Wikiversity:Colloquium

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      <comment>/* Proposal to rehost Wikinews here */ archive to [[Wikiversity:Colloquium/archives/June 2026#Proposal to rehost Wikinews here]] ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]])</comment>
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== Create an autopatrolled user group? ==

{{tracked|T428269|resolved}}
I would like to propose creating the user group &lt;code&gt;autopatrolled&lt;/code&gt; (autopatrolled user), in which for non-curators and non-custodians, their page creations and file uploads would be automatically marked as patrolled by the MediaWiki software. Custodians may grant the user group, at their discretion, to users who create good quality pages that do not need frequent patrolling.

On a side note, the term {{tq|autopatroller}} would be used, but because we don't have non-curator/custodian patrollers (as we rely on curators and custodians to patrol), I suggest on using the term {{tq|autopatrolled user}}. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:31, 29 May 2026 (UTC)

:'''Support''' re: the name, I don't really understand the reasoning, so I am '''neutral''' on that. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 15:45, 29 May 2026 (UTC)
:: Regarding the name, this is because as we don't have the patroller user group, we rely on curators and custodians to patrol new pages and file uploads. Does that make sense? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:39, 29 May 2026 (UTC)
:::Not really, but I don't think it's the most important thing. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 16:42, 29 May 2026 (UTC)
:::: We'll decide on the name later. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:48, 30 May 2026 (UTC)
:::::Oh, please don't let me stand in the way. I'm just not very smart, so don't hold up a matter on my account. I didn't want to derail the proposal, which is a fine and sensible one. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 04:16, 30 May 2026 (UTC)
: '''Support''' - sounds like a good idea
:* Suggest adding a draft section about this group to [[Wikiversity:Patrolling]]. There is a statement in the Introduction of the page that I'm not sure if its correct and at least could be improved: &quot;Wikiversity also uses an autopatrol right, meaning trusted users' contributions are automatically marked as checked so patrollers can focus on reviewing newer or anonymous editors.&quot;
:* Regarding autopatroller vs autropatrolled user, what terms are used on similar WMF wiki projects?
: -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:28, 30 May 2026 (UTC)
::# I would create a starting page about the user groups, with experienced editors expanding the page. A summarized part of that page would also be added to [[Wikiversity:Patrolling]].
::# For a similar example, English Wikipedia uses the term {{tq|Autopatrolled}}, just that term only.
:: [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:22, 30 May 2026 (UTC)
: @[[User:Jtneill|Jtneill]] and @[[User:Koavf|Koavf]]: the autopatroller user group has been implemented here. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:14, 8 June 2026 (UTC)
::Thanks. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:13, 9 June 2026 (UTC)

== How much of Wikiversity’s content is LLM slop? ==

Because it seems like a non-trivial amount, along with AI slop images as well. Is there some kind of AI cleanup project established yet? [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:20, 4 June 2026 (UTC)

:We have discussed AI but I don't know of any explicit initiative to find and delete AI-generated noise. Individual modules have been deleted for having been made by AI. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 08:50, 4 June 2026 (UTC)
:Recently agreed [[Wikiversity:Artificial intelligence|policy]] welcome users to tag AI generated pages. Me personally I am not against the use of AI. What is the difference in abstract schematic image created by a human and the same by an AI. If the users does not have finances to pay digital artest and you dont want to let them use AI, would you pay the artest for them? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:07, 8 June 2026 (UTC)
::Wikimedia has a lot of ''volunteer'' artists who can illustrate if asked. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:11, 9 June 2026 (UTC)
:::Interesting! That's good to know. Where can we find the volunteer artists for illustrating? [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 20:11, 9 June 2026 (UTC)
::::Wikimedia commons has [[commons:Commons:Graphic Lab/Illustration workshop]] [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 02:18, 10 June 2026 (UTC)

== Draft inactivity policy ==

I created [[Wikiversity:Inactivity policy]] as a start. Any experienced Wikiversity user may feel free to expand it. This is also one-to-two step(s) towards opting out of the [[m:Admin activity review|AAR process]].

However, I made a bold change to reduce the response timeframe from one month to two weeks. In addition, should we reduce the inactivity timeframe to one year? For the latter, most projects use that timeframe and I suggested this for consistency. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:57, 4 June 2026 (UTC)

:I support those suggestions. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:55, 4 June 2026 (UTC)
: Juandev has posted some comments on the [[Wikiversity talk:Inactivity policy|talk page]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:30, 12 June 2026 (UTC)
: Thanks for creating this draft. I've made some changes (including moving back to a one month response timeframe) and moved it from draft to proposed policy.
: Based on the discussion on the talk page, I think it is close to ready (or ready) to be formally proposed as a policy (by adding it to the sitenotice) to allow wider discussion and hopefully adoption. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 06:45, 21 July 2026 (UTC)

== Proposed user group and/or possible policy changes ==

{{tracked|T430416|fixed}}
I want to discuss about user group and possible policy changes.

# First, interface administrators. I don't think we should allow interface administrators to remove their permission from their own account, since we have multiple active bureaucrats and we can ask them to remove the permission when done, or for them to add a temporary grant. This is according to the [[Wikiversity:IA|current IA policy]]. I also left [[Wikiversity talk:Interface administrators#My thoughts about this user group|my thoughts on the relevant talk page]].
# Second, curators. Given that curators have some sensitive custodian rights (such as &lt;code&gt;delete&lt;/code&gt; [but not &lt;code&gt;undelete&lt;/code&gt; or similar rights that allow viewing deleted content, unless the curatorship process is RFA-like] and &lt;code&gt;protect&lt;/code&gt;), it would probably make more sense only for bureaucrats to grant and remove it, on par with them granting (but not removing) custodian permissions.
# Third, about probationary custodians. [[Wikiversity:Probationary custodians]] is currently marked as historical, and the process might still exist on [[Wikiversity:Custodianship]]. Therefore, to maintain consistency with [[Wikiversity:Curatorship#How does one become a curator?]], I propose that we repeal the probationary custodianship process and change it more or less to align with the curatorship process, effectively making probationary custodians permanent ones. However, custodian mentors would still be retained.

Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:55, 5 June 2026 (UTC)

:#Yes, I agree.
:#Thats a good point, but I dont know. At least I dont think its a good idea that both groups i.e. crats and custodiants can do that, it may create chaos.
:#Another good point. It seems to me that the current situation is somewhat unclear and should be clarified. I understand the original status of [[Wikiversity:Probationary custodians|Probationary custodians]] as a historicall and invalid, but at the same time I consider myself a probationary custodian, because on the Wikiversity:Custodianship page in the ''[[Wikiversity:Custodianship#How does one become a custodian?|How does one become a custodian?]]'' section it says, I quote, ''&quot;II ...then you will be approved as a probationary custodian for a period of at least four weeks&quot;''.
:::Mentors should definitely be kept, but for certain applicants the probation and mentorship should be abolished. For example, if someone was an active custodian for 5 years, then loses their rights or gives them up for a year and then wants to resume their custodial activities, there is no reason for them to undergo a training period. It burdens both the mentors and the community with double voting. The only exception could be a situation where policies or tools for custodians change significantly during that year, or the candidate wants to.
:[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 06:08, 9 June 2026 (UTC)

== New user what do I do here ==
I love wikipedia and the wikiversity project seems super interesting. However I know very little about wikiversity and would like to know how i can best contribute to the project. Also if there are forums or discord or reddit that would be very helpful.
(One last thing is it normal that my userboxes don't work here) {{unsigned|AUBSTRAWBS}}
:Hey {{ping|AUBSTRAWBS}} Welcome to Wikiversity! I've left a welcome message on your talk page so that should provide you a plethora of useful links for you to look at so you can familiarize yourself with the project. Also, feel free to create the userboxes you need. Wikiversity doesn't have as many userboxes as Wikipedia. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:45, 8 June 2026 (UTC)

:Thank you very much :) hope to contribute a lot.  [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 21:50, 8 June 2026 (UTC)

== Towards an Ethics policy ==

In connection with the [[Wikiversity:Community Review/Removal of Wikidebates|discussion of Wikidebates]], I said that it would be good to establish a policy on ethics, or rather a boundary between ethical and unethical content, so that we don't have to discuss individual cases. In addition, today we also have some global policies that prohibit, for example, attacks on members of the Wikimedia movement or undermining other projects.

However, at the very beginning, I would start by collecting your opinions. What content or what research should not be allowed on Wikiversity? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 05:52, 9 June 2026 (UTC)

:One ethical issue that I think should be non-controversial is related to good faith in the learning modules. So, learning materials should not be hoaxes or encourage behavior or methods that don't work or that misrepresent the facts or the likelihood of something occurring, etc. and authors should also not plagiarize or misrepresent authorship, etc. That was quite a run-on, but I hope that others can tease out what I mean here. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:39, 9 June 2026 (UTC)
::I look at it from a practical perspective. We can give that to the policy, but I see the problem in that we are not able to check it except plagiarism.
::Plagiarism can be partially detected during patrolling. I see a new text, I put part of it in Google and I check if it is copied from the web. It is a problem with copying from books or other offline sources, but sometimes it happens that someone finds out that something is copied from somewhere and it can be deleted.
::The biggest issue we have here is that we are missing Wikipedia's control mechanism: references. Only some types of resources on Wikiversity require references. In-line references are not often used in courses, exercises, lectures, etc. We are thus deprived of one of the excellent control mechanisms and the only option is for the increase in the number of members with various qualifications to check it for their colleagues. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:59, 9 June 2026 (UTC)
:::Having a policy and enforcing that policy are indeed two different things. If we are only concerned with issues that we can definitively enforce, then that will definitely change this conversation. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 08:06, 9 June 2026 (UTC)
::::ok [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:55, 13 June 2026 (UTC)
:AI generated content should not be allowed as it is inherently plagiarism. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:14, 9 June 2026 (UTC)
::And if the user mention it was generated by an AI? Note that there is something called as public domain, that is the author wave its rights. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:53, 9 June 2026 (UTC)
:::Plagiarism isn’t copyright violation. Crediting the AI is not crediting the authors the AI stole from without credit. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 10:18, 9 June 2026 (UTC)
::::I see, now I understand your point. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:56, 13 June 2026 (UTC)

== Deployment of Legal and Safety Contacts Link in the Footer of Your Wiki ==

Hello community,

The Wikimedia Foundation has provided [[foundation:Legal:Wikimedia Foundation Legal and Safety Contact Information|a single legal and safety contact page]], to be linked in the footer of your wiki, to ensure access to accurate legal information. This is a regulatory requirement.

We have already rolled out links to English, German, Italian, Spanish Wikipedias and other wikis and we will deploy to your wiki soon.

Please [[m:Wikimedia Foundation Legal and Safety Contacts FAQ|read more on the project page]] and leave any comments in this thread or on [[m:Talk:Wikimedia Foundation Legal and Safety Contacts FAQ|the talk page]]. –– [[User:STei (WMF)|STei (WMF)]] ([[User talk:STei (WMF)|discuss]] • [[Special:Contributions/STei (WMF)|contribs]]) 18:12, 9 June 2026 (UTC)

:Thanks for the notice. In case anyone is not clear, we cannot locally change the text at the footer, as it [[:mw:Manual:Footer|requires access to the server settings]]. If we locally needed to change it, we would have to file a ticket at [[:phab:]]. Since the above was sent by someone from the WMF, I think they are on it and it will be updated without any action from anyone here. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:24, 9 June 2026 (UTC)

== Image not displaying ==

Can anyone work out why this image isn't displaying?&lt;br&gt;
[[Educational Media Awareness Campaign/Physics/POTD 10]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:45, 11 June 2026 (UTC)

:Not sure, but it was an issue with the file itself and either way, it should be (and I have since done this) replaced with the SVG [[:File:Telescope-schematic.svg]]. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 13:59, 11 June 2026 (UTC)

== New nomination template(s) ==

I created {{tlx|Nomination}} when someone requests curator or custodian permissions, which often at least require mentorship. On the other hand, I might create {{tlx|Nomination 2}}, in which the latter does not have a section about mentorship (often used for bureaucrat or interface administrator nominations). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:29, 12 June 2026 (UTC)

== June 2026 Wikimedia Café meetups regarding the English Wikipedia Editor Reflections project ==

&lt;div class=&quot;border-box&quot; style=&quot;background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)&quot;&gt;
&lt;div class=&quot;box&quot; style=&quot;float:left; padding-top: 10px; padding-right: 10px; padding-left: 10px; padding-bottom: 10px;&quot;&gt;[[File:Wikimedia Café logo in plain SVG format.svg|60px|alt=The logo for the Wikimedia Café]]&lt;/div&gt;
Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities during the last weekend of June. Both sessions will focus on the [https://en.wikipedia.org/wiki/Wikipedia:Editor_reflections English Wikipedia Editor Reflections project]. The featured guest in the Café will be [https://en.wikipedia.org/wiki/User:Clovermoss User:Clovermoss]. Participants may attend either or both sessions.

