N.</math> A version of such a modification created the wikitable shown below. It illustrates the "wasted" efforts to see if 4 and 6 might be prime factors of 1550. And it illustrates the efficiency of dividing a candidate integer into the number <math>n</math>, which soon becomes much smaller than <math>N.</math> {| class="wikitable" style="text-align: center; vertical-align: bottom;" |+ <math>\text{Establishing } 1550=2\cdot 5^2 \cdot 31</math> |- ! n ! k ! k^2 ! k^2<n? ! n/k ! mod ! replace ! factor ! * |- | 1550 | 2 | 4 | TRUE | 775 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"|n: n/k | style="background:#ffffcc"|2 | |- | 775 | 2 | 4 | TRUE | 387 | 1 | k: k+1 | | |- | 775 | 3 | 9 | TRUE | 258 | 1 | k: k+1 | | |- | 775 | 4 | 16 | TRUE | 193 | 3 | k: k+1 | | |- | 775 | 5 | 25 | TRUE | 155 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 155 | 5 | 25 | TRUE | 31 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 31 | 5 | 25 | TRUE | 6 | 1 | k: k+1 | | |- | 31 | 6 | 36 | FALSE | - | - | - | | style="background:#ffffcc"| 31 |} ==Python code== ‎<syntaxhighlight lang="python" line="1"> ## In a python code, the function definitions precede the code. We need only 1 function: def prime_factorization(N): ## 12=(2**2)*(2**1)*(5**1), or equivalently, 12 = 2**2 * 3 * 5 ## The dictionary, dict, expresses k**v as a "key-value" pair. In other ## words, if we look up the key, k=5, the dictionary will return the ## value, v=1. In other words: ## k**v <=> key**value <=> base**exponent (for the factored form of N) ## Begin function: n=N #--------------------# n will change in the while loop (getting smaller.) dict = {} #--------------# Start with an empty dictionary if n==1: #---------------# The factorization of 1 is a special case. return {1: 1} # In other words 1 = 1**1 (but recall that 1 is not prime.) k = 2 #------------------# Our first candidate for factorizing any integer. while k**2 <= n: #-------# Do the following unless k is "too big": if n % k: #----------# . Equivalent to: "If k is not a factor of n:" k += 1 #---------# .. increase the value of k by 1 on next attempt. else: #--------------# . Equivalent to "If k is a factor of n:" n /= k #---------# .. Replace n by n/k (for next attempt) try: #-----------# .. If k is in the dictionary, this attempts to: dict[k] += 1 # ... increase the value of k by 1. except KeyError: # .. KeyError occurs if k is not in the dictionary dict[k] = 1 # ... Add k to dictionary and set exponent v=1 # end(while)#------------# (exits while k**2 <= n:) # Here I inserted a diagnostic to be discussed later: # Documented code continues: if n > 1: #--------------# It can be shown that n is either 1 or prime. try: #---------------# . Try to enter n into dictionary dict[n] += 1 #---# .. increase exponent v->v+1 if k in dictinary except KeyError: #---# . If n is not in the dictionary, dict[n] = 1 # .. then enter n into dictionary with v=1 return dict #------# python talk for ending function ################ End def prime_factorization | Begin program ########## #import sys again=True while again == 1: text="\nEnter a positive integer you wish to factor into prime numbers:\n" N=int(input(text)) dict=prime_factorization(N) print("\n"+str(int(N))+"=") text="(" for b in dict.keys(): text+=str(int(b))+"**"+str(int(dict.get(b)))+") * (" print(text.rstrip(" * (")+"\n") print("The dictionary is:\n") print(dict) again=int(input("\nType 1 for another integer, 0 to terminate.")) </syntaxhighlight> ==Comments on code== ===Line 15: while k**2 <= n:=== This is an important line because it permits the termination of the search before one might think. I leave it as an exercise for the student to learn why any value of k larger than square root of n will never be the next factor of N. It is also left to the student to understand the fact that the time to find the factors can be cut nearly in half if we modify line 17 and increase k by 2 (instead of 1), and why this larger step cannot be taken until after k=2 has been considered. ===Lines in the code that might seem confusing=== The code uses two cleverness tricks that might seem confusing. I lack the formal training in programming to know whether such cleverness would be perceived as a flaw. I know that programmers are allowed to be clever and I know that clear programs are better than confusing programs. The fact that these tricks confused me does not necessarily mean they should be avoided. And, the nice thing about both sources of confusion is that they taught me something about python. ====Line 16: if n%k might seem backwards==== Python follows a common practice of interpreting the number 1 to be True and 0 to be False. But python also accepts any non-zero number as True. Here n%k denotes nmod&nbsp;k, which is the remainder when on divides n by k. If there is no remainder the "if" condition is not satisfied and the code stipulates that the next value of k is attempted. Any number not equal to zero is interpreted as the "if" condition being satisfied, meaning that k is a divisor of n (in that case, we replace n by n/k.) This might be good coding, but it seems to me that it would have been better to construct it along the lines of "if n%k == 0" or even "if n%k !=0". ====Lines 21-23 and 29-31: A failed attempt to change the value of a key==== In this code fragment, we ask if a given value of k has already been identified as a factor. But instead of "asking" if k has been declared a key in the dictionary, an <u>attempt</u> is made to change the value of k. If that attempt fails (because k is not among the dictionary's keys), then k is added to the list of keys, with a value of 1 (a value of 1 indicates that at this point, k, but not k² has been established to be a factor of n.) Again, there might be nothing wrong with using a key error like this. But the code would have been easier for me to follow if the if statement simply called the question of whether k was a key in dict. ==Sample output (with a really big number)== Sometimes you have to wait a few minutes to factor a number this large. But usually it comes out in just a few seconds. I would imagine that a number that was the product of only two roughly equal prime numbers would take forever because k=k+1 must be repeated a virtually infinite number of times. This number came quickly: {{Font color|blue|Enter a positive integer you wish to factor into prime numbers:}} 99283428376482768916467328912876456416375474272263423467896879187289499287374899 &nbsp; <span style="color:blue>99283428376482768916467328912876456416375474272263423467896879187289499287374899=</span> {{Font color|blue|(619**1) * (688**1) * (768**3) * (1024**17) * (64853**1) * (5302548928**1)}} &nbsp; {{Font color|blue|The dictionary is:}} &nbsp; <span style="color:blue>{619: 1, 688: 1, 768: 3, 1024: 17, 64853: 1, 5302548928.0: 1}</span> &nbsp; {{Font color|blue|Type 1 for another integer, 0 to terminate.}} {{center|'''DO NOT TRY TO FACTOR A NUMBER THIS LARGE UNLESS YOU ARE WILLING TO INTERUPT YOUR PYTHON PROGRAM.'''}} {{center|Keep in mind that the universe is only <math>4.4\times10^{26}\text{nanoseconds}</math> old.<ref>http://astronomy.nmsu.edu/geas/lectures/lecture02/slide04.html</ref>}} The number shown rounds up to 10⁸⁰. It is not likely that any computer will ever count that high. If the number to be factored consists of two prime number that are close to 10⁸⁰, the code will iterate all the way up to k coming close to 10⁴⁰. The inability of any computer to perform that many iterations helps explain how cryptocurrency works. &nbsp; h8br2hu23x8yoc87u1ibgrsr2b9lh12 2410290 2410289 2022-07-29T20:00:16Z Guy vandegrift 813252 wikitext text/x-wiki [[File:Factorization flowchart.svg|thumb|420px|Flowchart for a simple prime factorization code]] '''Topics covered:''' ''prime factorization, flowcharts, python dictionaries'' Here we learn how a [[Python Concepts/Dictionaries|python dictionary]] can help express an integer as a product of prime numbers. There are a few things the student should know about the code you are about to study: * A serious python programmer would probably prefer to download a package that gives you a function that performs prime factorization. Two promising candidates are [https://pypi.org/project/primefac/ pypi.org/project/primefac] and [https://www.sympy.org/en/index.html www.sympy.org/]. * A simpler version of this program can be found at [[Wikipedia:Trial division#Method]].<ref>A working python script is at [[w:Special:Permalink/1086485306]]</ref>. * Lacking the mental wherewithal to seize upon either of these opportunities, I googled "python prime factorization", found many codes, and selected the [[#Python code|one we shall examine]] for two reasons:<ref>The code that follows is discussed at [https://stackoverflow.com/questions/15347174/python-finding-prime-factors stackoverflow question 15347174] and [https://scientific-python-101.readthedocs.io/_downloads/prime_factorization_solution.py scientific-python-101.readthedocs]</ref> :# I wanted a code that counts repeated factors in order to later write the factored form in wikitext with exponents.<ref>In the wikitext version, I had to add a few lines so that exponents of 1 would not appear. Example: <math>12=2^2\cdot 3^1</math> should instead read <math>12=2^2\cdot 3.</math> I ommitted the feature here in order to keep the lesson simple.</ref> :# I didn't understand the python code, and wanted master it in order to fully understand how python dictionaries are used. A good student exercise would be to modify this code so that it runs approximately twice as fast. Instructors could also task students with modifying this code so that the output can be shown in a table that displays each effort to see if any given integer <math>k</math> is a prime factor of <math>N.</math> A version of such a modification created the wikitable shown below. It illustrates the "wasted" efforts to see if 4 and 6 might be prime factors of 1550. And it illustrates the efficiency of dividing a candidate integer into the number <math>n</math>, which soon becomes much smaller than <math>N.</math> {| class="wikitable" style="text-align: center; vertical-align: bottom;" |+ <math>\text{Establishing } 1550=2\cdot 5^2 \cdot 31</math> |- ! n ! k ! k^2 ! k^2<n? ! n/k ! mod ! replace ! factor ! * |- | 1550 | 2 | 4 | TRUE | 775 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"|n: n/k | style="background:#ffffcc"|2 | |- | 775 | 2 | 4 | TRUE | 387 | 1 | k: k+1 | | |- | 775 | 3 | 9 | TRUE | 258 | 1 | k: k+1 | | |- | 775 | 4 | 16 | TRUE | 193 | 3 | k: k+1 | | |- | 775 | 5 | 25 | TRUE | 155 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 155 | 5 | 25 | TRUE | 31 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 31 | 5 | 25 | TRUE | 6 | 1 | k: k+1 | | |- | 31 | 6 | 36 | FALSE | - | - | - | | style="background:#ffffcc"| 31 |} ==Python code== ‎<syntaxhighlight lang="python" line="1"> ## In a python code, the function definitions precede the code. We need only 1 function: def prime_factorization(N): ## 12=(2**2)*(2**1)*(5**1), or equivalently, 12 = 2**2 * 3 * 5 ## The dictionary, dict, expresses k**v as a "key-value" pair. In other ## words, if we look up the key, k=5, the dictionary will return the ## value, v=1. In other words: ## k**v <=> key**value <=> base**exponent (for the factored form of N) ## Begin function: n=N #--------------------# n will change in the while loop (getting smaller.) dict = {} #--------------# Start with an empty dictionary if n==1: #---------------# The factorization of 1 is a special case. return {1: 1} # In other words 1 = 1**1 (but recall that 1 is not prime.) k = 2 #------------------# Our first candidate for factorizing any integer. while k**2 <= n: #-------# Do the following unless k is "too big": if n % k: #----------# . Equivalent to: "If k is not a factor of n:" k += 1 #---------# .. increase the value of k by 1 on next attempt. else: #--------------# . Equivalent to "If k is a factor of n:" n /= k #---------# .. Replace n by n/k (for next attempt) try: #-----------# .. If k is in the dictionary, this attempts to: dict[k] += 1 # ... increase the value of k by 1. except KeyError: # .. KeyError occurs if k is not in the dictionary dict[k] = 1 # ... Add k to dictionary and set exponent v=1 # end(while)#------------# (exits while k**2 <= n:) # Here I inserted a diagnostic to be discussed later: # Documented code continues: if n > 1: #--------------# It can be shown that n is either 1 or prime. try: #---------------# . Try to enter n into dictionary dict[n] += 1 #---# .. increase exponent v->v+1 if k in dictinary except KeyError: #---# . If n is not in the dictionary, dict[n] = 1 # .. then enter n into dictionary with v=1 return dict #------# python talk for ending function ################ End def prime_factorization | Begin program ########## #import sys again=True while again == 1: text="\nEnter a positive integer you wish to factor into prime numbers:\n" N=int(input(text)) dict=prime_factorization(N) print("\n"+str(int(N))+"=") text="(" for b in dict.keys(): text+=str(int(b))+"**"+str(int(dict.get(b)))+") * (" print(text.rstrip(" * (")+"\n") print("The dictionary is:\n") print(dict) again=int(input("\nType 1 for another integer, 0 to terminate.")) </syntaxhighlight> ==Comments on code== ===Line 15: while k**2 <= n:=== This is an important line because it permits the termination of the search before one might think. I leave it as an exercise for the student to learn why any value of k larger than square root of n will never be the next factor of N. It is also left to the student to understand the fact that the time to find the factors can be cut nearly in half if we modify line 17 and increase k by 2 (instead of 1), and why this larger step cannot be taken until after k=2 has been considered. ===Lines in the code that might seem confusing=== The code uses two cleverness tricks that might seem confusing. I lack the formal training in programming to know whether such cleverness would be perceived as a flaw. I know that programmers are allowed to be clever and I know that clear programs are better than confusing programs. The fact that these tricks confused me does not necessarily mean they should be avoided. And, the nice thing about both sources of confusion is that they taught me something about python. ====Line 16: if n%k might seem backwards==== Python follows a common practice of interpreting the number 1 to be True and 0 to be False. But python also accepts any non-zero number as True. Here n%k denotes nmod&nbsp;k, which is the remainder when on divides n by k. If there is no remainder the "if" condition is not satisfied and the code stipulates that the next value of k is attempted. Any number not equal to zero is interpreted as the "if" condition being satisfied, meaning that k is a divisor of n (in that case, we replace n by n/k.) This might be good coding, but it seems to me that it would have been better to construct it along the lines of "if n%k == 0" or even "if n%k !=0". ====Lines 21-23 and 29-31: A failed attempt to change the value of a key==== In this code fragment, we ask if a given value of k has already been identified as a factor. But instead of "asking" if k has been declared a key in the dictionary, an <u>attempt</u> is made to change the value of k. If that attempt fails (because k is not among the dictionary's keys), then k is added to the list of keys, with a value of 1 (a value of 1 indicates that at this point, k, but not k² has been established to be a factor of n.) Again, there might be nothing wrong with using a key error like this. But the code would have been easier for me to follow if the if statement simply called the question of whether k was a key in dict. ==Sample output (with a really big number)== Sometimes you have to wait a few minutes to factor a number this large. But usually it comes out in just a few seconds. I would imagine that a number that was the product of only two roughly equal prime numbers would take forever because k=k+1 must be repeated a virtually infinite number of times. This number came quickly: {{Font color|blue|Enter a positive integer you wish to factor into prime numbers:}} 99283428376482768916467328912876456416375474272263423467896879187289499287374899 &nbsp; <span style="color:blue>99283428376482768916467328912876456416375474272263423467896879187289499287374899=</span> {{Font color|blue|(619**1) * (688**1) * (768**3) * (1024**17) * (64853**1) * (5302548928**1)}} &nbsp; {{Font color|blue|The dictionary is:}} &nbsp; <span style="color:blue>{619: 1, 688: 1, 768: 3, 1024: 17, 64853: 1, 5302548928.0: 1}</span> &nbsp; {{Font color|blue|Type 1 for another integer, 0 to terminate.}} {{center|'''DO NOT TRY TO FACTOR A NUMBER THIS LARGE UNLESS YOU ARE WILLING TO INTERUPT YOUR PYTHON PROGRAM.'''}} {{center|Keep in mind that the universe is only <math>4.4\times10^{26}\text{nanoseconds}</math> old.<ref>http://astronomy.nmsu.edu/geas/lectures/lecture02/slide04.html</ref>}} The number shown rounds up to 10⁸⁰. It is not likely that any computer will ever count that high. If the number to be factored consists of two prime number that are close to 10⁸⁰, the code will iterate all the way up to k coming close to 10⁴⁰. The inability of any computer to perform that many iterations helps explain how cryptocurrency works. &nbsp; iahd3jxm3198wzn1eejaujl2ddis2q0 2410291 2410290 2022-07-29T20:01:25Z Guy vandegrift 813252 wikitext text/x-wiki [[File:Factorization flowchart.svg|thumb|420px|Flowchart for a simple prime factorization code]] '''Topics covered:''' ''prime factorization, flowcharts, python dictionaries'' Here we learn how a [[Python Concepts/Dictionaries|python dictionary]] can help express an integer as a [[w:Integer factorization|product of prime numbers]]. There are a few things the student should know about the code you are about to study: * A serious python programmer would probably prefer to download a package that gives you a function that performs prime factorization. Two promising candidates are [https://pypi.org/project/primefac/ pypi.org/project/primefac] and [https://www.sympy.org/en/index.html www.sympy.org/]. * A simpler version of this program can be found at [[Wikipedia:Trial division#Method]].<ref>A working python script is at [[w:Special:Permalink/1086485306]]</ref>. * Lacking the mental wherewithal to seize upon either of these opportunities, I googled "python prime factorization", found many codes, and selected the [[#Python code|one we shall examine]] for two reasons:<ref>The code that follows is discussed at [https://stackoverflow.com/questions/15347174/python-finding-prime-factors stackoverflow question 15347174] and [https://scientific-python-101.readthedocs.io/_downloads/prime_factorization_solution.py scientific-python-101.readthedocs]</ref> :# I wanted a code that counts repeated factors in order to later write the factored form in wikitext with exponents.<ref>In the wikitext version, I had to add a few lines so that exponents of 1 would not appear. Example: <math>12=2^2\cdot 3^1</math> should instead read <math>12=2^2\cdot 3.</math> I ommitted the feature here in order to keep the lesson simple.</ref> :# I didn't understand the python code, and wanted master it in order to fully understand how python dictionaries are used. A good student exercise would be to modify this code so that it runs approximately twice as fast. Instructors could also task students with modifying this code so that the output can be shown in a table that displays each effort to see if any given integer <math>k</math> is a prime factor of <math>N.</math> A version of such a modification created the wikitable shown below. It illustrates the "wasted" efforts to see if 4 and 6 might be prime factors of 1550. And it illustrates the efficiency of dividing a candidate integer into the number <math>n</math>, which soon becomes much smaller than <math>N.</math> {| class="wikitable" style="text-align: center; vertical-align: bottom;" |+ <math>\text{Establishing } 1550=2\cdot 5^2 \cdot 31</math> |- ! n ! k ! k^2 ! k^2<n? ! n/k ! mod ! replace ! factor ! * |- | 1550 | 2 | 4 | TRUE | 775 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"|n: n/k | style="background:#ffffcc"|2 | |- | 775 | 2 | 4 | TRUE | 387 | 1 | k: k+1 | | |- | 775 | 3 | 9 | TRUE | 258 | 1 | k: k+1 | | |- | 775 | 4 | 16 | TRUE | 193 | 3 | k: k+1 | | |- | 775 | 5 | 25 | TRUE | 155 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 155 | 5 | 25 | TRUE | 31 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 31 | 5 | 25 | TRUE | 6 | 1 | k: k+1 | | |- | 31 | 6 | 36 | FALSE | - | - | - | | style="background:#ffffcc"| 31 |} ==Python code== ‎<syntaxhighlight lang="python" line="1"> ## In a python code, the function definitions precede the code. We need only 1 function: def prime_factorization(N): ## 12=(2**2)*(2**1)*(5**1), or equivalently, 12 = 2**2 * 3 * 5 ## The dictionary, dict, expresses k**v as a "key-value" pair. In other ## words, if we look up the key, k=5, the dictionary will return the ## value, v=1. In other words: ## k**v <=> key**value <=> base**exponent (for the factored form of N) ## Begin function: n=N #--------------------# n will change in the while loop (getting smaller.) dict = {} #--------------# Start with an empty dictionary if n==1: #---------------# The factorization of 1 is a special case. return {1: 1} # In other words 1 = 1**1 (but recall that 1 is not prime.) k = 2 #------------------# Our first candidate for factorizing any integer. while k**2 <= n: #-------# Do the following unless k is "too big": if n % k: #----------# . Equivalent to: "If k is not a factor of n:" k += 1 #---------# .. increase the value of k by 1 on next attempt. else: #--------------# . Equivalent to "If k is a factor of n:" n /= k #---------# .. Replace n by n/k (for next attempt) try: #-----------# .. If k is in the dictionary, this attempts to: dict[k] += 1 # ... increase the value of k by 1. except KeyError: # .. KeyError occurs if k is not in the dictionary dict[k] = 1 # ... Add k to dictionary and set exponent v=1 # end(while)#------------# (exits while k**2 <= n:) # Here I inserted a diagnostic to be discussed later: # Documented code continues: if n > 1: #--------------# It can be shown that n is either 1 or prime. try: #---------------# . Try to enter n into dictionary dict[n] += 1 #---# .. increase exponent v->v+1 if k in dictinary except KeyError: #---# . If n is not in the dictionary, dict[n] = 1 # .. then enter n into dictionary with v=1 return dict #------# python talk for ending function ################ End def prime_factorization | Begin program ########## #import sys again=True while again == 1: text="\nEnter a positive integer you wish to factor into prime numbers:\n" N=int(input(text)) dict=prime_factorization(N) print("\n"+str(int(N))+"=") text="(" for b in dict.keys(): text+=str(int(b))+"**"+str(int(dict.get(b)))+") * (" print(text.rstrip(" * (")+"\n") print("The dictionary is:\n") print(dict) again=int(input("\nType 1 for another integer, 0 to terminate.")) </syntaxhighlight> ==Comments on code== ===Line 15: while k**2 <= n:=== This is an important line because it permits the termination of the search before one might think. I leave it as an exercise for the student to learn why any value of k larger than square root of n will never be the next factor of N. It is also left to the student to understand the fact that the time to find the factors can be cut nearly in half if we modify line 17 and increase k by 2 (instead of 1), and why this larger step cannot be taken until after k=2 has been considered. ===Lines in the code that might seem confusing=== The code uses two cleverness tricks that might seem confusing. I lack the formal training in programming to know whether such cleverness would be perceived as a flaw. I know that programmers are allowed to be clever and I know that clear programs are better than confusing programs. The fact that these tricks confused me does not necessarily mean they should be avoided. And, the nice thing about both sources of confusion is that they taught me something about python. ====Line 16: if n%k might seem backwards==== Python follows a common practice of interpreting the number 1 to be True and 0 to be False. But python also accepts any non-zero number as True. Here n%k denotes nmod&nbsp;k, which is the remainder when on divides n by k. If there is no remainder the "if" condition is not satisfied and the code stipulates that the next value of k is attempted. Any number not equal to zero is interpreted as the "if" condition being satisfied, meaning that k is a divisor of n (in that case, we replace n by n/k.) This might be good coding, but it seems to me that it would have been better to construct it along the lines of "if n%k == 0" or even "if n%k !=0". ====Lines 21-23 and 29-31: A failed attempt to change the value of a key==== In this code fragment, we ask if a given value of k has already been identified as a factor. But instead of "asking" if k has been declared a key in the dictionary, an <u>attempt</u> is made to change the value of k. If that attempt fails (because k is not among the dictionary's keys), then k is added to the list of keys, with a value of 1 (a value of 1 indicates that at this point, k, but not k² has been established to be a factor of n.) Again, there might be nothing wrong with using a key error like this. But the code would have been easier for me to follow if the if statement simply called the question of whether k was a key in dict. ==Sample output (with a really big number)== Sometimes you have to wait a few minutes to factor a number this large. But usually it comes out in just a few seconds. I would imagine that a number that was the product of only two roughly equal prime numbers would take forever because k=k+1 must be repeated a virtually infinite number of times. This number came quickly: {{Font color|blue|Enter a positive integer you wish to factor into prime numbers:}} 99283428376482768916467328912876456416375474272263423467896879187289499287374899 &nbsp; <span style="color:blue>99283428376482768916467328912876456416375474272263423467896879187289499287374899=</span> {{Font color|blue|(619**1) * (688**1) * (768**3) * (1024**17) * (64853**1) * (5302548928**1)}} &nbsp; {{Font color|blue|The dictionary is:}} &nbsp; <span style="color:blue>{619: 1, 688: 1, 768: 3, 1024: 17, 64853: 1, 5302548928.0: 1}</span> &nbsp; {{Font color|blue|Type 1 for another integer, 0 to terminate.}} {{center|'''DO NOT TRY TO FACTOR A NUMBER THIS LARGE UNLESS YOU ARE WILLING TO INTERUPT YOUR PYTHON PROGRAM.'''}} {{center|Keep in mind that the universe is only <math>4.4\times10^{26}\text{nanoseconds}</math> old.<ref>http://astronomy.nmsu.edu/geas/lectures/lecture02/slide04.html</ref>}} The number shown rounds up to 10⁸⁰. It is not likely that any computer will ever count that high. If the number to be factored consists of two prime number that are close to 10⁸⁰, the code will iterate all the way up to k coming close to 10⁴⁰. The inability of any computer to perform that many iterations helps explain how cryptocurrency works. &nbsp; jvl1ngkmuaj80upd148u0bvgqqnib7k 2410292 2410291 2022-07-29T20:01:51Z Guy vandegrift 813252 wikitext text/x-wiki [[File:Factorization flowchart.svg|thumb|420px|Flowchart for a simple prime factorization code]] '''Topics covered:''' ''prime factorization, flowcharts, python dictionaries'' Here we learn how a [[Python Concepts/Dictionaries|python dictionary]] can help express an integer as a [[w:Integer factorization|product of prime numbers]]. There are a few things the student should know about the code we are about to examine: * A serious python programmer would probably prefer to download a package that gives you a function that performs prime factorization. Two promising candidates are [https://pypi.org/project/primefac/ pypi.org/project/primefac] and [https://www.sympy.org/en/index.html www.sympy.org/]. * A simpler version of this program can be found at [[Wikipedia:Trial division#Method]].<ref>A working python script is at [[w:Special:Permalink/1086485306]]</ref>. * Lacking the mental wherewithal to seize upon either of these opportunities, I googled "python prime factorization", found many codes, and selected the [[#Python code|one we shall examine]] for two reasons:<ref>The code that follows is discussed at [https://stackoverflow.com/questions/15347174/python-finding-prime-factors stackoverflow question 15347174] and [https://scientific-python-101.readthedocs.io/_downloads/prime_factorization_solution.py scientific-python-101.readthedocs]</ref> :# I wanted a code that counts repeated factors in order to later write the factored form in wikitext with exponents.<ref>In the wikitext version, I had to add a few lines so that exponents of 1 would not appear. Example: <math>12=2^2\cdot 3^1</math> should instead read <math>12=2^2\cdot 3.</math> I ommitted the feature here in order to keep the lesson simple.</ref> :# I didn't understand the python code, and wanted master it in order to fully understand how python dictionaries are used. A good student exercise would be to modify this code so that it runs approximately twice as fast. Instructors could also task students with modifying this code so that the output can be shown in a table that displays each effort to see if any given integer <math>k</math> is a prime factor of <math>N.</math> A version of such a modification created the wikitable shown below. It illustrates the "wasted" efforts to see if 4 and 6 might be prime factors of 1550. And it illustrates the efficiency of dividing a candidate integer into the number <math>n</math>, which soon becomes much smaller than <math>N.</math> {| class="wikitable" style="text-align: center; vertical-align: bottom;" |+ <math>\text{Establishing } 1550=2\cdot 5^2 \cdot 31</math> |- ! n ! k ! k^2 ! k^2<n? ! n/k ! mod ! replace ! factor ! * |- | 1550 | 2 | 4 | TRUE | 775 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"|n: n/k | style="background:#ffffcc"|2 | |- | 775 | 2 | 4 | TRUE | 387 | 1 | k: k+1 | | |- | 775 | 3 | 9 | TRUE | 258 | 1 | k: k+1 | | |- | 775 | 4 | 16 | TRUE | 193 | 3 | k: k+1 | | |- | 775 | 5 | 25 | TRUE | 155 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 155 | 5 | 25 | TRUE | 31 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 31 | 5 | 25 | TRUE | 6 | 1 | k: k+1 | | |- | 31 | 6 | 36 | FALSE | - | - | - | | style="background:#ffffcc"| 31 |} ==Python code== ‎<syntaxhighlight lang="python" line="1"> ## In a python code, the function definitions precede the code. We need only 1 function: def prime_factorization(N): ## 12=(2**2)*(2**1)*(5**1), or equivalently, 12 = 2**2 * 3 * 5 ## The dictionary, dict, expresses k**v as a "key-value" pair. In other ## words, if we look up the key, k=5, the dictionary will return the ## value, v=1. In other words: ## k**v <=> key**value <=> base**exponent (for the factored form of N) ## Begin function: n=N #--------------------# n will change in the while loop (getting smaller.) dict = {} #--------------# Start with an empty dictionary if n==1: #---------------# The factorization of 1 is a special case. return {1: 1} # In other words 1 = 1**1 (but recall that 1 is not prime.) k = 2 #------------------# Our first candidate for factorizing any integer. while k**2 <= n: #-------# Do the following unless k is "too big": if n % k: #----------# . Equivalent to: "If k is not a factor of n:" k += 1 #---------# .. increase the value of k by 1 on next attempt. else: #--------------# . Equivalent to "If k is a factor of n:" n /= k #---------# .. Replace n by n/k (for next attempt) try: #-----------# .. If k is in the dictionary, this attempts to: dict[k] += 1 # ... increase the value of k by 1. except KeyError: # .. KeyError occurs if k is not in the dictionary dict[k] = 1 # ... Add k to dictionary and set exponent v=1 # end(while)#------------# (exits while k**2 <= n:) # Here I inserted a diagnostic to be discussed later: # Documented code continues: if n > 1: #--------------# It can be shown that n is either 1 or prime. try: #---------------# . Try to enter n into dictionary dict[n] += 1 #---# .. increase exponent v->v+1 if k in dictinary except KeyError: #---# . If n is not in the dictionary, dict[n] = 1 # .. then enter n into dictionary with v=1 return dict #------# python talk for ending function ################ End def prime_factorization | Begin program ########## #import sys again=True while again == 1: text="\nEnter a positive integer you wish to factor into prime numbers:\n" N=int(input(text)) dict=prime_factorization(N) print("\n"+str(int(N))+"=") text="(" for b in dict.keys(): text+=str(int(b))+"**"+str(int(dict.get(b)))+") * (" print(text.rstrip(" * (")+"\n") print("The dictionary is:\n") print(dict) again=int(input("\nType 1 for another integer, 0 to terminate.")) </syntaxhighlight> ==Comments on code== ===Line 15: while k**2 <= n:=== This is an important line because it permits the termination of the search before one might think. I leave it as an exercise for the student to learn why any value of k larger than square root of n will never be the next factor of N. It is also left to the student to understand the fact that the time to find the factors can be cut nearly in half if we modify line 17 and increase k by 2 (instead of 1), and why this larger step cannot be taken until after k=2 has been considered. ===Lines in the code that might seem confusing=== The code uses two cleverness tricks that might seem confusing. I lack the formal training in programming to know whether such cleverness would be perceived as a flaw. I know that programmers are allowed to be clever and I know that clear programs are better than confusing programs. The fact that these tricks confused me does not necessarily mean they should be avoided. And, the nice thing about both sources of confusion is that they taught me something about python. ====Line 16: if n%k might seem backwards==== Python follows a common practice of interpreting the number 1 to be True and 0 to be False. But python also accepts any non-zero number as True. Here n%k denotes nmod&nbsp;k, which is the remainder when on divides n by k. If there is no remainder the "if" condition is not satisfied and the code stipulates that the next value of k is attempted. Any number not equal to zero is interpreted as the "if" condition being satisfied, meaning that k is a divisor of n (in that case, we replace n by n/k.) This might be good coding, but it seems to me that it would have been better to construct it along the lines of "if n%k == 0" or even "if n%k !=0". ====Lines 21-23 and 29-31: A failed attempt to change the value of a key==== In this code fragment, we ask if a given value of k has already been identified as a factor. But instead of "asking" if k has been declared a key in the dictionary, an <u>attempt</u> is made to change the value of k. If that attempt fails (because k is not among the dictionary's keys), then k is added to the list of keys, with a value of 1 (a value of 1 indicates that at this point, k, but not k² has been established to be a factor of n.) Again, there might be nothing wrong with using a key error like this. But the code would have been easier for me to follow if the if statement simply called the question of whether k was a key in dict. ==Sample output (with a really big number)== Sometimes you have to wait a few minutes to factor a number this large. But usually it comes out in just a few seconds. I would imagine that a number that was the product of only two roughly equal prime numbers would take forever because k=k+1 must be repeated a virtually infinite number of times. This number came quickly: {{Font color|blue|Enter a positive integer you wish to factor into prime numbers:}} 99283428376482768916467328912876456416375474272263423467896879187289499287374899 &nbsp; <span style="color:blue>99283428376482768916467328912876456416375474272263423467896879187289499287374899=</span> {{Font color|blue|(619**1) * (688**1) * (768**3) * (1024**17) * (64853**1) * (5302548928**1)}} &nbsp; {{Font color|blue|The dictionary is:}} &nbsp; <span style="color:blue>{619: 1, 688: 1, 768: 3, 1024: 17, 64853: 1, 5302548928.0: 1}</span> &nbsp; {{Font color|blue|Type 1 for another integer, 0 to terminate.}} {{center|'''DO NOT TRY TO FACTOR A NUMBER THIS LARGE UNLESS YOU ARE WILLING TO INTERUPT YOUR PYTHON PROGRAM.'''}} {{center|Keep in mind that the universe is only <math>4.4\times10^{26}\text{nanoseconds}</math> old.<ref>http://astronomy.nmsu.edu/geas/lectures/lecture02/slide04.html</ref>}} The number shown rounds up to 10⁸⁰. It is not likely that any computer will ever count that high. If the number to be factored consists of two prime number that are close to 10⁸⁰, the code will iterate all the way up to k coming close to 10⁴⁰. The inability of any computer to perform that many iterations helps explain how cryptocurrency works. &nbsp; mmla9b99mvz73y9x1ocfs4816j8ixar 2410293 2410292 2022-07-29T20:03:51Z Guy vandegrift 813252 wikitext text/x-wiki [[File:Factorization flowchart.svg|thumb|420px|Flowchart for a simple prime factorization code]] '''Topics covered:''' ''prime factorization, flowcharts, python dictionaries'' Here we learn how a [[Python Concepts/Dictionaries|python dictionary]] can help express an integer as a [[w:Integer factorization|product of prime numbers]]. There are a few things the student should know about the code we are about to examine: * A serious python programmer would probably prefer to download a package that gives you a function that performs prime factorization. Two promising candidates are [https://pypi.org/project/primefac/ pypi.org/project/primefac] and [https://www.sympy.org/en/index.html www.sympy.org/]. * A simpler version of this same program does not use the dictionary.<ref>Outside of Python, these dictionaries are also called [[w:Associative lists]].</ref> It can be found at [[Wikipedia:Trial division#Method]].<ref>A working python script is at [[w:Special:Permalink/1086485306]]</ref>. * Lacking the mental wherewithal to seize upon either of these opportunities, I googled "python prime factorization", found many codes, and selected the [[#Python code|one we shall examine]] for two reasons:<ref>The code that follows is discussed at [https://stackoverflow.com/questions/15347174/python-finding-prime-factors stackoverflow question 15347174] and [https://scientific-python-101.readthedocs.io/_downloads/prime_factorization_solution.py scientific-python-101.readthedocs]</ref> :# I wanted a code that counts repeated factors in order to later write the factored form in wikitext with exponents.<ref>In the wikitext version, I had to add a few lines so that exponents of 1 would not appear. Example: <math>12=2^2\cdot 3^1</math> should instead read <math>12=2^2\cdot 3.</math> I ommitted the feature here in order to keep the lesson simple.</ref> :# I didn't understand the python code, and wanted master it in order to fully understand how python dictionaries are used. A good student exercise would be to modify this code so that it runs approximately twice as fast. Instructors could also task students with modifying this code so that the output can be shown in a table that displays each effort to see if any given integer <math>k</math> is a prime factor of <math>N.</math> A version of such a modification created the wikitable shown below. It illustrates the "wasted" efforts to see if 4 and 6 might be prime factors of 1550. And it illustrates the efficiency of dividing a candidate integer into the number <math>n</math>, which soon becomes much smaller than <math>N.</math> {| class="wikitable" style="text-align: center; vertical-align: bottom;" |+ <math>\text{Establishing } 1550=2\cdot 5^2 \cdot 31</math> |- ! n ! k ! k^2 ! k^2<n? ! n/k ! mod ! replace ! factor ! * |- | 1550 | 2 | 4 | TRUE | 775 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"|n: n/k | style="background:#ffffcc"|2 | |- | 775 | 2 | 4 | TRUE | 387 | 1 | k: k+1 | | |- | 775 | 3 | 9 | TRUE | 258 | 1 | k: k+1 | | |- | 775 | 4 | 16 | TRUE | 193 | 3 | k: k+1 | | |- | 775 | 5 | 25 | TRUE | 155 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 155 | 5 | 25 | TRUE | 31 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 31 | 5 | 25 | TRUE | 6 | 1 | k: k+1 | | |- | 31 | 6 | 36 | FALSE | - | - | - | | style="background:#ffffcc"| 31 |} ==Python code== ‎<syntaxhighlight lang="python" line="1"> ## In a python code, the function definitions precede the code. We need only 1 function: def prime_factorization(N): ## 12=(2**2)*(2**1)*(5**1), or equivalently, 12 = 2**2 * 3 * 5 ## The dictionary, dict, expresses k**v as a "key-value" pair. In other ## words, if we look up the key, k=5, the dictionary will return the ## value, v=1. In other words: ## k**v <=> key**value <=> base**exponent (for the factored form of N) ## Begin function: n=N #--------------------# n will change in the while loop (getting smaller.) dict = {} #--------------# Start with an empty dictionary if n==1: #---------------# The factorization of 1 is a special case. return {1: 1} # In other words 1 = 1**1 (but recall that 1 is not prime.) k = 2 #------------------# Our first candidate for factorizing any integer. while k**2 <= n: #-------# Do the following unless k is "too big": if n % k: #----------# . Equivalent to: "If k is not a factor of n:" k += 1 #---------# .. increase the value of k by 1 on next attempt. else: #--------------# . Equivalent to "If k is a factor of n:" n /= k #---------# .. Replace n by n/k (for next attempt) try: #-----------# .. If k is in the dictionary, this attempts to: dict[k] += 1 # ... increase the value of k by 1. except KeyError: # .. KeyError occurs if k is not in the dictionary dict[k] = 1 # ... Add k to dictionary and set exponent v=1 # end(while)#------------# (exits while k**2 <= n:) # Here I inserted a diagnostic to be discussed later: # Documented code continues: if n > 1: #--------------# It can be shown that n is either 1 or prime. try: #---------------# . Try to enter n into dictionary dict[n] += 1 #---# .. increase exponent v->v+1 if k in dictinary except KeyError: #---# . If n is not in the dictionary, dict[n] = 1 # .. then enter n into dictionary with v=1 return dict #------# python talk for ending function ################ End def prime_factorization | Begin program ########## #import sys again=True while again == 1: text="\nEnter a positive integer you wish to factor into prime numbers:\n" N=int(input(text)) dict=prime_factorization(N) print("\n"+str(int(N))+"=") text="(" for b in dict.keys(): text+=str(int(b))+"**"+str(int(dict.get(b)))+") * (" print(text.rstrip(" * (")+"\n") print("The dictionary is:\n") print(dict) again=int(input("\nType 1 for another integer, 0 to terminate.")) </syntaxhighlight> ==Comments on code== ===Line 15: while k**2 <= n:=== This is an important line because it permits the termination of the search before one might think. I leave it as an exercise for the student to learn why any value of k larger than square root of n will never be the next factor of N. It is also left to the student to understand the fact that the time to find the factors can be cut nearly in half if we modify line 17 and increase k by 2 (instead of 1), and why this larger step cannot be taken until after k=2 has been considered. ===Lines in the code that might seem confusing=== The code uses two cleverness tricks that might seem confusing. I lack the formal training in programming to know whether such cleverness would be perceived as a flaw. I know that programmers are allowed to be clever and I know that clear programs are better than confusing programs. The fact that these tricks confused me does not necessarily mean they should be avoided. And, the nice thing about both sources of confusion is that they taught me something about python. ====Line 16: if n%k might seem backwards==== Python follows a common practice of interpreting the number 1 to be True and 0 to be False. But python also accepts any non-zero number as True. Here n%k denotes nmod&nbsp;k, which is the remainder when on divides n by k. If there is no remainder the "if" condition is not satisfied and the code stipulates that the next value of k is attempted. Any number not equal to zero is interpreted as the "if" condition being satisfied, meaning that k is a divisor of n (in that case, we replace n by n/k.) This might be good coding, but it seems to me that it would have been better to construct it along the lines of "if n%k == 0" or even "if n%k !=0". ====Lines 21-23 and 29-31: A failed attempt to change the value of a key==== In this code fragment, we ask if a given value of k has already been identified as a factor. But instead of "asking" if k has been declared a key in the dictionary, an <u>attempt</u> is made to change the value of k. If that attempt fails (because k is not among the dictionary's keys), then k is added to the list of keys, with a value of 1 (a value of 1 indicates that at this point, k, but not k² has been established to be a factor of n.) Again, there might be nothing wrong with using a key error like this. But the code would have been easier for me to follow if the if statement simply called the question of whether k was a key in dict. ==Sample output (with a really big number)== Sometimes you have to wait a few minutes to factor a number this large. But usually it comes out in just a few seconds. I would imagine that a number that was the product of only two roughly equal prime numbers would take forever because k=k+1 must be repeated a virtually infinite number of times. This number came quickly: {{Font color|blue|Enter a positive integer you wish to factor into prime numbers:}} 99283428376482768916467328912876456416375474272263423467896879187289499287374899 &nbsp; <span style="color:blue>99283428376482768916467328912876456416375474272263423467896879187289499287374899=</span> {{Font color|blue|(619**1) * (688**1) * (768**3) * (1024**17) * (64853**1) * (5302548928**1)}} &nbsp; {{Font color|blue|The dictionary is:}} &nbsp; <span style="color:blue>{619: 1, 688: 1, 768: 3, 1024: 17, 64853: 1, 5302548928.0: 1}</span> &nbsp; {{Font color|blue|Type 1 for another integer, 0 to terminate.