#'''27 June 2026 15:00 UTC''' ([https://zonestamp.toolforge.org/1782572400 timestamp converter]), at a time friendly to the Americas, Africa, and Europe
#'''28 June 2026 03:00 UTC''' ([https://zonestamp.toolforge.org/1782615600 timestamp converter]), at a time friendly to Asia and the Pacific

Please see the Café page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]!
&lt;br /&gt;
[[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]&lt;/div&gt;
&lt;span style=&quot;white-space:nowrap;&quot;&gt;[[User:Pine|&lt;span style=&quot;color:#01796f; text-shadow:#00BFFF 0 0 1.0em&quot;&gt;↠Pine&lt;/span&gt;]] [[User talk:Pine|&lt;span style=&quot;color:DeepSkyBlue&quot;&gt;(&lt;b style=&quot;color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em&quot;&gt;✉&lt;/b&gt;)&lt;/span&gt;]]&lt;/span&gt; 04:00, 15 June 2026 (UTC)

== Mobile friendly main page ==

Hello, I have recently been using wikiversity on mobile and unlike wikipedia some images and boxes stick out instead of all having a set width which means you can scroll a little side to side, which makes the site feel a bit unfinished. Its just a suggestion but I think it will wake the user experience much better {{unsigned|AUBSTRAWBS}}
:{{Ping|AUBSTRAWBS}} I don't use a smartphone. Can you give me more details or even take some screenshots? You can upload them at [[:c:Category:English Wikiversity screenshots]]. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 13:30, 18 June 2026 (UTC)
::Hi i uploaded an image of the problem. Since some of the images are larger than the screen and not adjusted to fit they stick out and makes the page larger which lets you scroll right and have a big white rectangle on the side [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 14:03, 18 June 2026 (UTC)
:::Thanks. I agree that this is an issue, but it's a pretty minor-to-moderate one to me and I don't think I will be able to dedicate time to fix it myself. Showing it to others here is useful in case someone else wants to tinker with the CSS to resolve it. Thanks for bringing it to the community's attention. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 15:42, 18 June 2026 (UTC)
::::I do know CSS as I like to maintain a blog online so I could try and fix it but I don't know if I have the access to do that, would i need to be a curator/ custodian. Alternatively i could edit a sandbox version of the main page and then send it to someone. [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 20:00, 18 June 2026 (UTC)
:::::Oh great. There are a lot of draft versions of the main page like [[Wikiversity:Main Page/Draft version 0.2]], so you can make [[Wikiversity:Main Page/Sandbox]] if you want and edit there. If you can tinker it to your liking, I can edit the main page. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 20:14, 18 June 2026 (UTC)
::::::thank you, i'll check it out [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 22:16, 18 June 2026 (UTC)

== Main page titles ==

Currently, the title says &quot;Wikiversity:Main Page&quot;, but in my opinion, it's too basic. I would like to propose changing it with the following options (you may only pick one):
# Option 1: Set both [[MediaWiki:Mainpage-title]] and [[MediaWiki:Mainpage-title-loggedin]] to blank, giving the main page a portal-like design (as with English Wikipedia, English Wikibooks, etc.)
# Option 2: Modify [[MediaWiki:Mainpage-title]] to &lt;code&gt;Welcome to Wikiversity&lt;/code&gt; (for unregistered users), and [[MediaWiki:Mainpage-title-loggedin]] to &lt;code&gt;&lt;nowiki&gt;Welcome to Wikiversity, $1!&lt;/nowiki&gt;&lt;/code&gt;; the latter would display to me as &lt;code&gt;Welcome to Wikiversity, Codename Noreste!&lt;/code&gt;
Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:34, 18 June 2026 (UTC)
: Pinging @[[User:Jtneill|Jtneill]] and @[[User:Koavf|Koavf]] for input above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:03, 24 June 2026 (UTC)
:I'm afraid that I don't have strong feelings on this. Changing to either or staying with the status quo are all fine to me. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 16:10, 24 June 2026 (UTC)
: I like the option of being consistent with Wikipedia and Wikibooks -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:04, 25 June 2026 (UTC)
: {{done}}. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:34, 30 June 2026 (UTC)
: Thankyou - looks good. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:29, 2 July 2026 (UTC)

== Wiki x AI preconference day @ Wikimania ==
There will be a preconference day at Wikimania about [[meta:Artificial_intelligence/2026_Wiki_AI | Wiki AI]]. It will be mostly offline, but there will be at least one hybrid session for demos of community-developed AI tools and workflows. 

* If you've built something cool, that is a chance to show it off, list it on the gallery of tools in progress, and get feedback. 

* If you could ask the people shaping AI on the wikis (WMF, tool builders, model trainers, GLAM and policy folks) a question, what would it be? 

Cheers, &lt;span style=&quot;padding:0 2px 0 2px;background-color:white;color:#bbb;&quot;&gt;&amp;ndash;[[User:Sj|SJ]][[User Talk:Sj|&lt;span style=&quot;color:#ff9900;&quot;&gt;+&lt;/span&gt;]]&lt;/span&gt; 23:12, 20 June 2026 (UTC) and Alaexis&lt;br&gt;{{comment|1=Copied from https://en.wikiversity.org/w/index.php?title=Talk%3AMotivation_and_emotion%2FAssessment%2FUsing_generative_AI&amp;diff=2816357&amp;oldid=2807052}}

== RFC about AI-generated content in Wikimedia Commons ==

You are invited to participate in a [[c:Commons:Requests for comment/Policy update for AI content|request for comment on Wikimedia Commons about a policy update for AI content]]. This may affect files that are uploaded to Wikimedia Commons for use on this project. Thank you. [[m:User:Codename Noreste|Codename Noreste]] ([[m:User talk:Codename Noreste|discuss]]) 17:12, 23 June 2026 (UTC)
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== Deployment of Legal and Safety Contacts Link in the Footer of Your Wiki ==

&lt;section begin=&quot;Message&quot;/&gt;
'''Legal &amp; Safety Contacts'''

Hello community, the Wikimedia Foundation has provided a [[wmf:Special:MyLanguage/Legal:Wikimedia Foundation Legal and Safety Contact Information|single legal and safety contact page]], to be linked in the footer of your wiki, to ensure access to accurate legal information. This is a regulatory requirement. We have already rolled out links to English, German, Italian, Spanish and other wikis and we will deploy to your wiki soon. [[m:Special:MyLanguage/Wikimedia_Foundation_Legal_and_Safety_Contacts_FAQ|Please read more on the project page]] and leave any comments in this thread or on the [[m:Special:MyLanguage/Talk:Wikimedia Foundation Legal and Safety Contacts FAQ|talk page]].

&lt;section end=&quot;Message&quot;/&gt;

-- [[User:Sannita (WMF)|User:Sannita (WMF)]] ([[User talk:Sannita (WMF)|talk]]) 13:31, 25 June 2026 (UTC)
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== Preparing manuscript for submission to the ''WikiJournal of Humanities'' ==

I am working on preparing an article in Wikipedia for a dual goal of submitting it for a featured article candidacy in Wikipedia and submitting it to the ''[[WikiJournal of Humanities]]''.  I have an open request for pre-submission peer review at [[en:Wikipedia:Wikipedia:Peer review/Rei Ayanami/archive2|Wikipedia:Peer review/Rei Ayanami/archive2]], and I am asking for someone experienced with submitting journals to WikiJournals.  The article is not ready for submission, and I would like to know where I can get assistance from users who submitted articles to the journal, but did not necessarily review them.  Furthermore, I said there that submitting to the ''WikiJournal of Humanities'' depends on whether the article attains featured article status in Wikipedia, as I would like to use the featured article as a manuscript for a journal article. [[User:Z. Patterson|Z. Patterson]] ([[User talk:Z. Patterson|discuss]] • [[Special:Contributions/Z. Patterson|contribs]]) 12:04, 30 June 2026 (UTC)

== I could probably save many kilobytes by compressing my LLM chat history ==

I could probably save many kilobytes by compressing my LLM chat history. Would that be ok? I like how my new &quot;method&quot; looks: [[User:ThinkingScience/All General AI Prompt History Archive]] very compressed and neat. I like how to find new ways to make people using LLMs not become &quot;secondary citizens&quot;. [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 05:05, 7 July 2026 (UTC)

:{{replyto|ThinkingScience}} I do not see why not. People should be able to look in your page's history for LLM chats before your compression. [[User:Z. Patterson|Z. Patterson]] ([[User talk:Z. Patterson|discuss]] • [[Special:Contributions/Z. Patterson|contribs]]) 02:34, 14 July 2026 (UTC)
::That's good. Then an admin if they have been instructed to compress...then they can delete the edit history perhaps and save space, then it's up to the admin and I don't need to worry about taking up too many resources. That's good! [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 07:50, 14 July 2026 (UTC)

== July 2026 Wikimedia Café meetups regarding Wikimedia governance and options for reform ==

&lt;div class=&quot;border-box&quot; style=&quot;background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)&quot;&gt;
&lt;div class=&quot;box&quot; style=&quot;float:left; padding-top: 10px; padding-right: 10px; padding-left: 10px; padding-bottom: 10px;&quot;&gt;[[File:Wikimedia Café logo in plain SVG format.svg|60px|alt=The logo for the Wikimedia Café]]&lt;/div&gt;
Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities in July. Both sessions will focus on Wikimedia governance, including possible follow-ups to the [https://meta.wikimedia.org/wiki/Movement_Charter Movement Charter] and options for reform. Participants may attend either or both Café sessions. 

This month, to deconflict the Café meetups from Wikimania, the meetups will be held one day later than usual.

#'''26 July 2026 15:00 UTC''' ([https://zonestamp.toolforge.org/1785078000 timestamp converter]), at a time friendly to the Americas, Africa, and Europe
#'''27 July 2026 03:00 UTC''' ([https://zonestamp.toolforge.org/1785121200 timestamp converter]), at a time friendly to Asia and the Pacific

Please see the Café page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]!
&lt;br /&gt;
[[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]&lt;/div&gt;
&lt;span style=&quot;white-space:nowrap;&quot;&gt;[[User:Pine|&lt;span style=&quot;color:#01796f; text-shadow:#00BFFF 0 0 1.0em&quot;&gt;↠Pine&lt;/span&gt;]] [[User talk:Pine|&lt;span style=&quot;color:DeepSkyBlue&quot;&gt;(&lt;b style=&quot;color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em&quot;&gt;✉&lt;/b&gt;)&lt;/span&gt;]]&lt;/span&gt; 03:51, 13 July 2026 (UTC)

:@[[User:Pine|Pine]], Some wikimedians who may contribute less popular, but hopefully productive, comments,  are those who are currently blocked on Meta. I myself was blocked infinitely on Meta several years ago, but was allowed to contInue contributing on other wikimedia projects, such as this one. There are many long term contributors, some who have contrIbuted tens of thousands of edits  to wikimedia projects who are are now blocked on specific projects. 
:Those who happen to be blocked on Meta have,  to all intents and purposes, been removed from the wikimedia movement.
:Ottawahitech [[Special:Contributions/&amp;#126;2026-40514-81|&amp;#126;2026-40514-81]] ([[User talk:&amp;#126;2026-40514-81|talk]]) 21:41, 18 July 2026 (UTC)
:*Hi Ottawahitech, thanks for the comment. Reviewing local bans on Meta is outside of my current scope, but if you have comments that you'd like to share here in the Colloquium relating to Wikiversity, I can consider the comments in the scope of the Café discussions. &lt;span style=&quot;white-space:nowrap;&quot;&gt;[[User:Pine|&lt;span style=&quot;color:#01796f; text-shadow:#00BFFF 0 0 1.0em&quot;&gt;↠Pine&lt;/span&gt;]] [[User talk:Pine|&lt;span style=&quot;color:DeepSkyBlue&quot;&gt;(&lt;b style=&quot;color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em&quot;&gt;✉&lt;/b&gt;)&lt;/span&gt;]]&lt;/span&gt; 20:05, 25 July 2026 (UTC)

== Request for comment (the future of Abstract Wikipedia) ==

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot; class=&quot;mw-content-ltr&quot;&gt;

You are invited to voice your opinions in a [[:m:Requests for comment/The future of Abstract Wikipedia|request for comment about the future of Abstract Wikipedia]]. {{Int:Feedback-thanks-title}}
[[:m:User:Kowal2701|Kowal2701]] ([[:m:User talk:Kowal2701|talk]])  12:25, 24 July 2026 (UTC)
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      <sha1>qe3yw9211tsuw8ql9v9sihcdnde9o93</sha1>
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        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>/* Create an autopatrolled user group? */ archive to [[Wikiversity:Colloquium/archives/June 2026#Create an autopatrolled user group?]] ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]])</comment>
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== How much of Wikiversity’s content is LLM slop? ==

Because it seems like a non-trivial amount, along with AI slop images as well. Is there some kind of AI cleanup project established yet? [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:20, 4 June 2026 (UTC)

:We have discussed AI but I don't know of any explicit initiative to find and delete AI-generated noise. Individual modules have been deleted for having been made by AI. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 08:50, 4 June 2026 (UTC)
:Recently agreed [[Wikiversity:Artificial intelligence|policy]] welcome users to tag AI generated pages. Me personally I am not against the use of AI. What is the difference in abstract schematic image created by a human and the same by an AI. If the users does not have finances to pay digital artest and you dont want to let them use AI, would you pay the artest for them? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:07, 8 June 2026 (UTC)
::Wikimedia has a lot of ''volunteer'' artists who can illustrate if asked. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:11, 9 June 2026 (UTC)
:::Interesting! That's good to know. Where can we find the volunteer artists for illustrating? [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 20:11, 9 June 2026 (UTC)
::::Wikimedia commons has [[commons:Commons:Graphic Lab/Illustration workshop]] [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 02:18, 10 June 2026 (UTC)

== Draft inactivity policy ==

I created [[Wikiversity:Inactivity policy]] as a start. Any experienced Wikiversity user may feel free to expand it. This is also one-to-two step(s) towards opting out of the [[m:Admin activity review|AAR process]].

However, I made a bold change to reduce the response timeframe from one month to two weeks. In addition, should we reduce the inactivity timeframe to one year? For the latter, most projects use that timeframe and I suggested this for consistency. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:57, 4 June 2026 (UTC)

:I support those suggestions. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:55, 4 June 2026 (UTC)
: Juandev has posted some comments on the [[Wikiversity talk:Inactivity policy|talk page]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:30, 12 June 2026 (UTC)
: Thanks for creating this draft. I've made some changes (including moving back to a one month response timeframe) and moved it from draft to proposed policy.
: Based on the discussion on the talk page, I think it is close to ready (or ready) to be formally proposed as a policy (by adding it to the sitenotice) to allow wider discussion and hopefully adoption. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 06:45, 21 July 2026 (UTC)

== Proposed user group and/or possible policy changes ==

{{tracked|T430416|fixed}}
I want to discuss about user group and possible policy changes.

# First, interface administrators. I don't think we should allow interface administrators to remove their permission from their own account, since we have multiple active bureaucrats and we can ask them to remove the permission when done, or for them to add a temporary grant. This is according to the [[Wikiversity:IA|current IA policy]]. I also left [[Wikiversity talk:Interface administrators#My thoughts about this user group|my thoughts on the relevant talk page]].
# Second, curators. Given that curators have some sensitive custodian rights (such as &lt;code&gt;delete&lt;/code&gt; [but not &lt;code&gt;undelete&lt;/code&gt; or similar rights that allow viewing deleted content, unless the curatorship process is RFA-like] and &lt;code&gt;protect&lt;/code&gt;), it would probably make more sense only for bureaucrats to grant and remove it, on par with them granting (but not removing) custodian permissions.
# Third, about probationary custodians. [[Wikiversity:Probationary custodians]] is currently marked as historical, and the process might still exist on [[Wikiversity:Custodianship]]. Therefore, to maintain consistency with [[Wikiversity:Curatorship#How does one become a curator?]], I propose that we repeal the probationary custodianship process and change it more or less to align with the curatorship process, effectively making probationary custodians permanent ones. However, custodian mentors would still be retained.

Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:55, 5 June 2026 (UTC)

:#Yes, I agree.
:#Thats a good point, but I dont know. At least I dont think its a good idea that both groups i.e. crats and custodiants can do that, it may create chaos.
:#Another good point. It seems to me that the current situation is somewhat unclear and should be clarified. I understand the original status of [[Wikiversity:Probationary custodians|Probationary custodians]] as a historicall and invalid, but at the same time I consider myself a probationary custodian, because on the Wikiversity:Custodianship page in the ''[[Wikiversity:Custodianship#How does one become a custodian?|How does one become a custodian?]]'' section it says, I quote, ''&quot;II ...then you will be approved as a probationary custodian for a period of at least four weeks&quot;''.
:::Mentors should definitely be kept, but for certain applicants the probation and mentorship should be abolished. For example, if someone was an active custodian for 5 years, then loses their rights or gives them up for a year and then wants to resume their custodial activities, there is no reason for them to undergo a training period. It burdens both the mentors and the community with double voting. The only exception could be a situation where policies or tools for custodians change significantly during that year, or the candidate wants to.
:[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 06:08, 9 June 2026 (UTC)

== New user what do I do here ==
I love wikipedia and the wikiversity project seems super interesting. However I know very little about wikiversity and would like to know how i can best contribute to the project. Also if there are forums or discord or reddit that would be very helpful.
(One last thing is it normal that my userboxes don't work here) {{unsigned|AUBSTRAWBS}}
:Hey {{ping|AUBSTRAWBS}} Welcome to Wikiversity! I've left a welcome message on your talk page so that should provide you a plethora of useful links for you to look at so you can familiarize yourself with the project. Also, feel free to create the userboxes you need. Wikiversity doesn't have as many userboxes as Wikipedia. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:45, 8 June 2026 (UTC)

:Thank you very much :) hope to contribute a lot.  [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 21:50, 8 June 2026 (UTC)

== Towards an Ethics policy ==

In connection with the [[Wikiversity:Community Review/Removal of Wikidebates|discussion of Wikidebates]], I said that it would be good to establish a policy on ethics, or rather a boundary between ethical and unethical content, so that we don't have to discuss individual cases. In addition, today we also have some global policies that prohibit, for example, attacks on members of the Wikimedia movement or undermining other projects.

However, at the very beginning, I would start by collecting your opinions. What content or what research should not be allowed on Wikiversity? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 05:52, 9 June 2026 (UTC)

:One ethical issue that I think should be non-controversial is related to good faith in the learning modules. So, learning materials should not be hoaxes or encourage behavior or methods that don't work or that misrepresent the facts or the likelihood of something occurring, etc. and authors should also not plagiarize or misrepresent authorship, etc. That was quite a run-on, but I hope that others can tease out what I mean here. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:39, 9 June 2026 (UTC)
::I look at it from a practical perspective. We can give that to the policy, but I see the problem in that we are not able to check it except plagiarism.
::Plagiarism can be partially detected during patrolling. I see a new text, I put part of it in Google and I check if it is copied from the web. It is a problem with copying from books or other offline sources, but sometimes it happens that someone finds out that something is copied from somewhere and it can be deleted.
::The biggest issue we have here is that we are missing Wikipedia's control mechanism: references. Only some types of resources on Wikiversity require references. In-line references are not often used in courses, exercises, lectures, etc. We are thus deprived of one of the excellent control mechanisms and the only option is for the increase in the number of members with various qualifications to check it for their colleagues. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:59, 9 June 2026 (UTC)
:::Having a policy and enforcing that policy are indeed two different things. If we are only concerned with issues that we can definitively enforce, then that will definitely change this conversation. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 08:06, 9 June 2026 (UTC)
::::ok [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:55, 13 June 2026 (UTC)
:AI generated content should not be allowed as it is inherently plagiarism. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:14, 9 June 2026 (UTC)
::And if the user mention it was generated by an AI? Note that there is something called as public domain, that is the author wave its rights. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:53, 9 June 2026 (UTC)
:::Plagiarism isn’t copyright violation. Crediting the AI is not crediting the authors the AI stole from without credit. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 10:18, 9 June 2026 (UTC)
::::I see, now I understand your point. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:56, 13 June 2026 (UTC)

== Deployment of Legal and Safety Contacts Link in the Footer of Your Wiki ==

Hello community,

The Wikimedia Foundation has provided [[foundation:Legal:Wikimedia Foundation Legal and Safety Contact Information|a single legal and safety contact page]], to be linked in the footer of your wiki, to ensure access to accurate legal information. This is a regulatory requirement.

We have already rolled out links to English, German, Italian, Spanish Wikipedias and other wikis and we will deploy to your wiki soon.

Please [[m:Wikimedia Foundation Legal and Safety Contacts FAQ|read more on the project page]] and leave any comments in this thread or on [[m:Talk:Wikimedia Foundation Legal and Safety Contacts FAQ|the talk page]]. –– [[User:STei (WMF)|STei (WMF)]] ([[User talk:STei (WMF)|discuss]] • [[Special:Contributions/STei (WMF)|contribs]]) 18:12, 9 June 2026 (UTC)

:Thanks for the notice. In case anyone is not clear, we cannot locally change the text at the footer, as it [[:mw:Manual:Footer|requires access to the server settings]]. If we locally needed to change it, we would have to file a ticket at [[:phab:]]. Since the above was sent by someone from the WMF, I think they are on it and it will be updated without any action from anyone here. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:24, 9 June 2026 (UTC)

== Image not displaying ==

Can anyone work out why this image isn't displaying?&lt;br&gt;
[[Educational Media Awareness Campaign/Physics/POTD 10]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:45, 11 June 2026 (UTC)

:Not sure, but it was an issue with the file itself and either way, it should be (and I have since done this) replaced with the SVG [[:File:Telescope-schematic.svg]]. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 13:59, 11 June 2026 (UTC)

== New nomination template(s) ==

I created {{tlx|Nomination}} when someone requests curator or custodian permissions, which often at least require mentorship. On the other hand, I might create {{tlx|Nomination 2}}, in which the latter does not have a section about mentorship (often used for bureaucrat or interface administrator nominations). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:29, 12 June 2026 (UTC)

== June 2026 Wikimedia Café meetups regarding the English Wikipedia Editor Reflections project ==

&lt;div class=&quot;border-box&quot; style=&quot;background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)&quot;&gt;
&lt;div class=&quot;box&quot; style=&quot;float:left; padding-top: 10px; padding-right: 10px; padding-left: 10px; padding-bottom: 10px;&quot;&gt;[[File:Wikimedia Café logo in plain SVG format.svg|60px|alt=The logo for the Wikimedia Café]]&lt;/div&gt;
Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities during the last weekend of June. Both sessions will focus on the [https://en.wikipedia.org/wiki/Wikipedia:Editor_reflections English Wikipedia Editor Reflections project]. The featured guest in the Café will be [https://en.wikipedia.org/wiki/User:Clovermoss User:Clovermoss]. Participants may attend either or both sessions.