}} {{center|'''DO NOT TRY TO FACTOR A NUMBER THIS LARGE UNLESS YOU ARE WILLING TO INTERUPT YOUR PYTHON PROGRAM.'''}} {{center|Keep in mind that the universe is only <math>4.4\times10^{26}\text{nanoseconds}</math> old.<ref>http://astronomy.nmsu.edu/geas/lectures/lecture02/slide04.html</ref>}} The number shown rounds up to 10⁸⁰. It is not likely that any computer will ever count that high. If the number to be factored consists of two prime number that are close to 10⁸⁰, the code will iterate all the way up to k coming close to 10⁴⁰. The inability of any computer to perform that many iterations helps explain how cryptocurrency works. &nbsp; gf1g70zsj4twp1onlibq08i8kt7srkw 2410296 2410293 2022-07-29T20:06:36Z Guy vandegrift 813252 wikitext text/x-wiki [[File:Factorization flowchart.svg|thumb|420px|Flowchart for a simple prime factorization code]] '''Topics covered:''' ''prime factorization, flowcharts, python dictionaries'' Here we learn how a [[Python Concepts/Dictionaries|python dictionary]] can help express an integer as a [[w:Integer factorization|product of prime numbers]]. There are a few things the student should know about the code we are about to examine: * A serious python programmer would probably prefer to download a package that gives you a function that performs prime factorization. Two promising candidates are [https://pypi.org/project/primefac/ pypi.org/project/primefac] and [https://www.sympy.org/en/index.html www.sympy.org/]. * A simpler version of this same program does not use the "dictionary".<ref>Outside of Python a "dictionary" is also called an [[w:Associative array]].</ref> It can be found at [[Wikipedia:Trial division#Method]].<ref>A working python script is at [[w:Special:Permalink/1086485306]]</ref>. * Lacking the mental wherewithal to seize upon either of these opportunities, I googled "python prime factorization", found many codes, and selected the [[#Python code|one we shall examine]] for two reasons:<ref>The code that follows is discussed at [https://stackoverflow.com/questions/15347174/python-finding-prime-factors stackoverflow question 15347174] and [https://scientific-python-101.readthedocs.io/_downloads/prime_factorization_solution.py scientific-python-101.readthedocs]</ref> :# I wanted a code that counts repeated factors in order to later write the factored form in wikitext with exponents.<ref>In the wikitext version, I had to add a few lines so that exponents of 1 would not appear. Example: <math>12=2^2\cdot 3^1</math> should instead read <math>12=2^2\cdot 3.</math> I ommitted the feature here in order to keep the lesson simple.</ref> :# I didn't understand the python code, and wanted master it in order to fully understand how python dictionaries are used. A good student exercise would be to modify this code so that it runs approximately twice as fast. Instructors could also task students with modifying this code so that the output can be shown in a table that displays each effort to see if any given integer <math>k</math> is a prime factor of <math>N.</math> A version of such a modification created the wikitable shown below. It illustrates the "wasted" efforts to see if 4 and 6 might be prime factors of 1550. And it illustrates the efficiency of dividing a candidate integer into the number <math>n</math>, which soon becomes much smaller than <math>N.</math> {| class="wikitable" style="text-align: center; vertical-align: bottom;" |+ <math>\text{Establishing } 1550=2\cdot 5^2 \cdot 31</math> |- ! n ! k ! k^2 ! k^2<n? ! n/k ! mod ! replace ! factor ! * |- | 1550 | 2 | 4 | TRUE | 775 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"|n: n/k | style="background:#ffffcc"|2 | |- | 775 | 2 | 4 | TRUE | 387 | 1 | k: k+1 | | |- | 775 | 3 | 9 | TRUE | 258 | 1 | k: k+1 | | |- | 775 | 4 | 16 | TRUE | 193 | 3 | k: k+1 | | |- | 775 | 5 | 25 | TRUE | 155 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 155 | 5 | 25 | TRUE | 31 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 31 | 5 | 25 | TRUE | 6 | 1 | k: k+1 | | |- | 31 | 6 | 36 | FALSE | - | - | - | | style="background:#ffffcc"| 31 |} ==Python code== ‎<syntaxhighlight lang="python" line="1"> ## In a python code, the function definitions precede the code. We need only 1 function: def prime_factorization(N): ## 12=(2**2)*(2**1)*(5**1), or equivalently, 12 = 2**2 * 3 * 5 ## The dictionary, dict, expresses k**v as a "key-value" pair. In other ## words, if we look up the key, k=5, the dictionary will return the ## value, v=1. In other words: ## k**v <=> key**value <=> base**exponent (for the factored form of N) ## Begin function: n=N #--------------------# n will change in the while loop (getting smaller.) dict = {} #--------------# Start with an empty dictionary if n==1: #---------------# The factorization of 1 is a special case. return {1: 1} # In other words 1 = 1**1 (but recall that 1 is not prime.) k = 2 #------------------# Our first candidate for factorizing any integer. while k**2 <= n: #-------# Do the following unless k is "too big": if n % k: #----------# . Equivalent to: "If k is not a factor of n:" k += 1 #---------# .. increase the value of k by 1 on next attempt. else: #--------------# . Equivalent to "If k is a factor of n:" n /= k #---------# .. Replace n by n/k (for next attempt) try: #-----------# .. If k is in the dictionary, this attempts to: dict[k] += 1 # ... increase the value of k by 1. except KeyError: # .. KeyError occurs if k is not in the dictionary dict[k] = 1 # ... Add k to dictionary and set exponent v=1 # end(while)#------------# (exits while k**2 <= n:) # Here I inserted a diagnostic to be discussed later: # Documented code continues: if n > 1: #--------------# It can be shown that n is either 1 or prime. try: #---------------# . Try to enter n into dictionary dict[n] += 1 #---# .. increase exponent v->v+1 if k in dictinary except KeyError: #---# . If n is not in the dictionary, dict[n] = 1 # .. then enter n into dictionary with v=1 return dict #------# python talk for ending function ################ End def prime_factorization | Begin program ########## #import sys again=True while again == 1: text="\nEnter a positive integer you wish to factor into prime numbers:\n" N=int(input(text)) dict=prime_factorization(N) print("\n"+str(int(N))+"=") text="(" for b in dict.keys(): text+=str(int(b))+"**"+str(int(dict.get(b)))+") * (" print(text.rstrip(" * (")+"\n") print("The dictionary is:\n") print(dict) again=int(input("\nType 1 for another integer, 0 to terminate.")) </syntaxhighlight> ==Comments on code== ===Line 15: while k**2 <= n:=== This is an important line because it permits the termination of the search before one might think. I leave it as an exercise for the student to learn why any value of k larger than square root of n will never be the next factor of N. It is also left to the student to understand the fact that the time to find the factors can be cut nearly in half if we modify line 17 and increase k by 2 (instead of 1), and why this larger step cannot be taken until after k=2 has been considered. ===Lines in the code that might seem confusing=== The code uses two cleverness tricks that might seem confusing. I lack the formal training in programming to know whether such cleverness would be perceived as a flaw. I know that programmers are allowed to be clever and I know that clear programs are better than confusing programs. The fact that these tricks confused me does not necessarily mean they should be avoided. And, the nice thing about both sources of confusion is that they taught me something about python. ====Line 16: if n%k might seem backwards==== Python follows a common practice of interpreting the number 1 to be True and 0 to be False. But python also accepts any non-zero number as True. Here n%k denotes nmod&nbsp;k, which is the remainder when on divides n by k. If there is no remainder the "if" condition is not satisfied and the code stipulates that the next value of k is attempted. Any number not equal to zero is interpreted as the "if" condition being satisfied, meaning that k is a divisor of n (in that case, we replace n by n/k.) This might be good coding, but it seems to me that it would have been better to construct it along the lines of "if n%k == 0" or even "if n%k !=0". ====Lines 21-23 and 29-31: A failed attempt to change the value of a key==== In this code fragment, we ask if a given value of k has already been identified as a factor. But instead of "asking" if k has been declared a key in the dictionary, an <u>attempt</u> is made to change the value of k. If that attempt fails (because k is not among the dictionary's keys), then k is added to the list of keys, with a value of 1 (a value of 1 indicates that at this point, k, but not k² has been established to be a factor of n.) Again, there might be nothing wrong with using a key error like this. But the code would have been easier for me to follow if the if statement simply called the question of whether k was a key in dict. ==Sample output (with a really big number)== Sometimes you have to wait a few minutes to factor a number this large. But usually it comes out in just a few seconds. I would imagine that a number that was the product of only two roughly equal prime numbers would take forever because k=k+1 must be repeated a virtually infinite number of times. This number came quickly: {{Font color|blue|Enter a positive integer you wish to factor into prime numbers:}} 99283428376482768916467328912876456416375474272263423467896879187289499287374899 &nbsp; <span style="color:blue>99283428376482768916467328912876456416375474272263423467896879187289499287374899=</span> {{Font color|blue|(619**1) * (688**1) * (768**3) * (1024**17) * (64853**1) * (5302548928**1)}} &nbsp; {{Font color|blue|The dictionary is:}} &nbsp; <span style="color:blue>{619: 1, 688: 1, 768: 3, 1024: 17, 64853: 1, 5302548928.0: 1}</span> &nbsp; {{Font color|blue|Type 1 for another integer, 0 to terminate.}} {{center|'''DO NOT TRY TO FACTOR A NUMBER THIS LARGE UNLESS YOU ARE WILLING TO INTERUPT YOUR PYTHON PROGRAM.'''}} {{center|Keep in mind that the universe is only <math>4.4\times10^{26}\text{nanoseconds}</math> old.<ref>http://astronomy.nmsu.edu/geas/lectures/lecture02/slide04.html</ref>}} The number shown rounds up to 10⁸⁰. It is not likely that any computer will ever count that high. If the number to be factored consists of two prime number that are close to 10⁸⁰, the code will iterate all the way up to k coming close to 10⁴⁰. The inability of any computer to perform that many iterations helps explain how cryptocurrency works. &nbsp; fa0l4kbt8683nz0c6bz1sg5m6yhw7mb 2410299 2410296 2022-07-29T20:08:09Z Guy vandegrift 813252 /* Comments on code */ wikitext text/x-wiki [[File:Factorization flowchart.svg|thumb|420px|Flowchart for a simple prime factorization code]] '''Topics covered:''' ''prime factorization, flowcharts, python dictionaries'' Here we learn how a [[Python Concepts/Dictionaries|python dictionary]] can help express an integer as a [[w:Integer factorization|product of prime numbers]]. There are a few things the student should know about the code we are about to examine: * A serious python programmer would probably prefer to download a package that gives you a function that performs prime factorization. Two promising candidates are [https://pypi.org/project/primefac/ pypi.org/project/primefac] and [https://www.sympy.org/en/index.html www.sympy.org/]. * A simpler version of this same program does not use the "dictionary".<ref>Outside of Python a "dictionary" is also called an [[w:Associative array]].</ref> It can be found at [[Wikipedia:Trial division#Method]].<ref>A working python script is at [[w:Special:Permalink/1086485306]]</ref>. * Lacking the mental wherewithal to seize upon either of these opportunities, I googled "python prime factorization", found many codes, and selected the [[#Python code|one we shall examine]] for two reasons:<ref>The code that follows is discussed at [https://stackoverflow.com/questions/15347174/python-finding-prime-factors stackoverflow question 15347174] and [https://scientific-python-101.readthedocs.io/_downloads/prime_factorization_solution.py scientific-python-101.readthedocs]</ref> :# I wanted a code that counts repeated factors in order to later write the factored form in wikitext with exponents.<ref>In the wikitext version, I had to add a few lines so that exponents of 1 would not appear. Example: <math>12=2^2\cdot 3^1</math> should instead read <math>12=2^2\cdot 3.</math> I ommitted the feature here in order to keep the lesson simple.</ref> :# I didn't understand the python code, and wanted master it in order to fully understand how python dictionaries are used. A good student exercise would be to modify this code so that it runs approximately twice as fast. Instructors could also task students with modifying this code so that the output can be shown in a table that displays each effort to see if any given integer <math>k</math> is a prime factor of <math>N.</math> A version of such a modification created the wikitable shown below. It illustrates the "wasted" efforts to see if 4 and 6 might be prime factors of 1550. And it illustrates the efficiency of dividing a candidate integer into the number <math>n</math>, which soon becomes much smaller than <math>N.</math> {| class="wikitable" style="text-align: center; vertical-align: bottom;" |+ <math>\text{Establishing } 1550=2\cdot 5^2 \cdot 31</math> |- ! n ! k ! k^2 ! k^2<n? ! n/k ! mod ! replace ! factor ! * |- | 1550 | 2 | 4 | TRUE | 775 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"|n: n/k | style="background:#ffffcc"|2 | |- | 775 | 2 | 4 | TRUE | 387 | 1 | k: k+1 | | |- | 775 | 3 | 9 | TRUE | 258 | 1 | k: k+1 | | |- | 775 | 4 | 16 | TRUE | 193 | 3 | k: k+1 | | |- | 775 | 5 | 25 | TRUE | 155 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 155 | 5 | 25 | TRUE | 31 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 31 | 5 | 25 | TRUE | 6 | 1 | k: k+1 | | |- | 31 | 6 | 36 | FALSE | - | - | - | | style="background:#ffffcc"| 31 |} ==Python code== ‎<syntaxhighlight lang="python" line="1"> ## In a python code, the function definitions precede the code. We need only 1 function: def prime_factorization(N): ## 12=(2**2)*(2**1)*(5**1), or equivalently, 12 = 2**2 * 3 * 5 ## The dictionary, dict, expresses k**v as a "key-value" pair. In other ## words, if we look up the key, k=5, the dictionary will return the ## value, v=1. In other words: ## k**v <=> key**value <=> base**exponent (for the factored form of N) ## Begin function: n=N #--------------------# n will change in the while loop (getting smaller.) dict = {} #--------------# Start with an empty dictionary if n==1: #---------------# The factorization of 1 is a special case. return {1: 1} # In other words 1 = 1**1 (but recall that 1 is not prime.) k = 2 #------------------# Our first candidate for factorizing any integer. while k**2 <= n: #-------# Do the following unless k is "too big": if n % k: #----------# . Equivalent to: "If k is not a factor of n:" k += 1 #---------# .. increase the value of k by 1 on next attempt. else: #--------------# . Equivalent to "If k is a factor of n:" n /= k #---------# .. Replace n by n/k (for next attempt) try: #-----------# .. If k is in the dictionary, this attempts to: dict[k] += 1 # ... increase the value of k by 1. except KeyError: # .. KeyError occurs if k is not in the dictionary dict[k] = 1 # ... Add k to dictionary and set exponent v=1 # end(while)#------------# (exits while k**2 <= n:) # Here I inserted a diagnostic to be discussed later: # Documented code continues: if n > 1: #--------------# It can be shown that n is either 1 or prime. try: #---------------# . Try to enter n into dictionary dict[n] += 1 #---# .. increase exponent v->v+1 if k in dictinary except KeyError: #---# . If n is not in the dictionary, dict[n] = 1 # .. then enter n into dictionary with v=1 return dict #------# python talk for ending function ################ End def prime_factorization | Begin program ########## #import sys again=True while again == 1: text="\nEnter a positive integer you wish to factor into prime numbers:\n" N=int(input(text)) dict=prime_factorization(N) print("\n"+str(int(N))+"=") text="(" for b in dict.keys(): text+=str(int(b))+"**"+str(int(dict.get(b)))+") * (" print(text.rstrip(" * (")+"\n") print("The dictionary is:\n") print(dict) again=int(input("\nType 1 for another integer, 0 to terminate.")) </syntaxhighlight> ===Comments=== ===Line 15: while k**2 <= n:=== This is an important line because it permits the termination of the search before one might think. I leave it as an exercise for the student to learn why any value of k larger than square root of n will never be the next factor of N. It is also left to the student to understand the fact that the time to find the factors can be cut nearly in half if we modify line 17 and increase k by 2 (instead of 1), and why this larger step cannot be taken until after k=2 has been considered. ====Lines in the code that might seem confusing==== The code uses two cleverness tricks that might seem confusing. I lack the formal training in programming to know whether such cleverness would be perceived as a flaw. I know that programmers are allowed to be clever and I know that clear programs are better than confusing programs. The fact that these tricks confused me does not necessarily mean they should be avoided. And, the nice thing about both sources of confusion is that they taught me something about python. =====Line 16: if n%k might seem backwards===== Python follows a common practice of interpreting the number 1 to be True and 0 to be False. But python also accepts any non-zero number as True. Here n%k denotes nmod&nbsp;k, which is the remainder when on divides n by k. If there is no remainder the "if" condition is not satisfied and the code stipulates that the next value of k is attempted. Any number not equal to zero is interpreted as the "if" condition being satisfied, meaning that k is a divisor of n (in that case, we replace n by n/k.) This might be good coding, but it seems to me that it would have been better to construct it along the lines of "if n%k == 0" or even "if n%k !=0". =====Lines 21-23 and 29-31: A failed attempt to change the value of a key===== In this code fragment, we ask if a given value of k has already been identified as a factor. But instead of "asking" if k has been declared a key in the dictionary, an <u>attempt</u> is made to change the value of k. If that attempt fails (because k is not among the dictionary's keys), then k is added to the list of keys, with a value of 1 (a value of 1 indicates that at this point, k, but not k² has been established to be a factor of n.) Again, there might be nothing wrong with using a key error like this. But the code would have been easier for me to follow if the if statement simply called the question of whether k was a key in dict. ==Sample output (with a really big number)== Sometimes you have to wait a few minutes to factor a number this large. But usually it comes out in just a few seconds. I would imagine that a number that was the product of only two roughly equal prime numbers would take forever because k=k+1 must be repeated a virtually infinite number of times. This number came quickly: {{Font color|blue|Enter a positive integer you wish to factor into prime numbers:}} 99283428376482768916467328912876456416375474272263423467896879187289499287374899 &nbsp; <span style="color:blue>99283428376482768916467328912876456416375474272263423467896879187289499287374899=</span> {{Font color|blue|(619**1) * (688**1) * (768**3) * (1024**17) * (64853**1) * (5302548928**1)}} &nbsp; {{Font color|blue|The dictionary is:}} &nbsp; <span style="color:blue>{619: 1, 688: 1, 768: 3, 1024: 17, 64853: 1, 5302548928.0: 1}</span> &nbsp; {{Font color|blue|Type 1 for another integer, 0 to terminate.}} {{center|'''DO NOT TRY TO FACTOR A NUMBER THIS LARGE UNLESS YOU ARE WILLING TO INTERUPT YOUR PYTHON PROGRAM.'''}} {{center|Keep in mind that the universe is only <math>4.4\times10^{26}\text{nanoseconds}</math> old.<ref>http://astronomy.nmsu.edu/geas/lectures/lecture02/slide04.html</ref>}} The number shown rounds up to 10⁸⁰. It is not likely that any computer will ever count that high. If the number to be factored consists of two prime number that are close to 10⁸⁰, the code will iterate all the way up to k coming close to 10⁴⁰. The inability of any computer to perform that many iterations helps explain how cryptocurrency works. &nbsp; tdmb3cecx361w9qpejxzb3wuj5dqxbg 2410300 2410299 2022-07-29T20:08:54Z Guy vandegrift 813252 /* Python code */ wikitext text/x-wiki [[File:Factorization flowchart.svg|thumb|420px|Flowchart for a simple prime factorization code]] '''Topics covered:''' ''prime factorization, flowcharts, python dictionaries'' Here we learn how a [[Python Concepts/Dictionaries|python dictionary]] can help express an integer as a [[w:Integer factorization|product of prime numbers]]. There are a few things the student should know about the code we are about to examine: * A serious python programmer would probably prefer to download a package that gives you a function that performs prime factorization. Two promising candidates are [https://pypi.org/project/primefac/ pypi.org/project/primefac] and [https://www.sympy.org/en/index.html www.sympy.org/]. * A simpler version of this same program does not use the "dictionary".<ref>Outside of Python a "dictionary" is also called an [[w:Associative array]].</ref> It can be found at [[Wikipedia:Trial division#Method]].<ref>A working python script is at [[w:Special:Permalink/1086485306]]</ref>. * Lacking the mental wherewithal to seize upon either of these opportunities, I googled "python prime factorization", found many codes, and selected the [[#Python code|one we shall examine]] for two reasons:<ref>The code that follows is discussed at [https://stackoverflow.com/questions/15347174/python-finding-prime-factors stackoverflow question 15347174] and [https://scientific-python-101.readthedocs.io/_downloads/prime_factorization_solution.py scientific-python-101.readthedocs]</ref> :# I wanted a code that counts repeated factors in order to later write the factored form in wikitext with exponents.<ref>In the wikitext version, I had to add a few lines so that exponents of 1 would not appear. Example: <math>12=2^2\cdot 3^1</math> should instead read <math>12=2^2\cdot 3.</math> I ommitted the feature here in order to keep the lesson simple.</ref> :# I didn't understand the python code, and wanted master it in order to fully understand how python dictionaries are used. A good student exercise would be to modify this code so that it runs approximately twice as fast. Instructors could also task students with modifying this code so that the output can be shown in a table that displays each effort to see if any given integer <math>k</math> is a prime factor of <math>N.</math> A version of such a modification created the wikitable shown below. It illustrates the "wasted" efforts to see if 4 and 6 might be prime factors of 1550. And it illustrates the efficiency of dividing a candidate integer into the number <math>n</math>, which soon becomes much smaller than <math>N.</math> {| class="wikitable" style="text-align: center; vertical-align: bottom;" |+ <math>\text{Establishing } 1550=2\cdot 5^2 \cdot 31</math> |- ! n ! k ! k^2 ! k^2<n? ! n/k ! mod ! replace ! factor ! * |- | 1550 | 2 | 4 | TRUE | 775 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"|n: n/k | style="background:#ffffcc"|2 | |- | 775 | 2 | 4 | TRUE | 387 | 1 | k: k+1 | | |- | 775 | 3 | 9 | TRUE | 258 | 1 | k: k+1 | | |- | 775 | 4 | 16 | TRUE | 193 | 3 | k: k+1 | | |- | 775 | 5 | 25 | TRUE | 155 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 155 | 5 | 25 | TRUE | 31 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 31 | 5 | 25 | TRUE | 6 | 1 | k: k+1 | | |- | 31 | 6 | 36 | FALSE | - | - | - | | style="background:#ffffcc"| 31 |} ==Python code== ‎<syntaxhighlight lang="python" line="1"> ## In a python code, the function definitions precede the code. We need only 1 function: def prime_factorization(N): ## 12=(2**2)*(2**1)*(5**1), or equivalently, 12 = 2**2 * 3 * 5 ## The dictionary, dict, expresses k**v as a "key-value" pair. In other ## words, if we look up the key, k=5, the dictionary will return the ## value, v=1. In other words: ## k**v <=> key**value <=> base**exponent (for the factored form of N) ## Begin function: n=N #--------------------# n will change in the while loop (getting smaller.) dict = {} #--------------# Start with an empty dictionary if n==1: #---------------# The factorization of 1 is a special case. return {1: 1} # In other words 1 = 1**1 (but recall that 1 is not prime.) k = 2 #------------------# Our first candidate for factorizing any integer. while k**2 <= n: #-------# Do the following unless k is "too big": if n % k: #----------# . Equivalent to: "If k is not a factor of n:" k += 1 #---------# .. increase the value of k by 1 on next attempt. else: #--------------# . Equivalent to "If k is a factor of n:" n /= k #---------# .. Replace n by n/k (for next attempt) try: #-----------# .. If k is in the dictionary, this attempts to: dict[k] += 1 # ... increase the value of k by 1. except KeyError: # .. KeyError occurs if k is not in the dictionary dict[k] = 1 # ... Add k to dictionary and set exponent v=1 # end(while)#------------# (exits while k**2 <= n:) # Here I inserted a diagnostic to be discussed later: # Documented code continues: if n > 1: #--------------# It can be shown that n is either 1 or prime. try: #---------------# . Try to enter n into dictionary dict[n] += 1 #---# .. increase exponent v->v+1 if k in dictinary except KeyError: #---# . If n is not in the dictionary, dict[n] = 1 # .. then enter n into dictionary with v=1 return dict #------# python talk for ending function ################ End def prime_factorization | Begin program ########## #import sys again=True while again == 1: text="\nEnter a positive integer you wish to factor into prime numbers:\n" N=int(input(text)) dict=prime_factorization(N) print("\n"+str(int(N))+"=") text="(" for b in dict.keys(): text+=str(int(b))+"**"+str(int(dict.get(b)))+") * (" print(text.rstrip(" * (")+"\n") print("The dictionary is:\n") print(dict) again=int(input("\nType 1 for another integer, 0 to terminate.")) </syntaxhighlight> ===Comments=== ====Line 15: while k**2 <= n:==== This is an important line because it permits the termination of the search before one might think. I leave it as an exercise for the student to learn why any value of k larger than square root of n will never be the next factor of N. It is also left to the student to understand the fact that the time to find the factors can be cut nearly in half if we modify line 17 and increase k by 2 (instead of 1), and why this larger step cannot be taken until after k=2 has been considered. ====Lines in the code that might seem confusing==== The code uses two cleverness tricks that might seem confusing. I lack the formal training in programming to know whether such cleverness would be perceived as a flaw. I know that programmers are allowed to be clever and I know that clear programs are better than confusing programs. The fact that these tricks confused me does not necessarily mean they should be avoided. And, the nice thing about both sources of confusion is that they taught me something about python. ====Line 16: if n%k might seem backwards==== Python follows a common practice of interpreting the number 1 to be True and 0 to be False. But python also accepts any non-zero number as True. Here n%k denotes nmod&nbsp;k, which is the remainder when on divides n by k. If there is no remainder the "if" condition is not satisfied and the code stipulates that the next value of k is attempted. Any number not equal to zero is interpreted as the "if" condition being satisfied, meaning that k is a divisor of n (in that case, we replace n by n/k.) This might be good coding, but it seems to me that it would have been better to construct it along the lines of "if n%k == 0" or even "if n%k !=0". ====Lines 21-23 and 29-31: A failed attempt to change the value of a key==== In this code fragment, we ask if a given value of k has already been identified as a factor. But instead of "asking" if k has been declared a key in the dictionary, an <u>attempt</u> is made to change the value of k. If that attempt fails (because k is not among the dictionary's keys), then k is added to the list of keys, with a value of 1 (a value of 1 indicates that at this point, k, but not k² has been established to be a factor of n.) Again, there might be nothing wrong with using a key error like this. But the code would have been easier for me to follow if the if statement simply called the question of whether k was a key in dict. ==Sample output (with a really big number)== Sometimes you have to wait a few minutes to factor a number this large. But usually it comes out in just a few seconds. I would imagine that a number that was the product of only two roughly equal prime numbers would take forever because k=k+1 must be repeated a virtually infinite number of times. This number came quickly: {{Font color|blue|Enter a positive integer you wish to factor into prime numbers:}} 99283428376482768916467328912876456416375474272263423467896879187289499287374899 &nbsp; <span style="color:blue>99283428376482768916467328912876456416375474272263423467896879187289499287374899=</span> {{Font color|blue|(619**1) * (688**1) * (768**3) * (1024**17) * (64853**1) * (5302548928**1)}} &nbsp; {{Font color|blue|The dictionary is:}} &nbsp; <span style="color:blue>{619: 1, 688: 1, 768: 3, 1024: 17, 64853: 1, 5302548928.0: 1}</span> &nbsp; {{Font color|blue|Type 1 for another integer, 0 to terminate.}} {{center|'''DO NOT TRY TO FACTOR A NUMBER THIS LARGE UNLESS YOU ARE WILLING TO INTERUPT YOUR PYTHON PROGRAM.'''}} {{center|Keep in mind that the universe is only <math>4.4\times10^{26}\text{nanoseconds}</math> old.<ref>http://astronomy.nmsu.edu/geas/lectures/lecture02/slide04.html</ref>}} The number shown rounds up to 10⁸⁰. It is not likely that any computer will ever count that high. If the number to be factored consists of two prime number that are close to 10⁸⁰, the code will iterate all the way up to k coming close to 10⁴⁰. The inability of any computer to perform that many iterations helps explain how cryptocurrency works. &nbsp; pjx3rqejei2jekpksuvrw0g15pyg0nk 2410301 2410300 2022-07-29T20:09:20Z Guy vandegrift 813252 /* Sample output (with a really big number) */ wikitext text/x-wiki [[File:Factorization flowchart.svg|thumb|420px|Flowchart for a simple prime factorization code]] '''Topics covered:''' ''prime factorization, flowcharts, python dictionaries'' Here we learn how a [[Python Concepts/Dictionaries|python dictionary]] can help express an integer as a [[w:Integer factorization|product of prime numbers]]. There are a few things the student should know about the code we are about to examine: * A serious python programmer would probably prefer to download a package that gives you a function that performs prime factorization. Two promising candidates are [https://pypi.org/project/primefac/ pypi.org/project/primefac] and [https://www.sympy.org/en/index.html www.sympy.org/]. * A simpler version of this same program does not use the "dictionary".<ref>Outside of Python a "dictionary" is also called an [[w:Associative array]].</ref> It can be found at [[Wikipedia:Trial division#Method]].<ref>A working python script is at [[w:Special:Permalink/1086485306]]</ref>. * Lacking the mental wherewithal to seize upon either of these opportunities, I googled "python prime factorization", found many codes, and selected the [[#Python code|one we shall examine]] for two reasons:<ref>The code that follows is discussed at [https://stackoverflow.com/questions/15347174/python-finding-prime-factors stackoverflow question 15347174] and [https://scientific-python-101.readthedocs.io/_downloads/prime_factorization_solution.py scientific-python-101.readthedocs]</ref> :# I wanted a code that counts repeated factors in order to later write the factored form in wikitext with exponents.<ref>In the wikitext version, I had to add a few lines so that exponents of 1 would not appear. Example: <math>12=2^2\cdot 3^1</math> should instead read <math>12=2^2\cdot 3.</math> I ommitted the feature here in order to keep the lesson simple.</ref> :# I didn't understand the python code, and wanted master it in order to fully understand how python dictionaries are used. A good student exercise would be to modify this code so that it runs approximately twice as fast. Instructors could also task students with modifying this code so that the output can be shown in a table that displays each effort to see if any given integer <math>k</math> is a prime factor of <math>N.</math> A version of such a modification created the wikitable shown below. It illustrates the "wasted" efforts to see if 4 and 6 might be prime factors of 1550. And it illustrates the efficiency of dividing a candidate integer into the number <math>n</math>, which soon becomes much smaller than <math>N.</math> {| class="wikitable" style="text-align: center; vertical-align: bottom;" |+ <math>\text{Establishing } 1550=2\cdot 5^2 \cdot 31</math> |- ! n ! k ! k^2 ! k^2<n? ! n/k ! mod ! replace ! factor ! * |- | 1550 | 2 | 4 | TRUE | 775 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"|n: n/k | style="background:#ffffcc"|2 | |- | 775 | 2 | 4 | TRUE | 387 | 1 | k: k+1 | | |- | 775 | 3 | 9 | TRUE | 258 | 1 | k: k+1 | | |- | 775 | 4 | 16 | TRUE | 193 | 3 | k: k+1 | | |- | 775 | 5 | 25 | TRUE | 155 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 155 | 5 | 25 | TRUE | 31 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 31 | 5 | 25 | TRUE | 6 | 1 | k: k+1 | | |- | 31 | 6 | 36 | FALSE | - | - | - | | style="background:#ffffcc"| 31 |} ==Python code== ‎<syntaxhighlight lang="python" line="1"> ## In a python code, the function definitions precede the code. We need only 1 function: def prime_factorization(N): ## 12=(2**2)*(2**1)*(5**1), or equivalently, 12 = 2**2 * 3 * 5 ## The dictionary, dict, expresses k**v as a "key-value" pair. In other ## words, if we look up the key, k=5, the dictionary will return the ## value, v=1. In other words: ## k**v <=> key**value <=> base**exponent (for the factored form of N) ## Begin function: n=N #--------------------# n will change in the while loop (getting smaller.) dict = {} #--------------# Start with an empty dictionary if n==1: #---------------# The factorization of 1 is a special case. return {1: 1} # In other words 1 = 1**1 (but recall that 1 is not prime.) k = 2 #------------------# Our first candidate for factorizing any integer. while k**2 <= n: #-------# Do the following unless k is "too big": if n % k: #----------# . Equivalent to: "If k is not a factor of n:" k += 1 #---------# .. increase the value of k by 1 on next attempt. else: #--------------# . Equivalent to "If k is a factor of n:" n /= k #---------# .. Replace n by n/k (for next attempt) try: #-----------# .. If k is in the dictionary, this attempts to: dict[k] += 1 # ... increase the value of k by 1. except KeyError: # .. KeyError occurs if k is not in the dictionary dict[k] = 1 # ... Add k to dictionary and set exponent v=1 # end(while)#------------# (exits while k**2 <= n:) # Here I inserted a diagnostic to be discussed later: # Documented code continues: if n > 1: #--------------# It can be shown that n is either 1 or prime. try: #---------------# . Try to enter n into dictionary dict[n] += 1 #---# .. increase exponent v->v+1 if k in dictinary except KeyError: #---# . If n is not in the dictionary, dict[n] = 1 # .. then enter n into dictionary with v=1 return dict #------# python talk for ending function ################ End def prime_factorization | Begin program ########## #import sys again=True while again == 1: text="\nEnter a positive integer you wish to factor into prime numbers:\n" N=int(input(text)) dict=prime_factorization(N) print("\n"+str(int(N))+"=") text="(" for b in dict.keys(): text+=str(int(b))+"**"+str(int(dict.get(b)))+") * (" print(text.rstrip(" * (")+"\n") print("The dictionary is:\n") print(dict) again=int(input("\nType 1 for another integer, 0 to terminate.")) </syntaxhighlight> ===Comments=== ====Line 15: while k**2 <= n:==== This is an important line because it permits the termination of the search before one might think. I leave it as an exercise for the student to learn why any value of k larger than square root of n will never be the next factor of N. It is also left to the student to understand the fact that the time to find the factors can be cut nearly in half if we modify line 17 and increase k by 2 (instead of 1), and why this larger step cannot be taken until after k=2 has been considered. ====Lines in the code that might seem confusing==== The code uses two cleverness tricks that might seem confusing. I lack the formal training in programming to know whether such cleverness would be perceived as a flaw. I know that programmers are allowed to be clever and I know that clear programs are better than confusing programs. The fact that these tricks confused me does not necessarily mean they should be avoided. And, the nice thing about both sources of confusion is that they taught me something about python. ====Line 16: if n%k might seem backwards==== Python follows a common practice of interpreting the number 1 to be True and 0 to be False. But python also accepts any non-zero number as True. Here n%k denotes nmod&nbsp;k, which is the remainder when on divides n by k. If there is no remainder the "if" condition is not satisfied and the code stipulates that the next value of k is attempted. Any number not equal to zero is interpreted as the "if" condition being satisfied, meaning that k is a divisor of n (in that case, we replace n by n/k.) This might be good coding, but it seems to me that it would have been better to construct it along the lines of "if n%k == 0" or even "if n%k !=0". ====Lines 21-23 and 29-31: A failed attempt to change the value of a key==== In this code fragment, we ask if a given value of k has already been identified as a factor. But instead of "asking" if k has been declared a key in the dictionary, an <u>attempt</u> is made to change the value of k. If that attempt fails (because k is not among the dictionary's keys), then k is added to the list of keys, with a value of 1 (a value of 1 indicates that at this point, k, but not k² has been established to be a factor of n.) Again, there might be nothing wrong with using a key error like this. But the code would have been easier for me to follow if the if statement simply called the question of whether k was a key in dict. ==Sample output with a large number)== Sometimes you have to wait a few minutes to factor a number this large. But usually it comes out in just a few seconds. I would imagine that a number that was the product of only two roughly equal prime numbers would take forever because k=k+1 must be repeated a virtually infinite number of times. This number came quickly: {{Font color|blue|Enter a positive integer you wish to factor into prime numbers:}} 99283428376482768916467328912876456416375474272263423467896879187289499287374899 &nbsp; <span style="color:blue>99283428376482768916467328912876456416375474272263423467896879187289499287374899=</span> {{Font color|blue|(619**1) * (688**1) * (768**3) * (1024**17) * (64853**1) * (5302548928**1)}} &nbsp; {{Font color|blue|The dictionary is:}} &nbsp; <span style="color:blue>{619: 1, 688: 1, 768: 3, 1024: 17, 64853: 1, 5302548928.0: 1}</span> &nbsp; {{Font color|blue|Type 1 for another integer, 0 to terminate.}} {{center|'''DO NOT TRY TO FACTOR A NUMBER THIS LARGE UNLESS YOU ARE WILLING TO INTERUPT YOUR PYTHON PROGRAM.'''}} {{center|Keep in mind that the universe is only <math>4.4\times10^{26}\text{nanoseconds}</math> old.<ref>http://astronomy.nmsu.edu/geas/lectures/lecture02/slide04.html</ref>}} The number shown rounds up to 10⁸⁰. It is not likely that any computer will ever count that high. If the number to be factored consists of two prime number that are close to 10⁸⁰, the code will iterate all the way up to k coming close to 10⁴⁰. The inability of any computer to perform that many iterations helps explain how cryptocurrency works. &nbsp; 3hi9t8l5jeswvqfczeba4hyz5bx2k8x 2410302 2410301 2022-07-29T20:09:30Z Guy vandegrift 813252 /* Sample output with a large number) */ wikitext text/x-wiki [[File:Factorization flowchart.svg|thumb|420px|Flowchart for a simple prime factorization code]] '''Topics covered:''' ''prime factorization, flowcharts, python dictionaries'' Here we learn how a [[Python Concepts/Dictionaries|python dictionary]] can help express an integer as a [[w:Integer factorization|product of prime numbers]]. There are a few things the student should know about the code we are about to examine: * A serious python programmer would probably prefer to download a package that gives you a function that performs prime factorization. Two promising candidates are [https://pypi.org/project/primefac/ pypi.org/project/primefac] and [https://www.sympy.org/en/index.html www.sympy.org/]. * A simpler version of this same program does not use the "dictionary".<ref>Outside of Python a "dictionary" is also called an [[w:Associative array]].</ref> It can be found at [[Wikipedia:Trial division#Method]].<ref>A working python script is at [[w:Special:Permalink/1086485306]]</ref>. * Lacking the mental wherewithal to seize upon either of these opportunities, I googled "python prime factorization", found many codes, and selected the [[#Python code|one we shall examine]] for two reasons:<ref>The code that follows is discussed at [https://stackoverflow.com/questions/15347174/python-finding-prime-factors stackoverflow question 15347174] and [https://scientific-python-101.readthedocs.io/_downloads/prime_factorization_solution.py scientific-python-101.readthedocs]</ref> :# I wanted a code that counts repeated factors in order to later write the factored form in wikitext with exponents.<ref>In the wikitext version, I had to add a few lines so that exponents of 1 would not appear. Example: <math>12=2^2\cdot 3^1</math> should instead read <math>12=2^2\cdot 3.</math> I ommitted the feature here in order to keep the lesson simple.</ref> :# I didn't understand the python code, and wanted master it in order to fully understand how python dictionaries are used. A good student exercise would be to modify this code so that it runs approximately twice as fast. Instructors could also task students with modifying this code so that the output can be shown in a table that displays each effort to see if any given integer <math>k</math> is a prime factor of <math>N.</math> A version of such a modification created the wikitable shown below. It illustrates the "wasted" efforts to see if 4 and 6 might be prime factors of 1550. And it illustrates the efficiency of dividing a candidate integer into the number <math>n</math>, which soon becomes much smaller than <math>N.</math> {| class="wikitable" style="text-align: center; vertical-align: bottom;" |+ <math>\text{Establishing } 1550=2\cdot 5^2 \cdot 31</math> |- ! n ! k ! k^2 ! k^2<n? ! n/k ! mod ! replace ! factor ! * |- | 1550 | 2 | 4 | TRUE | 775 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"|n: n/k | style="background:#ffffcc"|2 | |- | 775 | 2 | 4 | TRUE | 387 | 1 | k: k+1 | | |- | 775 | 3 | 9 | TRUE | 258 | 1 | k: k+1 | | |- | 775 | 4 | 16 | TRUE | 193 | 3 | k: k+1 | | |- | 775 | 5 | 25 | TRUE | 155 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 155 | 5 | 25 | TRUE | 31 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 31 | 5 | 25 | TRUE | 6 | 1 | k: k+1 | | |- | 31 | 6 | 36 | FALSE | - | - | - | | style="background:#ffffcc"| 31 |} ==Python code== ‎<syntaxhighlight lang="python" line="1"> ## In a python code, the function definitions precede the code. We need only 1 function: def prime_factorization(N): ## 12=(2**2)*(2**1)*(5**1), or equivalently, 12 = 2**2 * 3 * 5 ## The dictionary, dict, expresses k**v as a "key-value" pair. In other ## words, if we look up the key, k=5, the dictionary will return the ## value, v=1. In other words: ## k**v <=> key**value <=> base**exponent (for the factored form of N) ## Begin function: n=N #--------------------# n will change in the while loop (getting smaller.) dict = {} #--------------# Start with an empty dictionary if n==1: #---------------# The factorization of 1 is a special case. return {1: 1} # In other words 1 = 1**1 (but recall that 1 is not prime.) k = 2 #------------------# Our first candidate for factorizing any integer. while k**2 <= n: #-------# Do the following unless k is "too big": if n % k: #----------# . Equivalent to: "If k is not a factor of n:" k += 1 #---------# .. increase the value of k by 1 on next attempt. else: #--------------# . Equivalent to "If k is a factor of n:" n /= k #---------# .. Replace n by n/k (for next attempt) try: #-----------# .. If k is in the dictionary, this attempts to: dict[k] += 1 # ... increase the value of k by 1. except KeyError: # .. KeyError occurs if k is not in the dictionary dict[k] = 1 # ... Add k to dictionary and set exponent v=1 # end(while)#------------# (exits while k**2 <= n:) # Here I inserted a diagnostic to be discussed later: # Documented code continues: if n > 1: #--------------# It can be shown that n is either 1 or prime. try: #---------------# . Try to enter n into dictionary dict[n] += 1 #---# .. increase exponent v->v+1 if k in dictinary except KeyError: #---# . If n is not in the dictionary, dict[n] = 1 # .. then enter n into dictionary with v=1 return dict #------# python talk for ending function ################ End def prime_factorization | Begin program ########## #import sys again=True while again == 1: text="\nEnter a positive integer you wish to factor into prime numbers:\n" N=int(input(text)) dict=prime_factorization(N) print("\n"+str(int(N))+"=") text="(" for b in dict.keys(): text+=str(int(b))+"**"+str(int(dict.get(b)))+") * (" print(text.rstrip(" * (")+"\n") print("The dictionary is:\n") print(dict) again=int(input("\nType 1 for another integer, 0 to terminate.")) </syntaxhighlight> ===Comments=== ====Line 15: while k**2 <= n:==== This is an important line because it permits the termination of the search before one might think. I leave it as an exercise for the student to learn why any value of k larger than square root of n will never be the next factor of N. It is also left to the student to understand the fact that the time to find the factors can be cut nearly in half if we modify line 17 and increase k by 2 (instead of 1), and why this larger step cannot be taken until after k=2 has been considered. ====Lines in the code that might seem confusing==== The code uses two cleverness tricks that might seem confusing. I lack the formal training in programming to know whether such cleverness would be perceived as a flaw. I know that programmers are allowed to be clever and I know that clear programs are better than confusing programs. The fact that these tricks confused me does not necessarily mean they should be avoided. And, the nice thing about both sources of confusion is that they taught me something about python. ====Line 16: if n%k might seem backwards==== Python follows a common practice of interpreting the number 1 to be True and 0 to be False. But python also accepts any non-zero number as True. Here n%k denotes nmod&nbsp;k, which is the remainder when on divides n by k. If there is no remainder the "if" condition is not satisfied and the code stipulates that the next value of k is attempted. Any number not equal to zero is interpreted as the "if" condition being satisfied, meaning that k is a divisor of n (in that case, we replace n by n/k.) This might be good coding, but it seems to me that it would have been better to construct it along the lines of "if n%k == 0" or even "if n%k !