#'''27 June 2026 15:00 UTC''' ([https://zonestamp.toolforge.org/1782572400 timestamp converter]), at a time friendly to the Americas, Africa, and Europe
#'''28 June 2026 03:00 UTC''' ([https://zonestamp.toolforge.org/1782615600 timestamp converter]), at a time friendly to Asia and the Pacific

Please see the Café page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]!
&lt;br /&gt;
[[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]&lt;/div&gt;
&lt;span style=&quot;white-space:nowrap;&quot;&gt;[[User:Pine|&lt;span style=&quot;color:#01796f; text-shadow:#00BFFF 0 0 1.0em&quot;&gt;↠Pine&lt;/span&gt;]] [[User talk:Pine|&lt;span style=&quot;color:DeepSkyBlue&quot;&gt;(&lt;b style=&quot;color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em&quot;&gt;✉&lt;/b&gt;)&lt;/span&gt;]]&lt;/span&gt; 04:00, 15 June 2026 (UTC)

== Mobile friendly main page ==

Hello, I have recently been using wikiversity on mobile and unlike wikipedia some images and boxes stick out instead of all having a set width which means you can scroll a little side to side, which makes the site feel a bit unfinished. Its just a suggestion but I think it will wake the user experience much better {{unsigned|AUBSTRAWBS}}
:{{Ping|AUBSTRAWBS}} I don't use a smartphone. Can you give me more details or even take some screenshots? You can upload them at [[:c:Category:English Wikiversity screenshots]]. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 13:30, 18 June 2026 (UTC)
::Hi i uploaded an image of the problem. Since some of the images are larger than the screen and not adjusted to fit they stick out and makes the page larger which lets you scroll right and have a big white rectangle on the side [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 14:03, 18 June 2026 (UTC)
:::Thanks. I agree that this is an issue, but it's a pretty minor-to-moderate one to me and I don't think I will be able to dedicate time to fix it myself. Showing it to others here is useful in case someone else wants to tinker with the CSS to resolve it. Thanks for bringing it to the community's attention. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 15:42, 18 June 2026 (UTC)
::::I do know CSS as I like to maintain a blog online so I could try and fix it but I don't know if I have the access to do that, would i need to be a curator/ custodian. Alternatively i could edit a sandbox version of the main page and then send it to someone. [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 20:00, 18 June 2026 (UTC)
:::::Oh great. There are a lot of draft versions of the main page like [[Wikiversity:Main Page/Draft version 0.2]], so you can make [[Wikiversity:Main Page/Sandbox]] if you want and edit there. If you can tinker it to your liking, I can edit the main page. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 20:14, 18 June 2026 (UTC)
::::::thank you, i'll check it out [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 22:16, 18 June 2026 (UTC)

== Main page titles ==

Currently, the title says &quot;Wikiversity:Main Page&quot;, but in my opinion, it's too basic. I would like to propose changing it with the following options (you may only pick one):
# Option 1: Set both [[MediaWiki:Mainpage-title]] and [[MediaWiki:Mainpage-title-loggedin]] to blank, giving the main page a portal-like design (as with English Wikipedia, English Wikibooks, etc.)
# Option 2: Modify [[MediaWiki:Mainpage-title]] to &lt;code&gt;Welcome to Wikiversity&lt;/code&gt; (for unregistered users), and [[MediaWiki:Mainpage-title-loggedin]] to &lt;code&gt;&lt;nowiki&gt;Welcome to Wikiversity, $1!&lt;/nowiki&gt;&lt;/code&gt;; the latter would display to me as &lt;code&gt;Welcome to Wikiversity, Codename Noreste!&lt;/code&gt;
Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:34, 18 June 2026 (UTC)
: Pinging @[[User:Jtneill|Jtneill]] and @[[User:Koavf|Koavf]] for input above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:03, 24 June 2026 (UTC)
:I'm afraid that I don't have strong feelings on this. Changing to either or staying with the status quo are all fine to me. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 16:10, 24 June 2026 (UTC)
: I like the option of being consistent with Wikipedia and Wikibooks -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:04, 25 June 2026 (UTC)
: {{done}}. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:34, 30 June 2026 (UTC)
: Thankyou - looks good. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:29, 2 July 2026 (UTC)

== Wiki x AI preconference day @ Wikimania ==
There will be a preconference day at Wikimania about [[meta:Artificial_intelligence/2026_Wiki_AI | Wiki AI]]. It will be mostly offline, but there will be at least one hybrid session for demos of community-developed AI tools and workflows. 

* If you've built something cool, that is a chance to show it off, list it on the gallery of tools in progress, and get feedback. 

* If you could ask the people shaping AI on the wikis (WMF, tool builders, model trainers, GLAM and policy folks) a question, what would it be? 

Cheers, &lt;span style=&quot;padding:0 2px 0 2px;background-color:white;color:#bbb;&quot;&gt;&amp;ndash;[[User:Sj|SJ]][[User Talk:Sj|&lt;span style=&quot;color:#ff9900;&quot;&gt;+&lt;/span&gt;]]&lt;/span&gt; 23:12, 20 June 2026 (UTC) and Alaexis&lt;br&gt;{{comment|1=Copied from https://en.wikiversity.org/w/index.php?title=Talk%3AMotivation_and_emotion%2FAssessment%2FUsing_generative_AI&amp;diff=2816357&amp;oldid=2807052}}

== RFC about AI-generated content in Wikimedia Commons ==

You are invited to participate in a [[c:Commons:Requests for comment/Policy update for AI content|request for comment on Wikimedia Commons about a policy update for AI content]]. This may affect files that are uploaded to Wikimedia Commons for use on this project. Thank you. [[m:User:Codename Noreste|Codename Noreste]] ([[m:User talk:Codename Noreste|discuss]]) 17:12, 23 June 2026 (UTC)
&lt;!-- Message sent by User:Codename Noreste@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&amp;oldid=30513860 --&gt;

== Deployment of Legal and Safety Contacts Link in the Footer of Your Wiki ==

&lt;section begin=&quot;Message&quot;/&gt;
'''Legal &amp; Safety Contacts'''

Hello community, the Wikimedia Foundation has provided a [[wmf:Special:MyLanguage/Legal:Wikimedia Foundation Legal and Safety Contact Information|single legal and safety contact page]], to be linked in the footer of your wiki, to ensure access to accurate legal information. This is a regulatory requirement. We have already rolled out links to English, German, Italian, Spanish and other wikis and we will deploy to your wiki soon. [[m:Special:MyLanguage/Wikimedia_Foundation_Legal_and_Safety_Contacts_FAQ|Please read more on the project page]] and leave any comments in this thread or on the [[m:Special:MyLanguage/Talk:Wikimedia Foundation Legal and Safety Contacts FAQ|talk page]].

&lt;section end=&quot;Message&quot;/&gt;

-- [[User:Sannita (WMF)|User:Sannita (WMF)]] ([[User talk:Sannita (WMF)|talk]]) 13:31, 25 June 2026 (UTC)
&lt;!-- Message sent by User:Sannita (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=User:Sannita_(WMF)/Mass_sending_test&amp;oldid=30731267 --&gt;

== Preparing manuscript for submission to the ''WikiJournal of Humanities'' ==

I am working on preparing an article in Wikipedia for a dual goal of submitting it for a featured article candidacy in Wikipedia and submitting it to the ''[[WikiJournal of Humanities]]''.  I have an open request for pre-submission peer review at [[en:Wikipedia:Wikipedia:Peer review/Rei Ayanami/archive2|Wikipedia:Peer review/Rei Ayanami/archive2]], and I am asking for someone experienced with submitting journals to WikiJournals.  The article is not ready for submission, and I would like to know where I can get assistance from users who submitted articles to the journal, but did not necessarily review them.  Furthermore, I said there that submitting to the ''WikiJournal of Humanities'' depends on whether the article attains featured article status in Wikipedia, as I would like to use the featured article as a manuscript for a journal article. [[User:Z. Patterson|Z. Patterson]] ([[User talk:Z. Patterson|discuss]] • [[Special:Contributions/Z. Patterson|contribs]]) 12:04, 30 June 2026 (UTC)

== I could probably save many kilobytes by compressing my LLM chat history ==

I could probably save many kilobytes by compressing my LLM chat history. Would that be ok? I like how my new &quot;method&quot; looks: [[User:ThinkingScience/All General AI Prompt History Archive]] very compressed and neat. I like how to find new ways to make people using LLMs not become &quot;secondary citizens&quot;. [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 05:05, 7 July 2026 (UTC)

:{{replyto|ThinkingScience}} I do not see why not. People should be able to look in your page's history for LLM chats before your compression. [[User:Z. Patterson|Z. Patterson]] ([[User talk:Z. Patterson|discuss]] • [[Special:Contributions/Z. Patterson|contribs]]) 02:34, 14 July 2026 (UTC)
::That's good. Then an admin if they have been instructed to compress...then they can delete the edit history perhaps and save space, then it's up to the admin and I don't need to worry about taking up too many resources. That's good! [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 07:50, 14 July 2026 (UTC)

== July 2026 Wikimedia Café meetups regarding Wikimedia governance and options for reform ==

&lt;div class=&quot;border-box&quot; style=&quot;background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)&quot;&gt;
&lt;div class=&quot;box&quot; style=&quot;float:left; padding-top: 10px; padding-right: 10px; padding-left: 10px; padding-bottom: 10px;&quot;&gt;[[File:Wikimedia Café logo in plain SVG format.svg|60px|alt=The logo for the Wikimedia Café]]&lt;/div&gt;
Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities in July. Both sessions will focus on Wikimedia governance, including possible follow-ups to the [https://meta.wikimedia.org/wiki/Movement_Charter Movement Charter] and options for reform. Participants may attend either or both Café sessions. 

This month, to deconflict the Café meetups from Wikimania, the meetups will be held one day later than usual.

#'''26 July 2026 15:00 UTC''' ([https://zonestamp.toolforge.org/1785078000 timestamp converter]), at a time friendly to the Americas, Africa, and Europe
#'''27 July 2026 03:00 UTC''' ([https://zonestamp.toolforge.org/1785121200 timestamp converter]), at a time friendly to Asia and the Pacific

Please see the Café page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]!
&lt;br /&gt;
[[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]&lt;/div&gt;
&lt;span style=&quot;white-space:nowrap;&quot;&gt;[[User:Pine|&lt;span style=&quot;color:#01796f; text-shadow:#00BFFF 0 0 1.0em&quot;&gt;↠Pine&lt;/span&gt;]] [[User talk:Pine|&lt;span style=&quot;color:DeepSkyBlue&quot;&gt;(&lt;b style=&quot;color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em&quot;&gt;✉&lt;/b&gt;)&lt;/span&gt;]]&lt;/span&gt; 03:51, 13 July 2026 (UTC)

:@[[User:Pine|Pine]], Some wikimedians who may contribute less popular, but hopefully productive, comments,  are those who are currently blocked on Meta. I myself was blocked infinitely on Meta several years ago, but was allowed to contInue contributing on other wikimedia projects, such as this one. There are many long term contributors, some who have contrIbuted tens of thousands of edits  to wikimedia projects who are are now blocked on specific projects. 
:Those who happen to be blocked on Meta have,  to all intents and purposes, been removed from the wikimedia movement.
:Ottawahitech [[Special:Contributions/&amp;#126;2026-40514-81|&amp;#126;2026-40514-81]] ([[User talk:&amp;#126;2026-40514-81|talk]]) 21:41, 18 July 2026 (UTC)
:*Hi Ottawahitech, thanks for the comment. Reviewing local bans on Meta is outside of my current scope, but if you have comments that you'd like to share here in the Colloquium relating to Wikiversity, I can consider the comments in the scope of the Café discussions. &lt;span style=&quot;white-space:nowrap;&quot;&gt;[[User:Pine|&lt;span style=&quot;color:#01796f; text-shadow:#00BFFF 0 0 1.0em&quot;&gt;↠Pine&lt;/span&gt;]] [[User talk:Pine|&lt;span style=&quot;color:DeepSkyBlue&quot;&gt;(&lt;b style=&quot;color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em&quot;&gt;✉&lt;/b&gt;)&lt;/span&gt;]]&lt;/span&gt; 20:05, 25 July 2026 (UTC)

== Request for comment (the future of Abstract Wikipedia) ==

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot; class=&quot;mw-content-ltr&quot;&gt;

You are invited to voice your opinions in a [[:m:Requests for comment/The future of Abstract Wikipedia|request for comment about the future of Abstract Wikipedia]]. {{Int:Feedback-thanks-title}}
[[:m:User:Kowal2701|Kowal2701]] ([[:m:User talk:Kowal2701|talk]])  12:25, 24 July 2026 (UTC)
&lt;/bdi&gt;
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      <sha1>0isia54kwqwa1kitolg5odsklu04ivs</sha1>
    </revision>
    <revision>
      <id>2819781</id>
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      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>/* How much of Wikiversity’s content is LLM slop? */ archive to [[Wikiversity:Colloquium/archives/June 2026#How much of Wikiversity’s content is LLM slop?]] ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]])</comment>
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      <text bytes="28030" sha1="h48xsb6qo2mlnzjgqngwzzj4tk96yka" xml:space="preserve">{{Wikiversity:Colloquium/Header}}
&lt;!-- MESSAGES GO BELOW --&gt;

== Draft inactivity policy ==

I created [[Wikiversity:Inactivity policy]] as a start. Any experienced Wikiversity user may feel free to expand it. This is also one-to-two step(s) towards opting out of the [[m:Admin activity review|AAR process]].