=0". ====Lines 21-23 and 29-31: A failed attempt to change the value of a key==== In this code fragment, we ask if a given value of k has already been identified as a factor. But instead of "asking" if k has been declared a key in the dictionary, an <u>attempt</u> is made to change the value of k. If that attempt fails (because k is not among the dictionary's keys), then k is added to the list of keys, with a value of 1 (a value of 1 indicates that at this point, k, but not k² has been established to be a factor of n.) Again, there might be nothing wrong with using a key error like this. But the code would have been easier for me to follow if the if statement simply called the question of whether k was a key in dict. ==Sample output with a large number== Sometimes you have to wait a few minutes to factor a number this large. But usually it comes out in just a few seconds. I would imagine that a number that was the product of only two roughly equal prime numbers would take forever because k=k+1 must be repeated a virtually infinite number of times. This number came quickly: {{Font color|blue|Enter a positive integer you wish to factor into prime numbers:}} 99283428376482768916467328912876456416375474272263423467896879187289499287374899 &nbsp; <span style="color:blue>99283428376482768916467328912876456416375474272263423467896879187289499287374899=</span> {{Font color|blue|(619**1) * (688**1) * (768**3) * (1024**17) * (64853**1) * (5302548928**1)}} &nbsp; {{Font color|blue|The dictionary is:}} &nbsp; <span style="color:blue>{619: 1, 688: 1, 768: 3, 1024: 17, 64853: 1, 5302548928.0: 1}</span> &nbsp; {{Font color|blue|Type 1 for another integer, 0 to terminate.}} {{center|'''DO NOT TRY TO FACTOR A NUMBER THIS LARGE UNLESS YOU ARE WILLING TO INTERUPT YOUR PYTHON PROGRAM.'''}} {{center|Keep in mind that the universe is only <math>4.4\times10^{26}\text{nanoseconds}</math> old.<ref>http://astronomy.nmsu.edu/geas/lectures/lecture02/slide04.html</ref>}} The number shown rounds up to 10⁸⁰. It is not likely that any computer will ever count that high. If the number to be factored consists of two prime number that are close to 10⁸⁰, the code will iterate all the way up to k coming close to 10⁴⁰. The inability of any computer to perform that many iterations helps explain how cryptocurrency works. &nbsp; h5i3yveqv80uizihd9wbg89zd84c7lv 2410303 2410302 2022-07-29T20:10:09Z Guy vandegrift 813252 wikitext text/x-wiki [[File:Factorization flowchart.svg|thumb|420px|Flowchart for a simple prime factorization code]] '''Topics covered:''' ''prime factorization, flowcharts, python dictionaries'' Here we learn how a [[Python Concepts/Dictionaries|python dictionary]] can help express an integer as a [[w:Integer factorization|product of prime numbers]]. There are a few things the student should know about the code we are about to examine: * A serious python programmer would probably prefer to download a package that gives you a function that performs prime factorization. Two promising candidates are [https://pypi.org/project/primefac/ pypi.org/project/primefac] and [https://www.sympy.org/en/index.html www.sympy.org/]. * A simpler version of this same program does not use the "dictionary".<ref>Outside of Python a "dictionary" is also called an [[w:Associative array]].</ref> It can be found at [[Wikipedia:Trial division#Method]].<ref>A working python script is at [[w:Special:Permalink/1086485306]]</ref>. * Lacking the mental wherewithal to seize upon either of these opportunities, I googled "python prime factorization", found many codes, and selected the [[#Python code|one we shall examine]] for two reasons:<ref>The code that follows is discussed at [https://stackoverflow.com/questions/15347174/python-finding-prime-factors stackoverflow question 15347174] and [https://scientific-python-101.readthedocs.io/_downloads/prime_factorization_solution.py scientific-python-101.readthedocs]</ref> :# I wanted a code that counts repeated factors in order to later write the factored form in wikitext with exponents.<ref>In the wikitext version, I had to add a few lines so that exponents of 1 would not appear. Example: <math>12=2^2\cdot 3^1</math> should instead read <math>12=2^2\cdot 3.</math> I ommitted the feature here in order to keep the lesson simple.</ref> :# I didn't understand the python code, and wanted master it in order to fully understand how python dictionaries are used. A good student exercise would be to modify this code so that it runs approximately twice as fast. Instructors could also task students with modifying this code so that the output can be shown in a table that displays each effort to see if any given integer <math>k</math> is a prime factor of <math>N.</math> One such a modification created the wikitable shown below. It illustrates the "wasted" efforts to see if 4 and 6 might be prime factors of 1550. And it illustrates the efficiency of dividing a candidate integer into the number <math>n</math>, which soon becomes much smaller than <math>N.</math> {| class="wikitable" style="text-align: center; vertical-align: bottom;" |+ <math>\text{Establishing } 1550=2\cdot 5^2 \cdot 31</math> |- ! n ! k ! k^2 ! k^2<n? ! n/k ! mod ! replace ! factor ! * |- | 1550 | 2 | 4 | TRUE | 775 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"|n: n/k | style="background:#ffffcc"|2 | |- | 775 | 2 | 4 | TRUE | 387 | 1 | k: k+1 | | |- | 775 | 3 | 9 | TRUE | 258 | 1 | k: k+1 | | |- | 775 | 4 | 16 | TRUE | 193 | 3 | k: k+1 | | |- | 775 | 5 | 25 | TRUE | 155 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 155 | 5 | 25 | TRUE | 31 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 31 | 5 | 25 | TRUE | 6 | 1 | k: k+1 | | |- | 31 | 6 | 36 | FALSE | - | - | - | | style="background:#ffffcc"| 31 |} ==Python code== ‎<syntaxhighlight lang="python" line="1"> ## In a python code, the function definitions precede the code. We need only 1 function: def prime_factorization(N): ## 12=(2**2)*(2**1)*(5**1), or equivalently, 12 = 2**2 * 3 * 5 ## The dictionary, dict, expresses k**v as a "key-value" pair. In other ## words, if we look up the key, k=5, the dictionary will return the ## value, v=1. In other words: ## k**v <=> key**value <=> base**exponent (for the factored form of N) ## Begin function: n=N #--------------------# n will change in the while loop (getting smaller.) dict = {} #--------------# Start with an empty dictionary if n==1: #---------------# The factorization of 1 is a special case. return {1: 1} # In other words 1 = 1**1 (but recall that 1 is not prime.) k = 2 #------------------# Our first candidate for factorizing any integer. while k**2 <= n: #-------# Do the following unless k is "too big": if n % k: #----------# . Equivalent to: "If k is not a factor of n:" k += 1 #---------# .. increase the value of k by 1 on next attempt. else: #--------------# . Equivalent to "If k is a factor of n:" n /= k #---------# .. Replace n by n/k (for next attempt) try: #-----------# .. If k is in the dictionary, this attempts to: dict[k] += 1 # ... increase the value of k by 1. except KeyError: # .. KeyError occurs if k is not in the dictionary dict[k] = 1 # ... Add k to dictionary and set exponent v=1 # end(while)#------------# (exits while k**2 <= n:) # Here I inserted a diagnostic to be discussed later: # Documented code continues: if n > 1: #--------------# It can be shown that n is either 1 or prime. try: #---------------# . Try to enter n into dictionary dict[n] += 1 #---# .. increase exponent v->v+1 if k in dictinary except KeyError: #---# . If n is not in the dictionary, dict[n] = 1 # .. then enter n into dictionary with v=1 return dict #------# python talk for ending function ################ End def prime_factorization | Begin program ########## #import sys again=True while again == 1: text="\nEnter a positive integer you wish to factor into prime numbers:\n" N=int(input(text)) dict=prime_factorization(N) print("\n"+str(int(N))+"=") text="(" for b in dict.keys(): text+=str(int(b))+"**"+str(int(dict.get(b)))+") * (" print(text.rstrip(" * (")+"\n") print("The dictionary is:\n") print(dict) again=int(input("\nType 1 for another integer, 0 to terminate.")) </syntaxhighlight> ===Comments=== ====Line 15: while k**2 <= n:==== This is an important line because it permits the termination of the search before one might think. I leave it as an exercise for the student to learn why any value of k larger than square root of n will never be the next factor of N. It is also left to the student to understand the fact that the time to find the factors can be cut nearly in half if we modify line 17 and increase k by 2 (instead of 1), and why this larger step cannot be taken until after k=2 has been considered. ====Lines in the code that might seem confusing==== The code uses two cleverness tricks that might seem confusing. I lack the formal training in programming to know whether such cleverness would be perceived as a flaw. I know that programmers are allowed to be clever and I know that clear programs are better than confusing programs. The fact that these tricks confused me does not necessarily mean they should be avoided. And, the nice thing about both sources of confusion is that they taught me something about python. ====Line 16: if n%k might seem backwards==== Python follows a common practice of interpreting the number 1 to be True and 0 to be False. But python also accepts any non-zero number as True. Here n%k denotes nmod&nbsp;k, which is the remainder when on divides n by k. If there is no remainder the "if" condition is not satisfied and the code stipulates that the next value of k is attempted. Any number not equal to zero is interpreted as the "if" condition being satisfied, meaning that k is a divisor of n (in that case, we replace n by n/k.) This might be good coding, but it seems to me that it would have been better to construct it along the lines of "if n%k == 0" or even "if n%k !=0". ====Lines 21-23 and 29-31: A failed attempt to change the value of a key==== In this code fragment, we ask if a given value of k has already been identified as a factor. But instead of "asking" if k has been declared a key in the dictionary, an <u>attempt</u> is made to change the value of k. If that attempt fails (because k is not among the dictionary's keys), then k is added to the list of keys, with a value of 1 (a value of 1 indicates that at this point, k, but not k² has been established to be a factor of n.) Again, there might be nothing wrong with using a key error like this. But the code would have been easier for me to follow if the if statement simply called the question of whether k was a key in dict. ==Sample output with a large number== Sometimes you have to wait a few minutes to factor a number this large. But usually it comes out in just a few seconds. I would imagine that a number that was the product of only two roughly equal prime numbers would take forever because k=k+1 must be repeated a virtually infinite number of times. This number came quickly: {{Font color|blue|Enter a positive integer you wish to factor into prime numbers:}} 99283428376482768916467328912876456416375474272263423467896879187289499287374899 &nbsp; <span style="color:blue>99283428376482768916467328912876456416375474272263423467896879187289499287374899=</span> {{Font color|blue|(619**1) * (688**1) * (768**3) * (1024**17) * (64853**1) * (5302548928**1)}} &nbsp; {{Font color|blue|The dictionary is:}} &nbsp; <span style="color:blue>{619: 1, 688: 1, 768: 3, 1024: 17, 64853: 1, 5302548928.0: 1}</span> &nbsp; {{Font color|blue|Type 1 for another integer, 0 to terminate.}} {{center|'''DO NOT TRY TO FACTOR A NUMBER THIS LARGE UNLESS YOU ARE WILLING TO INTERUPT YOUR PYTHON PROGRAM.'''}} {{center|Keep in mind that the universe is only <math>4.4\times10^{26}\text{nanoseconds}</math> old.<ref>http://astronomy.nmsu.edu/geas/lectures/lecture02/slide04.html</ref>}} The number shown rounds up to 10⁸⁰. It is not likely that any computer will ever count that high. If the number to be factored consists of two prime number that are close to 10⁸⁰, the code will iterate all the way up to k coming close to 10⁴⁰. The inability of any computer to perform that many iterations helps explain how cryptocurrency works. &nbsp; ia7btl0jf7uftcainwqgjosrms7haqs 2410304 2410303 2022-07-29T20:11:37Z Guy vandegrift 813252 wikitext text/x-wiki [[File:Factorization flowchart.svg|thumb|420px|Flowchart for a simple prime factorization code]] '''Topics covered:''' ''prime factorization, flowcharts, python dictionaries'' Here we learn how a [[Python Concepts/Dictionaries|python dictionary]] can help express an integer as a [[w:Integer factorization|product of prime numbers]]. There are a few things the student should know about the code we are about to examine: * A serious python programmer would probably prefer to download a package that gives you a function that performs prime factorization. Two promising candidates are [https://pypi.org/project/primefac/ pypi.org/project/primefac] and [https://www.sympy.org/en/index.html www.sympy.org/]. * A simpler version of this same program does not use the "dictionary".<ref>Outside of Python a "dictionary" is also called an [[w:Associative array]].</ref> It can be found at [[Wikipedia:Trial division#Method]].<ref>A working python script is at [[w:Special:Permalink/1086485306]]</ref>. * Lacking the mental wherewithal to seize upon either of these opportunities, I googled "python prime factorization", found many codes, and selected the [[#Python code|one we shall examine]] for two reasons:<ref>The code that follows is discussed at [https://stackoverflow.com/questions/15347174/python-finding-prime-factors stackoverflow question 15347174] and [https://scientific-python-101.readthedocs.io/_downloads/prime_factorization_solution.py scientific-python-101.readthedocs]</ref> :# I wanted a code that counts repeated factors in order to later write the factored form in wikitext with exponents.<ref>In the wikitext version, I had to add a few lines so that exponents of 1 would not appear. Example: <math>12=2^2\cdot 3^1</math> should instead read <math>12=2^2\cdot 3.</math> I ommitted the feature here in order to keep the lesson simple.</ref> :# I didn't understand the python code, and wanted master it in order to fully understand how python dictionaries are used. A good student exercise would be to modify this code so that it runs approximately twice as fast. Instructors could also task students with modifying this code so that the output can be shown in a table that displays each effort to see if any given integer <math>k</math> is a prime factor of <math>N.</math> One such a modification created the wikitable shown below. It illustrates the "wasted" efforts to see if 4 and 6 might be prime factors of 1550. And it illustrates the efficiency of dividing a candidate integer into the number <math>n</math>, which soon becomes much smaller than <math>N.</math> It took only eight (8) steps to factor {{math|1550}} into prime numbers. {| class="wikitable" style="text-align: center; vertical-align: bottom;" |+ <math>\text{Establishing } 1550=2\cdot 5^2 \cdot 31</math> |- ! n ! k ! k^2 ! k^2<n? ! n/k ! mod ! replace ! factor ! * |- | 1550 | 2 | 4 | TRUE | 775 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"|n: n/k | style="background:#ffffcc"|2 | |- | 775 | 2 | 4 | TRUE | 387 | 1 | k: k+1 | | |- | 775 | 3 | 9 | TRUE | 258 | 1 | k: k+1 | | |- | 775 | 4 | 16 | TRUE | 193 | 3 | k: k+1 | | |- | 775 | 5 | 25 | TRUE | 155 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 155 | 5 | 25 | TRUE | 31 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 31 | 5 | 25 | TRUE | 6 | 1 | k: k+1 | | |- | 31 | 6 | 36 | FALSE | - | - | - | | style="background:#ffffcc"| 31 |} ==Python code== ‎<syntaxhighlight lang="python" line="1"> ## In a python code, the function definitions precede the code. We need only 1 function: def prime_factorization(N): ## 12=(2**2)*(2**1)*(5**1), or equivalently, 12 = 2**2 * 3 * 5 ## The dictionary, dict, expresses k**v as a "key-value" pair. In other ## words, if we look up the key, k=5, the dictionary will return the ## value, v=1. In other words: ## k**v <=> key**value <=> base**exponent (for the factored form of N) ## Begin function: n=N #--------------------# n will change in the while loop (getting smaller.) dict = {} #--------------# Start with an empty dictionary if n==1: #---------------# The factorization of 1 is a special case. return {1: 1} # In other words 1 = 1**1 (but recall that 1 is not prime.) k = 2 #------------------# Our first candidate for factorizing any integer. while k**2 <= n: #-------# Do the following unless k is "too big": if n % k: #----------# . Equivalent to: "If k is not a factor of n:" k += 1 #---------# .. increase the value of k by 1 on next attempt. else: #--------------# . Equivalent to "If k is a factor of n:" n /= k #---------# .. Replace n by n/k (for next attempt) try: #-----------# .. If k is in the dictionary, this attempts to: dict[k] += 1 # ... increase the value of k by 1. except KeyError: # .. KeyError occurs if k is not in the dictionary dict[k] = 1 # ... Add k to dictionary and set exponent v=1 # end(while)#------------# (exits while k**2 <= n:) # Here I inserted a diagnostic to be discussed later: # Documented code continues: if n > 1: #--------------# It can be shown that n is either 1 or prime. try: #---------------# . Try to enter n into dictionary dict[n] += 1 #---# .. increase exponent v->v+1 if k in dictinary except KeyError: #---# . If n is not in the dictionary, dict[n] = 1 # .. then enter n into dictionary with v=1 return dict #------# python talk for ending function ################ End def prime_factorization | Begin program ########## #import sys again=True while again == 1: text="\nEnter a positive integer you wish to factor into prime numbers:\n" N=int(input(text)) dict=prime_factorization(N) print("\n"+str(int(N))+"=") text="(" for b in dict.keys(): text+=str(int(b))+"**"+str(int(dict.get(b)))+") * (" print(text.rstrip(" * (")+"\n") print("The dictionary is:\n") print(dict) again=int(input("\nType 1 for another integer, 0 to terminate.")) </syntaxhighlight> ===Comments=== ====Line 15: while k**2 <= n:==== This is an important line because it permits the termination of the search before one might think. I leave it as an exercise for the student to learn why any value of k larger than square root of n will never be the next factor of N. It is also left to the student to understand the fact that the time to find the factors can be cut nearly in half if we modify line 17 and increase k by 2 (instead of 1), and why this larger step cannot be taken until after k=2 has been considered. ====Lines in the code that might seem confusing==== The code uses two cleverness tricks that might seem confusing. I lack the formal training in programming to know whether such cleverness would be perceived as a flaw. I know that programmers are allowed to be clever and I know that clear programs are better than confusing programs. The fact that these tricks confused me does not necessarily mean they should be avoided. And, the nice thing about both sources of confusion is that they taught me something about python. ====Line 16: if n%k might seem backwards==== Python follows a common practice of interpreting the number 1 to be True and 0 to be False. But python also accepts any non-zero number as True. Here n%k denotes nmod&nbsp;k, which is the remainder when on divides n by k. If there is no remainder the "if" condition is not satisfied and the code stipulates that the next value of k is attempted. Any number not equal to zero is interpreted as the "if" condition being satisfied, meaning that k is a divisor of n (in that case, we replace n by n/k.) This might be good coding, but it seems to me that it would have been better to construct it along the lines of "if n%k == 0" or even "if n%k !=0". ====Lines 21-23 and 29-31: A failed attempt to change the value of a key==== In this code fragment, we ask if a given value of k has already been identified as a factor. But instead of "asking" if k has been declared a key in the dictionary, an <u>attempt</u> is made to change the value of k. If that attempt fails (because k is not among the dictionary's keys), then k is added to the list of keys, with a value of 1 (a value of 1 indicates that at this point, k, but not k² has been established to be a factor of n.) Again, there might be nothing wrong with using a key error like this. But the code would have been easier for me to follow if the if statement simply called the question of whether k was a key in dict. ==Sample output with a large number== Sometimes you have to wait a few minutes to factor a number this large. But usually it comes out in just a few seconds. I would imagine that a number that was the product of only two roughly equal prime numbers would take forever because k=k+1 must be repeated a virtually infinite number of times. This number came quickly: {{Font color|blue|Enter a positive integer you wish to factor into prime numbers:}} 99283428376482768916467328912876456416375474272263423467896879187289499287374899 &nbsp; <span style="color:blue>99283428376482768916467328912876456416375474272263423467896879187289499287374899=</span> {{Font color|blue|(619**1) * (688**1) * (768**3) * (1024**17) * (64853**1) * (5302548928**1)}} &nbsp; {{Font color|blue|The dictionary is:}} &nbsp; <span style="color:blue>{619: 1, 688: 1, 768: 3, 1024: 17, 64853: 1, 5302548928.0: 1}</span> &nbsp; {{Font color|blue|Type 1 for another integer, 0 to terminate.}} {{center|'''DO NOT TRY TO FACTOR A NUMBER THIS LARGE UNLESS YOU ARE WILLING TO INTERUPT YOUR PYTHON PROGRAM.'''}} {{center|Keep in mind that the universe is only <math>4.4\times10^{26}\text{nanoseconds}</math> old.<ref>http://astronomy.nmsu.edu/geas/lectures/lecture02/slide04.html</ref>}} The number shown rounds up to 10⁸⁰. It is not likely that any computer will ever count that high. If the number to be factored consists of two prime number that are close to 10⁸⁰, the code will iterate all the way up to k coming close to 10⁴⁰. The inability of any computer to perform that many iterations helps explain how cryptocurrency works. &nbsp; h8zz4k3keq26w4afxmd2122u2x0qhj9 2410371 2410304 2022-07-30T01:30:27Z Guy vandegrift 813252 /* Comments */ wikitext text/x-wiki [[File:Factorization flowchart.svg|thumb|420px|Flowchart for a simple prime factorization code]] '''Topics covered:''' ''prime factorization, flowcharts, python dictionaries'' Here we learn how a [[Python Concepts/Dictionaries|python dictionary]] can help express an integer as a [[w:Integer factorization|product of prime numbers]]. There are a few things the student should know about the code we are about to examine: * A serious python programmer would probably prefer to download a package that gives you a function that performs prime factorization. Two promising candidates are [https://pypi.org/project/primefac/ pypi.org/project/primefac] and [https://www.sympy.org/en/index.html www.sympy.org/]. * A simpler version of this same program does not use the "dictionary".<ref>Outside of Python a "dictionary" is also called an [[w:Associative array]].</ref> It can be found at [[Wikipedia:Trial division#Method]].<ref>A working python script is at [[w:Special:Permalink/1086485306]]</ref>. * Lacking the mental wherewithal to seize upon either of these opportunities, I googled "python prime factorization", found many codes, and selected the [[#Python code|one we shall examine]] for two reasons:<ref>The code that follows is discussed at [https://stackoverflow.com/questions/15347174/python-finding-prime-factors stackoverflow question 15347174] and [https://scientific-python-101.readthedocs.io/_downloads/prime_factorization_solution.py scientific-python-101.readthedocs]</ref> :# I wanted a code that counts repeated factors in order to later write the factored form in wikitext with exponents.<ref>In the wikitext version, I had to add a few lines so that exponents of 1 would not appear. Example: <math>12=2^2\cdot 3^1</math> should instead read <math>12=2^2\cdot 3.</math> I ommitted the feature here in order to keep the lesson simple.</ref> :# I didn't understand the python code, and wanted master it in order to fully understand how python dictionaries are used. A good student exercise would be to modify this code so that it runs approximately twice as fast. Instructors could also task students with modifying this code so that the output can be shown in a table that displays each effort to see if any given integer <math>k</math> is a prime factor of <math>N.</math> One such a modification created the wikitable shown below. It illustrates the "wasted" efforts to see if 4 and 6 might be prime factors of 1550. And it illustrates the efficiency of dividing a candidate integer into the number <math>n</math>, which soon becomes much smaller than <math>N.</math> It took only eight (8) steps to factor {{math|1550}} into prime numbers. {| class="wikitable" style="text-align: center; vertical-align: bottom;" |+ <math>\text{Establishing } 1550=2\cdot 5^2 \cdot 31</math> |- ! n ! k ! k^2 ! k^2<n? ! n/k ! mod ! replace ! factor ! * |- | 1550 | 2 | 4 | TRUE | 775 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"|n: n/k | style="background:#ffffcc"|2 | |- | 775 | 2 | 4 | TRUE | 387 | 1 | k: k+1 | | |- | 775 | 3 | 9 | TRUE | 258 | 1 | k: k+1 | | |- | 775 | 4 | 16 | TRUE | 193 | 3 | k: k+1 | | |- | 775 | 5 | 25 | TRUE | 155 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 155 | 5 | 25 | TRUE | 31 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 31 | 5 | 25 | TRUE | 6 | 1 | k: k+1 | | |- | 31 | 6 | 36 | FALSE | - | - | - | | style="background:#ffffcc"| 31 |} ==Python code== ‎<syntaxhighlight lang="python" line="1"> ## In a python code, the function definitions precede the code. We need only 1 function: def prime_factorization(N): ## 12=(2**2)*(2**1)*(5**1), or equivalently, 12 = 2**2 * 3 * 5 ## The dictionary, dict, expresses k**v as a "key-value" pair. In other ## words, if we look up the key, k=5, the dictionary will return the ## value, v=1. In other words: ## k**v <=> key**value <=> base**exponent (for the factored form of N) ## Begin function: n=N #--------------------# n will change in the while loop (getting smaller.) dict = {} #--------------# Start with an empty dictionary if n==1: #---------------# The factorization of 1 is a special case. return {1: 1} # In other words 1 = 1**1 (but recall that 1 is not prime.) k = 2 #------------------# Our first candidate for factorizing any integer. while k**2 <= n: #-------# Do the following unless k is "too big": if n % k: #----------# . Equivalent to: "If k is not a factor of n:" k += 1 #---------# .. increase the value of k by 1 on next attempt. else: #--------------# . Equivalent to "If k is a factor of n:" n /= k #---------# .. Replace n by n/k (for next attempt) try: #-----------# .. If k is in the dictionary, this attempts to: dict[k] += 1 # ... increase the value of k by 1. except KeyError: # .. KeyError occurs if k is not in the dictionary dict[k] = 1 # ... Add k to dictionary and set exponent v=1 # end(while)#------------# (exits while k**2 <= n:) # Here I inserted a diagnostic to be discussed later: # Documented code continues: if n > 1: #--------------# It can be shown that n is either 1 or prime. try: #---------------# . Try to enter n into dictionary dict[n] += 1 #---# .. increase exponent v->v+1 if k in dictinary except KeyError: #---# . If n is not in the dictionary, dict[n] = 1 # .. then enter n into dictionary with v=1 return dict #------# python talk for ending function ################ End def prime_factorization | Begin program ########## #import sys again=True while again == 1: text="\nEnter a positive integer you wish to factor into prime numbers:\n" N=int(input(text)) dict=prime_factorization(N) print("\n"+str(int(N))+"=") text="(" for b in dict.keys(): text+=str(int(b))+"**"+str(int(dict.get(b)))+") * (" print(text.rstrip(" * (")+"\n") print("The dictionary is:\n") print(dict) again=int(input("\nType 1 for another integer, 0 to terminate.")) </syntaxhighlight> ===Comments=== =====Line 15: while k**2 <= n:===== This is an important line because it permits the termination of the search before one might think. I leave it as an exercise for the student to learn why any value of k larger than square root of n will never be the next factor of N. It is also left to the student to understand the fact that the time to find the factors can be cut nearly in half if we modify line 17 and increase k by 2 (instead of 1), and why this larger step cannot be taken until after k=2 has been considered. ====Lines in the code that might seem confusing==== The code uses two cleverness tricks that might seem confusing. I lack the formal training in programming to know whether such cleverness would be perceived as a flaw. I know that programmers are allowed to be clever and I know that clear programs are better than confusing programs. The fact that these tricks confused me does not necessarily mean they should be avoided. And, the nice thing about both sources of confusion is that they taught me something about python. =====Line 16: if n%k might seem backwards===== Python follows a common practice of interpreting the number 1 to be True and 0 to be False. But python also accepts any non-zero number as True. Here n%k denotes nmod&nbsp;k, which is the remainder when on divides n by k. If there is no remainder the "if" condition is not satisfied and the code stipulates that the next value of k is attempted. Any number not equal to zero is interpreted as the "if" condition being satisfied, meaning that k is a divisor of n (in that case, we replace n by n/k.) This might be good coding, but it seems to me that it would have been better to construct it along the lines of "if n%k == 0" or even "if n%k !=0". =====Lines 21-23 and 29-31: A failed attempt to change the value of a key===== In this code fragment, we ask if a given value of k has already been identified as a factor. But instead of "asking" if k has been declared a key in the dictionary, an <u>attempt</u> is made to change the value of k. If that attempt fails (because k is not among the dictionary's keys), then k is added to the list of keys, with a value of 1 (a value of 1 indicates that at this point, k, but not k² has been established to be a factor of n.) Again, there might be nothing wrong with using a key error like this. But the code would have been easier for me to follow if the if statement simply called the question of whether k was a key in dict. ==Sample output with a large number== Sometimes you have to wait a few minutes to factor a number this large. But usually it comes out in just a few seconds. I would imagine that a number that was the product of only two roughly equal prime numbers would take forever because k=k+1 must be repeated a virtually infinite number of times. This number came quickly: {{Font color|blue|Enter a positive integer you wish to factor into prime numbers:}} 99283428376482768916467328912876456416375474272263423467896879187289499287374899 &nbsp; <span style="color:blue>99283428376482768916467328912876456416375474272263423467896879187289499287374899=</span> {{Font color|blue|(619**1) * (688**1) * (768**3) * (1024**17) * (64853**1) * (5302548928**1)}} &nbsp; {{Font color|blue|The dictionary is:}} &nbsp; <span style="color:blue>{619: 1, 688: 1, 768: 3, 1024: 17, 64853: 1, 5302548928.0: 1}</span> &nbsp; {{Font color|blue|Type 1 for another integer, 0 to terminate.}} {{center|'''DO NOT TRY TO FACTOR A NUMBER THIS LARGE UNLESS YOU ARE WILLING TO INTERUPT YOUR PYTHON PROGRAM.'''}} {{center|Keep in mind that the universe is only <math>4.4\times10^{26}\text{nanoseconds}</math> old.<ref>http://astronomy.nmsu.edu/geas/lectures/lecture02/slide04.html</ref>}} The number shown rounds up to 10⁸⁰. It is not likely that any computer will ever count that high. If the number to be factored consists of two prime number that are close to 10⁸⁰, the code will iterate all the way up to k coming close to 10⁴⁰. The inability of any computer to perform that many iterations helps explain how cryptocurrency works. &nbsp; e1gmirarbh7l7pac8idiz1mlg2cvemh 2410372 2410371 2022-07-30T01:45:29Z Guy vandegrift 813252 wikitext text/x-wiki [[File:Factorization flowchart.svg|thumb|420px|Flowchart for a simple prime factorization code]] '''Topics covered:''' ''prime factorization, flowcharts, python dictionaries'' Here we learn how a [[Python Concepts/Dictionaries|python dictionary]] can help express an integer as a [[w:Integer factorization|product of prime numbers]]. There are a few things the student should know about the code we are about to examine: * A serious python programmer would probably prefer to download a package that gives you a function that performs prime factorization. Two promising candidates are [https://pypi.org/project/primefac/ pypi.org/project/primefac] and [https://www.sympy.org/en/index.html www.sympy.org/]. * A simpler version of this same program does not use the "dictionary".<ref>Outside of Python a "dictionary" is also called an [[w:Associative array]].</ref> It can be found at [[Wikipedia:Trial division#Method]].<ref>A working python script is at [[w:Special:Permalink/1086485306]]</ref>. * Lacking the mental wherewithal to seize upon either of these opportunities, I googled "python prime factorization", found many codes, and selected the [[#Python code|one we shall examine]] for two reasons:<ref>The code that follows is discussed at [https://stackoverflow.com/questions/15347174/python-finding-prime-factors stackoverflow question 15347174] and [https://scientific-python-101.readthedocs.io/_downloads/prime_factorization_solution.py scientific-python-101.readthedocs]</ref> :# I wanted a code that counts repeated factors in order to later write the factored form in wikitext with exponents.<ref>In the wikitext version, I had to add a few lines so that exponents of 1 would not appear. Example: <math>12=2^2\cdot 3^1</math> should instead read <math>12=2^2\cdot 3.</math> I ommitted the feature here in order to keep the lesson simple.</ref> :# I didn't understand the python code, and wanted master it in order to fully understand how python dictionaries are used. A good student exercise would be to modify this code so that it runs approximately twice as fast. Instructors could also task students with modifying this code so that the output can be shown in a table that displays each effort to see if any given integer <math>k</math> is a prime factor of <math>N.</math> One such a modification created the wikitable shown below. It illustrates the "wasted" efforts to see if 4 and 6 might be prime factors of 1550. And it illustrates the efficiency of dividing a candidate integer into the number <math>n</math>, which soon becomes much smaller than <math>N.</math> It took only eight (8) steps to factor {{math|1550}} into prime numbers. {| class="wikitable" style="text-align: center; vertical-align: bottom;" |+ <math>\text{Establishing } 1550=2\cdot 5^2 \cdot 31</math> |- ! n ! k ! k^2 ! k^2<n? ! n/k ! mod ! replace ! factor ! * |- | 1550 | 2 | 4 | TRUE | 775 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"|n: n/k | style="background:#ffffcc"|2 | |- | 775 | 2 | 4 | TRUE | 387 | 1 | k: k+1 | | |- | 775 | 3 | 9 | TRUE | 258 | 1 | k: k+1 | | |- | 775 | 4 | 16 | TRUE | 193 | 3 | k: k+1 | | |- | 775 | 5 | 25 | TRUE | 155 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 155 | 5 | 25 | TRUE | 31 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 31 | 5 | 25 | TRUE | 6 | 1 | k: k+1 | | |- | 31 | 6 | 36 | FALSE | - | - | - | | style="background:#ffffcc"| 31 |} ==Python code== ‎<syntaxhighlight lang="python" line="1"> ## In a python code, the function definitions precede the code. We need only 1 function: def prime_factorization(N): ## 12=(2**2)*(2**1)*(5**1), or equivalently, 12 = 2**2 * 3 * 5 ## The dictionary, dict, expresses k**v as a "key-value" pair. In other ## words, if we look up the key, k=5, the dictionary will return the ## value, v=1. In other words: ## k**v <=> key**value <=> base**exponent (for the factored form of N) ## Begin function: n=N #--------------------# n will change in the while loop (getting smaller.) dict = {} #--------------# Start with an empty dictionary if n==1: #---------------# The factorization of 1 is a special case. return {1: 1} # In other words 1 = 1**1 (but recall that 1 is not prime.) k = 2 #------------------# Our first candidate for factorizing any integer. while k**2 <= n: #-------# Do the following unless k is "too big": if n % k: #----------# . Equivalent to: "If k is not a factor of n:" k += 1 #---------# .. increase the value of k by 1 on next attempt. else: #--------------# . Equivalent to "If k is a factor of n:" n /= k #---------# .. Replace n by n/k (for next attempt) try: #-----------# .. If k is in the dictionary, this attempts to: dict[k] += 1 # ... increase the value of k by 1. except KeyError: # .. KeyError occurs if k is not in the dictionary dict[k] = 1 # ... Add k to dictionary and set exponent v=1 # end(while)#------------# (exits while k**2 <= n:) # Here I inserted a diagnostic to be discussed later: # Documented code continues: if n > 1: #--------------# It can be shown that n is either 1 or prime. try: #---------------# . Try to enter n into dictionary dict[n] += 1 #---# .. increase exponent v->v+1 if k in dictinary except KeyError: #---# . If n is not in the dictionary, dict[n] = 1 # .. then enter n into dictionary with v=1 return dict #------# python talk for ending function ################ End def prime_factorization | Begin program ########## #import sys again=True while again == 1: text="\nEnter a positive integer you wish to factor into prime numbers:\n" N=int(input(text)) dict=prime_factorization(N) print("\n"+str(int(N))+"=") text="(" for b in dict.keys(): text+=str(int(b))+"**"+str(int(dict.get(b)))+") * (" print(text.rstrip(" * (")+"\n") print("The dictionary is:\n") print(dict) again=int(input("\nType 1 for another integer, 0 to terminate.")) </syntaxhighlight> With one extra line a print command will display the dictionary. From the fact that python decided to print {{math|31}} as {{math|31.0}}, we know that the values in this dictionary is taking the keys (2, 5, 31) to be floating point numbers. Python permits programmers to take a casual attitude as to whether a variable is an integer or a floating point number. This makes it easier to program python. But it also makes it more difficult to sort out such misunderstandings after a working program has been written. print(dict) {2: 1, 5: 2, 31.0: 1} ===Comments=== =====Line 15: while k**2 <= n:===== This is an important line because it permits the termination of the search before one might think. I leave it as an exercise for the student to learn why any value of k larger than square root of n will never be the next factor of N. It is also left to the student to understand the fact that the time to find the factors can be cut nearly in half if we modify line 17 and increase k by 2 (instead of 1), and why this larger step cannot be taken until after k=2 has been considered. ====Lines in the code that might seem confusing==== The code uses two cleverness tricks that might seem confusing. I lack the formal training in programming to know whether such cleverness would be perceived as a flaw. I know that programmers are allowed to be clever and I know that clear programs are better than confusing programs. The fact that these tricks confused me does not necessarily mean they should be avoided. And, the nice thing about both sources of confusion is that they taught me something about python. =====Line 16: if n%k might seem backwards===== Python follows a common practice of interpreting the number 1 to be True and 0 to be False. But python also accepts any non-zero number as True. Here n%k denotes nmod&nbsp;k, which is the remainder when on divides n by k. If there is no remainder the "if" condition is not satisfied and the code stipulates that the next value of k is attempted. Any number not equal to zero is interpreted as the "if" condition being satisfied, meaning that k is a divisor of n (in that case, we replace n by n/k.) This might be good coding, but it seems to me that it would have been better to construct it along the lines of "if n%k == 0" or even "if n%k !=0". =====Lines 21-23 and 29-31: A failed attempt to change the value of a key===== In this code fragment, we ask if a given value of k has already been identified as a factor. But instead of "asking" if k has been declared a key in the dictionary, an <u>attempt</u> is made to change the value of k. If that attempt fails (because k is not among the dictionary's keys), then k is added to the list of keys, with a value of 1 (a value of 1 indicates that at this point, k, but not k² has been established to be a factor of n.) Again, there might be nothing wrong with using a key error like this. But the code would have been easier for me to follow if the if statement simply called the question of whether k was a key in dict. ==Sample output with a large number== Sometimes you have to wait a few minutes to factor a number this large. But usually it comes out in just a few seconds. I would imagine that a number that was the product of only two roughly equal prime numbers would take forever because k=k+1 must be repeated a virtually infinite number of times. This number came quickly: {{Font color|blue|Enter a positive integer you wish to factor into prime numbers:}} 99283428376482768916467328912876456416375474272263423467896879187289499287374899 &nbsp; <span style="color:blue>99283428376482768916467328912876456416375474272263423467896879187289499287374899=</span> {{Font color|blue|(619**1) * (688**1) * (768**3) * (1024**17) * (64853**1) * (5302548928**1)}} &nbsp; {{Font color|blue|The dictionary is:}} &nbsp; <span style="color:blue>{619: 1, 688: 1, 768: 3, 1024: 17, 64853: 1, 5302548928.0: 1}</span> &nbsp; {{Font color|blue|Type 1 for another integer, 0 to terminate.}} {{center|'''DO NOT TRY TO FACTOR A NUMBER THIS LARGE UNLESS YOU ARE WILLING TO INTERUPT YOUR PYTHON PROGRAM.'''}} {{center|Keep in mind that the universe is only <math>4.4\times10^{26}\text{nanoseconds}</math> old.<ref>http://astronomy.nmsu.edu/geas/lectures/lecture02/slide04.html</ref>}} The number shown rounds up to 10⁸⁰. It is not likely that any computer will ever count that high. If the number to be factored consists of two prime number that are close to 10⁸⁰, the code will iterate all the way up to k coming close to 10⁴⁰. The inability of any computer to perform that many iterations helps explain how cryptocurrency works. &nbsp; o1xh22be7dyf5s9znen9qal37t0pcw0 2410373 2410372 2022-07-30T01:55:13Z Guy vandegrift 813252 /* Python code */ wikitext text/x-wiki [[File:Factorization flowchart.svg|thumb|420px|Flowchart for a simple prime factorization code]] '''Topics covered:''' ''prime factorization, flowcharts, python dictionaries'' Here we learn how a [[Python Concepts/Dictionaries|python dictionary]] can help express an integer as a [[w:Integer factorization|product of prime numbers]]. There are a few things the student should know about the code we are about to examine: * A serious python programmer would probably prefer to download a package that gives you a function that performs prime factorization. Two promising candidates are [https://pypi.org/project/primefac/ pypi.org/project/primefac] and [https://www.sympy.org/en/index.html www.sympy.org/]. * A simpler version of this same program does not use the "dictionary".<ref>Outside of Python a "dictionary" is also called an [[w:Associative array]].</ref> It can be found at [[Wikipedia:Trial division#Method]].<ref>A working python script is at [[w:Special:Permalink/1086485306]]</ref>. * Lacking the mental wherewithal to seize upon either of these opportunities, I googled "python prime factorization", found many codes, and selected the [[#Python code|one we shall examine]] for two reasons:<ref>The code that follows is discussed at [https://stackoverflow.com/questions/15347174/python-finding-prime-factors stackoverflow question 15347174] and [https://scientific-python-101.readthedocs.io/_downloads/prime_factorization_solution.py scientific-python-101.readthedocs]</ref> :# I wanted a code that counts repeated factors in order to later write the factored form in wikitext with exponents.<ref>In the wikitext version, I had to add a few lines so that exponents of 1 would not appear. Example: <math>12=2^2\cdot 3^1</math> should instead read <math>12=2^2\cdot 3.</math> I ommitted the feature here in order to keep the lesson simple.</ref> :# I didn't understand the python code, and wanted master it in order to fully understand how python dictionaries are used. A good student exercise would be to modify this code so that it runs approximately twice as fast. Instructors could also task students with modifying this code so that the output can be shown in a table that displays each effort to see if any given integer <math>k</math> is a prime factor of <math>N.</math> One such a modification created the wikitable shown below. It illustrates the "wasted" efforts to see if 4 and 6 might be prime factors of 1550. And it illustrates the efficiency of dividing a candidate integer into the number <math>n</math>, which soon becomes much smaller than <math>N.</math> It took only eight (8) steps to factor {{math|1550}} into prime numbers. {| class="wikitable" style="text-align: center; vertical-align: bottom;" |+ <math>\text{Establishing } 1550=2\cdot 5^2 \cdot 31</math> |- ! n ! k ! k^2 ! k^2<n? ! n/k ! mod ! replace ! factor ! * |- | 1550 | 2 | 4 | TRUE | 775 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"|n: n/k | style="background:#ffffcc"|2 | |- | 775 | 2 | 4 | TRUE | 387 | 1 | k: k+1 | | |- | 775 | 3 | 9 | TRUE | 258 | 1 | k: k+1 | | |- | 775 | 4 | 16 | TRUE | 193 | 3 | k: k+1 | | |- | 775 | 5 | 25 | TRUE | 155 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 155 | 5 | 25 | TRUE | 31 | style="background:#ffffcc"| '''0''' | style="background:#ffffcc"| n: n/k | style="background:#ffffcc"| 5 | |- | 31 | 5 | 25 | TRUE | 6 | 1 | k: k+1 | | |- | 31 | 6 | 36 | FALSE | - | - | - | | style="background:#ffffcc"| 31 |} ==Python code== ‎<syntaxhighlight lang="python" line="1"> ## In a python code, the function definitions precede the code. We need only 1 function: def prime_factorization(N): ## 12=(2**2)*(2**1)*(5**1), or equivalently, 12 = 2**2 * 3 * 5 ## The dictionary, dict, expresses k**v as a "key-value" pair. In other ## words, if we look up the key, k=5, the dictionary will return the ## value, v=1. In other words: ## k**v <=> key**value <=> base**exponent (for the factored form of N) ## Begin function: n=N #--------------------# n will change in the while loop (getting smaller.) dict = {} #--------------# Start with an empty dictionary if n==1: #---------------# The factorization of 1 is a special case. return {1: 1} # In other words 1 = 1**1 (but recall that 1 is not prime.) k = 2 #------------------# Our first candidate for factorizing any integer. while k**2 <= n: #-------# Do the following unless k is "too big": if n % k: #----------# . Equivalent to: "If k is not a factor of n:" k += 1 #---------# .. increase the value of k by 1 on next attempt. else: #--------------# . Equivalent to "If k is a factor of n:" n /= k #---------# .. Replace n by n/k (for next attempt) try: #-----------# .. If k is in the dictionary, this attempts to: dict[k] += 1 # ... increase the value of k by 1. except KeyError: # .. KeyError occurs if k is not in the dictionary dict[k] = 1 # ... Add k to dictionary and set exponent v=1 # end(while)#------------# (exits while k**2 <= n:) # Here I inserted a diagnostic to be discussed later: # Documented code continues: if n > 1: #--------------# It can be shown that n is either 1 or prime. try: #---------------# . Try to enter n into dictionary dict[n] += 1 #---# .. increase exponent v->v+1 if k in dictinary except KeyError: #---# . If n is not in the dictionary, dict[n] = 1 # .. then enter n into dictionary with v=1 return dict #------# python talk for ending function ################ End def prime_factorization | Begin program ########## #import sys again=True while again == 1: text="\nEnter a positive integer you wish to factor into prime numbers:\n" N=int(input(text)) dict=prime_factorization(N) print("\n"+str(int(N))+"=") text="(" for b in dict.keys(): text+=str(int(b))+"**"+str(int(dict.get(b)))+") * (" print(text.rstrip(" * (")+"\n") print("The dictionary is:\n") print(dict) again=int(input("\nType 1 for another integer, 0 to terminate.")) </syntaxhighlight> With one extra line a print command will display the dictionary. From the fact that python decided to print {{math|31}} as {{math|31.0}}, we know that this dictionary is taking the keys (2, 5, 31) to be floating point numbers. Python permits such a casual attitude regarding number types. This makes it easier to program python. But it more difficult to sort out such misunderstandings after the program has been written. print(dict) {2: 1, 5: 2, 31.0: 1} ===Comments=== =====Line 15: while k**2 <= n:===== This is an important line because it permits the termination of the search before one might think. I leave it as an exercise for the student to learn why any value of k larger than square root of n will never be the next factor of N. It is also left to the student to understand the fact that the time to find the factors can be cut nearly in half if we modify line 17 and increase k by 2 (instead of 1), and why this larger step cannot be taken until after k=2 has been considered. ====Lines in the code that might seem confusing==== The code uses two cleverness tricks that might seem confusing. I lack the formal training in programming to know whether such cleverness would be perceived as a flaw. I know that programmers are allowed to be clever and I know that clear programs are better than confusing programs. The fact that these tricks confused me does not necessarily mean they should be avoided. And, the nice thing about both sources of confusion is that they taught me something about python. =====Line 16: if n%k might seem backwards===== Python follows a common practice of interpreting the number 1 to be True and 0 to be False. But python also accepts any non-zero number as True. Here n%k denotes nmod&nbsp;k, which is the remainder when on divides n by k. If there is no remainder the "if" condition is not satisfied and the code stipulates that the next value of k is attempted. Any number not equal to zero is interpreted as the "if" condition being satisfied, meaning that k is a divisor of n (in that case, we replace n by n/k.) This might be good coding, but it seems to me that it would have been better to construct it along the lines of "if n%k == 0" or even "if n%k !=0". =====Lines 21-23 and 29-31: A failed attempt to change the value of a key===== In this code fragment, we ask if a given value of k has already been identified as a factor. But instead of "asking" if k has been declared a key in the dictionary, an <u>attempt</u> is made to change the value of k. If that attempt fails (because k is not among the dictionary's keys), then k is added to the list of keys, with a value of 1 (a value of 1 indicates that at this point, k, but not k² has been established to be a factor of n.) Again, there might be nothing wrong with using a key error like this. But the code would have been easier for me to follow if the if statement simply called the question of whether k was a key in dict. ==Sample output with a large number== Sometimes you have to wait a few minutes to factor a number this large. But usually it comes out in just a few seconds. I would imagine that a number that was the product of only two roughly equal prime numbers would take forever because k=k+1 must be repeated a virtually infinite number of times. This number came quickly: {{Font color|blue|Enter a positive integer you wish to factor into prime numbers:}} 99283428376482768916467328912876456416375474272263423467896879187289499287374899 &nbsp; <span style="color:blue>99283428376482768916467328912876456416375474272263423467896879187289499287374899=</span> {{Font color|blue|(619**1) * (688**1) * (768**3) * (1024**17) * (64853**1) * (5302548928**1)}} &nbsp; {{Font color|blue|The dictionary is:}} &nbsp; <span style="color:blue>{619: 1, 688: 1, 768: 3, 1024: 17, 64853: 1, 5302548928.0: 1}</span> &nbsp; {{Font color|blue|Type 1 for another integer, 0 to terminate.}} {{center|'''DO NOT TRY TO FACTOR A NUMBER THIS LARGE UNLESS YOU ARE WILLING TO INTERUPT YOUR PYTHON PROGRAM.'''}} {{center|Keep in mind that the universe is only about <math>4.4\times10^{26}\text{nanoseconds}</math> old.<ref>http://astronomy.nmsu.edu/geas/lectures/lecture02/slide04.html</ref>}} The number shown rounds up to 10⁸⁰. It is not likely that any computer will ever count that high. If the number to be factored consists of two prime number that are close to 10⁸⁰, the code will iterate all the way up to k coming close to 10⁴⁰. The inability of any computer to perform that many iterations helps explain how cryptocurrency works. The owner of cryptocurrency can all information to be made public by posting something with the properties of the product of two very large prime numbers. The product is made public and anybody with knowledge of one of the products can claim ownership of the funds. On the other hand, if you lose that information, nobody on earth can retrieve it for you. &nbsp; alydc5g69hkd07zr2riiz2j2fie667j 0 285730 2410345 2409783 2022-07-30T00:22:52Z Dave Braunschweig 426084 Advise wikitext text/x-wiki {{Advise|Ameloblastoma}} 96k5mm7kiicdms9cwwlkn4uegjxxecs 6 285772 2410229 2022-07-29T13:29:02Z Young1lim 21186 {{Information |Description=Laurent.5: Applications 6A (20220728 - 20220727) |Source={{own|Young1lim}} |Date=2022-07-29 |Author=Young W. Lim |Permission={{cc-by-sa-3.0,2.5,2.0,1.0}} }} wikitext text/x-wiki == Summary == {{Information |Description=Laurent.5: Applications 6A (20220728 - 20220727) |Source={{own|Young1lim}} |Date=2022-07-29 |Author=Young W. Lim |Permission={{cc-by-sa-3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} l9dr9t3npvyssofp43vzjn28p7j4zji 6 285773 2410230 2022-07-29T13:30:08Z Young1lim 21186 {{Information |Description=C04.Series.1: Arrays 1A (20220728 - 20220727) |Source={{own|Young1lim}} |Date=2022-07-29 |Author=Young W. Lim |Permission={{GFDL}} }} wikitext text/x-wiki == Summary == {{Information |Description=C04.Series.1: Arrays 1A (20220728 - 20220727) |Source={{own|Young1lim}} |Date=2022-07-29 |Author=Young W. Lim |Permission={{GFDL}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} noppi8ge0k5taouk30nqo6zln4c68x3 6 285774 2410231 2022-07-29T13:30:57Z Young1lim 21186 {{Information |Description=C04.Series.3: Array Pointers 1A (20220726 - 20220727) |Source={{own|Young1lim}} |Date=2022-07-29 |Author=Young W. Lim |Permission={{GFDL}} }} wikitext text/x-wiki == Summary == {{Information |Description=C04.Series.3: Array Pointers 1A (20220726 - 20220727) |Source={{own|Young1lim}} |Date=2022-07-29 |Author=Young W. Lim |Permission={{GFDL}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} 55pjo3nq6iqgvmnemc0906i2b01egje 2410234 2410231 2022-07-29T13:33:22Z Young1lim 21186 /* Summary */ wikitext text/x-wiki == Summary == {{Information |Description=C04.Series.3: Array Pointers 1A (20220728 - 20220727) |Source={{own|Young1lim}} |Date=2022-07-29 |Author=Young W. Lim |Permission={{GFDL}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} fn2eywxrvwj5s9gszclce35f6m0ruzi 6 285775 2410232 2022-07-29T13:31:53Z Young1lim 21186 {{Information |Description=C04.Series.1App: Applications of Arrays 1A (20220728 - 20220727) |Source={{own|Young1lim}} |Date=2022-07-29 |Author=Young W. Lim |Permission={{GFDL}} }} wikitext text/x-wiki == Summary == {{Information |Description=C04.Series.1App: Applications of Arrays 1A (20220728 - 20220727) |Source={{own|Young1lim}} |Date=2022-07-29 |Author=Young W. Lim |Permission={{GFDL}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} js3y3fxldzpvaak8ozzf1498kq7vla9 6 285776 2410233 2022-07-29T13:32:49Z Young1lim 21186 {{Information |Description=VLSI.Arith.1.A: Variable Block Adders (20220728 - 20220727) |Source={{own|Young1lim}} |Date=2022-07-29 |Author=Young W. Lim |Permission={{GFDL}} }} wikitext text/x-wiki == Summary == {{Information |Description=VLSI.Arith.1.A: Variable Block Adders (20220728 - 20220727) |Source={{own|Young1lim}} |Date=2022-07-29 |Author=Young W. Lim |Permission={{GFDL}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} dqmt6w3d9ecgahkbyecqmtfc9cra07l 2 285777 2410246 2022-07-29T16:11:25Z Mapunch 2947295 Page was created wikitext text/x-wiki ''Learning is very important they made it simple for us learning through internet with free basic skills thank you wikiversity'' 608s4u8kok9e7lljde9khnigutzjsia 3 285778 2410248 2022-07-29T16:30:52Z Mapunch 2947295 I need to get an EMR certificate .. I'm studying here in wikiversity and I got matric certificate only ...I want be an EMR wikitext text/x-wiki How to get certificate for your course e.g I'm metriculated I got certificate now Wikipedia is helping me with more free learning skills ..so how can I get certificate if I'm done studying here in wikiversity 90noiiqomsx5hckrlnkti0vwqt2ns3j 2410249 2410248 2022-07-29T16:37:27Z Mapunch 2947295 /* EMR */ new section wikitext text/x-wiki How to get certificate for your course e.g I'm metriculated I got certificate now Wikipedia is helping me with more free learning skills ..so how can I get certificate if I'm done studying here in wikiversity == EMR == I need to get more information about an EMR training course I want go for training don't know where I can find college that deal with an EMR training course so I can get certificate please help me if you can [[User:Mapunch|Mapunch]] ([[User talk:Mapunch|discuss]] • [[Special:Contributions/Mapunch|contribs]]) 16:37, 29 July 2022 (UTC) ca9gwoeewo4snuzwgtynyd4tz8mt605 6 285779 2410253 2022-07-29T18:59:32Z Young1lim 21186 {{Information |Description=ELF1.1E: Weak and Common Symbols (20220727 - 20220726) |Source={{own|Young1lim}} |Date=2022-07-29 |Author=Young W. Lim |Permission={{cc-by-sa-3.0,2.5,2.0,1.0}} }} wikitext text/x-wiki == Summary == {{Information |Description=ELF1.1E: Weak and Common Symbols (20220727 - 20220726) |Source={{own|Young1lim}} |Date=2022-07-29 |Author=Young W. Lim |Permission={{cc-by-sa-3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} t0jju0es5t714b7gwj7ipruqjwesl89 6 285780 2410255 2022-07-29T19:00:24Z Young1lim 21186 {{Information |Description=ELF1.1E: Weak and Common Symbols (20220728 - 20220727) |Source={{own|Young1lim}} |Date=2022-07-29 |Author=Young W. Lim |Permission={{cc-by-sa-3.0,2.5,2.0,1.0}} }} wikitext text/x-wiki == Summary == {{Information |Description=ELF1.1E: Weak and Common Symbols (20220728 - 20220727) |Source={{own|Young1lim}} |Date=2022-07-29 |Author=Young W. Lim |Permission={{cc-by-sa-3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} ahm6hb8jf3rxbk818v0yitzeqo3yn31 6 285781 2410295 2022-07-29T20:06:23Z Young1lim 21186 {{Information |Description=5MRV.3B: Stationary Random Process Examples (20220727 - 20220726) |Source={{own|Young1lim}} |Date=2022-07-29 |Author=Young W. Lim |Permission={{cc-by-sa-3.0,2.5,2.0,1.0}} }} wikitext text/x-wiki == Summary == {{Information |Description=5MRV.3B: Stationary Random Process Examples (20220727 - 20220726) |Source={{own|Young1lim}} |Date=2022-07-29 |Author=Young W. Lim |Permission={{cc-by-sa-3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} 5ojd6dbwzjhc7umy7mrafe5rpj39w0z 6 285782 2410298 2022-07-29T20:07:33Z Young1lim 21186 {{Information |Description=5MRV.3B: Stationary Random Process Examples (20220728 - 20220727) |Source={{own|Young1lim}} |Date=2022-07-29 |Author=Young W. Lim |Permission={{cc-by-sa-3.0,2.5,2.0,1.0}} }} wikitext text/x-wiki == Summary == {{Information |Description=5MRV.3B: Stationary Random Process Examples (20220728 - 20220727) |Source={{own|Young1lim}} |Date=2022-07-29 |Author=Young W. Lim |Permission={{cc-by-sa-3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} 3m8l4xpeigjh3z9z9gopyr96pv535z1 6 285783 2410309 2022-07-29T20:24:10Z Young1lim 21186 {{Information |Description=MP3.1F: Mutability - Lambda Calculus (20220728 - 20220727) |Source={{own|Young1lim}} |Date=2022-07-29 |Author=Young W. Lim |Permission={{GFDL}} }} wikitext text/x-wiki == Summary == {{Information |Description=MP3.1F: Mutability - Lambda Calculus (20220728 - 20220727) |Source={{own|Young1lim}} |Date=2022-07-29 |Author=Young W. Lim |Permission={{GFDL}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} o55myhyotjmz2dzzoleuq3c59eku53c 6 285784 2410311 2022-07-29T20:30:00Z Young1lim 21186 {{Information |Description=ARM.2ASM: Vector Interrupt Controller (20220728 - 20220727) |Source={{own|Young1lim}} |Date=2022-07-29 |Author=Young W. Lim |Permission={{GFDL}} }} wikitext text/x-wiki == Summary == {{Information |Description=ARM.2ASM: Vector Interrupt Controller (20220728 - 20220727) |Source={{own|Young1lim}} |Date=2022-07-29 |Author=Young W. Lim |Permission={{GFDL}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} 49gcmtfzzo52z4dgzat1o9i8bdo706t 0 285785 2410320 2022-07-29T22:21:38Z Aherman012 2943941 Creating extended page wikitext text/x-wiki <noinclude>{{Helping Give Away Psychological Science Banner}}</noinclude> {{medical disclaimer}} {{:{{BASEPAGENAME}}/Sidebar}} ==[[Evidence based assessment/Portfolio template/What is a "portfolio"|'''What is a "portfolio"?''']]== * For background information on what assessment portfolios are, click the link in the heading above. * Want even 'more' information about this topic? There's an extended version of this page [[Evidence-based assessment/Oppositional defiant disorder (assessment portfolio)/extended version|here]]. ==[[Evidence based assessment/Preparation phase|'''Preparation phase''']]== === Diagnostic criteria for oppositional defiant disorder=== {{blockquotetop}} <big>ICD-11 Diagnostic Criteria</big><br> <br> '''General Description:''' Oppositional defiant disorder is a persistent pattern (e.g., 6 months or more) of markedly defiant, disobedient, provocative or spiteful behaviour that occurs more frequently than is typically observed in individuals of comparable age and developmental level and that is not restricted to interaction with siblings. Oppositional defiant disorder may be manifest in prevailing, persistent angry or irritable mood, often accompanied by severe temper outbursts or in headstrong, argumentative and defiant behaviour. The behavior pattern is of sufficient severity to result in significant impairment in personal, family, social, educational, occupational or other important areas of functioning <br> <br> '''Oppositional Defiant Disorder With Chronic Irritability-Anger:''' All definitional requirements for oppositional defiant disorder are met. This form of oppositional defiant disorder is characterized by prevailing, persistent angry or irritable mood that may be present independent of any apparent provocation. The negative mood is often accompanied by regularly occurring severe temper outbursts that are grossly out of proportion in intensity or duration to the provocation. Chronic irritability and anger are characteristic of the individual’s functioning nearly every day, are observable across multiple settings or domains of functioning (e.g., home, school, social relationships), and are not restricted to the individual’s relationship with his/her parents or guardians. The pattern of chronic irritability and anger is not limited to occasional episodes (e.g., developmentally typical irritability) or discrete periods (e.g., irritable mood in the context of manic or depressive episodes). *Note: The ICD-11 lists 3 additional subcategories of oppositional defiant disorder with chronic irritability-anger (i.e., with limited prosocial emotions, with typical prosocial emotions, and unspecified). They can be found [https://icd.who.int/browse11/l-m/en#/http%3a%2f%2fid.who.int%2ficd%2fentity%2f818792113 here]. <br> '''Oppositional Defiant Disorder Without Chronic Irritability-Anger:''' Meets all definitional requirements for oppositional defiant disorder. This form of oppositional defiant disorder is not characterized by prevailing, persistent, angry or irritable mood, but does feature headstrong, argumentative, and defiant behavior. *Note: The ICD-11 lists 3 additional subcategories of oppositional defiant disorder without chronic irritability-anger (i.e., with limited prosocial emotions, with typical prosocial emotions, and unspecified). They can be found [https://icd.who.int/browse11/l-m/en#/http%3a%2f%2fid.who.int%2ficd%2fentity%2f736540987 here]. <br> <big>Changes in DSM-5</big> <br> The diagnostic criteria for ADHD changed slightly from DSM-IV to DSM-5. See the changes [https://www.ncbi.nlm.nih.gov/books/NBK519712/table/ch3.t14/ here]. {{blockquotebottom}}<ref>{{Cite web|url=https://icd.who.int/browse11/l-m/en#/http://id.who.int/icd/entity/1545599508|title=ICD-11 for Mortality and Morbidity Statistics|website=icd.who.int|access-date=2022-07-11}}</ref> ===Base rates of ODD in different clinical settings=== This section describes the demographic setting of the population(s) sampled, base rates of diagnosis, country/region sampled and the diagnostic method that was used. Using this information, clinicians will be able to anchor the rate of ODD that they are likely to see in their clinical practice. * '''''To see prevalence rates across multiple disorders, [[Evidence-based assessment/Preparation phase|click here.]]''''' {|class="wikitable sortable" border="1" |- ! Demography ! Setting !! Base Rate !! Diagnostic Method |- || Various locations across USA | Meta-analysis of 38 studies<ref>Canino G, Polanczyk G, Bauermeister JJ, et al. (2010) Does the prevalence of CD and ODD vary across cultures? Social Psychiatry Epidemiology, 45(7):695–704.</ref> || 3.3% || Varied |- || All of the U.S. | Nationally representative large-scale study (N = 3,119) <ref>Nock, M. K., Kazdin, A. E., Hiripi, E., & Kessler, R. C. (2007). Lifetime prevalence, correlates, and persistence of oppositional defiant disorder: results from the National Comorbidity Survey Replication. ''Journal of Child Psychology and Psychiatry, 48''(7), 703-713.</ref> || 10.2% (overall) 11.2% (males) 9.2% (females) || World Health Organization (WHO) Composite International Diagnostic Interview (CIDI)^r |- |Suburban and urban Colorado |Project to Learn about Youth-Mental health, school-based study for children from kindergarten to high-school (n = 236)<ref name=":1">{{Cite journal|last=Danielson|first=Melissa L.|last2=Bitsko|first2=Rebecca H.|last3=Holbrook|first3=Joseph R.|last4=Charania|first4=Sana N.|last5=Claussen|first5=Angelika H.|last6=McKeown|first6=Robert E.|last7=Cuffe|first7=Steven P.|last8=Owens|first8=Julie Sarno|last9=Evans|first9=Steven W.|date=2021-06-01|title=Community-Based Prevalence of Externalizing and Internalizing Disorders among School-Aged Children and Adolescents in Four Geographically Dispersed School Districts in the United States|url=https://doi.org/10.1007/s10578-020-01027-z|journal=Child Psychiatry & Human Development|language=en|volume=52|issue=3|pages=500–514|doi=10.1007/s10578-020-01027-z|issn=1573-3327|pmc=PMC8016018|pmid=32734339}}</ref> |6.8% |Diagnostic Interview Schedule for Children (DISC) |- |Urban and suburban Florida |Project to Learn about Youth-Mental health, school-based study for children from kindergarten to high-school (n = 289)<ref name=":1" /> |6.9% |Diagnostic Interview Schedule for Children (DISC) |- |Rural and suburban Ohio |Project to Learn about Youth-Mental health, school-based study for children from elementary to high-school (n = 152)<ref name=":1" /> |17.3% |Diagnostic Interview Schedule for Children (DISC) |- |Suburban and rural South Carolina |Project to Learn about Youth-Mental health, school-based study for children from elementary to high-school (n = 270)<ref name=":1" /> |5.7% |Diagnostic Interview Schedule for Children (DISC) |- || Semi-rural North Carolina | Preschool-aged children from pediatric practices (N = 306; age 2 - 5 years old)<ref>Egger, H.L., & Angold, A. (2006). Common emotional and behavioral disorders in preschool children: Presentation, nosology, and epidemiology. ''Journal of Child Psychiatry and Psychology, 47'', 313–337.</ref> || 6.6% || Preschool Age Psychiatric Assessment (PAPA)^p |- || Western North Carolina | The Great Smoky Mountains Study - longitudinal, population-based study of community sample<ref>Costello, E. J., Mustillo, S., Erkanli, A., Keeler, G., & Angold, A. (2003) Prevalence and development of psychiatric disorders in adolescence. ''Arch Gen Psychiatry, 60'', 837-844.</ref> || 2.33% (overall) 3.16% (males) 2.75% (females) || Child and Adolescent Psychiatric Assessment (CAPA)^{p, y} |- || Chicago | Preschool-aged children from inner city schools and pediatric practices (N = 796; age 2 - 5 years old)<ref>Lavigne, J. V., LeBailly, S. A., Hopkins, J., Gouze, K. R., & Binns, H. J. (2009). The prevalence of ADHD, ODD, depression, and anxiety in a community sample of 4-year-olds. ''Journal Of Clinical Child And Adolescent Psychology, 38''(3), 315-328. doi:10.1080/15374410902851382</ref> || 8.3% || Diagnostic Interview Schedule for Children–Parent Scale–Young Child Version (DISC-YC) ^p |- |Germany (Saarbrücken County) |All school-aged children were examined during a routine school-entry medical examination (N = 1676, mean age = 5.7)<ref>{{Cite journal|last=Niemczyk|first=Justine|last2=Equit|first2=Monika|last3=Braun-Bither|first3=Katrin|last4=Klein|first4=Anna-Maria|last5=von Gontard|first5=Alexander|date=2015-07-01|title=Prevalence of incontinence, attention deficit/hyperactivity disorder and oppositional defiant disorder in preschool children|url=https://doi.org/10.1007/s00787-014-0628-6|journal=European Child & Adolescent Psychiatry|language=en|volume=24|issue=7|pages=837–843|doi=10.1007/s00787-014-0628-6|issn=1435-165X}}</ref> |7.3% (males) 5.1%(females) |DISYPS-II |- |South Korea (Seoul) |Children were randomly surveyed across 6 school districts in Seoul (N = 1645, age 6 - 12 years old) |5.8% (males) 4.1%(females) |Diagnostic Interview Schedule for Children–Parent Scale IV (DISC-IV) |} ^p Parent interviewed as part of diagnostic assessment; ^y youth interviewed as part of diagnostic assessment, ^r adult interviewed for retrospective report as part of diagnostic assessment '''Note:''' Mash and Barkley note that prevalence rates of ODD must be qualified, because the definition of ODD has changed at a fast rate, the rates adolescents meeting criteria in any cross-sectional evaluation may be misleading because of the developmental progressions with and between ODD and Conduct Disorder, and categorical definitions of aggressive patterns may reflect arbitrary numbers of constituent estimates. These factors may lead to misleading prevalence rates. In addition, few studies have investigated the prevalence of ODD in preschool-aged children, and early onset of these behaviors is associated with more severe and stable impairment.<ref>Mash, E., & Barkley, R. (Eds.). (2003). ''Child Psychopathology''. 2nd Edition. New York: Guilford Press.</ref> ==[[Evidence based assessment/Prediction phase|'''Prediction phase''']]== The following section contains a list of screening and diagnostic instruments for ODD. The section includes administration information, psychometric data, and PDFs or links to the screenings. * Screenings are used as part of the [[Evidence based assessment/Prediction phase|prediction phase]] of assessment; for more information on interpretation of this data, or how screenings fit in to the assessment process, click [[Evidence based assessment/Prediction phase|here]]. * '''''For a list of more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&wteswitched=1#Psychometric_properties_of_common_screening_instruments click here.]''''' {| class="wikitable sortable" border="1" ! colspan="10" |Screening measures for ODD |- ! Measure ! Format (Reporter) ! Age Range ! Administration/ Completion Time ! Interrater Reliability ! Test-Retest Reliability ! Construct Validity ! Content Validity ! Highly Recommended !Where to access |- |Achenbach System of Empirically Based Assessments (ASEBA): Child Behavior Checklist (CBCL) |Parent report |6-18 <ref name=":9">{{Cite book|url=https://www.worldcat.org/oclc/1130319849|title=Assessment of disorders in childhood and adolescence|date=2020|others=Eric Arden Youngstrom, Mitchell J. Prinstein, Eric J. Mash, Russell A. Barkley|isbn=978-1-4625-4363-2|edition=Fifth edition|location=New York, NY|oclc=1130319849}}</ref> |10 - 15 minutes<ref name=":9" /> |A<ref name=":0">{{Cite book|url=https://www.worldcat.org/oclc/314222270|title=A guide to assessments that work|author=Hunsley, John|author2=Mash, Eric J.|date=2008|publisher=Oxford University Press|isbn=9780195310641|location=New York|oclc=314222270}}</ref> |E<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://store.aseba.org/School-Age-6-18-Materials/departments/11/ Purchase] |- |ASEBA: Teacher Report Form (TRF) |Teacher report |6-18 <ref name=":9" /> |10 - 15 minutes<ref name=":9" /> |A<ref name=":0" /> |E<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://store.aseba.org/School-Age-6-18-Materials/departments/11/ Purchase] |- |ASEBA: Youth Self-Report (YSR) |Youth self-report |11-18 <ref name=":9" /> |10 - 15 minutes<ref name=":9" /> |A<ref name=":0" /> |E<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://store.aseba.org/School-Age-6-18-Materials/departments/11/ Purchase] |- |Behavior Assessment for Children, Third Edition (BASC-3): Parent Rating Scale |Parent report |2-21 |10 - 20 minutes |A<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |E<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://www.pearsonassessments.com/store/usassessments/en/Store/Professional-Assessments/Behavior/Comprehensive/Behavior-Assessment-System-for-Children-%7C-Third-Edition-/p/100001402.html#:~:text=The%20BASC%E2%84%A2%20holds%20an,or%20adolescent's%20behavior%20and%20emotions. Purchase] |- |BASC-3: Teacher Rating Scale |Teacher report |2-21 |10 - 20 minutes |A<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |E<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://www.pearsonassessments.com/store/usassessments/en/Store/Professional-Assessments/Behavior/Comprehensive/Behavior-Assessment-System-for-Children-%7C-Third-Edition-/p/100001402.html#:~:text=The%20BASC%E2%84%A2%20holds%20an,or%20adolescent's%20behavior%20and%20emotions. Purchase] |- |BASC-3: Self-Report of Personality |Youth self-report |6 - college age |30 minutes |A<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |E<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://www.pearsonassessments.com/store/usassessments/en/Store/Professional-Assessments/Behavior/Comprehensive/Behavior-Assessment-System-for-Children-%7C-Third-Edition-/p/100001402.html#:~:text=The%20BASC%E2%84%A2%20holds%20an,or%20adolescent's%20behavior%20and%20emotions. Purchase] |- |Eyberg Child Behavior Inventory (ECBI) |Parent report |2-16 |5 minutes |A<ref name=":0" /> |G<ref name=":0" /> |E<ref name=":0" /> |E<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://www.parinc.com/products/pkey/97 Purchase] |- |Sutter-Eyberg Student Behavior Inventory - Revised (SESBI-R) |Teacher report |2-16 |5 minutes |A<ref name=":0" /> |G<ref name=":0" /> |E<ref name=":0" /> |E<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://www.parinc.com/products/pkey/97 Purchase] |- |Child and Adolescent Behavior Inventory (CABI) |Parent Report |3 - 18 |5 - 10 minutes | |A<ref>{{Cite journal|last=Burns|first=G. Leonard|last2=Preszler|first2=Jonathan|last3=Becker|first3=Stephen P.|date=2022-07-04|title=Psychometric and Normative Information on the Child and Adolescent Behavior Inventory in a Nationally Representative Sample of United States Children|url=https://doi.org/10.1080/15374416.2020.1852943|journal=Journal of Clinical Child & Adolescent Psychology|volume=51|issue=4|pages=443–452|doi=10.1080/15374416.2020.1852943|issn=1537-4416|pmc=PMC8272731|pmid=33428463}}</ref> | | | |PDF |- |Strengths and Difficulties Questionnaire (SDQ) |Parent report |3 - 16 |3 - 5 minutes |E<ref name=":10">{{Cite journal|last=Goodman|first=Robert|date=2001-11-01|title=Psychometric Properties of the Strengths and Difficulties Questionnaire|url=https://www.jaacap.org/article/S0890-8567(09)60543-8/abstract|journal=Journal of the American Academy of Child & Adolescent Psychiatry|language=English|volume=40|issue=11|pages=1337–1345|doi=10.1097/00004583-200111000-00015|issn=0890-8567|pmid=11699809}}</ref> |G<ref name=":10" /> | | | |[https://www.sdqinfo.org/a0.html SDQ Homepage][https://www.sdqinfo.org/py/sdqinfo/b0.py PDFs] |- |Disruptive Behavior Disorder Rating Scale |Parent report, teacher report |5 - 17 |5 - 10 minutes | | | | | |[https://web.archive.org/web/20151123022653/http://ccf.buffalo.edu/pdf/DBD_rating_scale.pdf PDF] |} '''Note:''' L = Less than adequate; A = Adequate; G = Good; E = Excellent; U = Unavailable; NA = Not applicable === Likelihood ratios and AUCs of screening instruments for ODD === * '''''For a list of the likelihood ratios for more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&wteswitched=1#Likelihood_ratios_and_AUCs_of_common_screening_instruments click here.]''''' {| class="wikitable sortable" border="1" ! Screening Measure (Primary Reference) ! Area Under Curve (AUC) ! LR+ (Score) ! LR- (Score) ! Clinical Generalizability !Where to Access |- |CBCL DSM-Oriented Scales<ref name=":7">Achenbach, T. M. (1991a). ''Manual for the Child Behavior Checklist/4–18 and 1991 Profile''. Burlington , VT : University of Vermont Department of Psychiatry.</ref> |.71 (N=475)<ref name=":3">Ebesutani, C., Bernstein, A., Nakamura, B. J., Chorpita, B. F., Higa-McMillan, C. K., & Weisz, J. R. (2010). Concurrent validity of the child behavior checklist DSM-oriented scales: Correspondence with DSM diagnoses and comparison to syndrome scales. ''Journal of Psychopathology and Behavioral Assessment, 32''(3), 373-384.</ref> | 2.80 (60+ to 70+) <ref name=":4">Warnick, E. M., Bracken, M. B., & Kasl, S. (2008). Screening efficiency of the Child Behavior Checklist and Strengths and Difficulties Questionnaire: a systematic review. ''Child and Adolescent Mental Health, 13''(3), 140-147.</ref> | .52 (60- to 70-)*<ref name=":4" /> |Youth aged 5 - 18 seeking out patient treatment across a variety of settings<ref name=":4" /> |[https://store.aseba.org/School-Age-6-18-Materials/departments/11/ Purchase] |- |SDQ<ref name=":6">{{Cite journal|last=Shabani|first=Amir|last2=Masoumian|first2=Samira|last3=Zamirinejad|first3=Somayeh|last4=Hejri|first4=Maryam|last5=Pirmorad|first5=Tahereh|last6=Yaghmaeezadeh|first6=Hooman|date=2021-05|title=Psychometric properties of Structured Clinical Interview for DSM‐5 Disorders‐Clinician Version (SCID‐5‐CV)|url=https://onlinelibrary.wiley.com/doi/10.1002/brb3.1894|journal=Brain and Behavior|language=en|volume=11|issue=5|doi=10.1002/brb3.1894|issn=2162-3279|pmc=PMC8119811|pmid=33729681}}</ref> | .81 -.88 (N = 18,416)<ref>{{Cite journal|last=Algorta|first=Guillermo Perez|last2=Dodd|first2=Alyson Lamont|last3=Stringaris|first3=Argyris|last4=Youngstrom|first4=Eric A.|date=2016-09|title=Diagnostic efficiency of the SDQ for parents to identify ADHD in the UK: a ROC analysis|url=http://link.springer.com/10.1007/s00787-015-0815-0|journal=European Child & Adolescent Psychiatry|language=en|volume=25|issue=9|pages=949–957|doi=10.1007/s00787-015-0815-0|issn=1018-8827|pmc=PMC4990620|pmid=26762184}}</ref> |7.00 <ref name=":4" /> |.55<ref name=":4" /> |Youth aged 5 - 18 seeking out patient treatment across a variety of settings<ref name=":4" /> |[https://www.sdqinfo.org/a0.html SDQ Homepage][https://www.sdqinfo.org/py/sdqinfo/b0.py PDFs] |- |SDQ - DSM -IV Conduct and ODD | |7.55 (3+)<ref name=":2">{{Cite journal|last=Goodman|first=Robert|date=2001-11-01|title=Psychometric Properties of the Strengths and Difficulties Questionnaire|url=https://www.jaacap.org/article/S0890-8567(09)60543-8/abstract|journal=Journal of the American Academy of Child & Adolescent Psychiatry|language=English|volume=40|issue=11|pages=1337–1345|doi=10.1097/00004583-200111000-00015|issn=0890-8567|pmid=11699809}}</ref> |.35 (3-)<ref name=":2" /> |Surveyed youth aged 5 - 15 in the UK <ref name=":2" /> |[https://www.sdqinfo.org/a0.html SDQ Homepage][https://www.sdqinfo.org/py/sdqinfo/b0.py PDFs] |- |ECBI- Intensity Scale<ref name=":8">Eyberg, S. M., & Robinson, E. A. (1983). Conduct problem behavior: Standardization of a behavioral rating scale with adolescents. Journal of Clinical Child Psychology, 12 (3), 347-354.</ref> | |6.92 (131+)<ref name=":5">{{Cite journal|last=LYNEHAM|first=HEIDI J.|last2=ABBOTT|first2=MAREE J.|last3=RAPEE|first3=RONALD M.|date=2007-06|title=Interrater Reliability of the Anxiety Disorders Interview Schedule for DSM-IV: Child and Parent Version|url=https://doi.org/10.1097/chi.0b013e3180465a09|journal=Journal of the American Academy of Child &amp; Adolescent Psychiatry|volume=46|issue=6|pages=731–736|doi=10.1097/chi.0b013e3180465a09|issn=0890-8567}}</ref> |.11 (131-)<ref name=":5" /> |Youth aged 7-16 had responses compared to diagnosis<ref name=":5" /> |[https://www.parinc.com/products/pkey/97 Purchase] |- |BASC-2 PRS - Aggression |.76 (N =156)<ref name=":12">{{Cite journal|last=Doyle|first=Alysa|last2=Ostrander|first2=Rick|last3=Skare|first3=Stacy|last4=Crosby|first4=Ross D.|last5=August|first5=Gerald J.|date=1997-09-01|title=Convergent and criterion-related validity of the behavior assessment system for children-parent rating scale|url=https://doi.org/10.1207/s15374424jccp2603_6|journal=Journal of Clinical Child Psychology|volume=26|issue=3|pages=276–284|doi=10.1207/s15374424jccp2603_6|issn=0047-228X|pmid=9292385}}</ref> |3.62 (65+)<ref name=":12" /> |.60 (65-)<ref name=":12" /> |Youth from first through fourth grade who were at risk for CD<ref name=":12" /> |[https://www.pearsonassessments.com/store/usassessments/en/Store/Professional-Assessments/Behavior/Comprehensive/Behavior-Assessment-System-for-Children-%7C-Third-Edition-/p/100001402.html#:~:text=The%20BASC%E2%84%A2%20holds%20an,or%20adolescent's%20behavior%20and%20emotions. Purchase] |} '''Note:''' “LR+” refers to the change in likelihood ratio associated with a positive test score, and “LR-” is the likelihood ratio for a low score. Likelihood ratios of 1 indicate that the test result did not change impressions at all. LRs larger than 10 or smaller than .10 are frequently clinically decisive; 5 or .20 are helpful, and between 2.0 and .5 are small enough that they rarely result in clinically meaningful changes of formulation<ref>Sackett, D. L., Straus, S. E., Richardson, W. S., Rosenberg, W., & Haynes, R. B. (2000). Evidence-based medicine: How to practice and teach EBM. Edinburgh: Churchill Livingstone.</ref>. '''Search terms:''' [Oppositional Defiant Disorder] AND [sensitivity OR specificity] in GoogleScholar and PsychINFO; === Interpreting ODD screening measure scores === * For information on interpreting screening measure scores, click [[Evidence based assessment/Prediction phase#Interpreting screening measure scores|here.]] == [[Evidence based assessment/Prescription phase|'''Prescription phase''']] == === Gold standard diagnostic interviews === * For a list of broad reaching diagnostic interviews sortable by disorder with PDFs (if applicable), [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prescription_phase&wteswitched=1#Common_Diagnostic_Interviews click here.] === Recommended diagnostic instruments for ODD === {| class="wikitable sortable" ! colspan="10" |Diagnostic instruments for ODD |- !Measure !Format (Reporter) !Age Range !Administration/ Completion Time !Interrater Reliability !Test-Retest Reliability !Construct Validity !Content Validity !Highly Recommended !Where to Access |- |Kiddie Schedule for Affective Disorders and Schizophrenia Present and Lifetime Version (KSADS-PL) |Structured interview |6-28 |45-75 minutes |E<ref name=":11">{{Cite journal|last=KAUFMAN|first=JOAN|last2=BIRMAHER|first2=BORIS|last3=BRENT|first3=DAVID|last4=RAO|first4=UMA|last5=FLYNN|first5=CYNTHIA|last6=MORECI|first6=PAULA|last7=WILLIAMSON|first7=DOUGLAS|last8=RYAN|first8=NEAL|date=1997-07|title=Schedule for Affective Disorders and Schizophrenia for School-Age Children-Present and Lifetime Version (K-SADS-PL): Initial Reliability and Validity Data|url=https://doi.org/10.1097/00004583-199707000-00021|journal=Journal of the American Academy of Child &amp; Adolescent Psychiatry|volume=36|issue=7|pages=980–988|doi=10.1097/00004583-199707000-00021|issn=0890-8567}}</ref> |G<ref name=":11" /> | | | |[https://www.pediatricbipolar.pitt.edu/resources/instruments Website to access] |- |Diagnostic Interview Schedule for Children (DISC-5) |Structured Interview (Self report and parent) |6-17 | | | | | |<ref name=":0" />[[File:Light green check.svg|center|frameless|36x36px]] |[https://telesage.com/netdisc-5/# Coming soon] |} '''Note:''' L = Less than adequate; A = Adequate; G = Good; E = Excellent; U = Unavailable; NA = Not applicable == [[Evidence based assessment/Process phase|'''Process phase''']] == The following section contains a list of process and outcome measures for oppositional defiant disorder. The section includes benchmarks based on published norms for several outcome and severity measures, as well as information about commonly used process measures. Process and outcome measures are used as part of the [[Evidence based assessment/Process phase|process phase]] of assessment. For more information of differences between process and outcome measures, see the page on the [[Evidence based assessment/Process phase|process phase of assessment]]. === Outcome and severity measures === * This table includes clinically significant benchmarks for generalized anxiety disorder specific outcome measures * Information on how to interpret this table can be [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase found here]. * Additionally, these [[Evidence based assessment/Vignettes|vignettes]] might be helpful resources for understanding appropriate adaptation of outcome measures in practice. * ''<u>For clinically significant change benchmarks for the CBCL, YSR, and TRF total, externalizing, internalizing, and attention benchmarks,</u>'' [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase&wteswitched=1#Clinically_significant_change_benchmarks_for_widely-used_outcome_measures see here.] {| class="wikitable sortable" border="1" |- | rowspan="1&quot;" style="text-align:center;font-size:130%;" | <b> Measure</b> | style="text-align:center;font-size:130%;" | <b> Subscale</b> | colspan="3" style="text-align:center;font-size:130%" width="300" | <b> Cut-off scores</b> | colspan="3" style="text-align:center;font-size:120%" | <b> Critical Change <br> (unstandardized scores)</b> |- | colspan="8" style="font-size:110%; text-align:center;" span | <b> Benchmarks Based on Published Norms</b> |- | colspan="2" | | style="text-align:center;font-size:110%" | <b> A</b> | style="text-align:center;font-size:110%" | <b> B</b> | style="text-align:center;font-size:110%" | <b> C</b> | style="text-align:center;font-size:110%" | <b> 95%</b> | style="text-align:center;font-size:110%" | <b> 90%</b> | style="text-align:center;font-size:110%" | <b> SE_difference</b> |- | rowspan="1" style="text-align:center;" | <b> CBCL T-scores <br> (2001 Norms)</b> | style="text-align:right;" | <i> Externalizing</i> | style="text-align:center;" | 49 | style="text-align:center;" | 70 | style="text-align:center;" | 58 | style="text-align:center;" | 7 | style="text-align:center;" | 6 | style="text-align:center;" | 3.4 |- | rowspan="1" style="text-align:center;" | <b> ECBI Scaled Scores <br> (1983 Norms)</b> | style="text-align:right;" | <i> Intensity</i> | style="text-align:center;" | 80.1 | style="text-align:center;" | 169.5 | style="text-align:center;" | 112.9 | style="text-align:center;" | 9.5 | style="text-align:center;" | 8 | style="text-align:center;" | 4.8 |- | rowspan="1" style="text-align:center;" | <b> ECBI Scaled Scores <br> (1983 Norms)</b> | style="text-align:right;" | <i> Problem</i> | style="text-align:center;" | 3.9 | style="text-align:center;" | 17.7 | style="text-align:center;" | 11.5 | style="text-align:center;" | 2.1 | style="text-align:center;" | 1.8 | style="text-align:center;" | 1.1 |} '''Note:''' “A” = Away from the clinical range, “B” = Back into the nonclinical range, “C” = Closer to the nonclinical than clinical mean. === Process measures=== See Section 1.1 for overview of evidence-based measures to use depending on etiology and symptomatology of Oppositional Defiant Disorder. ==Treatment== ===Behavioral parent training=== Behavioral Parent Training is considered the most effective treatment for childhood disruptive behavior disorders (e.g., Oppositional Defiant Disorder), especially for younger children (i.e., 3-8 year-olds). See http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/rule-breaking-defiance-and-acting-out/ a website sponsored by The Society for Child and Adolescent Psychology (APA, Division 53) and the Association for Behavioral and Cognitive Therapies (ABCT), for current summary of evidence-based treatments. ===Overview of recommendations for assessment and treatment=== See the [https://pathways.nice.org.uk/pathways/antisocial-behaviour-and-conduct-disorders-in-children-and-young-people#content=view-info-category%3Aview-about-menu National Institute for Health and Care Excellence (NICE) Practice Guidelines for Childhood Conduct Disorders], for an overview of recommendations for both assessment and treatment of Oppositional Defiant Disorder. == External Links == *[https://sccap53.org Society of Clinical Child and Adolescent Psychology] *http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/rule-breaking-defiance-and-acting-out/ == References == {{collapse top|Click here for references}} {{Reflist|30em}} [[Category:Psychological disorder portfolios|{{SUBPAGENAME}}]] {{collapse bottom}} [[Category:Psychological disorder portfolios|{{SUBPAGENAME}}]] oy9lh467ddh8ujhbzz7r3j5ym01xk45 2410321 2410320 2022-07-29T22:22:10Z Aherman012 2943941 Updated sidebar wikitext text/x-wiki <noinclude>{{Helping Give Away Psychological Science Banner}}</noinclude> {{medical disclaimer}} {{:{{ROOTPAGENAME}}/Sidebar}} ==[[Evidence based assessment/Portfolio template/What is a "portfolio"|'''What is a "portfolio"?''']]