However, I made a bold change to reduce the response timeframe from one month to two weeks. In addition, should we reduce the inactivity timeframe to one year? For the latter, most projects use that timeframe and I suggested this for consistency. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:57, 4 June 2026 (UTC)

:I support those suggestions. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:55, 4 June 2026 (UTC)
: Juandev has posted some comments on the [[Wikiversity talk:Inactivity policy|talk page]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:30, 12 June 2026 (UTC)
: Thanks for creating this draft. I've made some changes (including moving back to a one month response timeframe) and moved it from draft to proposed policy.
: Based on the discussion on the talk page, I think it is close to ready (or ready) to be formally proposed as a policy (by adding it to the sitenotice) to allow wider discussion and hopefully adoption. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 06:45, 21 July 2026 (UTC)

== Proposed user group and/or possible policy changes ==

{{tracked|T430416|fixed}}
I want to discuss about user group and possible policy changes.

# First, interface administrators. I don't think we should allow interface administrators to remove their permission from their own account, since we have multiple active bureaucrats and we can ask them to remove the permission when done, or for them to add a temporary grant. This is according to the [[Wikiversity:IA|current IA policy]]. I also left [[Wikiversity talk:Interface administrators#My thoughts about this user group|my thoughts on the relevant talk page]].
# Second, curators. Given that curators have some sensitive custodian rights (such as &lt;code&gt;delete&lt;/code&gt; [but not &lt;code&gt;undelete&lt;/code&gt; or similar rights that allow viewing deleted content, unless the curatorship process is RFA-like] and &lt;code&gt;protect&lt;/code&gt;), it would probably make more sense only for bureaucrats to grant and remove it, on par with them granting (but not removing) custodian permissions.
# Third, about probationary custodians. [[Wikiversity:Probationary custodians]] is currently marked as historical, and the process might still exist on [[Wikiversity:Custodianship]]. Therefore, to maintain consistency with [[Wikiversity:Curatorship#How does one become a curator?]], I propose that we repeal the probationary custodianship process and change it more or less to align with the curatorship process, effectively making probationary custodians permanent ones. However, custodian mentors would still be retained.

Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:55, 5 June 2026 (UTC)

:#Yes, I agree.
:#Thats a good point, but I dont know. At least I dont think its a good idea that both groups i.e. crats and custodiants can do that, it may create chaos.
:#Another good point. It seems to me that the current situation is somewhat unclear and should be clarified. I understand the original status of [[Wikiversity:Probationary custodians|Probationary custodians]] as a historicall and invalid, but at the same time I consider myself a probationary custodian, because on the Wikiversity:Custodianship page in the ''[[Wikiversity:Custodianship#How does one become a custodian?|How does one become a custodian?]]'' section it says, I quote, ''&quot;II ...then you will be approved as a probationary custodian for a period of at least four weeks&quot;''.
:::Mentors should definitely be kept, but for certain applicants the probation and mentorship should be abolished. For example, if someone was an active custodian for 5 years, then loses their rights or gives them up for a year and then wants to resume their custodial activities, there is no reason for them to undergo a training period. It burdens both the mentors and the community with double voting. The only exception could be a situation where policies or tools for custodians change significantly during that year, or the candidate wants to.
:[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 06:08, 9 June 2026 (UTC)

== New user what do I do here ==
I love wikipedia and the wikiversity project seems super interesting. However I know very little about wikiversity and would like to know how i can best contribute to the project. Also if there are forums or discord or reddit that would be very helpful.
(One last thing is it normal that my userboxes don't work here) {{unsigned|AUBSTRAWBS}}
:Hey {{ping|AUBSTRAWBS}} Welcome to Wikiversity! I've left a welcome message on your talk page so that should provide you a plethora of useful links for you to look at so you can familiarize yourself with the project. Also, feel free to create the userboxes you need. Wikiversity doesn't have as many userboxes as Wikipedia. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:45, 8 June 2026 (UTC)

:Thank you very much :) hope to contribute a lot.  [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 21:50, 8 June 2026 (UTC)

== Towards an Ethics policy ==

In connection with the [[Wikiversity:Community Review/Removal of Wikidebates|discussion of Wikidebates]], I said that it would be good to establish a policy on ethics, or rather a boundary between ethical and unethical content, so that we don't have to discuss individual cases. In addition, today we also have some global policies that prohibit, for example, attacks on members of the Wikimedia movement or undermining other projects.

However, at the very beginning, I would start by collecting your opinions. What content or what research should not be allowed on Wikiversity? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 05:52, 9 June 2026 (UTC)

:One ethical issue that I think should be non-controversial is related to good faith in the learning modules. So, learning materials should not be hoaxes or encourage behavior or methods that don't work or that misrepresent the facts or the likelihood of something occurring, etc. and authors should also not plagiarize or misrepresent authorship, etc. That was quite a run-on, but I hope that others can tease out what I mean here. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:39, 9 June 2026 (UTC)
::I look at it from a practical perspective. We can give that to the policy, but I see the problem in that we are not able to check it except plagiarism.
::Plagiarism can be partially detected during patrolling. I see a new text, I put part of it in Google and I check if it is copied from the web. It is a problem with copying from books or other offline sources, but sometimes it happens that someone finds out that something is copied from somewhere and it can be deleted.
::The biggest issue we have here is that we are missing Wikipedia's control mechanism: references. Only some types of resources on Wikiversity require references. In-line references are not often used in courses, exercises, lectures, etc. We are thus deprived of one of the excellent control mechanisms and the only option is for the increase in the number of members with various qualifications to check it for their colleagues. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:59, 9 June 2026 (UTC)
:::Having a policy and enforcing that policy are indeed two different things. If we are only concerned with issues that we can definitively enforce, then that will definitely change this conversation. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 08:06, 9 June 2026 (UTC)
::::ok [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:55, 13 June 2026 (UTC)
:AI generated content should not be allowed as it is inherently plagiarism. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:14, 9 June 2026 (UTC)
::And if the user mention it was generated by an AI? Note that there is something called as public domain, that is the author wave its rights. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:53, 9 June 2026 (UTC)
:::Plagiarism isn’t copyright violation. Crediting the AI is not crediting the authors the AI stole from without credit. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 10:18, 9 June 2026 (UTC)
::::I see, now I understand your point. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:56, 13 June 2026 (UTC)

== Deployment of Legal and Safety Contacts Link in the Footer of Your Wiki ==

Hello community,

The Wikimedia Foundation has provided [[foundation:Legal:Wikimedia Foundation Legal and Safety Contact Information|a single legal and safety contact page]], to be linked in the footer of your wiki, to ensure access to accurate legal information. This is a regulatory requirement.

We have already rolled out links to English, German, Italian, Spanish Wikipedias and other wikis and we will deploy to your wiki soon.

Please [[m:Wikimedia Foundation Legal and Safety Contacts FAQ|read more on the project page]] and leave any comments in this thread or on [[m:Talk:Wikimedia Foundation Legal and Safety Contacts FAQ|the talk page]]. –– [[User:STei (WMF)|STei (WMF)]] ([[User talk:STei (WMF)|discuss]] • [[Special:Contributions/STei (WMF)|contribs]]) 18:12, 9 June 2026 (UTC)

:Thanks for the notice. In case anyone is not clear, we cannot locally change the text at the footer, as it [[:mw:Manual:Footer|requires access to the server settings]]. If we locally needed to change it, we would have to file a ticket at [[:phab:]]. Since the above was sent by someone from the WMF, I think they are on it and it will be updated without any action from anyone here. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:24, 9 June 2026 (UTC)

== Image not displaying ==

Can anyone work out why this image isn't displaying?&lt;br&gt;
[[Educational Media Awareness Campaign/Physics/POTD 10]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:45, 11 June 2026 (UTC)

:Not sure, but it was an issue with the file itself and either way, it should be (and I have since done this) replaced with the SVG [[:File:Telescope-schematic.svg]]. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 13:59, 11 June 2026 (UTC)

== New nomination template(s) ==

I created {{tlx|Nomination}} when someone requests curator or custodian permissions, which often at least require mentorship. On the other hand, I might create {{tlx|Nomination 2}}, in which the latter does not have a section about mentorship (often used for bureaucrat or interface administrator nominations). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:29, 12 June 2026 (UTC)

== June 2026 Wikimedia Café meetups regarding the English Wikipedia Editor Reflections project ==

&lt;div class=&quot;border-box&quot; style=&quot;background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)&quot;&gt;
&lt;div class=&quot;box&quot; style=&quot;float:left; padding-top: 10px; padding-right: 10px; padding-left: 10px; padding-bottom: 10px;&quot;&gt;[[File:Wikimedia Café logo in plain SVG format.svg|60px|alt=The logo for the Wikimedia Café]]&lt;/div&gt;
Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities during the last weekend of June. Both sessions will focus on the [https://en.wikipedia.org/wiki/Wikipedia:Editor_reflections English Wikipedia Editor Reflections project]. The featured guest in the Café will be [https://en.wikipedia.org/wiki/User:Clovermoss User:Clovermoss]. Participants may attend either or both sessions.

#'''27 June 2026 15:00 UTC''' ([https://zonestamp.toolforge.org/1782572400 timestamp converter]), at a time friendly to the Americas, Africa, and Europe
#'''28 June 2026 03:00 UTC''' ([https://zonestamp.toolforge.org/1782615600 timestamp converter]), at a time friendly to Asia and the Pacific

Please see the Café page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]!
&lt;br /&gt;
[[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]&lt;/div&gt;
&lt;span style=&quot;white-space:nowrap;&quot;&gt;[[User:Pine|&lt;span style=&quot;color:#01796f; text-shadow:#00BFFF 0 0 1.0em&quot;&gt;↠Pine&lt;/span&gt;]] [[User talk:Pine|&lt;span style=&quot;color:DeepSkyBlue&quot;&gt;(&lt;b style=&quot;color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em&quot;&gt;✉&lt;/b&gt;)&lt;/span&gt;]]&lt;/span&gt; 04:00, 15 June 2026 (UTC)

== Mobile friendly main page ==

Hello, I have recently been using wikiversity on mobile and unlike wikipedia some images and boxes stick out instead of all having a set width which means you can scroll a little side to side, which makes the site feel a bit unfinished. Its just a suggestion but I think it will wake the user experience much better {{unsigned|AUBSTRAWBS}}
:{{Ping|AUBSTRAWBS}} I don't use a smartphone. Can you give me more details or even take some screenshots? You can upload them at [[:c:Category:English Wikiversity screenshots]]. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 13:30, 18 June 2026 (UTC)
::Hi i uploaded an image of the problem. Since some of the images are larger than the screen and not adjusted to fit they stick out and makes the page larger which lets you scroll right and have a big white rectangle on the side [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 14:03, 18 June 2026 (UTC)
:::Thanks. I agree that this is an issue, but it's a pretty minor-to-moderate one to me and I don't think I will be able to dedicate time to fix it myself. Showing it to others here is useful in case someone else wants to tinker with the CSS to resolve it. Thanks for bringing it to the community's attention. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 15:42, 18 June 2026 (UTC)
::::I do know CSS as I like to maintain a blog online so I could try and fix it but I don't know if I have the access to do that, would i need to be a curator/ custodian. Alternatively i could edit a sandbox version of the main page and then send it to someone. [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 20:00, 18 June 2026 (UTC)
:::::Oh great. There are a lot of draft versions of the main page like [[Wikiversity:Main Page/Draft version 0.2]], so you can make [[Wikiversity:Main Page/Sandbox]] if you want and edit there. If you can tinker it to your liking, I can edit the main page. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 20:14, 18 June 2026 (UTC)
::::::thank you, i'll check it out [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 22:16, 18 June 2026 (UTC)

== Main page titles ==

Currently, the title says &quot;Wikiversity:Main Page&quot;, but in my opinion, it's too basic. I would like to propose changing it with the following options (you may only pick one):
# Option 1: Set both [[MediaWiki:Mainpage-title]] and [[MediaWiki:Mainpage-title-loggedin]] to blank, giving the main page a portal-like design (as with English Wikipedia, English Wikibooks, etc.)
# Option 2: Modify [[MediaWiki:Mainpage-title]] to &lt;code&gt;Welcome to Wikiversity&lt;/code&gt; (for unregistered users), and [[MediaWiki:Mainpage-title-loggedin]] to &lt;code&gt;&lt;nowiki&gt;Welcome to Wikiversity, $1!&lt;/nowiki&gt;&lt;/code&gt;; the latter would display to me as &lt;code&gt;Welcome to Wikiversity, Codename Noreste!&lt;/code&gt;
Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:34, 18 June 2026 (UTC)
: Pinging @[[User:Jtneill|Jtneill]] and @[[User:Koavf|Koavf]] for input above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:03, 24 June 2026 (UTC)
:I'm afraid that I don't have strong feelings on this. Changing to either or staying with the status quo are all fine to me. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 16:10, 24 June 2026 (UTC)
: I like the option of being consistent with Wikipedia and Wikibooks -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:04, 25 June 2026 (UTC)
: {{done}}. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:34, 30 June 2026 (UTC)
: Thankyou - looks good. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:29, 2 July 2026 (UTC)

== Wiki x AI preconference day @ Wikimania ==
There will be a preconference day at Wikimania about [[meta:Artificial_intelligence/2026_Wiki_AI | Wiki AI]]. It will be mostly offline, but there will be at least one hybrid session for demos of community-developed AI tools and workflows. 