== * For background information on what assessment portfolios are, click the link in the heading above. * Want even 'more' information about this topic? There's an extended version of this page [[Evidence-based assessment/Oppositional defiant disorder (assessment portfolio)/extended version|here]]. ==[[Evidence based assessment/Preparation phase|'''Preparation phase''']]== === Diagnostic criteria for oppositional defiant disorder=== {{blockquotetop}} <big>ICD-11 Diagnostic Criteria</big><br> <br> '''General Description:''' Oppositional defiant disorder is a persistent pattern (e.g., 6 months or more) of markedly defiant, disobedient, provocative or spiteful behaviour that occurs more frequently than is typically observed in individuals of comparable age and developmental level and that is not restricted to interaction with siblings. Oppositional defiant disorder may be manifest in prevailing, persistent angry or irritable mood, often accompanied by severe temper outbursts or in headstrong, argumentative and defiant behaviour. The behavior pattern is of sufficient severity to result in significant impairment in personal, family, social, educational, occupational or other important areas of functioning <br> <br> '''Oppositional Defiant Disorder With Chronic Irritability-Anger:''' All definitional requirements for oppositional defiant disorder are met. This form of oppositional defiant disorder is characterized by prevailing, persistent angry or irritable mood that may be present independent of any apparent provocation. The negative mood is often accompanied by regularly occurring severe temper outbursts that are grossly out of proportion in intensity or duration to the provocation. Chronic irritability and anger are characteristic of the individual’s functioning nearly every day, are observable across multiple settings or domains of functioning (e.g., home, school, social relationships), and are not restricted to the individual’s relationship with his/her parents or guardians. The pattern of chronic irritability and anger is not limited to occasional episodes (e.g., developmentally typical irritability) or discrete periods (e.g., irritable mood in the context of manic or depressive episodes). *Note: The ICD-11 lists 3 additional subcategories of oppositional defiant disorder with chronic irritability-anger (i.e., with limited prosocial emotions, with typical prosocial emotions, and unspecified). They can be found [https://icd.who.int/browse11/l-m/en#/http%3a%2f%2fid.who.int%2ficd%2fentity%2f818792113 here]. <br> '''Oppositional Defiant Disorder Without Chronic Irritability-Anger:''' Meets all definitional requirements for oppositional defiant disorder. This form of oppositional defiant disorder is not characterized by prevailing, persistent, angry or irritable mood, but does feature headstrong, argumentative, and defiant behavior. *Note: The ICD-11 lists 3 additional subcategories of oppositional defiant disorder without chronic irritability-anger (i.e., with limited prosocial emotions, with typical prosocial emotions, and unspecified). They can be found [https://icd.who.int/browse11/l-m/en#/http%3a%2f%2fid.who.int%2ficd%2fentity%2f736540987 here]. <br> <big>Changes in DSM-5</big> <br> The diagnostic criteria for ADHD changed slightly from DSM-IV to DSM-5. See the changes [https://www.ncbi.nlm.nih.gov/books/NBK519712/table/ch3.t14/ here]. {{blockquotebottom}}<ref>{{Cite web|url=https://icd.who.int/browse11/l-m/en#/http://id.who.int/icd/entity/1545599508|title=ICD-11 for Mortality and Morbidity Statistics|website=icd.who.int|access-date=2022-07-11}}</ref> ===Base rates of ODD in different clinical settings=== This section describes the demographic setting of the population(s) sampled, base rates of diagnosis, country/region sampled and the diagnostic method that was used. Using this information, clinicians will be able to anchor the rate of ODD that they are likely to see in their clinical practice. * '''''To see prevalence rates across multiple disorders, [[Evidence-based assessment/Preparation phase|click here.]]''''' {|class="wikitable sortable" border="1" |- ! Demography ! Setting !! Base Rate !! Diagnostic Method |- || Various locations across USA | Meta-analysis of 38 studies<ref>Canino G, Polanczyk G, Bauermeister JJ, et al. (2010) Does the prevalence of CD and ODD vary across cultures? Social Psychiatry Epidemiology, 45(7):695–704.</ref> || 3.3% || Varied |- || All of the U.S. | Nationally representative large-scale study (N = 3,119) <ref>Nock, M. K., Kazdin, A. E., Hiripi, E., & Kessler, R. C. (2007). Lifetime prevalence, correlates, and persistence of oppositional defiant disorder: results from the National Comorbidity Survey Replication. ''Journal of Child Psychology and Psychiatry, 48''(7), 703-713.</ref> || 10.2% (overall) 11.2% (males) 9.2% (females) || World Health Organization (WHO) Composite International Diagnostic Interview (CIDI)^r |- |Suburban and urban Colorado |Project to Learn about Youth-Mental health, school-based study for children from kindergarten to high-school (n = 236)<ref name=":1">{{Cite journal|last=Danielson|first=Melissa L.|last2=Bitsko|first2=Rebecca H.|last3=Holbrook|first3=Joseph R.|last4=Charania|first4=Sana N.|last5=Claussen|first5=Angelika H.|last6=McKeown|first6=Robert E.|last7=Cuffe|first7=Steven P.|last8=Owens|first8=Julie Sarno|last9=Evans|first9=Steven W.|date=2021-06-01|title=Community-Based Prevalence of Externalizing and Internalizing Disorders among School-Aged Children and Adolescents in Four Geographically Dispersed School Districts in the United States|url=https://doi.org/10.1007/s10578-020-01027-z|journal=Child Psychiatry & Human Development|language=en|volume=52|issue=3|pages=500–514|doi=10.1007/s10578-020-01027-z|issn=1573-3327|pmc=PMC8016018|pmid=32734339}}</ref> |6.8% |Diagnostic Interview Schedule for Children (DISC) |- |Urban and suburban Florida |Project to Learn about Youth-Mental health, school-based study for children from kindergarten to high-school (n = 289)<ref name=":1" /> |6.9% |Diagnostic Interview Schedule for Children (DISC) |- |Rural and suburban Ohio |Project to Learn about Youth-Mental health, school-based study for children from elementary to high-school (n = 152)<ref name=":1" /> |17.3% |Diagnostic Interview Schedule for Children (DISC) |- |Suburban and rural South Carolina |Project to Learn about Youth-Mental health, school-based study for children from elementary to high-school (n = 270)<ref name=":1" /> |5.7% |Diagnostic Interview Schedule for Children (DISC) |- || Semi-rural North Carolina | Preschool-aged children from pediatric practices (N = 306; age 2 - 5 years old)<ref>Egger, H.L., & Angold, A. (2006). Common emotional and behavioral disorders in preschool children: Presentation, nosology, and epidemiology. ''Journal of Child Psychiatry and Psychology, 47'', 313–337.</ref> || 6.6% || Preschool Age Psychiatric Assessment (PAPA)^p |- || Western North Carolina | The Great Smoky Mountains Study - longitudinal, population-based study of community sample<ref>Costello, E. J., Mustillo, S., Erkanli, A., Keeler, G., & Angold, A. (2003) Prevalence and development of psychiatric disorders in adolescence. ''Arch Gen Psychiatry, 60'', 837-844.</ref> || 2.33% (overall) 3.16% (males) 2.75% (females) || Child and Adolescent Psychiatric Assessment (CAPA)^{p, y} |- || Chicago | Preschool-aged children from inner city schools and pediatric practices (N = 796; age 2 - 5 years old)<ref>Lavigne, J. V., LeBailly, S. A., Hopkins, J., Gouze, K. R., & Binns, H. J. (2009). The prevalence of ADHD, ODD, depression, and anxiety in a community sample of 4-year-olds. ''Journal Of Clinical Child And Adolescent Psychology, 38''(3), 315-328. doi:10.1080/15374410902851382</ref> || 8.3% || Diagnostic Interview Schedule for Children–Parent Scale–Young Child Version (DISC-YC) ^p |- |Germany (Saarbrücken County) |All school-aged children were examined during a routine school-entry medical examination (N = 1676, mean age = 5.7)<ref>{{Cite journal|last=Niemczyk|first=Justine|last2=Equit|first2=Monika|last3=Braun-Bither|first3=Katrin|last4=Klein|first4=Anna-Maria|last5=von Gontard|first5=Alexander|date=2015-07-01|title=Prevalence of incontinence, attention deficit/hyperactivity disorder and oppositional defiant disorder in preschool children|url=https://doi.org/10.1007/s00787-014-0628-6|journal=European Child & Adolescent Psychiatry|language=en|volume=24|issue=7|pages=837–843|doi=10.1007/s00787-014-0628-6|issn=1435-165X}}</ref> |7.3% (males) 5.1%(females) |DISYPS-II |- |South Korea (Seoul) |Children were randomly surveyed across 6 school districts in Seoul (N = 1645, age 6 - 12 years old) |5.8% (males) 4.1%(females) |Diagnostic Interview Schedule for Children–Parent Scale IV (DISC-IV) |} ^p Parent interviewed as part of diagnostic assessment; ^y youth interviewed as part of diagnostic assessment, ^r adult interviewed for retrospective report as part of diagnostic assessment '''Note:''' Mash and Barkley note that prevalence rates of ODD must be qualified, because the definition of ODD has changed at a fast rate, the rates adolescents meeting criteria in any cross-sectional evaluation may be misleading because of the developmental progressions with and between ODD and Conduct Disorder, and categorical definitions of aggressive patterns may reflect arbitrary numbers of constituent estimates. These factors may lead to misleading prevalence rates. In addition, few studies have investigated the prevalence of ODD in preschool-aged children, and early onset of these behaviors is associated with more severe and stable impairment.<ref>Mash, E., & Barkley, R. (Eds.). (2003). ''Child Psychopathology''. 2nd Edition. New York: Guilford Press.</ref> ==[[Evidence based assessment/Prediction phase|'''Prediction phase''']]== The following section contains a list of screening and diagnostic instruments for ODD. The section includes administration information, psychometric data, and PDFs or links to the screenings. * Screenings are used as part of the [[Evidence based assessment/Prediction phase|prediction phase]] of assessment; for more information on interpretation of this data, or how screenings fit in to the assessment process, click [[Evidence based assessment/Prediction phase|here]]. * '''''For a list of more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&wteswitched=1#Psychometric_properties_of_common_screening_instruments click here.]''''' {| class="wikitable sortable" border="1" ! colspan="10" |Screening measures for ODD |- ! Measure ! Format (Reporter) ! Age Range ! Administration/ Completion Time ! Interrater Reliability ! Test-Retest Reliability ! Construct Validity ! Content Validity ! Highly Recommended !Where to access |- |Achenbach System of Empirically Based Assessments (ASEBA): Child Behavior Checklist (CBCL) |Parent report |6-18 <ref name=":9">{{Cite book|url=https://www.worldcat.org/oclc/1130319849|title=Assessment of disorders in childhood and adolescence|date=2020|others=Eric Arden Youngstrom, Mitchell J. Prinstein, Eric J. Mash, Russell A. Barkley|isbn=978-1-4625-4363-2|edition=Fifth edition|location=New York, NY|oclc=1130319849}}</ref> |10 - 15 minutes<ref name=":9" /> |A<ref name=":0">{{Cite book|url=https://www.worldcat.org/oclc/314222270|title=A guide to assessments that work|author=Hunsley, John|author2=Mash, Eric J.|date=2008|publisher=Oxford University Press|isbn=9780195310641|location=New York|oclc=314222270}}</ref> |E<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://store.aseba.org/School-Age-6-18-Materials/departments/11/ Purchase] |- |ASEBA: Teacher Report Form (TRF) |Teacher report |6-18 <ref name=":9" /> |10 - 15 minutes<ref name=":9" /> |A<ref name=":0" /> |E<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://store.aseba.org/School-Age-6-18-Materials/departments/11/ Purchase] |- |ASEBA: Youth Self-Report (YSR) |Youth self-report |11-18 <ref name=":9" /> |10 - 15 minutes<ref name=":9" /> |A<ref name=":0" /> |E<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://store.aseba.org/School-Age-6-18-Materials/departments/11/ Purchase] |- |Behavior Assessment for Children, Third Edition (BASC-3): Parent Rating Scale |Parent report |2-21 |10 - 20 minutes |A<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |E<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://www.pearsonassessments.com/store/usassessments/en/Store/Professional-Assessments/Behavior/Comprehensive/Behavior-Assessment-System-for-Children-%7C-Third-Edition-/p/100001402.html#:~:text=The%20BASC%E2%84%A2%20holds%20an,or%20adolescent's%20behavior%20and%20emotions. Purchase] |- |BASC-3: Teacher Rating Scale |Teacher report |2-21 |10 - 20 minutes |A<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |E<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://www.pearsonassessments.com/store/usassessments/en/Store/Professional-Assessments/Behavior/Comprehensive/Behavior-Assessment-System-for-Children-%7C-Third-Edition-/p/100001402.html#:~:text=The%20BASC%E2%84%A2%20holds%20an,or%20adolescent's%20behavior%20and%20emotions. Purchase] |- |BASC-3: Self-Report of Personality |Youth self-report |6 - college age |30 minutes |A<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |E<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://www.pearsonassessments.com/store/usassessments/en/Store/Professional-Assessments/Behavior/Comprehensive/Behavior-Assessment-System-for-Children-%7C-Third-Edition-/p/100001402.html#:~:text=The%20BASC%E2%84%A2%20holds%20an,or%20adolescent's%20behavior%20and%20emotions. Purchase] |- |Eyberg Child Behavior Inventory (ECBI) |Parent report |2-16 |5 minutes |A<ref name=":0" /> |G<ref name=":0" /> |E<ref name=":0" /> |E<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://www.parinc.com/products/pkey/97 Purchase] |- |Sutter-Eyberg Student Behavior Inventory - Revised (SESBI-R) |Teacher report |2-16 |5 minutes |A<ref name=":0" /> |G<ref name=":0" /> |E<ref name=":0" /> |E<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://www.parinc.com/products/pkey/97 Purchase] |- |Child and Adolescent Behavior Inventory (CABI) |Parent Report |3 - 18 |5 - 10 minutes | |A<ref>{{Cite journal|last=Burns|first=G. Leonard|last2=Preszler|first2=Jonathan|last3=Becker|first3=Stephen P.|date=2022-07-04|title=Psychometric and Normative Information on the Child and Adolescent Behavior Inventory in a Nationally Representative Sample of United States Children|url=https://doi.org/10.1080/15374416.2020.1852943|journal=Journal of Clinical Child & Adolescent Psychology|volume=51|issue=4|pages=443–452|doi=10.1080/15374416.2020.1852943|issn=1537-4416|pmc=PMC8272731|pmid=33428463}}</ref> | | | |PDF |- |Strengths and Difficulties Questionnaire (SDQ) |Parent report |3 - 16 |3 - 5 minutes |E<ref name=":10">{{Cite journal|last=Goodman|first=Robert|date=2001-11-01|title=Psychometric Properties of the Strengths and Difficulties Questionnaire|url=https://www.jaacap.org/article/S0890-8567(09)60543-8/abstract|journal=Journal of the American Academy of Child & Adolescent Psychiatry|language=English|volume=40|issue=11|pages=1337–1345|doi=10.1097/00004583-200111000-00015|issn=0890-8567|pmid=11699809}}</ref> |G<ref name=":10" /> | | | |[https://www.sdqinfo.org/a0.html SDQ Homepage][https://www.sdqinfo.org/py/sdqinfo/b0.py PDFs] |- |Disruptive Behavior Disorder Rating Scale |Parent report, teacher report |5 - 17 |5 - 10 minutes | | | | | |[https://web.archive.org/web/20151123022653/http://ccf.buffalo.edu/pdf/DBD_rating_scale.pdf PDF] |} '''Note:''' L = Less than adequate; A = Adequate; G = Good; E = Excellent; U = Unavailable; NA = Not applicable === Likelihood ratios and AUCs of screening instruments for ODD === * '''''For a list of the likelihood ratios for more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&wteswitched=1#Likelihood_ratios_and_AUCs_of_common_screening_instruments click here.]''''' {| class="wikitable sortable" border="1" ! Screening Measure (Primary Reference) ! Area Under Curve (AUC) ! LR+ (Score) ! LR- (Score) ! Clinical Generalizability !Where to Access |- |CBCL DSM-Oriented Scales<ref name=":7">Achenbach, T. M. (1991a). ''Manual for the Child Behavior Checklist/4–18 and 1991 Profile''. Burlington , VT : University of Vermont Department of Psychiatry.</ref> |.71 (N=475)<ref name=":3">Ebesutani, C., Bernstein, A., Nakamura, B. J., Chorpita, B. F., Higa-McMillan, C. K., & Weisz, J. R. (2010). Concurrent validity of the child behavior checklist DSM-oriented scales: Correspondence with DSM diagnoses and comparison to syndrome scales. ''Journal of Psychopathology and Behavioral Assessment, 32''(3), 373-384.</ref> | 2.80 (60+ to 70+) <ref name=":4">Warnick, E. M., Bracken, M. B., & Kasl, S. (2008). Screening efficiency of the Child Behavior Checklist and Strengths and Difficulties Questionnaire: a systematic review. ''Child and Adolescent Mental Health, 13''(3), 140-147.</ref> | .52 (60- to 70-)*<ref name=":4" /> |Youth aged 5 - 18 seeking out patient treatment across a variety of settings<ref name=":4" /> |[https://store.aseba.org/School-Age-6-18-Materials/departments/11/ Purchase] |- |SDQ<ref name=":6">{{Cite journal|last=Shabani|first=Amir|last2=Masoumian|first2=Samira|last3=Zamirinejad|first3=Somayeh|last4=Hejri|first4=Maryam|last5=Pirmorad|first5=Tahereh|last6=Yaghmaeezadeh|first6=Hooman|date=2021-05|title=Psychometric properties of Structured Clinical Interview for DSM‐5 Disorders‐Clinician Version (SCID‐5‐CV)|url=https://onlinelibrary.wiley.com/doi/10.1002/brb3.1894|journal=Brain and Behavior|language=en|volume=11|issue=5|doi=10.1002/brb3.1894|issn=2162-3279|pmc=PMC8119811|pmid=33729681}}</ref> | .81 -.88 (N = 18,416)<ref>{{Cite journal|last=Algorta|first=Guillermo Perez|last2=Dodd|first2=Alyson Lamont|last3=Stringaris|first3=Argyris|last4=Youngstrom|first4=Eric A.|date=2016-09|title=Diagnostic efficiency of the SDQ for parents to identify ADHD in the UK: a ROC analysis|url=http://link.springer.com/10.1007/s00787-015-0815-0|journal=European Child & Adolescent Psychiatry|language=en|volume=25|issue=9|pages=949–957|doi=10.1007/s00787-015-0815-0|issn=1018-8827|pmc=PMC4990620|pmid=26762184}}</ref> |7.00 <ref name=":4" /> |.55<ref name=":4" /> |Youth aged 5 - 18 seeking out patient treatment across a variety of settings<ref name=":4" /> |[https://www.sdqinfo.org/a0.html SDQ Homepage][https://www.sdqinfo.org/py/sdqinfo/b0.py PDFs] |- |SDQ - DSM -IV Conduct and ODD | |7.55 (3+)<ref name=":2">{{Cite journal|last=Goodman|first=Robert|date=2001-11-01|title=Psychometric Properties of the Strengths and Difficulties Questionnaire|url=https://www.jaacap.org/article/S0890-8567(09)60543-8/abstract|journal=Journal of the American Academy of Child & Adolescent Psychiatry|language=English|volume=40|issue=11|pages=1337–1345|doi=10.1097/00004583-200111000-00015|issn=0890-8567|pmid=11699809}}</ref> |.35 (3-)<ref name=":2" /> |Surveyed youth aged 5 - 15 in the UK <ref name=":2" /> |[https://www.sdqinfo.org/a0.html SDQ Homepage][https://www.sdqinfo.org/py/sdqinfo/b0.py PDFs] |- |ECBI- Intensity Scale<ref name=":8">Eyberg, S. M., & Robinson, E. A. (1983). Conduct problem behavior: Standardization of a behavioral rating scale with adolescents. Journal of Clinical Child Psychology, 12 (3), 347-354.</ref> | |6.92 (131+)<ref name=":5">{{Cite journal|last=LYNEHAM|first=HEIDI J.|last2=ABBOTT|first2=MAREE J.|last3=RAPEE|first3=RONALD M.|date=2007-06|title=Interrater Reliability of the Anxiety Disorders Interview Schedule for DSM-IV: Child and Parent Version|url=https://doi.org/10.1097/chi.0b013e3180465a09|journal=Journal of the American Academy of Child &amp; Adolescent Psychiatry|volume=46|issue=6|pages=731–736|doi=10.1097/chi.0b013e3180465a09|issn=0890-8567}}</ref> |.11 (131-)<ref name=":5" /> |Youth aged 7-16 had responses compared to diagnosis<ref name=":5" /> |[https://www.parinc.com/products/pkey/97 Purchase] |- |BASC-2 PRS - Aggression |.76 (N =156)<ref name=":12">{{Cite journal|last=Doyle|first=Alysa|last2=Ostrander|first2=Rick|last3=Skare|first3=Stacy|last4=Crosby|first4=Ross D.|last5=August|first5=Gerald J.|date=1997-09-01|title=Convergent and criterion-related validity of the behavior assessment system for children-parent rating scale|url=https://doi.org/10.1207/s15374424jccp2603_6|journal=Journal of Clinical Child Psychology|volume=26|issue=3|pages=276–284|doi=10.1207/s15374424jccp2603_6|issn=0047-228X|pmid=9292385}}</ref> |3.62 (65+)<ref name=":12" /> |.60 (65-)<ref name=":12" /> |Youth from first through fourth grade who were at risk for CD<ref name=":12" /> |[https://www.pearsonassessments.com/store/usassessments/en/Store/Professional-Assessments/Behavior/Comprehensive/Behavior-Assessment-System-for-Children-%7C-Third-Edition-/p/100001402.html#:~:text=The%20BASC%E2%84%A2%20holds%20an,or%20adolescent's%20behavior%20and%20emotions. Purchase] |} '''Note:''' “LR+” refers to the change in likelihood ratio associated with a positive test score, and “LR-” is the likelihood ratio for a low score. Likelihood ratios of 1 indicate that the test result did not change impressions at all. LRs larger than 10 or smaller than .10 are frequently clinically decisive; 5 or .20 are helpful, and between 2.0 and .5 are small enough that they rarely result in clinically meaningful changes of formulation<ref>Sackett, D. L., Straus, S. E., Richardson, W. S., Rosenberg, W., & Haynes, R. B. (2000). Evidence-based medicine: How to practice and teach EBM. Edinburgh: Churchill Livingstone.</ref>. '''Search terms:''' [Oppositional Defiant Disorder] AND [sensitivity OR specificity] in GoogleScholar and PsychINFO; === Interpreting ODD screening measure scores === * For information on interpreting screening measure scores, click [[Evidence based assessment/Prediction phase#Interpreting screening measure scores|here.]] == [[Evidence based assessment/Prescription phase|'''Prescription phase''']] == === Gold standard diagnostic interviews === * For a list of broad reaching diagnostic interviews sortable by disorder with PDFs (if applicable), [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prescription_phase&wteswitched=1#Common_Diagnostic_Interviews click here.] === Recommended diagnostic instruments for ODD === {| class="wikitable sortable" ! colspan="10" |Diagnostic instruments for ODD |- !Measure !Format (Reporter) !Age Range !Administration/ Completion Time !Interrater Reliability !Test-Retest Reliability !Construct Validity !Content Validity !Highly Recommended !Where to Access |- |Kiddie Schedule for Affective Disorders and Schizophrenia Present and Lifetime Version (KSADS-PL) |Structured interview |6-28 |45-75 minutes |E<ref name=":11">{{Cite journal|last=KAUFMAN|first=JOAN|last2=BIRMAHER|first2=BORIS|last3=BRENT|first3=DAVID|last4=RAO|first4=UMA|last5=FLYNN|first5=CYNTHIA|last6=MORECI|first6=PAULA|last7=WILLIAMSON|first7=DOUGLAS|last8=RYAN|first8=NEAL|date=1997-07|title=Schedule for Affective Disorders and Schizophrenia for School-Age Children-Present and Lifetime Version (K-SADS-PL): Initial Reliability and Validity Data|url=https://doi.org/10.1097/00004583-199707000-00021|journal=Journal of the American Academy of Child &amp; Adolescent Psychiatry|volume=36|issue=7|pages=980–988|doi=10.1097/00004583-199707000-00021|issn=0890-8567}}</ref> |G<ref name=":11" /> | | | |[https://www.pediatricbipolar.pitt.edu/resources/instruments Website to access] |- |Diagnostic Interview Schedule for Children (DISC-5) |Structured Interview (Self report and parent) |6-17 | | | | | |<ref name=":0" />[[File:Light green check.svg|center|frameless|36x36px]] |[https://telesage.com/netdisc-5/# Coming soon] |} '''Note:''' L = Less than adequate; A = Adequate; G = Good; E = Excellent; U = Unavailable; NA = Not applicable == [[Evidence based assessment/Process phase|'''Process phase''']] == The following section contains a list of process and outcome measures for oppositional defiant disorder. The section includes benchmarks based on published norms for several outcome and severity measures, as well as information about commonly used process measures. Process and outcome measures are used as part of the [[Evidence based assessment/Process phase|process phase]] of assessment. For more information of differences between process and outcome measures, see the page on the [[Evidence based assessment/Process phase|process phase of assessment]]. === Outcome and severity measures === * This table includes clinically significant benchmarks for generalized anxiety disorder specific outcome measures * Information on how to interpret this table can be [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase found here]. * Additionally, these [[Evidence based assessment/Vignettes|vignettes]] might be helpful resources for understanding appropriate adaptation of outcome measures in practice. * ''<u>For clinically significant change benchmarks for the CBCL, YSR, and TRF total, externalizing, internalizing, and attention benchmarks,</u>'' [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase&wteswitched=1#Clinically_significant_change_benchmarks_for_widely-used_outcome_measures see here.] {| class="wikitable sortable" border="1" |- | rowspan="1&quot;" style="text-align:center;font-size:130%;" | <b> Measure</b> | style="text-align:center;font-size:130%;" | <b> Subscale</b> | colspan="3" style="text-align:center;font-size:130%" width="300" | <b> Cut-off scores</b> | colspan="3" style="text-align:center;font-size:120%" | <b> Critical Change <br> (unstandardized scores)</b> |- | colspan="8" style="font-size:110%; text-align:center;" span | <b> Benchmarks Based on Published Norms</b> |- | colspan="2" | | style="text-align:center;font-size:110%" | <b> A</b> | style="text-align:center;font-size:110%" | <b> B</b> | style="text-align:center;font-size:110%" | <b> C</b> | style="text-align:center;font-size:110%" | <b> 95%</b> | style="text-align:center;font-size:110%" | <b> 90%</b> | style="text-align:center;font-size:110%" | <b> SE_difference</b> |- | rowspan="1" style="text-align:center;" | <b> CBCL T-scores <br> (2001 Norms)</b> | style="text-align:right;" | <i> Externalizing</i> | style="text-align:center;" | 49 | style="text-align:center;" | 70 | style="text-align:center;" | 58 | style="text-align:center;" | 7 | style="text-align:center;" | 6 | style="text-align:center;" | 3.4 |- | rowspan="1" style="text-align:center;" | <b> ECBI Scaled Scores <br> (1983 Norms)</b> | style="text-align:right;" | <i> Intensity</i> | style="text-align:center;" | 80.1 | style="text-align:center;" | 169.5 | style="text-align:center;" | 112.9 | style="text-align:center;" | 9.5 | style="text-align:center;" | 8 | style="text-align:center;" | 4.8 |- | rowspan="1" style="text-align:center;" | <b> ECBI Scaled Scores <br> (1983 Norms)</b> | style="text-align:right;" | <i> Problem</i> | style="text-align:center;" | 3.9 | style="text-align:center;" | 17.7 | style="text-align:center;" | 11.5 | style="text-align:center;" | 2.1 | style="text-align:center;" | 1.8 | style="text-align:center;" | 1.1 |} '''Note:''' “A” = Away from the clinical range, “B” = Back into the nonclinical range, “C” = Closer to the nonclinical than clinical mean. === Process measures=== See Section 1.1 for overview of evidence-based measures to use depending on etiology and symptomatology of Oppositional Defiant Disorder. ==Treatment== ===Behavioral parent training=== Behavioral Parent Training is considered the most effective treatment for childhood disruptive behavior disorders (e.g., Oppositional Defiant Disorder), especially for younger children (i.e., 3-8 year-olds). See http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/rule-breaking-defiance-and-acting-out/ a website sponsored by The Society for Child and Adolescent Psychology (APA, Division 53) and the Association for Behavioral and Cognitive Therapies (ABCT), for current summary of evidence-based treatments. ===Overview of recommendations for assessment and treatment=== See the [https://pathways.nice.org.uk/pathways/antisocial-behaviour-and-conduct-disorders-in-children-and-young-people#content=view-info-category%3Aview-about-menu National Institute for Health and Care Excellence (NICE) Practice Guidelines for Childhood Conduct Disorders], for an overview of recommendations for both assessment and treatment of Oppositional Defiant Disorder. == External Links == *[https://sccap53.org Society of Clinical Child and Adolescent Psychology] *http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/rule-breaking-defiance-and-acting-out/ == References == {{collapse top|Click here for references}} {{Reflist|30em}} [[Category:Psychological disorder portfolios|{{SUBPAGENAME}}]] {{collapse bottom}} [[Category:Psychological disorder portfolios|{{SUBPAGENAME}}]] cfm40gv6fudxcvk966ncfntmlz24bqg 2410322 2410321 2022-07-29T22:23:26Z Aherman012 2943941 /* What is a "portfolio"? */ linking to condensed wikitext text/x-wiki <noinclude>{{Helping Give Away Psychological Science Banner}}</noinclude> {{medical disclaimer}} {{:{{ROOTPAGENAME}}/Sidebar}} ==[[Evidence based assessment/Portfolio template/What is a "portfolio"|'''What is a "portfolio"?''']]== For background information on what assessment portfolios are, click the link in the heading above. Does this page feel like too much information? Click [[Evidence-based assessment/Oppositional defiant disorder (assessment portfolio)|here]] for the condensed version. ==[[Evidence based assessment/Preparation phase|'''Preparation phase''']]== === Diagnostic criteria for oppositional defiant disorder=== {{blockquotetop}} <big>ICD-11 Diagnostic Criteria</big><br> <br> '''General Description:''' Oppositional defiant disorder is a persistent pattern (e.g., 6 months or more) of markedly defiant, disobedient, provocative or spiteful behaviour that occurs more frequently than is typically observed in individuals of comparable age and developmental level and that is not restricted to interaction with siblings. Oppositional defiant disorder may be manifest in prevailing, persistent angry or irritable mood, often accompanied by severe temper outbursts or in headstrong, argumentative and defiant behaviour. The behavior pattern is of sufficient severity to result in significant impairment in personal, family, social, educational, occupational or other important areas of functioning <br> <br> '''Oppositional Defiant Disorder With Chronic Irritability-Anger:''' All definitional requirements for oppositional defiant disorder are met. This form of oppositional defiant disorder is characterized by prevailing, persistent angry or irritable mood that may be present independent of any apparent provocation. The negative mood is often accompanied by regularly occurring severe temper outbursts that are grossly out of proportion in intensity or duration to the provocation. Chronic irritability and anger are characteristic of the individual’s functioning nearly every day, are observable across multiple settings or domains of functioning (e.g., home, school, social relationships), and are not restricted to the individual’s relationship with his/her parents or guardians. The pattern of chronic irritability and anger is not limited to occasional episodes (e.g., developmentally typical irritability) or discrete periods (e.g., irritable mood in the context of manic or depressive episodes). *Note: The ICD-11 lists 3 additional subcategories of oppositional defiant disorder with chronic irritability-anger (i.e., with limited prosocial emotions, with typical prosocial emotions, and unspecified). They can be found [https://icd.who.int/browse11/l-m/en#/http%3a%2f%2fid.who.int%2ficd%2fentity%2f818792113 here]. <br> '''Oppositional Defiant Disorder Without Chronic Irritability-Anger:''' Meets all definitional requirements for oppositional defiant disorder. This form of oppositional defiant disorder is not characterized by prevailing, persistent, angry or irritable mood, but does feature headstrong, argumentative, and defiant behavior. *Note: The ICD-11 lists 3 additional subcategories of oppositional defiant disorder without chronic irritability-anger (i.e., with limited prosocial emotions, with typical prosocial emotions, and unspecified). They can be found [https://icd.who.int/browse11/l-m/en#/http%3a%2f%2fid.who.int%2ficd%2fentity%2f736540987 here]. <br> <big>Changes in DSM-5</big> <br> The diagnostic criteria for ADHD changed slightly from DSM-IV to DSM-5. See the changes [https://www.ncbi.nlm.nih.gov/books/NBK519712/table/ch3.t14/ here]. {{blockquotebottom}}<ref>{{Cite web|url=https://icd.who.int/browse11/l-m/en#/http://id.who.int/icd/entity/1545599508|title=ICD-11 for Mortality and Morbidity Statistics|website=icd.who.int|access-date=2022-07-11}}</ref> ===Base rates of ODD in different clinical settings=== This section describes the demographic setting of the population(s) sampled, base rates of diagnosis, country/region sampled and the diagnostic method that was used. Using this information, clinicians will be able to anchor the rate of ODD that they are likely to see in their clinical practice. * '''''To see prevalence rates across multiple disorders, [[Evidence-based assessment/Preparation phase|click here.]]''''' {|class="wikitable sortable" border="1" |- ! Demography ! Setting !! Base Rate !! Diagnostic Method |- || Various locations across USA | Meta-analysis of 38 studies<ref>Canino G, Polanczyk G, Bauermeister JJ, et al. (2010) Does the prevalence of CD and ODD vary across cultures? Social Psychiatry Epidemiology, 45(7):695–704.</ref> || 3.3% || Varied |- || All of the U.S. | Nationally representative large-scale study (N = 3,119) <ref>Nock, M. K., Kazdin, A. E., Hiripi, E., & Kessler, R. C. (2007). Lifetime prevalence, correlates, and persistence of oppositional defiant disorder: results from the National Comorbidity Survey Replication. ''Journal of Child Psychology and Psychiatry, 48''(7), 703-713.</ref> || 10.2% (overall) 11.2% (males) 9.2% (females) || World Health Organization (WHO) Composite International Diagnostic Interview (CIDI)^r |- |Suburban and urban Colorado |Project to Learn about Youth-Mental health, school-based study for children from kindergarten to high-school (n = 236)<ref name=":1">{{Cite journal|last=Danielson|first=Melissa L.|last2=Bitsko|first2=Rebecca H.|last3=Holbrook|first3=Joseph R.|last4=Charania|first4=Sana N.|last5=Claussen|first5=Angelika H.|last6=McKeown|first6=Robert E.|last7=Cuffe|first7=Steven P.|last8=Owens|first8=Julie Sarno|last9=Evans|first9=Steven W.|date=2021-06-01|title=Community-Based Prevalence of Externalizing and Internalizing Disorders among School-Aged Children and Adolescents in Four Geographically Dispersed School Districts in the United States|url=https://doi.org/10.1007/s10578-020-01027-z|journal=Child Psychiatry & Human Development|language=en|volume=52|issue=3|pages=500–514|doi=10.1007/s10578-020-01027-z|issn=1573-3327|pmc=PMC8016018|pmid=32734339}}</ref> |6.8% |Diagnostic Interview Schedule for Children (DISC) |- |Urban and suburban Florida |Project to Learn about Youth-Mental health, school-based study for children from kindergarten to high-school (n = 289)<ref name=":1" /> |6.9% |Diagnostic Interview Schedule for Children (DISC) |- |Rural and suburban Ohio |Project to Learn about Youth-Mental health, school-based study for children from elementary to high-school (n = 152)<ref name=":1" /> |17.3% |Diagnostic Interview Schedule for Children (DISC) |- |Suburban and rural South Carolina |Project to Learn about Youth-Mental health, school-based study for children from elementary to high-school (n = 270)<ref name=":1" /> |5.7% |Diagnostic Interview Schedule for Children (DISC) |- || Semi-rural North Carolina | Preschool-aged children from pediatric practices (N = 306; age 2 - 5 years old)<ref>Egger, H.L., & Angold, A. (2006). Common emotional and behavioral disorders in preschool children: Presentation, nosology, and epidemiology. ''Journal of Child Psychiatry and Psychology, 47'', 313–337.</ref> || 6.6% || Preschool Age Psychiatric Assessment (PAPA)^p |- || Western North Carolina | The Great Smoky Mountains Study - longitudinal, population-based study of community sample<ref>Costello, E. J., Mustillo, S., Erkanli, A., Keeler, G., & Angold, A. (2003) Prevalence and development of psychiatric disorders in adolescence. ''Arch Gen Psychiatry, 60'', 837-844.</ref> || 2.33% (overall) 3.16% (males) 2.75% (females) || Child and Adolescent Psychiatric Assessment (CAPA)^{p, y} |- || Chicago | Preschool-aged children from inner city schools and pediatric practices (N = 796; age 2 - 5 years old)<ref>Lavigne, J. V., LeBailly, S. A., Hopkins, J., Gouze, K. R., & Binns, H. J. (2009). The prevalence of ADHD, ODD, depression, and anxiety in a community sample of 4-year-olds. ''Journal Of Clinical Child And Adolescent Psychology, 38''(3), 315-328. doi:10.1080/15374410902851382</ref> || 8.3% || Diagnostic Interview Schedule for Children–Parent Scale–Young Child Version (DISC-YC) ^p |- |Germany (Saarbrücken County) |All school-aged children were examined during a routine school-entry medical examination (N = 1676, mean age = 5.7)<ref>{{Cite journal|last=Niemczyk|first=Justine|last2=Equit|first2=Monika|last3=Braun-Bither|first3=Katrin|last4=Klein|first4=Anna-Maria|last5=von Gontard|first5=Alexander|date=2015-07-01|title=Prevalence of incontinence, attention deficit/hyperactivity disorder and oppositional defiant disorder in preschool children|url=https://doi.org/10.1007/s00787-014-0628-6|journal=European Child & Adolescent Psychiatry|language=en|volume=24|issue=7|pages=837–843|doi=10.1007/s00787-014-0628-6|issn=1435-165X}}</ref> |7.3% (males) 5.1%(females) |DISYPS-II |- |South Korea (Seoul) |Children were randomly surveyed across 6 school districts in Seoul (N = 1645, age 6 - 12 years old) |5.8% (males) 4.1%(females) |Diagnostic Interview Schedule for Children–Parent Scale IV (DISC-IV) |} ^p Parent interviewed as part of diagnostic assessment; ^y youth interviewed as part of diagnostic assessment, ^r adult interviewed for retrospective report as part of diagnostic assessment '''Note:''' Mash and Barkley note that prevalence rates of ODD must be qualified, because the definition of ODD has changed at a fast rate, the rates adolescents meeting criteria in any cross-sectional evaluation may be misleading because of the developmental progressions with and between ODD and Conduct Disorder, and categorical definitions of aggressive patterns may reflect arbitrary numbers of constituent estimates. These factors may lead to misleading prevalence rates. In addition, few studies have investigated the prevalence of ODD in preschool-aged children, and early onset of these behaviors is associated with more severe and stable impairment.<ref>Mash, E., & Barkley, R. (Eds.). (2003). ''Child Psychopathology''. 2nd Edition. New York: Guilford Press.</ref> ==[[Evidence based assessment/Prediction phase|'''Prediction phase''']]== The following section contains a list of screening and diagnostic instruments for ODD. The section includes administration information, psychometric data, and PDFs or links to the screenings. * Screenings are used as part of the [[Evidence based assessment/Prediction phase|prediction phase]] of assessment; for more information on interpretation of this data, or how screenings fit in to the assessment process, click [[Evidence based assessment/Prediction phase|here]]. * '''''For a list of more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&wteswitched=1#Psychometric_properties_of_common_screening_instruments click here.]''''' {| class="wikitable sortable" border="1" ! colspan="10" |Screening measures for ODD |- ! Measure ! Format (Reporter) ! Age Range ! Administration/ Completion Time ! Interrater Reliability ! Test-Retest Reliability ! Construct Validity ! Content Validity ! Highly Recommended !Where to access |- |Achenbach System of Empirically Based Assessments (ASEBA): Child Behavior Checklist (CBCL) |Parent report |6-18 <ref name=":9">{{Cite book|url=https://www.worldcat.org/oclc/1130319849|title=Assessment of disorders in childhood and adolescence|date=2020|others=Eric Arden Youngstrom, Mitchell J. Prinstein, Eric J. Mash, Russell A. Barkley|isbn=978-1-4625-4363-2|edition=Fifth edition|location=New York, NY|oclc=1130319849}}</ref> |10 - 15 minutes<ref name=":9" /> |A<ref name=":0">{{Cite book|url=https://www.worldcat.org/oclc/314222270|title=A guide to assessments that work|author=Hunsley, John|author2=Mash, Eric J.|date=2008|publisher=Oxford University Press|isbn=9780195310641|location=New York|oclc=314222270}}</ref> |E<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://store.aseba.org/School-Age-6-18-Materials/departments/11/ Purchase] |- |ASEBA: Teacher Report Form (TRF) |Teacher report |6-18 <ref name=":9" /> |10 - 15 minutes<ref name=":9" /> |A<ref name=":0" /> |E<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://store.aseba.org/School-Age-6-18-Materials/departments/11/ Purchase] |- |ASEBA: Youth Self-Report (YSR) |Youth self-report |11-18 <ref name=":9" /> |10 - 15 minutes<ref name=":9" /> |A<ref name=":0" /> |E<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://store.aseba.org/School-Age-6-18-Materials/departments/11/ Purchase] |- |Behavior Assessment for Children, Third Edition (BASC-3): Parent Rating Scale |Parent report |2-21 |10 - 20 minutes |A<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |E<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://www.pearsonassessments.com/store/usassessments/en/Store/Professional-Assessments/Behavior/Comprehensive/Behavior-Assessment-System-for-Children-%7C-Third-Edition-/p/100001402.html#:~:text=The%20BASC%E2%84%A2%20holds%20an,or%20adolescent's%20behavior%20and%20emotions. Purchase] |- |BASC-3: Teacher Rating Scale |Teacher report |2-21 |10 - 20 minutes |A<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |E<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://www.pearsonassessments.com/store/usassessments/en/Store/Professional-Assessments/Behavior/Comprehensive/Behavior-Assessment-System-for-Children-%7C-Third-Edition-/p/100001402.html#:~:text=The%20BASC%E2%84%A2%20holds%20an,or%20adolescent's%20behavior%20and%20emotions. Purchase] |- |BASC-3: Self-Report of Personality |Youth self-report |6 - college age |30 minutes |A<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |E<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://www.pearsonassessments.com/store/usassessments/en/Store/Professional-Assessments/Behavior/Comprehensive/Behavior-Assessment-System-for-Children-%7C-Third-Edition-/p/100001402.html#:~:text=The%20BASC%E2%84%A2%20holds%20an,or%20adolescent's%20behavior%20and%20emotions. Purchase] |- |Eyberg Child Behavior Inventory (ECBI) |Parent report |2-16 |5 minutes |A<ref name=":0" /> |G<ref name=":0" /> |E<ref name=":0" /> |E<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://www.parinc.com/products/pkey/97 Purchase] |- |Sutter-Eyberg Student Behavior Inventory - Revised (SESBI-R) |Teacher report |2-16 |5 minutes |A<ref name=":0" /> |G<ref name=":0" /> |E<ref name=":0" /> |E<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://www.parinc.com/products/pkey/97 Purchase] |- |Child and Adolescent Behavior Inventory (CABI) |Parent Report |3 - 18 |5 - 10 minutes | |A<ref>{{Cite journal|last=Burns|first=G. Leonard|last2=Preszler|first2=Jonathan|last3=Becker|first3=Stephen P.|date=2022-07-04|title=Psychometric and Normative Information on the Child and Adolescent Behavior Inventory in a Nationally Representative Sample of United States Children|url=https://doi.org/10.1080/15374416.2020.1852943|journal=Journal of Clinical Child & Adolescent Psychology|volume=51|issue=4|pages=443–452|doi=10.1080/15374416.2020.1852943|issn=1537-4416|pmc=PMC8272731|pmid=33428463}}</ref> | | | |PDF |- |Strengths and Difficulties Questionnaire (SDQ) |Parent report |3 - 16 |3 - 5 minutes |E<ref name=":10">{{Cite journal|last=Goodman|first=Robert|date=2001-11-01|title=Psychometric Properties of the Strengths and Difficulties Questionnaire|url=https://www.jaacap.org/article/S0890-8567(09)60543-8/abstract|journal=Journal of the American Academy of Child & Adolescent Psychiatry|language=English|volume=40|issue=11|pages=1337–1345|doi=10.1097/00004583-200111000-00015|issn=0890-8567|pmid=11699809}}</ref> |G<ref name=":10" /> | | | |[https://www.sdqinfo.org/a0.html SDQ Homepage][https://www.sdqinfo.org/py/sdqinfo/b0.py PDFs] |- |Disruptive Behavior Disorder Rating Scale |Parent report, teacher report |5 - 17 |5 - 10 minutes | | | | | |[https://web.archive.org/web/20151123022653/http://ccf.buffalo.edu/pdf/DBD_rating_scale.pdf PDF] |} '''Note:''' L = Less than adequate; A = Adequate; G = Good; E = Excellent; U = Unavailable; NA = Not applicable === Likelihood ratios and AUCs of screening instruments for ODD === * '''''For a list of the likelihood ratios for more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&wteswitched=1#Likelihood_ratios_and_AUCs_of_common_screening_instruments click here.]''''' {| class="wikitable sortable" border="1" ! Screening Measure (Primary Reference) ! Area Under Curve (AUC) ! LR+ (Score) ! LR- (Score) ! Clinical Generalizability !Where to Access |- |CBCL DSM-Oriented Scales<ref name=":7">Achenbach, T. M. (1991a). ''Manual for the Child Behavior Checklist/4–18 and 1991 Profile''. Burlington , VT : University of Vermont Department of Psychiatry.</ref> |.71 (N=475)<ref name=":3">Ebesutani, C., Bernstein, A., Nakamura, B. J., Chorpita, B. F., Higa-McMillan, C. K., & Weisz, J. R. (2010). Concurrent validity of the child behavior checklist DSM-oriented scales: Correspondence with DSM diagnoses and comparison to syndrome scales. ''Journal of Psychopathology and Behavioral Assessment, 32''(3), 373-384.</ref> | 2.80 (60+ to 70+) <ref name=":4">Warnick, E. M., Bracken, M. B., & Kasl, S. (2008). Screening efficiency of the Child Behavior Checklist and Strengths and Difficulties Questionnaire: a systematic review. ''Child and Adolescent Mental Health, 13''(3), 140-147.</ref> | .52 (60- to 70-)*<ref name=":4" /> |Youth aged 5 - 18 seeking out patient treatment across a variety of settings<ref name=":4" /> |[https://store.aseba.org/School-Age-6-18-Materials/departments/11/ Purchase] |- |SDQ<ref name=":6">{{Cite journal|last=Shabani|first=Amir|last2=Masoumian|first2=Samira|last3=Zamirinejad|first3=Somayeh|last4=Hejri|first4=Maryam|last5=Pirmorad|first5=Tahereh|last6=Yaghmaeezadeh|first6=Hooman|date=2021-05|title=Psychometric properties of Structured Clinical Interview for DSM‐5 Disorders‐Clinician Version (SCID‐5‐CV)|url=https://onlinelibrary.wiley.com/doi/10.1002/brb3.1894|journal=Brain and Behavior|language=en|volume=11|issue=5|doi=10.1002/brb3.1894|issn=2162-3279|pmc=PMC8119811|pmid=33729681}}</ref> | .81 -.88 (N = 18,416)<ref>{{Cite journal|last=Algorta|first=Guillermo Perez|last2=Dodd|first2=Alyson Lamont|last3=Stringaris|first3=Argyris|last4=Youngstrom|first4=Eric A.|date=2016-09|title=Diagnostic efficiency of the SDQ for parents to identify ADHD in the UK: a ROC analysis|url=http://link.springer.com/10.1007/s00787-015-0815-0|journal=European Child & Adolescent Psychiatry|language=en|volume=25|issue=9|pages=949–957|doi=10.1007/s00787-015-0815-0|issn=1018-8827|pmc=PMC4990620|pmid=26762184}}</ref> |7.00 <ref name=":4" /> |.55<ref name=":4" /> |Youth aged 5 - 18 seeking out patient treatment across a variety of settings<ref name=":4" /> |[https://www.sdqinfo.org/a0.html SDQ Homepage][https://www.sdqinfo.org/py/sdqinfo/b0.py PDFs] |- |SDQ - DSM -IV Conduct and ODD | |7.55 (3+)<ref name=":2">{{Cite journal|last=Goodman|first=Robert|date=2001-11-01|title=Psychometric Properties of the Strengths and Difficulties Questionnaire|url=https://www.jaacap.org/article/S0890-8567(09)60543-8/abstract|journal=Journal of the American Academy of Child & Adolescent Psychiatry|language=English|volume=40|issue=11|pages=1337–1345|doi=10.1097/00004583-200111000-00015|issn=0890-8567|pmid=11699809}}</ref> |.35 (3-)<ref name=":2" /> |Surveyed youth aged 5 - 15 in the UK <ref name=":2" /> |[https://www.sdqinfo.org/a0.html SDQ Homepage][https://www.sdqinfo.org/py/sdqinfo/b0.py PDFs] |- |ECBI- Intensity Scale<ref name=":8">Eyberg, S. M., & Robinson, E. A. (1983). Conduct problem behavior: Standardization of a behavioral rating scale with adolescents. Journal of Clinical Child Psychology, 12 (3), 347-354.</ref> | |6.92 (131+)<ref name=":5">{{Cite journal|last=LYNEHAM|first=HEIDI J.|last2=ABBOTT|first2=MAREE J.|last3=RAPEE|first3=RONALD M.|date=2007-06|title=Interrater Reliability of the Anxiety Disorders Interview Schedule for DSM-IV: Child and Parent Version|url=https://doi.org/10.1097/chi.0b013e3180465a09|journal=Journal of the American Academy of Child &amp; Adolescent Psychiatry|volume=46|issue=6|pages=731–736|doi=10.1097/chi.0b013e3180465a09|issn=0890-8567}}</ref> |.11 (131-)<ref name=":5" /> |Youth aged 7-16 had responses compared to diagnosis<ref name=":5" /> |[https://www.parinc.com/products/pkey/97 Purchase] |- |BASC-2 PRS - Aggression |.76 (N =156)<ref name=":12">{{Cite journal|last=Doyle|first=Alysa|last2=Ostrander|first2=Rick|last3=Skare|first3=Stacy|last4=Crosby|first4=Ross D.|last5=August|first5=Gerald J.|date=1997-09-01|title=Convergent and criterion-related validity of the behavior assessment system for children-parent rating scale|url=https://doi.org/10.1207/s15374424jccp2603_6|journal=Journal of Clinical Child Psychology|volume=26|issue=3|pages=276–284|doi=10.1207/s15374424jccp2603_6|issn=0047-228X|pmid=9292385}}</ref> |3.62 (65+)<ref name=":12" /> |.60 (65-)<ref name=":12" /> |Youth from first through fourth grade who were at risk for CD<ref name=":12" /> |[https://www.pearsonassessments.com/store/usassessments/en/Store/Professional-Assessments/Behavior/Comprehensive/Behavior-Assessment-System-for-Children-%7C-Third-Edition-/p/100001402.html#:~:text=The%20BASC%E2%84%A2%20holds%20an,or%20adolescent's%20behavior%20and%20emotions. Purchase] |} '''Note:''' “LR+” refers to the change in likelihood ratio associated with a positive test score, and “LR-” is the likelihood ratio for a low score. Likelihood ratios of 1 indicate that the test result did not change impressions at all. LRs larger than 10 or smaller than .10 are frequently clinically decisive; 5 or .20 are helpful, and between 2.0 and .5 are small enough that they rarely result in clinically meaningful changes of formulation<ref>Sackett, D. L., Straus, S. E., Richardson, W. S., Rosenberg, W., & Haynes, R. B. (2000). Evidence-based medicine: How to practice and teach EBM. Edinburgh: Churchill Livingstone.</ref>. '''Search terms:''' [Oppositional Defiant Disorder] AND [sensitivity OR specificity] in GoogleScholar and PsychINFO; === Interpreting ODD screening measure scores === * For information on interpreting screening measure scores, click [[Evidence based assessment/Prediction phase#Interpreting screening measure scores|here.]] == [[Evidence based assessment/Prescription phase|'''Prescription phase''']] == === Gold standard diagnostic interviews === * For a list of broad reaching diagnostic interviews sortable by disorder with PDFs (if applicable), [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prescription_phase&wteswitched=1#Common_Diagnostic_Interviews click here.] === Recommended diagnostic instruments for ODD === {| class="wikitable sortable" ! colspan="10" |Diagnostic instruments for ODD |- !Measure !Format (Reporter) !Age Range !Administration/ Completion Time !Interrater Reliability !Test-Retest Reliability !Construct Validity !Content Validity !Highly Recommended !Where to Access |- |Kiddie Schedule for Affective Disorders and Schizophrenia Present and Lifetime Version (KSADS-PL) |Structured interview |6-28 |45-75 minutes |E<ref name=":11">{{Cite journal|last=KAUFMAN|first=JOAN|last2=BIRMAHER|first2=BORIS|last3=BRENT|first3=DAVID|last4=RAO|first4=UMA|last5=FLYNN|first5=CYNTHIA|last6=MORECI|first6=PAULA|last7=WILLIAMSON|first7=DOUGLAS|last8=RYAN|first8=NEAL|date=1997-07|title=Schedule for Affective Disorders and Schizophrenia for School-Age Children-Present and Lifetime Version (K-SADS-PL): Initial Reliability and Validity Data|url=https://doi.org/10.1097/00004583-199707000-00021|journal=Journal of the American Academy of Child &amp; Adolescent Psychiatry|volume=36|issue=7|pages=980–988|doi=10.1097/00004583-199707000-00021|issn=0890-8567}}</ref> |G<ref name=":11" /> | | | |[https://www.pediatricbipolar.pitt.edu/resources/instruments Website to access] |- |Diagnostic Interview Schedule for Children (DISC-5) |Structured Interview (Self report and parent) |6-17 | | | | | |<ref name=":0" />[[File:Light green check.svg|center|frameless|36x36px]] |[https://telesage.com/netdisc-5/# Coming soon] |} '''Note:''' L = Less than adequate; A = Adequate; G = Good; E = Excellent; U = Unavailable; NA = Not applicable == [[Evidence based assessment/Process phase|'''Process phase''']] == The following section contains a list of process and outcome measures for oppositional defiant disorder. The section includes benchmarks based on published norms for several outcome and severity measures, as well as information about commonly used process measures. Process and outcome measures are used as part of the [[Evidence based assessment/Process phase|process phase]] of assessment. For more information of differences between process and outcome measures, see the page on the [[Evidence based assessment/Process phase|process phase of assessment]]. === Outcome and severity measures === * This table includes clinically significant benchmarks for generalized anxiety disorder specific outcome measures * Information on how to interpret this table can be [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase found here]. * Additionally, these [[Evidence based assessment/Vignettes|vignettes]] might be helpful resources for understanding appropriate adaptation of outcome measures in practice. * ''<u>For clinically significant change benchmarks for the CBCL, YSR, and TRF total, externalizing, internalizing, and attention benchmarks,</u>'' [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase&wteswitched=1#Clinically_significant_change_benchmarks_for_widely-used_outcome_measures see here.] {| class="wikitable sortable" border="1" |- | rowspan="1&quot;" style="text-align:center;font-size:130%;" | <b> Measure</b> | style="text-align:center;font-size:130%;" | <b> Subscale</b> | colspan="3" style="text-align:center;font-size:130%" width="300" | <b> Cut-off scores</b> | colspan="3" style="text-align:center;font-size:120%" | <b> Critical Change <br> (unstandardized scores)</b> |- | colspan="8" style="font-size:110%; text-align:center;" span | <b> Benchmarks Based on Published Norms</b> |- | colspan="2" | | style="text-align:center;font-size:110%" | <b> A</b> | style="text-align:center;font-size:110%" | <b> B</b> | style="text-align:center;font-size:110%" | <b> C</b> | style="text-align:center;font-size:110%" | <b> 95%</b> | style="text-align:center;font-size:110%" | <b> 90%</b> | style="text-align:center;font-size:110%" | <b> SE_difference</b> |- | rowspan="1" style="text-align:center;" | <b> CBCL T-scores <br> (2001 Norms)</b> | style="text-align:right;" | <i> Externalizing</i> | style="text-align:center;" | 49 | style="text-align:center;" | 70 | style="text-align:center;" | 58 | style="text-align:center;" | 7 | style="text-align:center;" | 6 | style="text-align:center;" | 3.4 |- | rowspan="1" style="text-align:center;" | <b> ECBI Scaled Scores <br> (1983 Norms)</b> | style="text-align:right;" | <i> Intensity</i> | style="text-align:center;" | 80.1 | style="text-align:center;" | 169.5 | style="text-align:center;" | 112.9 | style="text-align:center;" | 9.5 | style="text-align:center;" | 8 | style="text-align:center;" | 4.8 |- | rowspan="1" style="text-align:center;" | <b> ECBI Scaled Scores <br> (1983 Norms)</b> | style="text-align:right;" | <i> Problem</i> | style="text-align:center;" | 3.9 | style="text-align:center;" | 17.7 | style="text-align:center;" | 11.5 | style="text-align:center;" | 2.1 | style="text-align:center;" | 1.8 | style="text-align:center;" | 1.1 |} '''Note:''' “A” = Away from the clinical range, “B” = Back into the nonclinical range, “C” = Closer to the nonclinical than clinical mean. === Process measures=== See Section 1.1 for overview of evidence-based measures to use depending on etiology and symptomatology of Oppositional Defiant Disorder. ==Treatment== ===Behavioral parent training=== Behavioral Parent Training is considered the most effective treatment for childhood disruptive behavior disorders (e.g., Oppositional Defiant Disorder), especially for younger children (i.e., 3-8 year-olds). See http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/rule-breaking-defiance-and-acting-out/ a website sponsored by The Society for Child and Adolescent Psychology (APA, Division 53) and the Association for Behavioral and Cognitive Therapies (ABCT), for current summary of evidence-based treatments. ===Overview of recommendations for assessment and treatment=== See the [https://pathways.nice.org.uk/pathways/antisocial-behaviour-and-conduct-disorders-in-children-and-young-people#content=view-info-category%3Aview-about-menu National Institute for Health and Care Excellence (NICE) Practice Guidelines for Childhood Conduct Disorders], for an overview of recommendations for both assessment and treatment of Oppositional Defiant Disorder. == External Links == *[https://sccap53.org Society of Clinical Child and Adolescent Psychology] *http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/rule-breaking-defiance-and-acting-out/ == References == {{collapse top|Click here for references}} {{Reflist|30em}} [[Category:Psychological disorder portfolios|{{SUBPAGENAME}}]] {{collapse bottom}} [[Category:Psychological disorder portfolios|{{SUBPAGENAME}}]] ksm8c6l890e1memyjgp2cvevc7ifox2 2410323 2410322 2022-07-29T22:25:00Z Aherman012 2943941 /* What is a "portfolio"? */ wikitext text/x-wiki <noinclude>{{Helping Give Away Psychological Science Banner}}</noinclude> {{medical disclaimer}} {{:{{ROOTPAGENAME}}/Sidebar}} ==[[Evidence based assessment/Portfolio template/What is a "portfolio"|'''What is a "portfolio"?''']]== For background information on what assessment portfolios are, click the link in the heading above. Does this page feel like too much information? Click [[Evidence-based assessment/Oppositional defiant disorder (disorder portfolio)|here]] for the condensed version. ==[[Evidence based assessment/Preparation phase|'''Preparation phase''']]== === Diagnostic criteria for oppositional defiant disorder=== {{blockquotetop}} <big>ICD-11 Diagnostic Criteria</big><br> <br> '''General Description:''' Oppositional defiant disorder is a persistent pattern (e.g., 6 months or more) of markedly defiant, disobedient, provocative or spiteful behaviour that occurs more frequently than is typically observed in individuals of comparable age and developmental level and that is not restricted to interaction with siblings. Oppositional defiant disorder may be manifest in prevailing, persistent angry or irritable mood, often accompanied by severe temper outbursts or in headstrong, argumentative and defiant behaviour. The behavior pattern is of sufficient severity to result in significant impairment in personal, family, social, educational, occupational or other important areas of functioning <br> <br> '''Oppositional Defiant Disorder With Chronic Irritability-Anger:''' All definitional requirements for oppositional defiant disorder are met. This form of oppositional defiant disorder is characterized by prevailing, persistent angry or irritable mood that may be present independent of any apparent provocation. The negative mood is often accompanied by regularly occurring severe temper outbursts that are grossly out of proportion in intensity or duration to the provocation. Chronic irritability and anger are characteristic of the individual’s functioning nearly every day, are observable across multiple settings or domains of functioning (e.g., home, school, social relationships), and are not restricted to the individual’s relationship with his/her parents or guardians. The pattern of chronic irritability and anger is not limited to occasional episodes (e.g., developmentally typical irritability) or discrete periods (e.g., irritable mood in the context of manic or depressive episodes). *Note: The ICD-11 lists 3 additional subcategories of oppositional defiant disorder with chronic irritability-anger (i.e., with limited prosocial emotions, with typical prosocial emotions, and unspecified). They can be found [https://icd.who.int/browse11/l-m/en#/http%3a%2f%2fid.who.int%2ficd%2fentity%2f818792113 here]. <br> '''Oppositional Defiant Disorder Without Chronic Irritability-Anger:''' Meets all definitional requirements for oppositional defiant disorder. This form of oppositional defiant disorder is not characterized by prevailing, persistent, angry or irritable mood, but does feature headstrong, argumentative, and defiant behavior. *Note: The ICD-11 lists 3 additional subcategories of oppositional defiant disorder without chronic irritability-anger (i.e., with limited prosocial emotions, with typical prosocial emotions, and unspecified). They can be found [https://icd.who.int/browse11/l-m/en#/http%3a%2f%2fid.who.int%2ficd%2fentity%2f736540987 here]. <br> <big>Changes in DSM-5</big> <br> The diagnostic criteria for ADHD changed slightly from DSM-IV to DSM-5. See the changes [https://www.ncbi.nlm.nih.gov/books/NBK519712/table/ch3.t14/ here]. {{blockquotebottom}}<ref>{{Cite web|url=https://icd.who.int/browse11/l-m/en#/http://id.who.int/icd/entity/1545599508|title=ICD-11 for Mortality and Morbidity Statistics|website=icd.who.int|access-date=2022-07-11}}</ref> ===Base rates of ODD in different clinical settings=== This section describes the demographic setting of the population(s) sampled, base rates of diagnosis, country/region sampled and the diagnostic method that was used. Using this information, clinicians will be able to anchor the rate of ODD that they are likely to see in their clinical practice. * '''''To see prevalence rates across multiple disorders, [[Evidence-based assessment/Preparation phase|click here.]]''''' {|class="wikitable sortable" border="1" |- ! Demography ! Setting !! Base Rate !! Diagnostic Method |- || Various locations across USA | Meta-analysis of 38 studies<ref>Canino G, Polanczyk G, Bauermeister JJ, et al. (2010) Does the prevalence of CD and ODD vary across cultures? Social Psychiatry Epidemiology, 45(7):695–704.</ref> || 3.3% || Varied |- || All of the U.S. | Nationally representative large-scale study (N = 3,119) <ref>Nock, M. K., Kazdin, A. E., Hiripi, E., & Kessler, R. C. (2007). Lifetime prevalence, correlates, and persistence of oppositional defiant disorder: results from the National Comorbidity Survey Replication. ''Journal of Child Psychology and Psychiatry, 48''(7), 703-713.</ref> || 10.2% (overall) 11.2% (males) 9.2% (females) || World Health Organization (WHO) Composite International Diagnostic Interview (CIDI)^r |- |Suburban and urban Colorado |Project to Learn about Youth-Mental health, school-based study for children from kindergarten to high-school (n = 236)<ref name=":1">{{Cite journal|last=Danielson|first=Melissa L.|last2=Bitsko|first2=Rebecca H.|last3=Holbrook|first3=Joseph R.|last4=Charania|first4=Sana N.|last5=Claussen|first5=Angelika H.|last6=McKeown|first6=Robert E.|last7=Cuffe|first7=Steven P.|last8=Owens|first8=Julie Sarno|last9=Evans|first9=Steven W.|date=2021-06-01|title=Community-Based Prevalence of Externalizing and Internalizing Disorders among School-Aged Children and Adolescents in Four Geographically Dispersed School Districts in the United States|url=https://doi.org/10.1007/s10578-020-01027-z|journal=Child Psychiatry & Human Development|language=en|volume=52|issue=3|pages=500–514|doi=10.1007/s10578-020-01027-z|issn=1573-3327|pmc=PMC8016018|pmid=32734339}}</ref> |6.8% |Diagnostic Interview Schedule for Children (DISC) |- |Urban and suburban Florida |Project to Learn about Youth-Mental health, school-based study for children from kindergarten to high-school (n = 289)<ref name=":1" /> |6.9% |Diagnostic Interview Schedule for Children (DISC) |- |Rural and suburban Ohio |Project to Learn about Youth-Mental health, school-based study for children from elementary to high-school (n = 152)<ref name=":1" /> |17.3% |Diagnostic Interview Schedule for Children (DISC) |- |Suburban and rural South Carolina |Project to Learn about Youth-Mental health, school-based study for children from elementary to high-school (n = 270)<ref name=":1" /> |5.7% |Diagnostic Interview Schedule for Children (DISC) |- || Semi-rural North Carolina | Preschool-aged children from pediatric practices (N = 306; age 2 - 5 years old)<ref>Egger, H.L., & Angold, A. (2006). Common emotional and behavioral disorders in preschool children: Presentation, nosology, and epidemiology. ''Journal of Child Psychiatry and Psychology, 47'', 313–337.</ref> || 6.6% || Preschool Age Psychiatric Assessment (PAPA)^p |- || Western North Carolina | The Great Smoky Mountains Study - longitudinal, population-based study of community sample<ref>Costello, E. J., Mustillo, S., Erkanli, A., Keeler, G., & Angold, A. (2003) Prevalence and development of psychiatric disorders in adolescence. ''Arch Gen Psychiatry, 60'', 837-844.</ref> || 2.33% (overall) 3.16% (males) 2.75% (females) || Child and Adolescent Psychiatric Assessment (CAPA)^{p, y} |- || Chicago | Preschool-aged children from inner city schools and pediatric practices (N = 796; age 2 - 5 years old)<ref>Lavigne, J. V., LeBailly, S. A., Hopkins, J., Gouze, K. R., & Binns, H. J. (2009). The prevalence of ADHD, ODD, depression, and anxiety in a community sample of 4-year-olds. ''Journal Of Clinical Child And Adolescent Psychology, 38''(3), 315-328. doi:10.1080/15374410902851382</ref> || 8.3% || Diagnostic Interview Schedule for Children–Parent Scale–Young Child Version (DISC-YC) ^p |- |Germany (Saarbrücken County) |All school-aged children were examined during a routine school-entry medical examination (N = 1676, mean age = 5.7)<ref>{{Cite journal|last=Niemczyk|first=Justine|last2=Equit|first2=Monika|last3=Braun-Bither|first3=Katrin|last4=Klein|first4=Anna-Maria|last5=von Gontard|first5=Alexander|date=2015-07-01|title=Prevalence of incontinence, attention deficit/hyperactivity disorder and oppositional defiant disorder in preschool children|url=https://doi.org/10.1007/s00787-014-0628-6|journal=European Child & Adolescent Psychiatry|language=en|volume=24|issue=7|pages=837–843|doi=10.1007/s00787-014-0628-6|issn=1435-165X}}</ref> |7.3% (males) 5.1%(females) |DISYPS-II |- |South Korea (Seoul) |Children were randomly surveyed across 6 school districts in Seoul (N = 1645, age 6 - 12 years old) |5.8% (males) 4.1%(females) |Diagnostic Interview Schedule for Children–Parent Scale IV (DISC-IV) |} ^p Parent interviewed as part of diagnostic assessment; ^y youth interviewed as part of diagnostic assessment, ^r adult interviewed for retrospective report as part of diagnostic assessment '''Note:''' Mash and Barkley note that prevalence rates of ODD must be qualified, because the definition of ODD has changed at a fast rate, the rates adolescents meeting criteria in any cross-sectional evaluation may be misleading because of the developmental progressions with and between ODD and Conduct Disorder, and categorical definitions of aggressive patterns may reflect arbitrary numbers of constituent estimates. These factors may lead to misleading prevalence rates. In addition, few studies have investigated the prevalence of ODD in preschool-aged children, and early onset of these behaviors is associated with more severe and stable impairment.<ref>Mash, E., & Barkley, R. (Eds.). (2003). ''Child Psychopathology''. 2nd Edition. New York: Guilford Press.</ref> ==[[Evidence based assessment/Prediction phase|'''Prediction phase''']]== The following section contains a list of screening and diagnostic instruments for ODD. The section includes administration information, psychometric data, and PDFs or links to the screenings. * Screenings are used as part of the [[Evidence based assessment/Prediction phase|prediction phase]] of assessment; for more information on interpretation of this data, or how screenings fit in to the assessment process, click [[Evidence based assessment/Prediction phase|here]]. * '''''For a list of more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&wteswitched=1#Psychometric_properties_of_common_screening_instruments click here.]''''' {| class="wikitable sortable" border="1" ! colspan="10" |Screening measures for ODD |- ! Measure ! Format (Reporter) ! Age Range ! Administration/ Completion Time ! Interrater Reliability ! Test-Retest Reliability ! Construct Validity ! Content Validity ! Highly Recommended !Where to access |- |Achenbach System of Empirically Based Assessments (ASEBA): Child Behavior Checklist (CBCL) |Parent report |6-18 <ref name=":9">{{Cite book|url=https://www.worldcat.org/oclc/1130319849|title=Assessment of disorders in childhood and adolescence|date=2020|others=Eric Arden Youngstrom, Mitchell J. Prinstein, Eric J. Mash, Russell A. Barkley|isbn=978-1-4625-4363-2|edition=Fifth edition|location=New York, NY|oclc=1130319849}}</ref> |10 - 15 minutes<ref name=":9" /> |A<ref name=":0">{{Cite book|url=https://www.worldcat.org/oclc/314222270|title=A guide to assessments that work|author=Hunsley, John|author2=Mash, Eric J.|date=2008|publisher=Oxford University Press|isbn=9780195310641|location=New York|oclc=314222270}}</ref> |E<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://store.aseba.org/School-Age-6-18-Materials/departments/11/ Purchase] |- |ASEBA: Teacher Report Form (TRF) |Teacher report |6-18 <ref name=":9" /> |10 - 15 minutes<ref name=":9" /> |A<ref name=":0" /> |E<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://store.aseba.org/School-Age-6-18-Materials/departments/11/ Purchase] |- |ASEBA: Youth Self-Report (YSR) |Youth self-report |11-18 <ref name=":9" /> |10 - 15 minutes<ref name=":9" /> |A<ref name=":0" /> |E<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://store.aseba.org/School-Age-6-18-Materials/departments/11/ Purchase] |- |Behavior Assessment for Children, Third Edition (BASC-3): Parent Rating Scale |Parent report |2-21 |10 - 20 minutes |A<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |E<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://www.pearsonassessments.com/store/usassessments/en/Store/Professional-Assessments/Behavior/Comprehensive/Behavior-Assessment-System-for-Children-%7C-Third-Edition-/p/100001402.html#:~:text=The%20BASC%E2%84%A2%20holds%20an,or%20adolescent's%20behavior%20and%20emotions. Purchase] |- |BASC-3: Teacher Rating Scale |Teacher report |2-21 |10 - 20 minutes |A<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |E<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://www.pearsonassessments.com/store/usassessments/en/Store/Professional-Assessments/Behavior/Comprehensive/Behavior-Assessment-System-for-Children-%7C-Third-Edition-/p/100001402.html#:~:text=The%20BASC%E2%84%A2%20holds%20an,or%20adolescent's%20behavior%20and%20emotions. Purchase] |- |BASC-3: Self-Report of Personality |Youth self-report |6 - college age |30 minutes |A<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |E<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://www.pearsonassessments.com/store/usassessments/en/Store/Professional-Assessments/Behavior/Comprehensive/Behavior-Assessment-System-for-Children-%7C-Third-Edition-/p/100001402.html#:~:text=The%20BASC%E2%84%A2%20holds%20an,or%20adolescent's%20behavior%20and%20emotions. Purchase] |- |Eyberg Child Behavior Inventory (ECBI) |Parent report |2-16 |5 minutes |A<ref name=":0" /> |G<ref name=":0" /> |E<ref name=":0" /> |E<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://www.parinc.com/products/pkey/97 Purchase] |- |Sutter-Eyberg Student Behavior Inventory - Revised (SESBI-R) |Teacher report |2-16 |5 minutes |A<ref name=":0" /> |G<ref name=":0" /> |E<ref name=":0" /> |E<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://www.parinc.com/products/pkey/97 Purchase] |- |Child and Adolescent Behavior Inventory (CABI) |Parent Report |3 - 18 |5 - 10 minutes | |A<ref>{{Cite journal|last=Burns|first=G. Leonard|last2=Preszler|first2=Jonathan|last3=Becker|first3=Stephen P.|date=2022-07-04|title=Psychometric and Normative Information on the Child and Adolescent Behavior Inventory in a Nationally Representative Sample of United States Children|url=https://doi.org/10.1080/15374416.2020.1852943|journal=Journal of Clinical Child & Adolescent Psychology|volume=51|issue=4|pages=443–452|doi=10.1080/15374416.2020.1852943|issn=1537-4416|pmc=PMC8272731|pmid=33428463}}</ref> | | | |PDF |- |Strengths and Difficulties Questionnaire (SDQ) |Parent report |3 - 16 |3 - 5 minutes |E<ref name=":10">{{Cite journal|last=Goodman|first=Robert|date=2001-11-01|title=Psychometric Properties of the Strengths and Difficulties Questionnaire|url=https://www.jaacap.org/article/S0890-8567(09)60543-8/abstract|journal=Journal of the American Academy of Child & Adolescent Psychiatry|language=English|volume=40|issue=11|pages=1337–1345|doi=10.1097/00004583-200111000-00015|issn=0890-8567|pmid=11699809}}</ref> |G<ref name=":10" /> | | | |[https://www.sdqinfo.org/a0.html SDQ Homepage][https://www.sdqinfo.org/py/sdqinfo/b0.py PDFs] |- |Disruptive Behavior Disorder Rating Scale |Parent report, teacher report |5 - 17 |5 - 10 minutes | | | | | |[https://web.archive.org/web/20151123022653/http://ccf.buffalo.edu/pdf/DBD_rating_scale.pdf PDF] |} '''Note:''' L = Less than adequate; A = Adequate; G = Good; E = Excellent; U = Unavailable; NA = Not applicable === Likelihood ratios and AUCs of screening instruments for ODD === * '''''For a list of the likelihood ratios for more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&wteswitched=1#Likelihood_ratios_and_AUCs_of_common_screening_instruments click here.]''''' {| class="wikitable sortable" border="1" ! Screening Measure (Primary Reference) ! Area Under Curve (AUC) ! LR+ (Score) ! LR- (Score) ! Clinical Generalizability !Where to Access |- |CBCL DSM-Oriented Scales<ref name=":7">Achenbach, T. M. (1991a). ''Manual for the Child Behavior Checklist/4–18 and 1991 Profile''. Burlington , VT : University of Vermont Department of Psychiatry.</ref> |.71 (N=475)<ref name=":3">Ebesutani, C., Bernstein, A., Nakamura, B. J., Chorpita, B. F., Higa-McMillan, C. K., & Weisz, J. R. (2010). Concurrent validity of the child behavior checklist DSM-oriented scales: Correspondence with DSM diagnoses and comparison to syndrome scales. ''Journal of Psychopathology and Behavioral Assessment, 32''(3), 373-384.</ref> | 2.80 (60+ to 70+) <ref name=":4">Warnick, E. M., Bracken, M. B., & Kasl, S. (2008). Screening efficiency of the Child Behavior Checklist and Strengths and Difficulties Questionnaire: a systematic review. ''Child and Adolescent Mental Health, 13''(3), 140-147.</ref> | .52 (60- to 70-)*<ref name=":4" /> |Youth aged 5 - 18 seeking out patient treatment across a variety of settings<ref name=":4" /> |[https://store.aseba.org/School-Age-6-18-Materials/departments/11/ Purchase] |- |SDQ<ref name=":6">{{Cite journal|last=Shabani|first=Amir|last2=Masoumian|first2=Samira|last3=Zamirinejad|first3=Somayeh|last4=Hejri|first4=Maryam|last5=Pirmorad|first5=Tahereh|last6=Yaghmaeezadeh|first6=Hooman|date=2021-05|title=Psychometric properties of Structured Clinical Interview for DSM‐5 Disorders‐Clinician Version (SCID‐5‐CV)|url=https://onlinelibrary.wiley.com/doi/10.1002/brb3.1894|journal=Brain and Behavior|language=en|volume=11|issue=5|doi=10.1002/brb3.1894|issn=2162-3279|pmc=PMC8119811|pmid=33729681}}</ref> | .81 -.88 (N = 18,416)<ref>{{Cite journal|last=Algorta|first=Guillermo Perez|last2=Dodd|first2=Alyson Lamont|last3=Stringaris|first3=Argyris|last4=Youngstrom|first4=Eric A.|date=2016-09|title=Diagnostic efficiency of the SDQ for parents to identify ADHD in the UK: a ROC analysis|url=http://link.springer.com/10.1007/s00787-015-0815-0|journal=European Child & Adolescent Psychiatry|language=en|volume=25|issue=9|pages=949–957|doi=10.1007/s00787-015-0815-0|issn=1018-8827|pmc=PMC4990620|pmid=26762184}}</ref> |7.00 <ref name=":4" /> |.55<ref name=":4" /> |Youth aged 5 - 18 seeking out patient treatment across a variety of settings<ref name=":4" /> |[https://www.sdqinfo.org/a0.html SDQ Homepage][https://www.sdqinfo.org/py/sdqinfo/b0.py PDFs] |- |SDQ - DSM -IV Conduct and ODD | |7.55 (3+)<ref name=":2">{{Cite journal|last=Goodman|first=Robert|date=2001-11-01|title=Psychometric Properties of the Strengths and Difficulties Questionnaire|url=https://www.jaacap.org/article/S0890-8567(09)60543-8/abstract|journal=Journal of the American Academy of Child & Adolescent Psychiatry|language=English|volume=40|issue=11|pages=1337–1345|doi=10.1097/00004583-200111000-00015|issn=0890-8567|pmid=11699809}}</ref> |.35 (3-)<ref name=":2" /> |Surveyed youth aged 5 - 15 in the UK <ref name=":2" /> |[https://www.sdqinfo.org/a0.html SDQ Homepage][https://www.sdqinfo.org/py/sdqinfo/b0.py PDFs] |- |ECBI- Intensity Scale<ref name=":8">Eyberg, S. M., & Robinson, E. A. (1983). Conduct problem behavior: Standardization of a behavioral rating scale with adolescents. Journal of Clinical Child Psychology, 12 (3), 347-354.</ref> | |6.92 (131+)<ref name=":5">{{Cite journal|last=LYNEHAM|first=HEIDI J.|last2=ABBOTT|first2=MAREE J.|last3=RAPEE|first3=RONALD M.|date=2007-06|title=Interrater Reliability of the Anxiety Disorders Interview Schedule for DSM-IV: Child and Parent Version|url=https://doi.org/10.1097/chi.0b013e3180465a09|journal=Journal of the American Academy of Child &amp; Adolescent Psychiatry|volume=46|issue=6|pages=731–736|doi=10.1097/chi.0b013e3180465a09|issn=0890-8567}}</ref> |.11 (131-)<ref name=":5" /> |Youth aged 7-16 had responses compared to diagnosis<ref name=":5" /> |[https://www.parinc.com/products/pkey/97 Purchase] |- |BASC-2 PRS - Aggression |.76 (N =156)<ref name=":12">{{Cite journal|last=Doyle|first=Alysa|last2=Ostrander|first2=Rick|last3=Skare|first3=Stacy|last4=Crosby|first4=Ross D.|last5=August|first5=Gerald J.|date=1997-09-01|title=Convergent and criterion-related validity of the behavior assessment system for children-parent rating scale|url=https://doi.org/10.1207/s15374424jccp2603_6|journal=Journal of Clinical Child Psychology|volume=26|issue=3|pages=276–284|doi=10.1207/s15374424jccp2603_6|issn=0047-228X|pmid=9292385}}</ref> |3.62 (65+)<ref name=":12" /> |.60 (65-)<ref name=":12" /> |Youth from first through fourth grade who were at risk for CD<ref name=":12" /> |[https://www.pearsonassessments.com/store/usassessments/en/Store/Professional-Assessments/Behavior/Comprehensive/Behavior-Assessment-System-for-Children-%7C-Third-Edition-/p/100001402.html#:~:text=The%20BASC%E2%84%A2%20holds%20an,or%20adolescent's%20behavior%20and%20emotions. Purchase] |} '''Note:''' “LR+” refers to the change in likelihood ratio associated with a positive test score, and “LR-” is the likelihood ratio for a low score. Likelihood ratios of 1 indicate that the test result did not change impressions at all. LRs larger than 10 or smaller than .10 are frequently clinically decisive; 5 or .20 are helpful, and between 2.0 and .5 are small enough that they rarely result in clinically meaningful changes of formulation<ref>Sackett, D. L., Straus, S. E., Richardson, W. S., Rosenberg, W., & Haynes, R. B. (2000). Evidence-based medicine: How to practice and teach EBM. Edinburgh: Churchill Livingstone.</ref>. '''Search terms:''' [Oppositional Defiant Disorder] AND [sensitivity OR specificity] in GoogleScholar and PsychINFO; === Interpreting ODD screening measure scores === * For information on interpreting screening measure scores, click [[Evidence based assessment/Prediction phase#Interpreting screening measure scores|here.]] == [[Evidence based assessment/Prescription phase|'''Prescription phase''']] == === Gold standard diagnostic interviews === * For a list of broad reaching diagnostic interviews sortable by disorder with PDFs (if applicable), [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prescription_phase&wteswitched=1#Common_Diagnostic_Interviews click here.] === Recommended diagnostic instruments for ODD === {| class="wikitable sortable" ! colspan="10" |Diagnostic instruments for ODD |- !Measure !Format (Reporter) !Age Range !Administration/ Completion Time !Interrater Reliability !Test-Retest Reliability !Construct Validity !Content Validity !Highly Recommended !Where to Access |- |Kiddie Schedule for Affective Disorders and Schizophrenia Present and Lifetime Version (KSADS-PL) |Structured interview |6-28 |45-75 minutes |E<ref name=":11">{{Cite journal|last=KAUFMAN|first=JOAN|last2=BIRMAHER|first2=BORIS|last3=BRENT|first3=DAVID|last4=RAO|first4=UMA|last5=FLYNN|first5=CYNTHIA|last6=MORECI|first6=PAULA|last7=WILLIAMSON|first7=DOUGLAS|last8=RYAN|first8=NEAL|date=1997-07|title=Schedule for Affective Disorders and Schizophrenia for School-Age Children-Present and Lifetime Version (K-SADS-PL): Initial Reliability and Validity Data|url=https://doi.org/10.1097/00004583-199707000-00021|journal=Journal of the American Academy of Child &amp; Adolescent Psychiatry|volume=36|issue=7|pages=980–988|doi=10.1097/00004583-199707000-00021|issn=0890-8567}}</ref> |G<ref name=":11" /> | | | |[https://www.pediatricbipolar.pitt.edu/resources/instruments Website to access] |- |Diagnostic Interview Schedule for Children (DISC-5) |Structured Interview (Self report and parent) |6-17 | | | | | |<ref name=":0" />[[File:Light green check.svg|center|frameless|36x36px]] |[https://telesage.com/netdisc-5/# Coming soon] |} '''Note:''' L = Less than adequate; A = Adequate; G = Good; E = Excellent; U = Unavailable; NA = Not applicable == [[Evidence based assessment/Process phase|'''Process phase''']] == The following section contains a list of process and outcome measures for oppositional defiant disorder. The section includes benchmarks based on published norms for several outcome and severity measures, as well as information about commonly used process measures. Process and outcome measures are used as part of the [[Evidence based assessment/Process phase|process phase]] of assessment. For more information of differences between process and outcome measures, see the page on the [[Evidence based assessment/Process phase|process phase of assessment]]. === Outcome and severity measures === * This table includes clinically significant benchmarks for generalized anxiety disorder specific outcome measures * Information on how to interpret this table can be [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase found here]. * Additionally, these [[Evidence based assessment/Vignettes|vignettes]] might be helpful resources for understanding appropriate adaptation of outcome measures in practice. * ''<u>For clinically significant change benchmarks for the CBCL, YSR, and TRF total, externalizing, internalizing, and attention benchmarks,</u>'' [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase&wteswitched=1#Clinically_significant_change_benchmarks_for_widely-used_outcome_measures see here.] {| class="wikitable sortable" border="1" |- | rowspan="1&quot;" style="text-align:center;font-size:130%;" | <b> Measure</b> | style="text-align:center;font-size:130%;" | <b> Subscale</b> | colspan="3" style="text-align:center;font-size:130%" width="300" | <b> Cut-off scores</b> | colspan="3" style="text-align:center;font-size:120%" | <b> Critical Change <br> (unstandardized scores)</b> |- | colspan="8" style="font-size:110%; text-align:center;" span | <b> Benchmarks Based on Published Norms</b> |- | colspan="2" | | style="text-align:center;font-size:110%" | <b> A</b> | style="text-align:center;font-size:110%" | <b> B</b> | style="text-align:center;font-size:110%" | <b> C</b> | style="text-align:center;font-size:110%" | <b> 95%</b> | style="text-align:center;font-size:110%" | <b> 90%</b> | style="text-align:center;font-size:110%" | <b> SE_difference</b> |- | rowspan="1" style="text-align:center;" | <b> CBCL T-scores <br> (2001 Norms)</b> | style="text-align:right;" | <i> Externalizing</i> | style="text-align:center;" | 49 | style="text-align:center;" | 70 | style="text-align:center;" | 58 | style="text-align:center;" | 7 | style="text-align:center;" | 6 | style="text-align:center;" | 3.4 |- | rowspan="1" style="text-align:center;" | <b> ECBI Scaled Scores <br> (1983 Norms)</b> | style="text-align:right;" | <i> Intensity</i> | style="text-align:center;" | 80.1 | style="text-align:center;" | 169.5 | style="text-align:center;" | 112.9 | style="text-align:center;" | 9.5 | style="text-align:center;" | 8 | style="text-align:center;" | 4.8 |- | rowspan="1" style="text-align:center;" | <b> ECBI Scaled Scores <br> (1983 Norms)</b> | style="text-align:right;" | <i> Problem</i> | style="text-align:center;" | 3.9 | style="text-align:center;" | 17.7 | style="text-align:center;" | 11.5 | style="text-align:center;" | 2.1 | style="text-align:center;" | 1.8 | style="text-align:center;" | 1.1 |} '''Note:''' “A” = Away from the clinical range, “B” = Back into the nonclinical range, “C” = Closer to the nonclinical than clinical mean. === Process measures=== See Section 1.1 for overview of evidence-based measures to use depending on etiology and symptomatology of Oppositional Defiant Disorder. ==Treatment== ===Behavioral parent training=== Behavioral Parent Training is considered the most effective treatment for childhood disruptive behavior disorders (e.g., Oppositional Defiant Disorder), especially for younger children (i.e., 3-8 year-olds). See http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/rule-breaking-defiance-and-acting-out/ a website sponsored by The Society for Child and Adolescent Psychology (APA, Division 53) and the Association for Behavioral and Cognitive Therapies (ABCT), for current summary of evidence-based treatments. ===Overview of recommendations for assessment and treatment=== See the [https://pathways.nice.org.uk/pathways/antisocial-behaviour-and-conduct-disorders-in-children-and-young-people#content=view-info-category%3Aview-about-menu National Institute for Health and Care Excellence (NICE) Practice Guidelines for Childhood Conduct Disorders], for an overview of recommendations for both assessment and treatment of Oppositional Defiant Disorder. == External Links == *[https://sccap53.org Society of Clinical Child and Adolescent Psychology] *http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/rule-breaking-defiance-and-acting-out/ == References == {{collapse top|Click here for references}} {{Reflist|30em}} [[Category:Psychological disorder portfolios|{{SUBPAGENAME}}]] {{collapse bottom}} [[Category:Psychological disorder portfolios|{{SUBPAGENAME}}]] d5c41o9j4ysfs17os80oirtqkrdfv8z 2410324 2410323 2022-07-29T22:25:19Z Aherman012 2943941 /* What is a "portfolio"? */ wikitext text/x-wiki <noinclude>{{Helping Give Away Psychological Science Banner}}</noinclude> {{medical disclaimer}} {{:{{ROOTPAGENAME}}/Sidebar}} ==[[Evidence based assessment/Portfolio template/What is a "portfolio"|'''What is a "portfolio"?''']]== For background information on what assessment portfolios are, click the link in the heading above. Does this page feel like too much information? Click [[Evidence-based assessment/Oppositional defiant disorder (disorder portfolio)|here]] for the condensed version. ==[[Evidence based assessment/Preparation phase|'''Preparation phase''']]== === Diagnostic criteria for oppositional defiant disorder=== {{blockquotetop}} <big>ICD-11 Diagnostic Criteria</big><br> <br> '''General Description:''' Oppositional defiant disorder is a persistent pattern (e.g., 6 months or more) of markedly defiant, disobedient, provocative or spiteful behaviour that occurs more frequently than is typically observed in individuals of comparable age and developmental level and that is not restricted to interaction with siblings. Oppositional defiant disorder may be manifest in prevailing, persistent angry or irritable mood, often accompanied by severe temper outbursts or in headstrong, argumentative and defiant behaviour. The behavior pattern is of sufficient severity to result in significant impairment in personal, family, social, educational, occupational or other important areas of functioning <br> <br> '''Oppositional Defiant Disorder With Chronic Irritability-Anger:''' All definitional requirements for oppositional defiant disorder are met. This form of oppositional defiant disorder is characterized by prevailing, persistent angry or irritable mood that may be present independent of any apparent provocation. The negative mood is often accompanied by regularly occurring severe temper outbursts that are grossly out of proportion in intensity or duration to the provocation. Chronic irritability and anger are characteristic of the individual’s functioning nearly every day, are observable across multiple settings or domains of functioning (e.g., home, school, social relationships), and are not restricted to the individual’s relationship with his/her parents or guardians. The pattern of chronic irritability and anger is not limited to occasional episodes (e.g., developmentally typical irritability) or discrete periods (e.g., irritable mood in the context of manic or depressive episodes). *Note: The ICD-11 lists 3 additional subcategories of oppositional defiant disorder with chronic irritability-anger (i.e., with limited prosocial emotions, with typical prosocial emotions, and unspecified). They can be found [https://icd.who.int/browse11/l-m/en#/http%3a%2f%2fid.who.int%2ficd%2fentity%2f818792113 here]. <br> '''Oppositional Defiant Disorder Without Chronic Irritability-Anger:''' Meets all definitional requirements for oppositional defiant disorder. This form of oppositional defiant disorder is not characterized by prevailing, persistent, angry or irritable mood, but does feature headstrong, argumentative, and defiant behavior. *Note: The ICD-11 lists 3 additional subcategories of oppositional defiant disorder without chronic irritability-anger (i.e., with limited prosocial emotions, with typical prosocial emotions, and unspecified). They can be found [https://icd.who.int/browse11/l-m/en#/http%3a%2f%2fid.who.int%2ficd%2fentity%2f736540987 here]. <br> <big>Changes in DSM-5</big> <br> The diagnostic criteria for ADHD changed slightly from DSM-IV to DSM-5. See the changes [https://www.ncbi.nlm.nih.gov/books/NBK519712/table/ch3.t14/ here]. {{blockquotebottom}}<ref>{{Cite web|url=https://icd.who.int/browse11/l-m/en#/http://id.who.int/icd/entity/1545599508|title=ICD-11 for Mortality and Morbidity Statistics|website=icd.who.int|access-date=2022-07-11}}</ref> ===Base rates of ODD in different clinical settings=== This section describes the demographic setting of the population(s) sampled, base rates of diagnosis, country/region sampled and the diagnostic method that was used. Using this information, clinicians will be able to anchor the rate of ODD that they are likely to see in their clinical practice. * '''''To see prevalence rates across multiple disorders, [[Evidence-based assessment/Preparation phase|click here.]]''''' {|class="wikitable sortable" border="1" |- ! Demography ! Setting !! Base Rate !! Diagnostic Method |- || Various locations across USA | Meta-analysis of 38 studies<ref>Canino G, Polanczyk G, Bauermeister JJ, et al. (2010) Does the prevalence of CD and ODD vary across cultures? Social Psychiatry Epidemiology, 45(7):695–704.