* If you've built something cool, that is a chance to show it off, list it on the gallery of tools in progress, and get feedback. 

* If you could ask the people shaping AI on the wikis (WMF, tool builders, model trainers, GLAM and policy folks) a question, what would it be? 

Cheers, &lt;span style=&quot;padding:0 2px 0 2px;background-color:white;color:#bbb;&quot;&gt;&amp;ndash;[[User:Sj|SJ]][[User Talk:Sj|&lt;span style=&quot;color:#ff9900;&quot;&gt;+&lt;/span&gt;]]&lt;/span&gt; 23:12, 20 June 2026 (UTC) and Alaexis&lt;br&gt;{{comment|1=Copied from https://en.wikiversity.org/w/index.php?title=Talk%3AMotivation_and_emotion%2FAssessment%2FUsing_generative_AI&amp;diff=2816357&amp;oldid=2807052}}

== RFC about AI-generated content in Wikimedia Commons ==

You are invited to participate in a [[c:Commons:Requests for comment/Policy update for AI content|request for comment on Wikimedia Commons about a policy update for AI content]]. This may affect files that are uploaded to Wikimedia Commons for use on this project. Thank you. [[m:User:Codename Noreste|Codename Noreste]] ([[m:User talk:Codename Noreste|discuss]]) 17:12, 23 June 2026 (UTC)
&lt;!-- Message sent by User:Codename Noreste@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&amp;oldid=30513860 --&gt;

== Deployment of Legal and Safety Contacts Link in the Footer of Your Wiki ==

&lt;section begin=&quot;Message&quot;/&gt;
'''Legal &amp; Safety Contacts'''

Hello community, the Wikimedia Foundation has provided a [[wmf:Special:MyLanguage/Legal:Wikimedia Foundation Legal and Safety Contact Information|single legal and safety contact page]], to be linked in the footer of your wiki, to ensure access to accurate legal information. This is a regulatory requirement. We have already rolled out links to English, German, Italian, Spanish and other wikis and we will deploy to your wiki soon. [[m:Special:MyLanguage/Wikimedia_Foundation_Legal_and_Safety_Contacts_FAQ|Please read more on the project page]] and leave any comments in this thread or on the [[m:Special:MyLanguage/Talk:Wikimedia Foundation Legal and Safety Contacts FAQ|talk page]].

&lt;section end=&quot;Message&quot;/&gt;

-- [[User:Sannita (WMF)|User:Sannita (WMF)]] ([[User talk:Sannita (WMF)|talk]]) 13:31, 25 June 2026 (UTC)
&lt;!-- Message sent by User:Sannita (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=User:Sannita_(WMF)/Mass_sending_test&amp;oldid=30731267 --&gt;

== Preparing manuscript for submission to the ''WikiJournal of Humanities'' ==

I am working on preparing an article in Wikipedia for a dual goal of submitting it for a featured article candidacy in Wikipedia and submitting it to the ''[[WikiJournal of Humanities]]''.  I have an open request for pre-submission peer review at [[en:Wikipedia:Wikipedia:Peer review/Rei Ayanami/archive2|Wikipedia:Peer review/Rei Ayanami/archive2]], and I am asking for someone experienced with submitting journals to WikiJournals.  The article is not ready for submission, and I would like to know where I can get assistance from users who submitted articles to the journal, but did not necessarily review them.  Furthermore, I said there that submitting to the ''WikiJournal of Humanities'' depends on whether the article attains featured article status in Wikipedia, as I would like to use the featured article as a manuscript for a journal article. [[User:Z. Patterson|Z. Patterson]] ([[User talk:Z. Patterson|discuss]] • [[Special:Contributions/Z. Patterson|contribs]]) 12:04, 30 June 2026 (UTC)

== I could probably save many kilobytes by compressing my LLM chat history ==

I could probably save many kilobytes by compressing my LLM chat history. Would that be ok? I like how my new &quot;method&quot; looks: [[User:ThinkingScience/All General AI Prompt History Archive]] very compressed and neat. I like how to find new ways to make people using LLMs not become &quot;secondary citizens&quot;. [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 05:05, 7 July 2026 (UTC)

:{{replyto|ThinkingScience}} I do not see why not. People should be able to look in your page's history for LLM chats before your compression. [[User:Z. Patterson|Z. Patterson]] ([[User talk:Z. Patterson|discuss]] • [[Special:Contributions/Z. Patterson|contribs]]) 02:34, 14 July 2026 (UTC)
::That's good. Then an admin if they have been instructed to compress...then they can delete the edit history perhaps and save space, then it's up to the admin and I don't need to worry about taking up too many resources. That's good! [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 07:50, 14 July 2026 (UTC)

== July 2026 Wikimedia Café meetups regarding Wikimedia governance and options for reform ==

&lt;div class=&quot;border-box&quot; style=&quot;background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)&quot;&gt;
&lt;div class=&quot;box&quot; style=&quot;float:left; padding-top: 10px; padding-right: 10px; padding-left: 10px; padding-bottom: 10px;&quot;&gt;[[File:Wikimedia Café logo in plain SVG format.svg|60px|alt=The logo for the Wikimedia Café]]&lt;/div&gt;
Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities in July. Both sessions will focus on Wikimedia governance, including possible follow-ups to the [https://meta.wikimedia.org/wiki/Movement_Charter Movement Charter] and options for reform. Participants may attend either or both Café sessions. 

This month, to deconflict the Café meetups from Wikimania, the meetups will be held one day later than usual.

#'''26 July 2026 15:00 UTC''' ([https://zonestamp.toolforge.org/1785078000 timestamp converter]), at a time friendly to the Americas, Africa, and Europe
#'''27 July 2026 03:00 UTC''' ([https://zonestamp.toolforge.org/1785121200 timestamp converter]), at a time friendly to Asia and the Pacific

Please see the Café page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]!
&lt;br /&gt;
[[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]&lt;/div&gt;
&lt;span style=&quot;white-space:nowrap;&quot;&gt;[[User:Pine|&lt;span style=&quot;color:#01796f; text-shadow:#00BFFF 0 0 1.0em&quot;&gt;↠Pine&lt;/span&gt;]] [[User talk:Pine|&lt;span style=&quot;color:DeepSkyBlue&quot;&gt;(&lt;b style=&quot;color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em&quot;&gt;✉&lt;/b&gt;)&lt;/span&gt;]]&lt;/span&gt; 03:51, 13 July 2026 (UTC)

:@[[User:Pine|Pine]], Some wikimedians who may contribute less popular, but hopefully productive, comments,  are those who are currently blocked on Meta. I myself was blocked infinitely on Meta several years ago, but was allowed to contInue contributing on other wikimedia projects, such as this one. There are many long term contributors, some who have contrIbuted tens of thousands of edits  to wikimedia projects who are are now blocked on specific projects. 
:Those who happen to be blocked on Meta have,  to all intents and purposes, been removed from the wikimedia movement.
:Ottawahitech [[Special:Contributions/&amp;#126;2026-40514-81|&amp;#126;2026-40514-81]] ([[User talk:&amp;#126;2026-40514-81|talk]]) 21:41, 18 July 2026 (UTC)
:*Hi Ottawahitech, thanks for the comment. Reviewing local bans on Meta is outside of my current scope, but if you have comments that you'd like to share here in the Colloquium relating to Wikiversity, I can consider the comments in the scope of the Café discussions. &lt;span style=&quot;white-space:nowrap;&quot;&gt;[[User:Pine|&lt;span style=&quot;color:#01796f; text-shadow:#00BFFF 0 0 1.0em&quot;&gt;↠Pine&lt;/span&gt;]] [[User talk:Pine|&lt;span style=&quot;color:DeepSkyBlue&quot;&gt;(&lt;b style=&quot;color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em&quot;&gt;✉&lt;/b&gt;)&lt;/span&gt;]]&lt;/span&gt; 20:05, 25 July 2026 (UTC)

== Request for comment (the future of Abstract Wikipedia) ==

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot; class=&quot;mw-content-ltr&quot;&gt;

You are invited to voice your opinions in a [[:m:Requests for comment/The future of Abstract Wikipedia|request for comment about the future of Abstract Wikipedia]]. {{Int:Feedback-thanks-title}}
[[:m:User:Kowal2701|Kowal2701]] ([[:m:User talk:Kowal2701|talk]])  12:25, 24 July 2026 (UTC)
&lt;/bdi&gt;
&lt;!-- Message sent by User:DreamRimmer@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&amp;oldid=30513860 --&gt;</text>
      <sha1>h48xsb6qo2mlnzjgqngwzzj4tk96yka</sha1>
    </revision>
    <revision>
      <id>2819783</id>
      <parentid>2819781</parentid>
      <timestamp>2026-07-30T04:47:55Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>/* New user what do I do here */ archive to [[Wikiversity:Colloquium/archives/June 2026#New user what do I do here]] ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]])</comment>
      <origin>2819783</origin>
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&lt;!-- MESSAGES GO BELOW --&gt;

== Draft inactivity policy ==

I created [[Wikiversity:Inactivity policy]] as a start. Any experienced Wikiversity user may feel free to expand it. This is also one-to-two step(s) towards opting out of the [[m:Admin activity review|AAR process]].

However, I made a bold change to reduce the response timeframe from one month to two weeks. In addition, should we reduce the inactivity timeframe to one year? For the latter, most projects use that timeframe and I suggested this for consistency. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:57, 4 June 2026 (UTC)

:I support those suggestions. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:55, 4 June 2026 (UTC)
: Juandev has posted some comments on the [[Wikiversity talk:Inactivity policy|talk page]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:30, 12 June 2026 (UTC)
: Thanks for creating this draft. I've made some changes (including moving back to a one month response timeframe) and moved it from draft to proposed policy.
: Based on the discussion on the talk page, I think it is close to ready (or ready) to be formally proposed as a policy (by adding it to the sitenotice) to allow wider discussion and hopefully adoption. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 06:45, 21 July 2026 (UTC)

== Proposed user group and/or possible policy changes ==

{{tracked|T430416|fixed}}
I want to discuss about user group and possible policy changes.

# First, interface administrators. I don't think we should allow interface administrators to remove their permission from their own account, since we have multiple active bureaucrats and we can ask them to remove the permission when done, or for them to add a temporary grant. This is according to the [[Wikiversity:IA|current IA policy]]. I also left [[Wikiversity talk:Interface administrators#My thoughts about this user group|my thoughts on the relevant talk page]].
# Second, curators. Given that curators have some sensitive custodian rights (such as &lt;code&gt;delete&lt;/code&gt; [but not &lt;code&gt;undelete&lt;/code&gt; or similar rights that allow viewing deleted content, unless the curatorship process is RFA-like] and &lt;code&gt;protect&lt;/code&gt;), it would probably make more sense only for bureaucrats to grant and remove it, on par with them granting (but not removing) custodian permissions.
# Third, about probationary custodians. [[Wikiversity:Probationary custodians]] is currently marked as historical, and the process might still exist on [[Wikiversity:Custodianship]]. Therefore, to maintain consistency with [[Wikiversity:Curatorship#How does one become a curator?]], I propose that we repeal the probationary custodianship process and change it more or less to align with the curatorship process, effectively making probationary custodians permanent ones. However, custodian mentors would still be retained.

Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:55, 5 June 2026 (UTC)

:#Yes, I agree.
:#Thats a good point, but I dont know. At least I dont think its a good idea that both groups i.e. crats and custodiants can do that, it may create chaos.
:#Another good point. It seems to me that the current situation is somewhat unclear and should be clarified. I understand the original status of [[Wikiversity:Probationary custodians|Probationary custodians]] as a historicall and invalid, but at the same time I consider myself a probationary custodian, because on the Wikiversity:Custodianship page in the ''[[Wikiversity:Custodianship#How does one become a custodian?|How does one become a custodian?]]'' section it says, I quote, ''&quot;II ...then you will be approved as a probationary custodian for a period of at least four weeks&quot;''.
:::Mentors should definitely be kept, but for certain applicants the probation and mentorship should be abolished. For example, if someone was an active custodian for 5 years, then loses their rights or gives them up for a year and then wants to resume their custodial activities, there is no reason for them to undergo a training period. It burdens both the mentors and the community with double voting. The only exception could be a situation where policies or tools for custodians change significantly during that year, or the candidate wants to.
:[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 06:08, 9 June 2026 (UTC)

== Towards an Ethics policy ==

In connection with the [[Wikiversity:Community Review/Removal of Wikidebates|discussion of Wikidebates]], I said that it would be good to establish a policy on ethics, or rather a boundary between ethical and unethical content, so that we don't have to discuss individual cases. In addition, today we also have some global policies that prohibit, for example, attacks on members of the Wikimedia movement or undermining other projects.

However, at the very beginning, I would start by collecting your opinions. What content or what research should not be allowed on Wikiversity? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 05:52, 9 June 2026 (UTC)

:One ethical issue that I think should be non-controversial is related to good faith in the learning modules. So, learning materials should not be hoaxes or encourage behavior or methods that don't work or that misrepresent the facts or the likelihood of something occurring, etc. and authors should also not plagiarize or misrepresent authorship, etc. That was quite a run-on, but I hope that others can tease out what I mean here. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:39, 9 June 2026 (UTC)
::I look at it from a practical perspective. We can give that to the policy, but I see the problem in that we are not able to check it except plagiarism.
::Plagiarism can be partially detected during patrolling. I see a new text, I put part of it in Google and I check if it is copied from the web. It is a problem with copying from books or other offline sources, but sometimes it happens that someone finds out that something is copied from somewhere and it can be deleted.
::The biggest issue we have here is that we are missing Wikipedia's control mechanism: references. Only some types of resources on Wikiversity require references. In-line references are not often used in courses, exercises, lectures, etc. We are thus deprived of one of the excellent control mechanisms and the only option is for the increase in the number of members with various qualifications to check it for their colleagues. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:59, 9 June 2026 (UTC)
:::Having a policy and enforcing that policy are indeed two different things. If we are only concerned with issues that we can definitively enforce, then that will definitely change this conversation. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 08:06, 9 June 2026 (UTC)
::::ok [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:55, 13 June 2026 (UTC)
:AI generated content should not be allowed as it is inherently plagiarism. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:14, 9 June 2026 (UTC)
::And if the user mention it was generated by an AI? Note that there is something called as public domain, that is the author wave its rights. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:53, 9 June 2026 (UTC)
:::Plagiarism isn’t copyright violation. Crediting the AI is not crediting the authors the AI stole from without credit. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 10:18, 9 June 2026 (UTC)
::::I see, now I understand your point. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:56, 13 June 2026 (UTC)

== Deployment of Legal and Safety Contacts Link in the Footer of Your Wiki ==

Hello community,

The Wikimedia Foundation has provided [[foundation:Legal:Wikimedia Foundation Legal and Safety Contact Information|a single legal and safety contact page]], to be linked in the footer of your wiki, to ensure access to accurate legal information. This is a regulatory requirement.

We have already rolled out links to English, German, Italian, Spanish Wikipedias and other wikis and we will deploy to your wiki soon.

Please [[m:Wikimedia Foundation Legal and Safety Contacts FAQ|read more on the project page]] and leave any comments in this thread or on [[m:Talk:Wikimedia Foundation Legal and Safety Contacts FAQ|the talk page]]. –– [[User:STei (WMF)|STei (WMF)]] ([[User talk:STei (WMF)|discuss]] • [[Special:Contributions/STei (WMF)|contribs]]) 18:12, 9 June 2026 (UTC)

:Thanks for the notice. In case anyone is not clear, we cannot locally change the text at the footer, as it [[:mw:Manual:Footer|requires access to the server settings]]. If we locally needed to change it, we would have to file a ticket at [[:phab:]]. Since the above was sent by someone from the WMF, I think they are on it and it will be updated without any action from anyone here. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:24, 9 June 2026 (UTC)

== Image not displaying ==

Can anyone work out why this image isn't displaying?&lt;br&gt;
[[Educational Media Awareness Campaign/Physics/POTD 10]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:45, 11 June 2026 (UTC)

:Not sure, but it was an issue with the file itself and either way, it should be (and I have since done this) replaced with the SVG [[:File:Telescope-schematic.svg]]. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 13:59, 11 June 2026 (UTC)

== New nomination template(s) ==

I created {{tlx|Nomination}} when someone requests curator or custodian permissions, which often at least require mentorship. On the other hand, I might create {{tlx|Nomination 2}}, in which the latter does not have a section about mentorship (often used for bureaucrat or interface administrator nominations). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:29, 12 June 2026 (UTC)

== June 2026 Wikimedia Café meetups regarding the English Wikipedia Editor Reflections project ==

&lt;div class=&quot;border-box&quot; style=&quot;background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)&quot;&gt;
&lt;div class=&quot;box&quot; style=&quot;float:left; padding-top: 10px; padding-right: 10px; padding-left: 10px; padding-bottom: 10px;&quot;&gt;[[File:Wikimedia Café logo in plain SVG format.svg|60px|alt=The logo for the Wikimedia Café]]&lt;/div&gt;
Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities during the last weekend of June. Both sessions will focus on the [https://en.wikipedia.org/wiki/Wikipedia:Editor_reflections English Wikipedia Editor Reflections project]. The featured guest in the Café will be [https://en.wikipedia.org/wiki/User:Clovermoss User:Clovermoss]. Participants may attend either or both sessions.

#'''27 June 2026 15:00 UTC''' ([https://zonestamp.toolforge.org/1782572400 timestamp converter]), at a time friendly to the Americas, Africa, and Europe
#'''28 June 2026 03:00 UTC''' ([https://zonestamp.toolforge.org/1782615600 timestamp converter]), at a time friendly to Asia and the Pacific

Please see the Café page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]!
&lt;br /&gt;
[[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]&lt;/div&gt;
&lt;span style=&quot;white-space:nowrap;&quot;&gt;[[User:Pine|&lt;span style=&quot;color:#01796f; text-shadow:#00BFFF 0 0 1.0em&quot;&gt;↠Pine&lt;/span&gt;]] [[User talk:Pine|&lt;span style=&quot;color:DeepSkyBlue&quot;&gt;(&lt;b style=&quot;color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em&quot;&gt;✉&lt;/b&gt;)&lt;/span&gt;]]&lt;/span&gt; 04:00, 15 June 2026 (UTC)

== Mobile friendly main page ==

Hello, I have recently been using wikiversity on mobile and unlike wikipedia some images and boxes stick out instead of all having a set width which means you can scroll a little side to side, which makes the site feel a bit unfinished. Its just a suggestion but I think it will wake the user experience much better {{unsigned|AUBSTRAWBS}}
:{{Ping|AUBSTRAWBS}} I don't use a smartphone. Can you give me more details or even take some screenshots? You can upload them at [[:c:Category:English Wikiversity screenshots]]. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 13:30, 18 June 2026 (UTC)
::Hi i uploaded an image of the problem. Since some of the images are larger than the screen and not adjusted to fit they stick out and makes the page larger which lets you scroll right and have a big white rectangle on the side [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 14:03, 18 June 2026 (UTC)
:::Thanks. I agree that this is an issue, but it's a pretty minor-to-moderate one to me and I don't think I will be able to dedicate time to fix it myself. Showing it to others here is useful in case someone else wants to tinker with the CSS to resolve it. Thanks for bringing it to the community's attention. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 15:42, 18 June 2026 (UTC)
::::I do know CSS as I like to maintain a blog online so I could try and fix it but I don't know if I have the access to do that, would i need to be a curator/ custodian. Alternatively i could edit a sandbox version of the main page and then send it to someone. [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 20:00, 18 June 2026 (UTC)
:::::Oh great. There are a lot of draft versions of the main page like [[Wikiversity:Main Page/Draft version 0.2]], so you can make [[Wikiversity:Main Page/Sandbox]] if you want and edit there. If you can tinker it to your liking, I can edit the main page. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 20:14, 18 June 2026 (UTC)
::::::thank you, i'll check it out [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 22:16, 18 June 2026 (UTC)

== Main page titles ==

Currently, the title says &quot;Wikiversity:Main Page&quot;, but in my opinion, it's too basic. I would like to propose changing it with the following options (you may only pick one):
# Option 1: Set both [[MediaWiki:Mainpage-title]] and [[MediaWiki:Mainpage-title-loggedin]] to blank, giving the main page a portal-like design (as with English Wikipedia, English Wikibooks, etc.)
# Option 2: Modify [[MediaWiki:Mainpage-title]] to &lt;code&gt;Welcome to Wikiversity&lt;/code&gt; (for unregistered users), and [[MediaWiki:Mainpage-title-loggedin]] to &lt;code&gt;&lt;nowiki&gt;Welcome to Wikiversity, $1!&lt;/nowiki&gt;&lt;/code&gt;; the latter would display to me as &lt;code&gt;Welcome to Wikiversity, Codename Noreste!&lt;/code&gt;
Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:34, 18 June 2026 (UTC)
: Pinging @[[User:Jtneill|Jtneill]] and @[[User:Koavf|Koavf]] for input above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:03, 24 June 2026 (UTC)
:I'm afraid that I don't have strong feelings on this. Changing to either or staying with the status quo are all fine to me. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 16:10, 24 June 2026 (UTC)
: I like the option of being consistent with Wikipedia and Wikibooks -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:04, 25 June 2026 (UTC)
: {{done}}. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:34, 30 June 2026 (UTC)
: Thankyou - looks good. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:29, 2 July 2026 (UTC)

== Wiki x AI preconference day @ Wikimania ==
There will be a preconference day at Wikimania about [[meta:Artificial_intelligence/2026_Wiki_AI | Wiki AI]]. It will be mostly offline, but there will be at least one hybrid session for demos of community-developed AI tools and workflows. 

* If you've built something cool, that is a chance to show it off, list it on the gallery of tools in progress, and get feedback. 

* If you could ask the people shaping AI on the wikis (WMF, tool builders, model trainers, GLAM and policy folks) a question, what would it be? 

Cheers, &lt;span style=&quot;padding:0 2px 0 2px;background-color:white;color:#bbb;&quot;&gt;&amp;ndash;[[User:Sj|SJ]][[User Talk:Sj|&lt;span style=&quot;color:#ff9900;&quot;&gt;+&lt;/span&gt;]]&lt;/span&gt; 23:12, 20 June 2026 (UTC) and Alaexis&lt;br&gt;{{comment|1=Copied from https://en.wikiversity.org/w/index.php?title=Talk%3AMotivation_and_emotion%2FAssessment%2FUsing_generative_AI&amp;diff=2816357&amp;oldid=2807052}}

== RFC about AI-generated content in Wikimedia Commons ==

You are invited to participate in a [[c:Commons:Requests for comment/Policy update for AI content|request for comment on Wikimedia Commons about a policy update for AI content]]. This may affect files that are uploaded to Wikimedia Commons for use on this project. Thank you. [[m:User:Codename Noreste|Codename Noreste]] ([[m:User talk:Codename Noreste|discuss]]) 17:12, 23 June 2026 (UTC)
&lt;!-- Message sent by User:Codename Noreste@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&amp;oldid=30513860 --&gt;

== Deployment of Legal and Safety Contacts Link in the Footer of Your Wiki ==

&lt;section begin=&quot;Message&quot;/&gt;
'''Legal &amp; Safety Contacts'''

Hello community, the Wikimedia Foundation has provided a [[wmf:Special:MyLanguage/Legal:Wikimedia Foundation Legal and Safety Contact Information|single legal and safety contact page]], to be linked in the footer of your wiki, to ensure access to accurate legal information. This is a regulatory requirement. We have already rolled out links to English, German, Italian, Spanish and other wikis and we will deploy to your wiki soon. [[m:Special:MyLanguage/Wikimedia_Foundation_Legal_and_Safety_Contacts_FAQ|Please read more on the project page]] and leave any comments in this thread or on the [[m:Special:MyLanguage/Talk:Wikimedia Foundation Legal and Safety Contacts FAQ|talk page]].