</ref> || 3.3% || Varied |- || All of the U.S. | Nationally representative large-scale study (N = 3,119) <ref>Nock, M. K., Kazdin, A. E., Hiripi, E., & Kessler, R. C. (2007). Lifetime prevalence, correlates, and persistence of oppositional defiant disorder: results from the National Comorbidity Survey Replication. ''Journal of Child Psychology and Psychiatry, 48''(7), 703-713.</ref> || 10.2% (overall) 11.2% (males) 9.2% (females) || World Health Organization (WHO) Composite International Diagnostic Interview (CIDI)^r |- |Suburban and urban Colorado |Project to Learn about Youth-Mental health, school-based study for children from kindergarten to high-school (n = 236)<ref name=":1">{{Cite journal|last=Danielson|first=Melissa L.|last2=Bitsko|first2=Rebecca H.|last3=Holbrook|first3=Joseph R.|last4=Charania|first4=Sana N.|last5=Claussen|first5=Angelika H.|last6=McKeown|first6=Robert E.|last7=Cuffe|first7=Steven P.|last8=Owens|first8=Julie Sarno|last9=Evans|first9=Steven W.|date=2021-06-01|title=Community-Based Prevalence of Externalizing and Internalizing Disorders among School-Aged Children and Adolescents in Four Geographically Dispersed School Districts in the United States|url=https://doi.org/10.1007/s10578-020-01027-z|journal=Child Psychiatry & Human Development|language=en|volume=52|issue=3|pages=500–514|doi=10.1007/s10578-020-01027-z|issn=1573-3327|pmc=PMC8016018|pmid=32734339}}</ref> |6.8% |Diagnostic Interview Schedule for Children (DISC) |- |Urban and suburban Florida |Project to Learn about Youth-Mental health, school-based study for children from kindergarten to high-school (n = 289)<ref name=":1" /> |6.9% |Diagnostic Interview Schedule for Children (DISC) |- |Rural and suburban Ohio |Project to Learn about Youth-Mental health, school-based study for children from elementary to high-school (n = 152)<ref name=":1" /> |17.3% |Diagnostic Interview Schedule for Children (DISC) |- |Suburban and rural South Carolina |Project to Learn about Youth-Mental health, school-based study for children from elementary to high-school (n = 270)<ref name=":1" /> |5.7% |Diagnostic Interview Schedule for Children (DISC) |- || Semi-rural North Carolina | Preschool-aged children from pediatric practices (N = 306; age 2 - 5 years old)<ref>Egger, H.L., & Angold, A. (2006). Common emotional and behavioral disorders in preschool children: Presentation, nosology, and epidemiology. ''Journal of Child Psychiatry and Psychology, 47'', 313–337.</ref> || 6.6% || Preschool Age Psychiatric Assessment (PAPA)^p |- || Western North Carolina | The Great Smoky Mountains Study - longitudinal, population-based study of community sample<ref>Costello, E. J., Mustillo, S., Erkanli, A., Keeler, G., & Angold, A. (2003) Prevalence and development of psychiatric disorders in adolescence. ''Arch Gen Psychiatry, 60'', 837-844.</ref> || 2.33% (overall) 3.16% (males) 2.75% (females) || Child and Adolescent Psychiatric Assessment (CAPA)^{p, y} |- || Chicago | Preschool-aged children from inner city schools and pediatric practices (N = 796; age 2 - 5 years old)<ref>Lavigne, J. V., LeBailly, S. A., Hopkins, J., Gouze, K. R., & Binns, H. J. (2009). The prevalence of ADHD, ODD, depression, and anxiety in a community sample of 4-year-olds. ''Journal Of Clinical Child And Adolescent Psychology, 38''(3), 315-328. doi:10.1080/15374410902851382</ref> || 8.3% || Diagnostic Interview Schedule for Children–Parent Scale–Young Child Version (DISC-YC) ^p |- |Germany (Saarbrücken County) |All school-aged children were examined during a routine school-entry medical examination (N = 1676, mean age = 5.7)<ref>{{Cite journal|last=Niemczyk|first=Justine|last2=Equit|first2=Monika|last3=Braun-Bither|first3=Katrin|last4=Klein|first4=Anna-Maria|last5=von Gontard|first5=Alexander|date=2015-07-01|title=Prevalence of incontinence, attention deficit/hyperactivity disorder and oppositional defiant disorder in preschool children|url=https://doi.org/10.1007/s00787-014-0628-6|journal=European Child & Adolescent Psychiatry|language=en|volume=24|issue=7|pages=837–843|doi=10.1007/s00787-014-0628-6|issn=1435-165X}}</ref> |7.3% (males) 5.1%(females) |DISYPS-II |- |South Korea (Seoul) |Children were randomly surveyed across 6 school districts in Seoul (N = 1645, age 6 - 12 years old) |5.8% (males) 4.1%(females) |Diagnostic Interview Schedule for Children–Parent Scale IV (DISC-IV) |} ^p Parent interviewed as part of diagnostic assessment; ^y youth interviewed as part of diagnostic assessment, ^r adult interviewed for retrospective report as part of diagnostic assessment '''Note:''' Mash and Barkley note that prevalence rates of ODD must be qualified, because the definition of ODD has changed at a fast rate, the rates adolescents meeting criteria in any cross-sectional evaluation may be misleading because of the developmental progressions with and between ODD and Conduct Disorder, and categorical definitions of aggressive patterns may reflect arbitrary numbers of constituent estimates. These factors may lead to misleading prevalence rates. In addition, few studies have investigated the prevalence of ODD in preschool-aged children, and early onset of these behaviors is associated with more severe and stable impairment.<ref>Mash, E., & Barkley, R. (Eds.). (2003). ''Child Psychopathology''. 2nd Edition. New York: Guilford Press.</ref> ==[[Evidence based assessment/Prediction phase|'''Prediction phase''']]== The following section contains a list of screening and diagnostic instruments for ODD. The section includes administration information, psychometric data, and PDFs or links to the screenings. * Screenings are used as part of the [[Evidence based assessment/Prediction phase|prediction phase]] of assessment; for more information on interpretation of this data, or how screenings fit in to the assessment process, click [[Evidence based assessment/Prediction phase|here]]. * '''''For a list of more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&wteswitched=1#Psychometric_properties_of_common_screening_instruments click here.]''''' {| class="wikitable sortable" border="1" ! colspan="10" |Screening measures for ODD |- ! Measure ! Format (Reporter) ! Age Range ! Administration/ Completion Time ! Interrater Reliability ! Test-Retest Reliability ! Construct Validity ! Content Validity ! Highly Recommended !Where to access |- |Achenbach System of Empirically Based Assessments (ASEBA): Child Behavior Checklist (CBCL) |Parent report |6-18 <ref name=":9">{{Cite book|url=https://www.worldcat.org/oclc/1130319849|title=Assessment of disorders in childhood and adolescence|date=2020|others=Eric Arden Youngstrom, Mitchell J. Prinstein, Eric J. Mash, Russell A. Barkley|isbn=978-1-4625-4363-2|edition=Fifth edition|location=New York, NY|oclc=1130319849}}</ref> |10 - 15 minutes<ref name=":9" /> |A<ref name=":0">{{Cite book|url=https://www.worldcat.org/oclc/314222270|title=A guide to assessments that work|author=Hunsley, John|author2=Mash, Eric J.|date=2008|publisher=Oxford University Press|isbn=9780195310641|location=New York|oclc=314222270}}</ref> |E<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://store.aseba.org/School-Age-6-18-Materials/departments/11/ Purchase] |- |ASEBA: Teacher Report Form (TRF) |Teacher report |6-18 <ref name=":9" /> |10 - 15 minutes<ref name=":9" /> |A<ref name=":0" /> |E<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://store.aseba.org/School-Age-6-18-Materials/departments/11/ Purchase] |- |ASEBA: Youth Self-Report (YSR) |Youth self-report |11-18 <ref name=":9" /> |10 - 15 minutes<ref name=":9" /> |A<ref name=":0" /> |E<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://store.aseba.org/School-Age-6-18-Materials/departments/11/ Purchase] |- |Behavior Assessment for Children, Third Edition (BASC-3): Parent Rating Scale |Parent report |2-21 |10 - 20 minutes |A<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |E<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://www.pearsonassessments.com/store/usassessments/en/Store/Professional-Assessments/Behavior/Comprehensive/Behavior-Assessment-System-for-Children-%7C-Third-Edition-/p/100001402.html#:~:text=The%20BASC%E2%84%A2%20holds%20an,or%20adolescent's%20behavior%20and%20emotions. Purchase] |- |BASC-3: Teacher Rating Scale |Teacher report |2-21 |10 - 20 minutes |A<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |E<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://www.pearsonassessments.com/store/usassessments/en/Store/Professional-Assessments/Behavior/Comprehensive/Behavior-Assessment-System-for-Children-%7C-Third-Edition-/p/100001402.html#:~:text=The%20BASC%E2%84%A2%20holds%20an,or%20adolescent's%20behavior%20and%20emotions. Purchase] |- |BASC-3: Self-Report of Personality |Youth self-report |6 - college age |30 minutes |A<ref name=":0" /> |E<ref name=":0" /> |G<ref name=":0" /> |E<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://www.pearsonassessments.com/store/usassessments/en/Store/Professional-Assessments/Behavior/Comprehensive/Behavior-Assessment-System-for-Children-%7C-Third-Edition-/p/100001402.html#:~:text=The%20BASC%E2%84%A2%20holds%20an,or%20adolescent's%20behavior%20and%20emotions. Purchase] |- |Eyberg Child Behavior Inventory (ECBI) |Parent report |2-16 |5 minutes |A<ref name=":0" /> |G<ref name=":0" /> |E<ref name=":0" /> |E<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://www.parinc.com/products/pkey/97 Purchase] |- |Sutter-Eyberg Student Behavior Inventory - Revised (SESBI-R) |Teacher report |2-16 |5 minutes |A<ref name=":0" /> |G<ref name=":0" /> |E<ref name=":0" /> |E<ref name=":0" /> |[[File:Light green check.svg|center|frameless|36x36px]] |[https://www.parinc.com/products/pkey/97 Purchase] |- |Child and Adolescent Behavior Inventory (CABI) |Parent Report |3 - 18 |5 - 10 minutes | |A<ref>{{Cite journal|last=Burns|first=G. Leonard|last2=Preszler|first2=Jonathan|last3=Becker|first3=Stephen P.|date=2022-07-04|title=Psychometric and Normative Information on the Child and Adolescent Behavior Inventory in a Nationally Representative Sample of United States Children|url=https://doi.org/10.1080/15374416.2020.1852943|journal=Journal of Clinical Child & Adolescent Psychology|volume=51|issue=4|pages=443–452|doi=10.1080/15374416.2020.1852943|issn=1537-4416|pmc=PMC8272731|pmid=33428463}}</ref> | | | |PDF |- |Strengths and Difficulties Questionnaire (SDQ) |Parent report |3 - 16 |3 - 5 minutes |E<ref name=":10">{{Cite journal|last=Goodman|first=Robert|date=2001-11-01|title=Psychometric Properties of the Strengths and Difficulties Questionnaire|url=https://www.jaacap.org/article/S0890-8567(09)60543-8/abstract|journal=Journal of the American Academy of Child & Adolescent Psychiatry|language=English|volume=40|issue=11|pages=1337–1345|doi=10.1097/00004583-200111000-00015|issn=0890-8567|pmid=11699809}}</ref> |G<ref name=":10" /> | | | |[https://www.sdqinfo.org/a0.html SDQ Homepage][https://www.sdqinfo.org/py/sdqinfo/b0.py PDFs] |- |Disruptive Behavior Disorder Rating Scale |Parent report, teacher report |5 - 17 |5 - 10 minutes | | | | | |[https://web.archive.org/web/20151123022653/http://ccf.buffalo.edu/pdf/DBD_rating_scale.pdf PDF] |} '''Note:''' L = Less than adequate; A = Adequate; G = Good; E = Excellent; U = Unavailable; NA = Not applicable === Likelihood ratios and AUCs of screening instruments for ODD === * '''''For a list of the likelihood ratios for more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&wteswitched=1#Likelihood_ratios_and_AUCs_of_common_screening_instruments click here.]''''' {| class="wikitable sortable" border="1" ! Screening Measure (Primary Reference) ! Area Under Curve (AUC) ! LR+ (Score) ! LR- (Score) ! Clinical Generalizability !Where to Access |- |CBCL DSM-Oriented Scales<ref name=":7">Achenbach, T. M. (1991a). ''Manual for the Child Behavior Checklist/4–18 and 1991 Profile''. Burlington , VT : University of Vermont Department of Psychiatry.</ref> |.71 (N=475)<ref name=":3">Ebesutani, C., Bernstein, A., Nakamura, B. J., Chorpita, B. F., Higa-McMillan, C. K., & Weisz, J. R. (2010). Concurrent validity of the child behavior checklist DSM-oriented scales: Correspondence with DSM diagnoses and comparison to syndrome scales. ''Journal of Psychopathology and Behavioral Assessment, 32''(3), 373-384.</ref> | 2.80 (60+ to 70+) <ref name=":4">Warnick, E. M., Bracken, M. B., & Kasl, S. (2008). Screening efficiency of the Child Behavior Checklist and Strengths and Difficulties Questionnaire: a systematic review. ''Child and Adolescent Mental Health, 13''(3), 140-147.</ref> | .52 (60- to 70-)*<ref name=":4" /> |Youth aged 5 - 18 seeking out patient treatment across a variety of settings<ref name=":4" /> |[https://store.aseba.org/School-Age-6-18-Materials/departments/11/ Purchase] |- |SDQ<ref name=":6">{{Cite journal|last=Shabani|first=Amir|last2=Masoumian|first2=Samira|last3=Zamirinejad|first3=Somayeh|last4=Hejri|first4=Maryam|last5=Pirmorad|first5=Tahereh|last6=Yaghmaeezadeh|first6=Hooman|date=2021-05|title=Psychometric properties of Structured Clinical Interview for DSM‐5 Disorders‐Clinician Version (SCID‐5‐CV)|url=https://onlinelibrary.wiley.com/doi/10.1002/brb3.1894|journal=Brain and Behavior|language=en|volume=11|issue=5|doi=10.1002/brb3.1894|issn=2162-3279|pmc=PMC8119811|pmid=33729681}}</ref> | .81 -.88 (N = 18,416)<ref>{{Cite journal|last=Algorta|first=Guillermo Perez|last2=Dodd|first2=Alyson Lamont|last3=Stringaris|first3=Argyris|last4=Youngstrom|first4=Eric A.|date=2016-09|title=Diagnostic efficiency of the SDQ for parents to identify ADHD in the UK: a ROC analysis|url=http://link.springer.com/10.1007/s00787-015-0815-0|journal=European Child & Adolescent Psychiatry|language=en|volume=25|issue=9|pages=949–957|doi=10.1007/s00787-015-0815-0|issn=1018-8827|pmc=PMC4990620|pmid=26762184}}</ref> |7.00 <ref name=":4" /> |.55<ref name=":4" /> |Youth aged 5 - 18 seeking out patient treatment across a variety of settings<ref name=":4" /> |[https://www.sdqinfo.org/a0.html SDQ Homepage][https://www.sdqinfo.org/py/sdqinfo/b0.py PDFs] |- |SDQ - DSM -IV Conduct and ODD | |7.55 (3+)<ref name=":2">{{Cite journal|last=Goodman|first=Robert|date=2001-11-01|title=Psychometric Properties of the Strengths and Difficulties Questionnaire|url=https://www.jaacap.org/article/S0890-8567(09)60543-8/abstract|journal=Journal of the American Academy of Child & Adolescent Psychiatry|language=English|volume=40|issue=11|pages=1337–1345|doi=10.1097/00004583-200111000-00015|issn=0890-8567|pmid=11699809}}</ref> |.35 (3-)<ref name=":2" /> |Surveyed youth aged 5 - 15 in the UK <ref name=":2" /> |[https://www.sdqinfo.org/a0.html SDQ Homepage][https://www.sdqinfo.org/py/sdqinfo/b0.py PDFs] |- |ECBI- Intensity Scale<ref name=":8">Eyberg, S. M., & Robinson, E. A. (1983). Conduct problem behavior: Standardization of a behavioral rating scale with adolescents. Journal of Clinical Child Psychology, 12 (3), 347-354.</ref> | |6.92 (131+)<ref name=":5">{{Cite journal|last=LYNEHAM|first=HEIDI J.|last2=ABBOTT|first2=MAREE J.|last3=RAPEE|first3=RONALD M.|date=2007-06|title=Interrater Reliability of the Anxiety Disorders Interview Schedule for DSM-IV: Child and Parent Version|url=https://doi.org/10.1097/chi.0b013e3180465a09|journal=Journal of the American Academy of Child &amp; Adolescent Psychiatry|volume=46|issue=6|pages=731–736|doi=10.1097/chi.0b013e3180465a09|issn=0890-8567}}</ref> |.11 (131-)<ref name=":5" /> |Youth aged 7-16 had responses compared to diagnosis<ref name=":5" /> |[https://www.parinc.com/products/pkey/97 Purchase] |- |BASC-2 PRS - Aggression |.76 (N =156)<ref name=":12">{{Cite journal|last=Doyle|first=Alysa|last2=Ostrander|first2=Rick|last3=Skare|first3=Stacy|last4=Crosby|first4=Ross D.|last5=August|first5=Gerald J.|date=1997-09-01|title=Convergent and criterion-related validity of the behavior assessment system for children-parent rating scale|url=https://doi.org/10.1207/s15374424jccp2603_6|journal=Journal of Clinical Child Psychology|volume=26|issue=3|pages=276–284|doi=10.1207/s15374424jccp2603_6|issn=0047-228X|pmid=9292385}}</ref> |3.62 (65+)<ref name=":12" /> |.60 (65-)<ref name=":12" /> |Youth from first through fourth grade who were at risk for CD<ref name=":12" /> |[https://www.pearsonassessments.com/store/usassessments/en/Store/Professional-Assessments/Behavior/Comprehensive/Behavior-Assessment-System-for-Children-%7C-Third-Edition-/p/100001402.html#:~:text=The%20BASC%E2%84%A2%20holds%20an,or%20adolescent's%20behavior%20and%20emotions. Purchase] |} '''Note:''' “LR+” refers to the change in likelihood ratio associated with a positive test score, and “LR-” is the likelihood ratio for a low score. Likelihood ratios of 1 indicate that the test result did not change impressions at all. LRs larger than 10 or smaller than .10 are frequently clinically decisive; 5 or .20 are helpful, and between 2.0 and .5 are small enough that they rarely result in clinically meaningful changes of formulation<ref>Sackett, D. L., Straus, S. E., Richardson, W. S., Rosenberg, W., & Haynes, R. B. (2000). Evidence-based medicine: How to practice and teach EBM. Edinburgh: Churchill Livingstone.</ref>. '''Search terms:''' [Oppositional Defiant Disorder] AND [sensitivity OR specificity] in GoogleScholar and PsychINFO; === Interpreting ODD screening measure scores === * For information on interpreting screening measure scores, click [[Evidence based assessment/Prediction phase#Interpreting screening measure scores|here.]] == [[Evidence based assessment/Prescription phase|'''Prescription phase''']] == === Gold standard diagnostic interviews === * For a list of broad reaching diagnostic interviews sortable by disorder with PDFs (if applicable), [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prescription_phase&wteswitched=1#Common_Diagnostic_Interviews click here.] === Recommended diagnostic instruments for ODD === {| class="wikitable sortable" ! colspan="10" |Diagnostic instruments for ODD |- !Measure !Format (Reporter) !Age Range !Administration/ Completion Time !Interrater Reliability !Test-Retest Reliability !Construct Validity !Content Validity !Highly Recommended !Where to Access |- |Kiddie Schedule for Affective Disorders and Schizophrenia Present and Lifetime Version (KSADS-PL) |Structured interview |6-28 |45-75 minutes |E<ref name=":11">{{Cite journal|last=KAUFMAN|first=JOAN|last2=BIRMAHER|first2=BORIS|last3=BRENT|first3=DAVID|last4=RAO|first4=UMA|last5=FLYNN|first5=CYNTHIA|last6=MORECI|first6=PAULA|last7=WILLIAMSON|first7=DOUGLAS|last8=RYAN|first8=NEAL|date=1997-07|title=Schedule for Affective Disorders and Schizophrenia for School-Age Children-Present and Lifetime Version (K-SADS-PL): Initial Reliability and Validity Data|url=https://doi.org/10.1097/00004583-199707000-00021|journal=Journal of the American Academy of Child &amp; Adolescent Psychiatry|volume=36|issue=7|pages=980–988|doi=10.1097/00004583-199707000-00021|issn=0890-8567}}</ref> |G<ref name=":11" /> | | | |[https://www.pediatricbipolar.pitt.edu/resources/instruments Website to access] |- |Diagnostic Interview Schedule for Children (DISC-5) |Structured Interview (Self report and parent) |6-17 | | | | | |<ref name=":0" />[[File:Light green check.svg|center|frameless|36x36px]] |[https://telesage.com/netdisc-5/# Coming soon] |} '''Note:''' L = Less than adequate; A = Adequate; G = Good; E = Excellent; U = Unavailable; NA = Not applicable == [[Evidence based assessment/Process phase|'''Process phase''']] == The following section contains a list of process and outcome measures for oppositional defiant disorder. The section includes benchmarks based on published norms for several outcome and severity measures, as well as information about commonly used process measures. Process and outcome measures are used as part of the [[Evidence based assessment/Process phase|process phase]] of assessment. For more information of differences between process and outcome measures, see the page on the [[Evidence based assessment/Process phase|process phase of assessment]]. === Outcome and severity measures === * This table includes clinically significant benchmarks for generalized anxiety disorder specific outcome measures * Information on how to interpret this table can be [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase found here]. * Additionally, these [[Evidence based assessment/Vignettes|vignettes]] might be helpful resources for understanding appropriate adaptation of outcome measures in practice. * ''<u>For clinically significant change benchmarks for the CBCL, YSR, and TRF total, externalizing, internalizing, and attention benchmarks,</u>'' [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase&wteswitched=1#Clinically_significant_change_benchmarks_for_widely-used_outcome_measures see here.] {| class="wikitable sortable" border="1" |- | rowspan="1&quot;" style="text-align:center;font-size:130%;" | <b> Measure</b> | style="text-align:center;font-size:130%;" | <b> Subscale</b> | colspan="3" style="text-align:center;font-size:130%" width="300" | <b> Cut-off scores</b> | colspan="3" style="text-align:center;font-size:120%" | <b> Critical Change <br> (unstandardized scores)</b> |- | colspan="8" style="font-size:110%; text-align:center;" span | <b> Benchmarks Based on Published Norms</b> |- | colspan="2" | | style="text-align:center;font-size:110%" | <b> A</b> | style="text-align:center;font-size:110%" | <b> B</b> | style="text-align:center;font-size:110%" | <b> C</b> | style="text-align:center;font-size:110%" | <b> 95%</b> | style="text-align:center;font-size:110%" | <b> 90%</b> | style="text-align:center;font-size:110%" | <b> SE_difference</b> |- | rowspan="1" style="text-align:center;" | <b> CBCL T-scores <br> (2001 Norms)</b> | style="text-align:right;" | <i> Externalizing</i> | style="text-align:center;" | 49 | style="text-align:center;" | 70 | style="text-align:center;" | 58 | style="text-align:center;" | 7 | style="text-align:center;" | 6 | style="text-align:center;" | 3.4 |- | rowspan="1" style="text-align:center;" | <b> ECBI Scaled Scores <br> (1983 Norms)</b> | style="text-align:right;" | <i> Intensity</i> | style="text-align:center;" | 80.1 | style="text-align:center;" | 169.5 | style="text-align:center;" | 112.9 | style="text-align:center;" | 9.5 | style="text-align:center;" | 8 | style="text-align:center;" | 4.8 |- | rowspan="1" style="text-align:center;" | <b> ECBI Scaled Scores <br> (1983 Norms)</b> | style="text-align:right;" | <i> Problem</i> | style="text-align:center;" | 3.9 | style="text-align:center;" | 17.7 | style="text-align:center;" | 11.5 | style="text-align:center;" | 2.1 | style="text-align:center;" | 1.8 | style="text-align:center;" | 1.1 |} '''Note:''' “A” = Away from the clinical range, “B” = Back into the nonclinical range, “C” = Closer to the nonclinical than clinical mean. === Process measures=== See Section 1.1 for overview of evidence-based measures to use depending on etiology and symptomatology of Oppositional Defiant Disorder. ==Treatment== ===Behavioral parent training=== Behavioral Parent Training is considered the most effective treatment for childhood disruptive behavior disorders (e.g., Oppositional Defiant Disorder), especially for younger children (i.e., 3-8 year-olds). See http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/rule-breaking-defiance-and-acting-out/ a website sponsored by The Society for Child and Adolescent Psychology (APA, Division 53) and the Association for Behavioral and Cognitive Therapies (ABCT), for current summary of evidence-based treatments. ===Overview of recommendations for assessment and treatment=== See the [https://pathways.nice.org.uk/pathways/antisocial-behaviour-and-conduct-disorders-in-children-and-young-people#content=view-info-category%3Aview-about-menu National Institute for Health and Care Excellence (NICE) Practice Guidelines for Childhood Conduct Disorders], for an overview of recommendations for both assessment and treatment of Oppositional Defiant Disorder. == External Links == *[https://sccap53.org Society of Clinical Child and Adolescent Psychology] *http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/rule-breaking-defiance-and-acting-out/ == References == {{collapse top|Click here for references}} {{Reflist|30em}} [[Category:Psychological disorder portfolios|{{SUBPAGENAME}}]] {{collapse bottom}} [[Category:Psychological disorder portfolios|{{SUBPAGENAME}}]] pseorl4zpcpdm61x4zefv9aw4rqzt18 2 285786 2410326 2022-07-29T23:00:31Z IncarnationOfTarnation 2947317 Created Basic Page wikitext text/x-wiki A user interested in Computer Science, Cybersecurity, and Genetics. 7qrfpsbl116q60tejk3k5p8ogql62mr 2 285787 2410328 2022-07-29T23:45:03Z IncarnationOfTarnation 2947317 New resource with "<h1>Hello</h1> Peasants" wikitext text/x-wiki <h1>Hello</h1> Peasants jxt1qwixe66gfxi7ejgq9l5bl2nhfqd 2410360 2410328 2022-07-30T00:31:00Z IncarnationOfTarnation 2947317 wikitext text/x-wiki '''Bold text'''<h1>Hello</h1> Peasants * tot [[TestingSandbox]] toast ==== bacon ==== = Bacon = rkpe1lkuxrlc1i9vful0qqf7zwd804h 3 285788 2410331 2022-07-29T23:53:46Z PhantomTech 1293967 Message re. [[Introduction to Calculus/Differentiation]] ([[Project:Huggle|HG]]) (3.4.10) wikitext text/x-wiki == July 2022 == [[File:Information.svg|25px|alt=Information icon]] Hello, I'm [[User:PhantomTech|PhantomTech]]. I wanted to let you know that I undid one of [[Special:Contributions/24.52.159.126|your recent contributions]],&nbsp;such as the one you made with <span class="plainlinks">[https://en.wikiversity.org/w/index.php?title=Introduction%20to%20Calculus%2FDifferentiation&diff=2410329 this edit]</span> to [[:Introduction to Calculus/Differentiation]], because it didn’t appear constructive to me. If you think I made a mistake, or if you have any questions, you can leave me a message on [[User_talk:PhantomTech|my talk page]]. Thanks. <!-- Template:Huggle/warn-1 --><!-- Template:uw-vandalism1 -->[[User:PhantomTech|PhantomTech]] ([[User talk:PhantomTech|discuss]] • [[Special:Contributions/PhantomTech|contribs]]) 23:53, 29 July 2022 (UTC) 1zobn9lstioop9rh2unyw5q178f20vd 3 285789 2410337 2022-07-30T00:12:19Z Dave Braunschweig 426084 Welcome wikitext text/x-wiki {{Robelbox|theme=9|title=Welcome!|width=100%}} <div style="{{Robelbox/pad}}"> '''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] HolyNetworkAdapter!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Dave Braunschweig|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. 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To [[Wikiversity:Introduction|get started]], you may <!-- The Left column --> <div style="width:50.0%; float:left"> * [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]]. * Visit a (kind of) [[Wikiversity:Random|random project]]. * [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]]. * Find out about [[Wikiversity:Research|research]] activities on Wikiversity. * [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left. </div> <!-- The Right column --> <div style="width:50.0%; float:left"> * Enable VisualEditor under [[Special:Preferences#mw-prefsection-betafeatures|Beta]] settings to make article editing easier. * Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity. * Give [[Wikiversity:Feedback|feedback]] about your initial observations. * Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]]. * [[Wikiversity:Chat|Chat]] with other Wikiversitans on [[:freenode:wikiversity|<kbd>#wikiversity</kbd>]]. </div> <br clear="both"/> You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 00:12, 30 July 2022 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} on1cdkye17xlau3qzbfxgmjntcu5292 3 285790 2410339 2022-07-30T00:16:05Z Dave Braunschweig 426084 Welcome wikitext text/x-wiki {{Robelbox|theme=9|title=Welcome!|width=100%}} <div style="{{Robelbox/pad}}"> '''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] U3203655!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Dave Braunschweig|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOjs UI icon signature-ltr.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity. To [[Wikiversity:Introduction|get started]], you may <!-- The Left column --> <div style="width:50.0%; float:left"> * [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]]. * Visit a (kind of) [[Wikiversity:Random|random project]]. * [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]]. * Find out about [[Wikiversity:Research|research]] activities on Wikiversity. * [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left. </div> <!-- The Right column --> <div style="width:50.0%; float:left"> * Enable VisualEditor under [[Special:Preferences#mw-prefsection-betafeatures|Beta]] settings to make article editing easier. * Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity. * Give [[Wikiversity:Feedback|feedback]] about your initial observations. * Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]]. * [[Wikiversity:Chat|Chat]] with other Wikiversitans on [[:freenode:wikiversity|<kbd>#wikiversity</kbd>]]. </div> <br clear="both"/> You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 00:16, 30 July 2022 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} rowxg5hi7j3ljcx4rbcwrbqf0ydcnoh 3 285791 2410340 2022-07-30T00:16:17Z Dave Braunschweig 426084 Welcome wikitext text/x-wiki {{Robelbox|theme=9|title=Welcome!|width=100%}} <div style="{{Robelbox/pad}}"> '''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] U3086459!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Dave Braunschweig|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOjs UI icon signature-ltr.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity. To [[Wikiversity:Introduction|get started]], you may <!-- The Left column --> <div style="width:50.0%; float:left"> * [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]]. * Visit a (kind of) [[Wikiversity:Random|random project]]. * [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]]. * Find out about [[Wikiversity:Research|research]] activities on Wikiversity. * [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left. </div> <!-- The Right column --> <div style="width:50.0%; float:left"> * Enable VisualEditor under [[Special:Preferences#mw-prefsection-betafeatures|Beta]] settings to make article editing easier. * Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity. * Give [[Wikiversity:Feedback|feedback]] about your initial observations. * Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]]. * [[Wikiversity:Chat|Chat]] with other Wikiversitans on [[:freenode:wikiversity|<kbd>#wikiversity</kbd>]]. </div> <br clear="both"/> You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 00:16, 30 July 2022 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} oots1jcxpo1hsdymgzye5fr95du7aob 3 285792 2410341 2022-07-30T00:17:02Z Dave Braunschweig 426084 Welcome wikitext text/x-wiki {{Robelbox|theme=9|title=Welcome!|width=100%}} <div style="{{Robelbox/pad}}"> '''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] Almanta!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Dave Braunschweig|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOjs UI icon signature-ltr.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity. To [[Wikiversity:Introduction|get started]], you may <!-- The Left column --> <div style="width:50.0%; float:left"> * [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]]. * Visit a (kind of) [[Wikiversity:Random|random project]]. * [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]]. * Find out about [[Wikiversity:Research|research]] activities on Wikiversity. * [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left. </div> <!-- The Right column --> <div style="width:50.0%; float:left"> * Enable VisualEditor under [[Special:Preferences#mw-prefsection-betafeatures|Beta]] settings to make article editing easier. * Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity. * Give [[Wikiversity:Feedback|feedback]] about your initial observations. * Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]]. * [[Wikiversity:Chat|Chat]] with other Wikiversitans on [[:freenode:wikiversity|<kbd>#wikiversity</kbd>]]. </div> <br clear="both"/> You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 00:17, 30 July 2022 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} sy7ll6kweuszkcqpstxr64azc5ongcj 3 285793 2410342 2022-07-30T00:17:24Z Dave Braunschweig 426084 Welcome wikitext text/x-wiki {{Robelbox|theme=9|title=Welcome!|width=100%}} <div style="{{Robelbox/pad}}"> '''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] U3214117!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Dave Braunschweig|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOjs UI icon signature-ltr.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity. To [[Wikiversity:Introduction|get started]], you may <!-- The Left column --> <div style="width:50.0%; float:left"> * [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]]. * Visit a (kind of) [[Wikiversity:Random|random project]]. * [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]]. * Find out about [[Wikiversity:Research|research]] activities on Wikiversity. * [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left. </div> <!-- The Right column --> <div style="width:50.0%; float:left"> * Enable VisualEditor under [[Special:Preferences#mw-prefsection-betafeatures|Beta]] settings to make article editing easier. * Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity. * Give [[Wikiversity:Feedback|feedback]] about your initial observations. * Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]]. * [[Wikiversity:Chat|Chat]] with other Wikiversitans on [[:freenode:wikiversity|<kbd>#wikiversity</kbd>]]. </div> <br clear="both"/> You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 00:17, 30 July 2022 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} 8cttwvns0uwxncb4oc2jsxm77wvb4tz 3 285794 2410343 2022-07-30T00:19:10Z Dave Braunschweig 426084 Welcome wikitext text/x-wiki {{Robelbox|theme=9|title=Welcome!|width=100%}} <div style="{{Robelbox/pad}}"> '''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] Magog the Ogre!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Dave Braunschweig|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOjs UI icon signature-ltr.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity. To [[Wikiversity:Introduction|get started]], you may <!-- The Left column --> <div style="width:50.0%; float:left"> * [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]]. * Visit a (kind of) [[Wikiversity:Random|random project]]. * [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]]. * Find out about [[Wikiversity:Research|research]] activities on Wikiversity. * [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left. </div> <!-- The Right column --> <div style="width:50.0%; float:left"> * Enable VisualEditor under [[Special:Preferences#mw-prefsection-betafeatures|Beta]] settings to make article editing easier. * Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity. * Give [[Wikiversity:Feedback|feedback]] about your initial observations. * Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]]. * [[Wikiversity:Chat|Chat]] with other Wikiversitans on [[:freenode:wikiversity|<kbd>#wikiversity</kbd>]]. </div> <br clear="both"/> You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 00:19, 30 July 2022 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} r7wl00xjoxqas4w6ncywpqdvmsnuy8q 3 285795 2410344 2022-07-30T00:22:06Z Dave Braunschweig 426084 Welcome wikitext text/x-wiki {{Robelbox|theme=9|title=Welcome!|width=100%}} <div style="{{Robelbox/pad}}"> '''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] Ajeofula22!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Dave Braunschweig|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOjs UI icon signature-ltr.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity. To [[Wikiversity:Introduction|get started]], you may <!-- The Left column --> <div style="width:50.0%; float:left"> * [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]]. * Visit a (kind of) [[Wikiversity:Random|random project]]. * [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]]. * Find out about [[Wikiversity:Research|research]] activities on Wikiversity. * [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left. </div> <!-- The Right column --> <div style="width:50.0%; float:left"> * Enable VisualEditor under [[Special:Preferences#mw-prefsection-betafeatures|Beta]] settings to make article editing easier. * Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity. * Give [[Wikiversity:Feedback|feedback]] about your initial observations. * Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]]. * [[Wikiversity:Chat|Chat]] with other Wikiversitans on [[:freenode:wikiversity|<kbd>#wikiversity</kbd>]]. </div> <br clear="both"/> You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 00:22, 30 July 2022 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} k1aps6ew4zdvhm6c556eu9nnn2nv39h 3 285796 2410346 2022-07-30T00:23:27Z Dave Braunschweig 426084 Welcome wikitext text/x-wiki {{Robelbox|theme=9|title=Welcome!|width=100%}} <div style="{{Robelbox/pad}}"> '''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] Drgrajeshwar!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Dave Braunschweig|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOjs UI icon signature-ltr.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity. To [[Wikiversity:Introduction|get started]], you may <!-- The Left column --> <div style="width:50.0%; float:left"> * [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]]. * Visit a (kind of) [[Wikiversity:Random|random project]]. * [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]]. * Find out about [[Wikiversity:Research|research]] activities on Wikiversity. * [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left. </div> <!-- The Right column --> <div style="width:50.0%; float:left"> * Enable VisualEditor under [[Special:Preferences#mw-prefsection-betafeatures|Beta]] settings to make article editing easier. * Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity. * Give [[Wikiversity:Feedback|feedback]] about your initial observations. * Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]]. * [[Wikiversity:Chat|Chat]] with other Wikiversitans on [[:freenode:wikiversity|<kbd>#wikiversity</kbd>]]. </div> <br clear="both"/> You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 00:23, 30 July 2022 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} ijjf63f9mcb3dxw5ny7x4g41wrrrgqv 0 285797 2410349 2022-07-30T00:26:13Z Dave Braunschweig 426084 Dave Braunschweig moved page [[InsertionSortMiddle]] to [[Data Structures and Algorithms/InsertionSortMiddle]]: Moving under learning project wikitext text/x-wiki #REDIRECT [[Data Structures and Algorithms/InsertionSortMiddle]] s8qrdwjcc5683q3n5zf0111rbzon06v 0 285798 2410351 2022-07-30T00:26:26Z Dave Braunschweig 426084 Dave Braunschweig moved page [[SortedListMergingSort]] to [[Data Structures and Algorithms/SortedListMergingSort]]: Moving under learning project wikitext text/x-wiki #REDIRECT [[Data Structures and Algorithms/SortedListMergingSort]] rcg3wu7td9ejiva9mnq811qltzl1vgf 0 285799 2410353 2022-07-30T00:26:44Z Dave Braunschweig 426084 Dave Braunschweig moved page [[PyramidSelectionSort]] to [[Data Structures and Algorithms/PyramidSelectionSort]]: Moving under learning project wikitext text/x-wiki #REDIRECT [[Data Structures and Algorithms/PyramidSelectionSort]] ip0j14dlmje1ti4f3foecyy1nown2eu 104 285800 2410358 2022-07-30T00:29:40Z IncarnationOfTarnation 2947317 Initialized the page. Setup basic stuff. I will continue to update and expand this content, and intend to make pages and links. Simply short on time 😭 wikitext text/x-wiki = Physical Security = <h1>Introduction</h1> Physical Security is the implementation of security measures for the protection of assets, information, or people. It aims to prevent as much unauthorized or malicious access as possible, while still allowing those authorized or permitted access to what they need. This topic aims to outline and define aspects of physical security, alongside providing necessary resources for further research. Terminologies are defined in as standard a way as possible considering the multi-resource nature of the topic, but have a preference to the terminology of other Wikiversity topic or lecture pages. Physical Security measures rely on a combination of physical and psychological methodologies for preventing unwanted access to assets. <h1>Physical Security Terminology</h1> === General Terminologies === * '''Security System''' - a linked or related set of physical security implementations. May apply to: an individual object, such as a lock; A complex object or set of objects, such as a car, computer, or engine; Or some other large scale system, such as a facility and its grounds, a series of complex objects, or a complex of buildings or rooms. === The Five Components of Physical Security === The core ideas of physical security can be organized into five components: The three key components - Access Control, Surveillance, and Security testing - alongside Deterrence and Detection. * '''Access Control''' - Access Control is the concept of managing and restricting the accessibility and use of certain assets and areas, either to a whitelist of individuals, or from a blacklist of individuals. Those who need a resource, such as a ledger, account number, or stored paperwork, should have access to it, and those who don't should not. * '''Surveillance''' - Surveillance is the concept of watching, monitoring, observing, or recording an area and/or asset, alongside the entities that access it. For an area or asset to be considered surveilled in some form, authorized individuals should be capable of retrieving information about recent accesses and/or interactions with the surveilled entity. * '''Security Testing''' - Security Testing refers to the act and practice of testing the implementation of physical security methods, the practices of agents involved in the enforcement of physical security methods, and the grading or assessment of the capability of a system of security implementations. Security Testing is an active, repeated practice, and should often result in changes, reimplementation, and enhancements alongside evolving information and a changing threat landscape. * '''Deterrence''' - Deterrence as a concept refers to methods to prevent an intrusion attempt in the first place, and is typically the first layer of defense against intruders. Deterrence can take a variety of forms, and may not be a physically implemented system so much as an implied, warned or suggested concept or response. * '''Detection''' - Detection is the concept of identifying a physical intrusion, and possibly the intruders involved. For a method to classify as detection it must be capable of, often 'autonomously' (agents such as human guards, dogs, or Computer-Vision systems classify here), indicating a breach or attempt against a security implementation, site, or asset. None of the above methods are, nor are expected to be, full-proof, which applies further to the definitions. Methods to evade Detection, Access-Control, Surveillance, and Deterrence do exist and will likely be discussed further under Security Testing. The above components experience overlap/ Access Control may serve as Deterrence, Surveillance may perform a Detection role, Detection may be required for Access Control, and so on; These components are conceptual, and not hard limitations. Beyond the components of security, a security system must be applied to their own areas of a physical system or environment, defined below. === The Three Levels of Physical Security * '''Outer Perimeter Security''' - Refers to the marginal outer layer of a security system such as: The physical property boundary lines of a site; The entrance buildings or hubs to a large site such as Disney World or other multi-building site; The room or area around a device such as a server box, car, or cash-register. * '''Inner Perimeter Security''' - Refers to the direct outer layer of a security system such as: The walls, doors, windows or entry points of a building; The accessible areas of a device such as a laptop or car in standard use (not reverse engineered, disassembled, etc;). * '''Interior Security''' - Refers to the internal operations layer of a security system such as: inner spaces, cubbies, and cabinets of an office, building, or room; Internal mechanics of a device such as a lock, engine, or computer; Not every system of security will be capable of realistically or permissively implementing protections on every level listed above, however, every level should still be held in consideration as a vector of intrusion, or consideration for secondary protections. <h1>Research Links</h1> hwsqp8tdvt0i59my2vfsga2mddf4n3d 2410359 2410358 2022-07-30T00:29:55Z IncarnationOfTarnation 2947317 wikitext text/x-wiki = Physical Security = <h1>Introduction</h1> Physical Security is the implementation of security measures for the protection of assets, information, or people. It aims to prevent as much unauthorized or malicious access as possible, while still allowing those authorized or permitted access to what they need. This topic aims to outline and define aspects of physical security, alongside providing necessary resources for further research. Terminologies are defined in as standard a way as possible considering the multi-resource nature of the topic, but have a preference to the terminology of other Wikiversity topic or lecture pages. Physical Security measures rely on a combination of physical and psychological methodologies for preventing unwanted access to assets. <h1>Physical Security Terminology</h1> === General Terminologies === * '''Security System''' - a linked or related set of physical security implementations. May apply to: an individual object, such as a lock; A complex object or set of objects, such as a car, computer, or engine; Or some other large scale system, such as a facility and its grounds, a series of complex objects, or a complex of buildings or rooms. === The Five Components of Physical Security === The core ideas of physical security can be organized into five components: The three key components - Access Control, Surveillance, and Security testing - alongside Deterrence and Detection. * '''Access Control''' - Access Control is the concept of managing and restricting the accessibility and use of certain assets and areas, either to a whitelist of individuals, or from a blacklist of individuals. Those who need a resource, such as a ledger, account number, or stored paperwork, should have access to it, and those who don't should not. * '''Surveillance''' - Surveillance is the concept of watching, monitoring, observing, or recording an area and/or asset, alongside the entities that access it. For an area or asset to be considered surveilled in some form, authorized individuals should be capable of retrieving information about recent accesses and/or interactions with the surveilled entity. * '''Security Testing''' - Security Testing refers to the act and practice of testing the implementation of physical security methods, the practices of agents involved in the enforcement of physical security methods, and the grading or assessment of the capability of a system of security implementations. Security Testing is an active, repeated practice, and should often result in changes, reimplementation, and enhancements alongside evolving information and a changing threat landscape. * '''Deterrence''' - Deterrence as a concept refers to methods to prevent an intrusion attempt in the first place, and is typically the first layer of defense against intruders. Deterrence can take a variety of forms, and may not be a physically implemented system so much as an implied, warned or suggested concept or response. * '''Detection''' - Detection is the concept of identifying a physical intrusion, and possibly the intruders involved. For a method to classify as detection it must be capable of, often 'autonomously' (agents such as human guards, dogs, or Computer-Vision systems classify here), indicating a breach or attempt against a security implementation, site, or asset. None of the above methods are, nor are expected to be, full-proof, which applies further to the definitions. Methods to evade Detection, Access-Control, Surveillance, and Deterrence do exist and will likely be discussed further under Security Testing. The above components experience overlap/ Access Control may serve as Deterrence, Surveillance may perform a Detection role, Detection may be required for Access Control, and so on; These components are conceptual, and not hard limitations. Beyond the components of security, a security system must be applied to their own areas of a physical system or environment, defined below. === The Three Levels of Physical Security === * '''Outer Perimeter Security''' - Refers to the marginal outer layer of a security system such as: The physical property boundary lines of a site; The entrance buildings or hubs to a large site such as Disney World or other multi-building site; The room or area around a device such as a server box, car, or cash-register. * '''Inner Perimeter Security''' - Refers to the direct outer layer of a security system such as: The walls, doors, windows or entry points of a building; The accessible areas of a device such as a laptop or car in standard use (not reverse engineered, disassembled, etc;). * '''Interior Security''' - Refers to the internal operations layer of a security system such as: inner spaces, cubbies, and cabinets of an office, building, or room; Internal mechanics of a device such as a lock, engine, or computer; Not every system of security will be capable of realistically or permissively implementing protections on every level listed above, however, every level should still be held in consideration as a vector of intrusion, or consideration for secondary protections. <h1>Research Links</h1> 3xbn7zo6rt7c7nu9um6ugbj01d4kb5u 2410361 2410359 2022-07-30T00:34:36Z IncarnationOfTarnation 2947317 wikitext text/x-wiki = Physical Security = <h1>Introduction</h1> Physical Security is the implementation of security measures for the protection of assets, information, or people. It aims to prevent as much unauthorized or malicious access as possible, while still allowing those authorized or permitted access to what they need. This topic aims to outline and define aspects of physical security, alongside providing necessary resources for further research. Terminologies are defined in as standard a way as possible considering the multi-resource nature of the topic, but have a preference to the terminology of other Wikiversity topic or lecture pages. Physical Security measures rely on a combination of physical and psychological methodologies for preventing unwanted access to assets. <h1>Physical Security Terminology</h1> === General Terminologies === * '''Security System''' - a linked or related set of physical security implementations. May apply to: an individual object, such as a lock; A complex object or set of objects, such as a car, computer, or engine; Or some other large scale system, such as a facility and its grounds, a series of complex objects, or a complex of buildings or rooms. * '''CIA Triad''' - A concept from information security that applies well to physical security. The CIA triad is a triangular representation of compromises between the Confidentiality, Integrity, and Availability of an asset (in the typical case, information). When considering Access Controls, particularly to things such as devices for Security Testing, Surveillance, or any other component of the security system, the CIA Triad is of particular relevance. The Triad is not an absolute representation, since some systems can contain high levels of multiple categories, but an important conceptual representation. === The Five Components of Physical Security === The core ideas of physical security can be organized into five components: The three key components - Access Control, Surveillance, and Security testing - alongside Deterrence and Detection. * '''Access Control''' - Access Control is the concept of managing and restricting the accessibility and use of certain assets and areas, either to a whitelist of individuals, or from a blacklist of individuals. Those who need a resource, such as a ledger, account number, or stored paperwork, should have access to it, and those who don't should not. * '''Surveillance''' - Surveillance is the concept of watching, monitoring, observing, or recording an area and/or asset, alongside the entities that access it. For an area or asset to be considered surveilled in some form, authorized individuals should be capable of retrieving information about recent accesses and/or interactions with the surveilled entity. * '''Security Testing''' - Security Testing refers to the act and practice of testing the implementation of physical security methods, the practices of agents involved in the enforcement of physical security methods, and the grading or assessment of the capability of a system of security implementations. Security Testing is an active, repeated practice, and should often result in changes, reimplementation, and enhancements alongside evolving information and a changing threat landscape. * '''Deterrence''' - Deterrence as a concept refers to methods to prevent an intrusion attempt in the first place, and is typically the first layer of defense against intruders. Deterrence can take a variety of forms, and may not be a physically implemented system so much as an implied, warned or suggested concept or response. * '''Detection''' - Detection is the concept of identifying a physical intrusion, and possibly the intruders involved. For a method to classify as detection it must be capable of, often 'autonomously' (agents such as human guards, dogs, or Computer-Vision systems classify here), indicating a breach or attempt against a security implementation, site, or asset. None of the above methods are, nor are expected to be, full-proof, which applies further to the definitions. Methods to evade Detection, Access-Control, Surveillance, and Deterrence do exist and will likely be discussed further under Security Testing. The above components experience overlap/ Access Control may serve as Deterrence, Surveillance may perform a Detection role, Detection may be required for Access Control, and so on; These components are conceptual, and not hard limitations. Beyond the components of security, a security system must be applied to their own areas of a physical system or environment, defined below. === The Three Levels of Physical Security === * '''Outer Perimeter Security''' - Refers to the marginal outer layer of a security system such as: The physical property boundary lines of a site; The entrance buildings or hubs to a large site such as Disney World or other multi-building site; The room or area around a device such as a server box, car, or cash-register. * '''Inner Perimeter Security''' - Refers to the direct outer layer of a security system such as: The walls, doors, windows or entry points of a building; The accessible areas of a device such as a laptop or car in standard use (not reverse engineered, disassembled, etc;). * '''Interior Security''' - Refers to the internal operations layer of a security system such as: inner spaces, cubbies, and cabinets of an office, building, or room; Internal mechanics of a device such as a lock, engine, or computer; Not every system of security will be capable of realistically or permissively implementing protections on every level listed above, however, every level should still be held in consideration as a vector of intrusion, or consideration for secondary protections. <h1>Research Links</h1> hkj2c0hnumq4bcxdegjv1l1myydhx1w 2410364 2410361 2022-07-30T00:37:11Z IncarnationOfTarnation 2947317 wikitext text/x-wiki = Physical Security = <h1>Introduction</h1> Physical Security is the implementation of security measures for the protection of assets, information, or people. It aims to prevent as much unauthorized or malicious access as possible, while still allowing those authorized or permitted access to what they need. This topic aims to outline and define aspects of physical security, alongside providing necessary resources for further research. Terminologies are defined in as standard a way as possible considering the multi-resource nature of the topic, but have a preference to the terminology of other Wikiversity topic or lecture pages. Physical Security measures rely on a combination of physical and psychological methodologies for preventing unwanted access to assets. <h1>Physical Security Terminology</h1> === General Terminologies === * '''Security System''' - a linked or related set of physical security implementations. May apply to: an individual object, such as a lock; A complex object or set of objects, such as a car, computer, or engine; Or some other large scale system, such as a facility and its grounds, a series of complex objects, or a complex of buildings or rooms. * '''CIA Triad''' - A concept from information security that applies well to physical security. The CIA triad is a triangular representation of compromises between the Confidentiality, Integrity, and Availability of an asset (in the typical case, information). When considering Access Controls, particularly to things such as devices for Security Testing, Surveillance, or any other component of the security system, the CIA Triad is of particular relevance. The Triad is not an absolute representation, since some systems can contain high levels of multiple categories, but an important conceptual representation. === The Five Components of Physical Security === The core ideas of physical security can be organized into five components: The three key components - Access Control, Surveillance, and Security testing - alongside Deterrence and Detection. * '''Access Control''' - Access Control is the concept of managing and restricting the accessibility and use of certain assets and areas, either to a whitelist of individuals, or from a blacklist of individuals. Those who need a resource, such as a ledger, account number, or stored paperwork, should have access to it, and those who don't should not. * '''Surveillance''' - Surveillance is the concept of watching, monitoring, observing, or recording an area and/or asset, alongside the entities that access it. For an area or asset to be considered surveilled in some form, authorized individuals should be capable of retrieving information about recent accesses and/or interactions with the surveilled entity. * '''Security Testing''' - Security Testing refers to the act and practice of testing the implementation of physical security methods, the practices of agents involved in the enforcement of physical security methods, and the grading or assessment of the capability of a system of security implementations. Security Testing is an active, repeated practice, and should often result in changes, reimplementation, and enhancements alongside evolving information and a changing threat landscape. * '''Deterrence''' - Deterrence as a concept refers to methods to prevent an intrusion attempt in the first place, and is typically the first layer of defense against intruders. Deterrence can take a variety of forms, and may not be a physically implemented system so much as an implied, warned or suggested concept or response. * '''Detection''' - Detection is the concept of identifying a physical intrusion, and possibly the intruders involved. For a method to classify as detection it must be capable of, often 'autonomously' (agents such as human guards, dogs, or Computer-Vision systems classify here), indicating a breach or attempt against a security implementation, site, or asset. None of the above methods are, nor are expected to be, full-proof, which applies further to the definitions. Methods to evade Detection, Access-Control, Surveillance, and Deterrence do exist and will likely be discussed further under Security Testing. The above components experience overlap/ Access Control may serve as Deterrence, Surveillance may perform a Detection role, Detection may be required for Access Control, and so on; These components are conceptual, and not hard limitations. Beyond the components of security, a security system must be applied to their own areas of a physical system or environment, defined below. === The Three Levels of Physical Security === * '''Outer Perimeter Security''' - Refers to the marginal outer layer of a security system such as: The physical property boundary lines of a site; The entrance buildings or hubs to a large site such as Disney World or other multi-building site; The room or area around a device such as a server box, car, or cash-register. * '''Inner Perimeter Security''' - Refers to the direct outer layer of a security system such as: The walls, doors, windows or entry points of a building; The accessible areas of a device such as a laptop or car in standard use (not reverse engineered, disassembled, etc;). * '''Interior Security''' - Refers to the internal operations layer of a security system such as: inner spaces, cubbies, and cabinets of an office, building, or room; Internal mechanics of a device such as a lock, engine, or computer; Not every system of security will be capable of realistically or permissively implementing protections on every level listed above, however, every level should still be held in consideration as a vector of intrusion, or consideration for secondary protections. <h1>Research Links</h1> * Example of the results of poor physical security: https://www.youtube.com/watch?v=R5RE0mVbJ3s * Reference building codes in your area or government for further information. Often a security section concerning exact rules and techniques for preventing intrusion is contained within, ie: https://www.kcmo.gov/home/showpublisheddocument/6420/637509790534730000 7oiy23eyx2mpo768jzgoz5cva9ji5m7 2410365 2410364 2022-07-30T00:40:42Z IncarnationOfTarnation 2947317 wikitext text/x-wiki = Physical Security = <h1>Introduction</h1> Physical Security is the implementation of security measures for the protection of assets, information, or people. It aims to prevent as much unauthorized or malicious access as possible, while still allowing those authorized or permitted access to what they need. This topic aims to outline and define aspects of physical security, alongside providing necessary resources for further research. Terminologies are defined in as standard a way as possible considering the multi-resource nature of the topic, but have a preference to the terminology of other Wikiversity topic or lecture pages. Physical Security measures rely on a combination of physical and psychological methodologies for preventing unwanted access to assets. <h1>Physical Security Terminology</h1> === General Terminologies === * '''Security System''' - a linked or related set of physical security implementations. May apply to: an individual object, such as a lock; A complex object or set of objects, such as a car, computer, or engine; Or some other large scale system, such as a facility and its grounds, a series of complex objects, or a complex of buildings or rooms. * '''CIA Triad''' - A concept from information security that applies well to physical security. The CIA triad is a triangular representation of compromises between the Confidentiality, Integrity, and Availability of an asset (in the typical case, information). When considering Access Controls, particularly to things such as devices for Security Testing, Surveillance, or any other component of the security system, the CIA Triad is of particular relevance. The Triad is not an absolute representation, since some systems can contain high levels of multiple categories, but an important conceptual representation. === The Five Components of Physical Security === The core ideas of physical security can be organized into five components: The three key components - Access Control, Surveillance, and Security testing - alongside Deterrence and Detection. * '''Access Control''' - Access Control is the concept of managing and restricting the accessibility and use of certain assets and areas, either to a whitelist of individuals, or from a blacklist of individuals. Those who need a resource, such as a ledger, account number, or stored paperwork, should have access to it, and those who don't should not. * '''Surveillance''' - Surveillance is the concept of watching, monitoring, observing, or recording an area and/or asset, alongside the entities that access it. For an area or asset to be considered surveilled in some form, authorized individuals should be capable of retrieving information about recent accesses and/or interactions with the surveilled entity. * '''Security Testing''' - Security Testing refers to the act and practice of testing the implementation of physical security methods, the practices of agents involved in the enforcement of physical security methods, and the grading or assessment of the capability of a system of security implementations. Security Testing is an active, repeated practice, and should often result in changes, reimplementation, and enhancements alongside evolving information and a changing threat landscape. * '''Deterrence''' - Deterrence as a concept refers to methods to prevent an intrusion attempt in the first place, and is typically the first layer of defense against intruders. Deterrence can take a variety of forms, and may not be a physically implemented system so much as an implied, warned or suggested concept or response. * '''Detection''' - Detection is the concept of identifying a physical intrusion, and possibly the intruders involved. For a method to classify as detection it must be capable of, often 'autonomously' (agents such as human guards, dogs, or Computer-Vision systems classify here), indicating a breach or attempt against a security implementation, site, or asset. None of the above methods are, nor are expected to be, full-proof, which applies further to the definitions. Methods to evade Detection, Access-Control, Surveillance, and Deterrence do exist and will likely be discussed further under Security Testing. The above components experience overlap/ Access Control may serve as Deterrence, Surveillance may perform a Detection role, Detection may be required for Access Control, and so on; These components are conceptual, and not hard limitations. Beyond the components of security, a security system must be applied to their own areas of a physical system or environment, defined below. === The Three Levels of Physical Security === * '''Outer Perimeter Security''' - Refers to the marginal outer layer of a security system such as: The physical property boundary lines of a site; The entrance buildings or hubs to a large site such as Disney World or other multi-building site; The room or area around a device such as a server box, car, or cash-register. * '''Inner Perimeter Security''' - Refers to the direct outer layer of a security system such as: The walls, doors, windows or entry points of a building; The accessible areas of a device such as a laptop or car in standard use (not reverse engineered, disassembled, etc;). * '''Interior Security''' - Refers to the internal operations layer of a security system such as: inner spaces, cubbies, and cabinets of an office, building, or room; Internal mechanics of a device such as a lock, engine, or computer; Not every system of security will be capable of realistically or permissively implementing protections on every level listed above, however, every level should still be held in consideration as a vector of intrusion, or consideration for secondary protections. Additionally, some texts and references may include grades of security (often 5 grades) for physical security implementation. These grades are omitted simply due to the variance in definition across sources, combined with their lack of necessity for the understanding of the content. <h1>Research Links</h1> * Example of the results of poor physical security: https://www.youtube.com/watch?v=R5RE0mVbJ3s * Reference building codes in your area or government for further information. Often a security section concerning exact rules and techniques for preventing intrusion is contained within, ie: https://www.kcmo.gov/home/showpublisheddocument/6420/637509790534730000 o25g1nxffd3noico8b3l66df9410nsj 2410368 2410365 2022-07-30T00:51:24Z IncarnationOfTarnation 2947317 wikitext text/x-wiki = Physical Security = <h1>Introduction</h1> Physical Security is the implementation of security measures for the protection of assets, information, or people. It aims to prevent as much unauthorized or malicious access as possible, while still allowing those authorized or permitted access to what they need. This topic aims to outline and define aspects of physical security, alongside providing necessary resources for further research. Terminologies are defined in as standard a way as possible considering the multi-resource nature of the topic, but have a preference to the terminology of other Wikiversity topic or lecture pages. Physical Security measures rely on a combination of physical and psychological methodologies for preventing unwanted access to assets. <h1>Physical Security Terminology</h1> === General Terminologies === * '''Security System''' - a linked or related set of physical security implementations. May apply to: an individual object, such as a lock; A complex object or set of objects, such as a car, computer, or engine; Or some other large scale system, such as a facility and its grounds, a series of complex objects, or a complex of buildings or rooms. * '''CIA Triad''' - A concept from information security that applies well to physical security. The CIA triad is a triangular representation of compromises between the Confidentiality, Integrity, and Availability of an asset (in the typical case, information). When considering Access Controls, particularly to things such as devices for Security Testing, Surveillance, or any other component of the security system, the CIA Triad is of particular relevance. The Triad is not an absolute representation, since some systems can contain high levels of multiple categories, but an important conceptual representation. === The Five Components of Physical Security === The core ideas of physical security can be organized into five components: The three key components - Access Control, Surveillance, and Security testing - alongside Deterrence and Detection. * '''Access Control''' - Access Control is the concept of managing and restricting the accessibility and use of certain assets and areas, either to a whitelist of individuals, or from a blacklist of individuals. Those who need a resource, such as a ledger, account number, or stored paperwork, should have access to it, and those who don't should not. * '''Surveillance''' - Surveillance is the concept of watching, monitoring, observing, or recording an area and/or asset, alongside the entities that access it. For an area or asset to be considered surveilled in some form, authorized individuals should be capable of retrieving information about recent accesses and/or interactions with the surveilled entity. * '''Security Testing''' - Security Testing refers to the act and practice of testing the implementation of physical security methods, the practices of agents involved in the enforcement of physical security methods, and the grading or assessment of the capability of a system of security implementations. Security Testing is an active, repeated practice, and should often result in changes, reimplementation, and enhancements alongside evolving information and a changing threat landscape. * '''Deterrence''' - Deterrence as a concept refers to methods to prevent an intrusion attempt in the first place, and is typically the first layer of defense against intruders. Deterrence can take a variety of forms, and may not be a physically implemented system so much as an implied, warned or suggested concept or response. * '''Detection''' - Detection is the concept of identifying a physical intrusion, and possibly the intruders involved. For a method to classify as detection it must be capable of, often 'autonomously' (agents such as human guards, dogs, or Computer-Vision systems classify here), indicating a breach or attempt against a security implementation, site, or asset. None of the above methods are, nor are expected to be, full-proof, which applies further to the definitions. Methods to evade Detection, Access-Control, Surveillance, and Deterrence do exist and will likely be discussed further under Security Testing. The above components experience overlap/ Access Control may serve as Deterrence, Surveillance may perform a Detection role, Detection may be required for Access Control, and so on; These components are conceptual, and not hard limitations. Beyond the components of security, a security system must be applied to their own areas of a physical system or environment, defined below. === The Three Levels of Physical Security === * '''Outer Perimeter Security''' - Refers to the marginal outer layer of a security system such as: The physical property boundary lines of a site; The entrance buildings or hubs to a large site such as Disney World or other multi-building site; The room or area around a device such as a server box, car, or cash-register. * '''Inner Perimeter Security''' - Refers to the direct outer layer of a security system such as: The walls, doors, windows or entry points of a building; The accessible areas of a device such as a laptop or car in standard use (not reverse engineered, disassembled, etc;). * '''Interior Security''' - Refers to the internal operations layer of a security system such as: inner spaces, cubbies, and cabinets of an office, building, or room; Internal mechanics of a device such as a lock, engine, or computer; Not every system of security will be capable of realistically or permissively implementing protections on every level listed above, however, every level should still be held in consideration as a vector of intrusion, or consideration for secondary protections. Additionally, some texts and references may include grades of security (often 5 grades) for physical security implementation. These grades are omitted simply due to the variance in definition across sources, combined with their lack of necessity for the understanding of the content. <h1>Physical Security as a Practice</h1> Various concepts of Physical Security will be broken down, linked to, or explained below. === Planning a System of Security === Before creating a system of security, one must create a plan or set of plans, with an underlying goal. This is particularly the case in regards to large organizations or security teams, which must standardize and communicate ideas clearly and quickly for an efficient and correct implementation. * [[Physical Threat Landscaping]] - The first aspect of planning a system of security, identifying threats and considerations that a physical security system must defend against. <h1>Research Links</h1> * Example of the results of poor physical security: https://www.youtube.com/watch?v=R5RE0mVbJ3s * Reference building codes in your area or government for further information. Often a security section concerning exact rules and techniques for preventing intrusion is contained within, ie: https://www.kcmo.gov/home/showpublisheddocument/6420/637509790534730000 pb1e26e6uadzgxsy4dg1ixb2nlfh0o5 2410369 2410368 2022-07-30T00:51:39Z IncarnationOfTarnation 2947317 wikitext text/x-wiki = Physical Security = <h1>Introduction</h1> Physical Security is the implementation of security measures for the protection of assets, information, or people. It aims to prevent as much unauthorized or malicious access as possible, while still allowing those authorized or permitted access to what they need. This topic aims to outline and define aspects of physical security, alongside providing necessary resources for further research. Terminologies are defined in as standard a way as possible considering the multi-resource nature of the topic, but have a preference to the terminology of other Wikiversity topic or lecture pages. Physical Security measures rely on a combination of physical and psychological methodologies for preventing unwanted access to assets. <h1>Physical Security Terminology</h1> === General Terminologies === * '''Security System''' - a linked or related set of physical security implementations. May apply to: an individual object, such as a lock; A complex object or set of objects, such as a car, computer, or engine; Or some other large scale system, such as a facility and its grounds, a series of complex objects, or a complex of buildings or rooms. * '''CIA Triad''' - A concept from information security that applies well to physical security. The CIA triad is a triangular representation of compromises between the Confidentiality, Integrity, and Availability of an asset (in the typical case, information). When considering Access Controls, particularly to things such as devices for Security Testing, Surveillance, or any other component of the security system, the CIA Triad is of particular relevance. The Triad is not an absolute representation, since some systems can contain high levels of multiple categories, but an important conceptual representation. === The Five Components of Physical Security === The core ideas of physical security can be organized into five components: The three key components - Access Control, Surveillance, and Security testing - alongside Deterrence and Detection. * '''Access Control''' - Access Control is the concept of managing and restricting the accessibility and use of certain assets and areas, either to a whitelist of individuals, or from a blacklist of individuals. Those who need a resource, such as a ledger, account number, or stored paperwork, should have access to it, and those who don't should not. * '''Surveillance''' - Surveillance is the concept of watching, monitoring, observing, or recording an area and/or asset, alongside the entities that access it. For an area or asset to be considered surveilled in some form, authorized individuals should be capable of retrieving information about recent accesses and/or interactions with the surveilled entity. * '''Security Testing''' - Security Testing refers to the act and practice of testing the implementation of physical security methods, the practices of agents involved in the enforcement of physical security methods, and the grading or assessment of the capability of a system of security implementations. Security Testing is an active, repeated practice, and should often result in changes, reimplementation, and enhancements alongside evolving information and a changing threat landscape. * '''Deterrence''' - Deterrence as a concept refers to methods to prevent an intrusion attempt in the first place, and is typically the first layer of defense against intruders. Deterrence can take a variety of forms, and may not be a physically implemented system so much as an implied, warned or suggested concept or response. * '''Detection''' - Detection is the concept of identifying a physical intrusion, and possibly the intruders involved. For a method to classify as detection it must be capable of, often 'autonomously' (agents such as human guards, dogs, or Computer-Vision systems classify here), indicating a breach or attempt against a security implementation, site, or asset. None of the above methods are, nor are expected to be, full-proof, which applies further to the definitions. Methods to evade Detection, Access-Control, Surveillance, and Deterrence do exist and will likely be discussed further under Security Testing. The above components experience overlap/ Access Control may serve as Deterrence, Surveillance may perform a Detection role, Detection may be required for Access Control, and so on; These components are conceptual, and not hard limitations. Beyond the components of security, a security system must be applied to their own areas of a physical system or environment, defined below. === The Three Levels of Physical Security === * '''Outer Perimeter Security''' - Refers to the marginal outer layer of a security system such as: The physical property boundary lines of a site; The entrance buildings or hubs to a large site such as Disney World or other multi-building site; The room or area around a device such as a server box, car, or cash-register. * '''Inner Perimeter Security''' - Refers to the direct outer layer of a security system such as: The walls, doors, windows or entry points of a building; The accessible areas of a device such as a laptop or car in standard use (not reverse engineered, disassembled, etc;). * '''Interior Security''' - Refers to the internal operations layer of a security system such as: inner spaces, cubbies, and cabinets of an office, building, or room; Internal mechanics of a device such as a lock, engine, or computer; Not every system of security will be capable of realistically or permissively implementing protections on every level listed above, however, every level should still be held in consideration as a vector of intrusion, or consideration for secondary protections. Additionally, some texts and references may include grades of security (often 5 grades) for physical security implementation. These grades are omitted simply due to the variance in definition across sources, combined with their lack of necessity for the understanding of the content. <h1>Physical Security as a Practice</h1> Various concepts of Physical Security will be broken down, linked to, or explained below. === Planning a System of Security === Before creating a system of security, one must create a plan or set of plans, with an underlying goal. This is particularly the case in regards to large organizations or security teams, which must standardize and communicate ideas clearly and quickly for an efficient and correct implementation. * '''[[Physical Threat Landscaping]]''' - The first aspect of planning a system of security, identifying threats and considerations that a physical security system must defend against. <h1>Research Links</h1> * Example of the results of poor physical security: https://www.youtube.com/watch?v=R5RE0mVbJ3s * Reference building codes in your area or government for further information. Often a security section concerning exact rules and techniques for preventing intrusion is contained within, ie: https://www.kcmo.gov/home/showpublisheddocument/6420/637509790534730000 04lorqhc8q4hg6iw1qh532up8rfj5oh 2410370 2410369 2022-07-30T00:53:29Z IncarnationOfTarnation 2947317 wikitext text/x-wiki = Physical Security = === Introduction === Physical Security is the implementation of security measures for the protection of assets, information, or people. It aims to prevent as much unauthorized or malicious access as possible, while still allowing those authorized or permitted access to what they need. This topic aims to outline and define aspects of physical security, alongside providing necessary resources for further research. Terminologies are defined in as standard a way as possible considering the multi-resource nature of the topic, but have a preference to the terminology of other Wikiversity topic or lecture pages. Physical Security measures rely on a combination of physical and psychological methodologies for preventing unwanted access to assets. <h1>Physical Security Terminology</h1> === General Terminologies === * '''Security System''' - a linked or related set of physical security implementations. May apply to: an individual object, such as a lock; A complex object or set of objects, such as a car, computer, or engine; Or some other large scale system, such as a facility and its grounds, a series of complex objects, or a complex of buildings or rooms. * '''CIA Triad''' - A concept from information security that applies well to physical security. The CIA triad is a triangular representation of compromises between the Confidentiality, Integrity, and Availability of an asset (in the typical case, information). When considering Access Controls, particularly to things such as devices for Security Testing, Surveillance, or any other component of the security system, the CIA Triad is of particular relevance. The Triad is not an absolute representation, since some systems can contain high levels of multiple categories, but an important conceptual representation. === The Five Components of Physical Security === The core ideas of physical security can be organized into five components: The three key components - Access Control, Surveillance, and Security testing - alongside Deterrence and Detection. * '''Access Control''' - Access Control is the concept of managing and restricting the accessibility and use of certain assets and areas, either to a whitelist of individuals, or from a blacklist of individuals. Those who need a resource, such as a ledger, account number, or stored paperwork, should have access to it, and those who don't should not. * '''Surveillance''' - Surveillance is the concept of watching, monitoring, observing, or recording an area and/or asset, alongside the entities that access it. For an area or asset to be considered surveilled in some form, authorized individuals should be capable of retrieving information about recent accesses and/or interactions with the surveilled entity. * '''Security Testing''' - Security Testing refers to the act and practice of testing the implementation of physical security methods, the practices of agents involved in the enforcement of physical security methods, and the grading or assessment of the capability of a system of security implementations. Security Testing is an active, repeated practice, and should often result in changes, reimplementation, and enhancements alongside evolving information and a changing threat landscape. * '''Deterrence''' - Deterrence as a concept refers to methods to prevent an intrusion attempt in the first place, and is typically the first layer of defense against intruders. Deterrence can take a variety of forms, and may not be a physically implemented system so much as an implied, warned or suggested concept or response. * '''Detection''' - Detection is the concept of identifying a physical intrusion, and possibly the intruders involved. For a method to classify as detection it must be capable of, often 'autonomously' (agents such as human guards, dogs, or Computer-Vision systems classify here), indicating a breach or attempt against a security implementation, site, or asset. None of the above methods are, nor are expected to be, full-proof, which applies further to the definitions. Methods to evade Detection, Access-Control, Surveillance, and Deterrence do exist and will likely be discussed further under Security Testing. The above components experience overlap/ Access Control may serve as Deterrence, Surveillance may perform a Detection role, Detection may be required for Access Control, and so on; These components are conceptual, and not hard limitations. Beyond the components of security, a security system must be applied to their own areas of a physical system or environment, defined below. === The Three Levels of Physical Security === * '''Outer Perimeter Security''' - Refers to the marginal outer layer of a security system such as: The physical property boundary lines of a site; The entrance buildings or hubs to a large site such as Disney World or other multi-building site; The room or area around a device such as a server box, car, or cash-register. * '''Inner Perimeter Security''' - Refers to the direct outer layer of a security system such as: The walls, doors, windows or entry points of a building; The accessible areas of a device such as a laptop or car in standard use (not reverse engineered, disassembled, etc;). * '''Interior Security''' - Refers to the internal operations layer of a security system such as: inner spaces, cubbies, and cabinets of an office, building, or room; Internal mechanics of a device such as a lock, engine, or computer; Not every system of security will be capable of realistically or permissively implementing protections on every level listed above, however, every level should still be held in consideration as a vector of intrusion, or consideration for secondary protections. Additionally, some texts and references may include grades of security (often 5 grades) for physical security implementation. These grades are omitted simply due to the variance in definition across sources, combined with their lack of necessity for the understanding of the content. <h1>Physical Security as a Practice</h1> Various concepts of Physical Security will be broken down, linked to, or explained below. === Planning a System of Security === Before creating a system of security, one must create a plan or set of plans, with an underlying goal. This is particularly the case in regards to large organizations or security teams, which must standardize and communicate ideas clearly and quickly for an efficient and correct implementation. * '''[[Physical Threat Landscaping]]''' - The first aspect of planning a system of security, identifying threats and considerations that a physical security system must defend against. <h1>Research Links</h1> * Example of the results of poor physical security: https://www.youtube.com/watch?v=R5RE0mVbJ3s * Reference building codes in your area or government for further information. Often a security section concerning exact rules and techniques for preventing intrusion is contained within, ie: https://www.kcmo.gov/home/showpublisheddocument/6420/637509790534730000 oy18pnlvlp0w8t9p1sg819ui7ta7nvo 3 285801 2410363 2022-07-30T00:36:49Z Dave Braunschweig 426084 Welcome wikitext text/x-wiki {{Robelbox|theme=9|title=Welcome!|width=100%}} <div style="{{Robelbox/pad}}"> '''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] ReynardTF!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Dave Braunschweig|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. 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Lim |Permission={{cc-by-sa-3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} fjickn3vclac9nt1854b0yn8rjvi5u4 2 285803 2410378 2022-07-30T02:31:29Z KingMob221 2947327 New resource with "'''[[Template:Motivation and emotion/Book chapter structure|<nowiki>{{subst:ME/BCS}}</nowiki>]]'''" wikitext text/x-wiki '''[[Template:Motivation and emotion/Book chapter structure|<nowiki>{{subst:ME/BCS}}</nowiki>]]''' t3054y0f423djt1wolgdhxzd8r64uyi 2410379 2410378 2022-07-30T02:38:16Z KingMob221 2947327 Undo all revisions. Resource is empty, but not [[Wikiversity:Deletions|deleted]]. wikitext text/x-wiki phoiac9h4m842xq45sp7s6u21eteeq1