&lt;section end=&quot;Message&quot;/&gt;

-- [[User:Sannita (WMF)|User:Sannita (WMF)]] ([[User talk:Sannita (WMF)|talk]]) 13:31, 25 June 2026 (UTC)
&lt;!-- Message sent by User:Sannita (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=User:Sannita_(WMF)/Mass_sending_test&amp;oldid=30731267 --&gt;

== Preparing manuscript for submission to the ''WikiJournal of Humanities'' ==

I am working on preparing an article in Wikipedia for a dual goal of submitting it for a featured article candidacy in Wikipedia and submitting it to the ''[[WikiJournal of Humanities]]''.  I have an open request for pre-submission peer review at [[en:Wikipedia:Wikipedia:Peer review/Rei Ayanami/archive2|Wikipedia:Peer review/Rei Ayanami/archive2]], and I am asking for someone experienced with submitting journals to WikiJournals.  The article is not ready for submission, and I would like to know where I can get assistance from users who submitted articles to the journal, but did not necessarily review them.  Furthermore, I said there that submitting to the ''WikiJournal of Humanities'' depends on whether the article attains featured article status in Wikipedia, as I would like to use the featured article as a manuscript for a journal article. [[User:Z. Patterson|Z. Patterson]] ([[User talk:Z. Patterson|discuss]] • [[Special:Contributions/Z. Patterson|contribs]]) 12:04, 30 June 2026 (UTC)

== I could probably save many kilobytes by compressing my LLM chat history ==

I could probably save many kilobytes by compressing my LLM chat history. Would that be ok? I like how my new &quot;method&quot; looks: [[User:ThinkingScience/All General AI Prompt History Archive]] very compressed and neat. I like how to find new ways to make people using LLMs not become &quot;secondary citizens&quot;. [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 05:05, 7 July 2026 (UTC)

:{{replyto|ThinkingScience}} I do not see why not. People should be able to look in your page's history for LLM chats before your compression. [[User:Z. Patterson|Z. Patterson]] ([[User talk:Z. Patterson|discuss]] • [[Special:Contributions/Z. Patterson|contribs]]) 02:34, 14 July 2026 (UTC)
::That's good. Then an admin if they have been instructed to compress...then they can delete the edit history perhaps and save space, then it's up to the admin and I don't need to worry about taking up too many resources. That's good! [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 07:50, 14 July 2026 (UTC)

== July 2026 Wikimedia Café meetups regarding Wikimedia governance and options for reform ==

&lt;div class=&quot;border-box&quot; style=&quot;background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)&quot;&gt;
&lt;div class=&quot;box&quot; style=&quot;float:left; padding-top: 10px; padding-right: 10px; padding-left: 10px; padding-bottom: 10px;&quot;&gt;[[File:Wikimedia Café logo in plain SVG format.svg|60px|alt=The logo for the Wikimedia Café]]&lt;/div&gt;
Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities in July. Both sessions will focus on Wikimedia governance, including possible follow-ups to the [https://meta.wikimedia.org/wiki/Movement_Charter Movement Charter] and options for reform. Participants may attend either or both Café sessions. 

This month, to deconflict the Café meetups from Wikimania, the meetups will be held one day later than usual.

#'''26 July 2026 15:00 UTC''' ([https://zonestamp.toolforge.org/1785078000 timestamp converter]), at a time friendly to the Americas, Africa, and Europe
#'''27 July 2026 03:00 UTC''' ([https://zonestamp.toolforge.org/1785121200 timestamp converter]), at a time friendly to Asia and the Pacific

Please see the Café page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]!
&lt;br /&gt;
[[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]&lt;/div&gt;
&lt;span style=&quot;white-space:nowrap;&quot;&gt;[[User:Pine|&lt;span style=&quot;color:#01796f; text-shadow:#00BFFF 0 0 1.0em&quot;&gt;↠Pine&lt;/span&gt;]] [[User talk:Pine|&lt;span style=&quot;color:DeepSkyBlue&quot;&gt;(&lt;b style=&quot;color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em&quot;&gt;✉&lt;/b&gt;)&lt;/span&gt;]]&lt;/span&gt; 03:51, 13 July 2026 (UTC)

:@[[User:Pine|Pine]], Some wikimedians who may contribute less popular, but hopefully productive, comments,  are those who are currently blocked on Meta. I myself was blocked infinitely on Meta several years ago, but was allowed to contInue contributing on other wikimedia projects, such as this one. There are many long term contributors, some who have contrIbuted tens of thousands of edits  to wikimedia projects who are are now blocked on specific projects. 
:Those who happen to be blocked on Meta have,  to all intents and purposes, been removed from the wikimedia movement.
:Ottawahitech [[Special:Contributions/&amp;#126;2026-40514-81|&amp;#126;2026-40514-81]] ([[User talk:&amp;#126;2026-40514-81|talk]]) 21:41, 18 July 2026 (UTC)
:*Hi Ottawahitech, thanks for the comment. Reviewing local bans on Meta is outside of my current scope, but if you have comments that you'd like to share here in the Colloquium relating to Wikiversity, I can consider the comments in the scope of the Café discussions. &lt;span style=&quot;white-space:nowrap;&quot;&gt;[[User:Pine|&lt;span style=&quot;color:#01796f; text-shadow:#00BFFF 0 0 1.0em&quot;&gt;↠Pine&lt;/span&gt;]] [[User talk:Pine|&lt;span style=&quot;color:DeepSkyBlue&quot;&gt;(&lt;b style=&quot;color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em&quot;&gt;✉&lt;/b&gt;)&lt;/span&gt;]]&lt;/span&gt; 20:05, 25 July 2026 (UTC)

== Request for comment (the future of Abstract Wikipedia) ==

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot; class=&quot;mw-content-ltr&quot;&gt;

You are invited to voice your opinions in a [[:m:Requests for comment/The future of Abstract Wikipedia|request for comment about the future of Abstract Wikipedia]]. {{Int:Feedback-thanks-title}}
[[:m:User:Kowal2701|Kowal2701]] ([[:m:User talk:Kowal2701|talk]])  12:25, 24 July 2026 (UTC)
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      <sha1>bwdsz82e9xvwi49odrvntcia2mefea9</sha1>
    </revision>
    <revision>
      <id>2819785</id>
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      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>/* Deployment of Legal and Safety Contacts Link in the Footer of Your Wiki */ archive to [[Wikiversity:Colloquium/archives/June 2026#Deployment of Legal and Safety Contacts Link in the Footer of Your Wiki]] ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]])</comment>
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      <text bytes="25495" sha1="ppoqwfbxtv99y9tgf7wdrxhjlef6suw" xml:space="preserve">{{Wikiversity:Colloquium/Header}}
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== Draft inactivity policy ==

I created [[Wikiversity:Inactivity policy]] as a start. Any experienced Wikiversity user may feel free to expand it. This is also one-to-two step(s) towards opting out of the [[m:Admin activity review|AAR process]].

However, I made a bold change to reduce the response timeframe from one month to two weeks. In addition, should we reduce the inactivity timeframe to one year? For the latter, most projects use that timeframe and I suggested this for consistency. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:57, 4 June 2026 (UTC)

:I support those suggestions. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:55, 4 June 2026 (UTC)
: Juandev has posted some comments on the [[Wikiversity talk:Inactivity policy|talk page]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:30, 12 June 2026 (UTC)
: Thanks for creating this draft. I've made some changes (including moving back to a one month response timeframe) and moved it from draft to proposed policy.
: Based on the discussion on the talk page, I think it is close to ready (or ready) to be formally proposed as a policy (by adding it to the sitenotice) to allow wider discussion and hopefully adoption. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 06:45, 21 July 2026 (UTC)

== Proposed user group and/or possible policy changes ==

{{tracked|T430416|fixed}}
I want to discuss about user group and possible policy changes.

# First, interface administrators. I don't think we should allow interface administrators to remove their permission from their own account, since we have multiple active bureaucrats and we can ask them to remove the permission when done, or for them to add a temporary grant. This is according to the [[Wikiversity:IA|current IA policy]]. I also left [[Wikiversity talk:Interface administrators#My thoughts about this user group|my thoughts on the relevant talk page]].
# Second, curators. Given that curators have some sensitive custodian rights (such as &lt;code&gt;delete&lt;/code&gt; [but not &lt;code&gt;undelete&lt;/code&gt; or similar rights that allow viewing deleted content, unless the curatorship process is RFA-like] and &lt;code&gt;protect&lt;/code&gt;), it would probably make more sense only for bureaucrats to grant and remove it, on par with them granting (but not removing) custodian permissions.
# Third, about probationary custodians. [[Wikiversity:Probationary custodians]] is currently marked as historical, and the process might still exist on [[Wikiversity:Custodianship]]. Therefore, to maintain consistency with [[Wikiversity:Curatorship#How does one become a curator?]], I propose that we repeal the probationary custodianship process and change it more or less to align with the curatorship process, effectively making probationary custodians permanent ones. However, custodian mentors would still be retained.

Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:55, 5 June 2026 (UTC)

:#Yes, I agree.
:#Thats a good point, but I dont know. At least I dont think its a good idea that both groups i.e. crats and custodiants can do that, it may create chaos.
:#Another good point. It seems to me that the current situation is somewhat unclear and should be clarified. I understand the original status of [[Wikiversity:Probationary custodians|Probationary custodians]] as a historicall and invalid, but at the same time I consider myself a probationary custodian, because on the Wikiversity:Custodianship page in the ''[[Wikiversity:Custodianship#How does one become a custodian?|How does one become a custodian?]]'' section it says, I quote, ''&quot;II ...then you will be approved as a probationary custodian for a period of at least four weeks&quot;''.
:::Mentors should definitely be kept, but for certain applicants the probation and mentorship should be abolished. For example, if someone was an active custodian for 5 years, then loses their rights or gives them up for a year and then wants to resume their custodial activities, there is no reason for them to undergo a training period. It burdens both the mentors and the community with double voting. The only exception could be a situation where policies or tools for custodians change significantly during that year, or the candidate wants to.
:[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 06:08, 9 June 2026 (UTC)

== Towards an Ethics policy ==

In connection with the [[Wikiversity:Community Review/Removal of Wikidebates|discussion of Wikidebates]], I said that it would be good to establish a policy on ethics, or rather a boundary between ethical and unethical content, so that we don't have to discuss individual cases. In addition, today we also have some global policies that prohibit, for example, attacks on members of the Wikimedia movement or undermining other projects.

However, at the very beginning, I would start by collecting your opinions. What content or what research should not be allowed on Wikiversity? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 05:52, 9 June 2026 (UTC)

:One ethical issue that I think should be non-controversial is related to good faith in the learning modules. So, learning materials should not be hoaxes or encourage behavior or methods that don't work or that misrepresent the facts or the likelihood of something occurring, etc. and authors should also not plagiarize or misrepresent authorship, etc. That was quite a run-on, but I hope that others can tease out what I mean here. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:39, 9 June 2026 (UTC)
::I look at it from a practical perspective. We can give that to the policy, but I see the problem in that we are not able to check it except plagiarism.
::Plagiarism can be partially detected during patrolling. I see a new text, I put part of it in Google and I check if it is copied from the web. It is a problem with copying from books or other offline sources, but sometimes it happens that someone finds out that something is copied from somewhere and it can be deleted.
::The biggest issue we have here is that we are missing Wikipedia's control mechanism: references. Only some types of resources on Wikiversity require references. In-line references are not often used in courses, exercises, lectures, etc. We are thus deprived of one of the excellent control mechanisms and the only option is for the increase in the number of members with various qualifications to check it for their colleagues. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:59, 9 June 2026 (UTC)
:::Having a policy and enforcing that policy are indeed two different things. If we are only concerned with issues that we can definitively enforce, then that will definitely change this conversation. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 08:06, 9 June 2026 (UTC)
::::ok [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:55, 13 June 2026 (UTC)
:AI generated content should not be allowed as it is inherently plagiarism. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:14, 9 June 2026 (UTC)
::And if the user mention it was generated by an AI? Note that there is something called as public domain, that is the author wave its rights. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:53, 9 June 2026 (UTC)
:::Plagiarism isn’t copyright violation. Crediting the AI is not crediting the authors the AI stole from without credit. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 10:18, 9 June 2026 (UTC)
::::I see, now I understand your point. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:56, 13 June 2026 (UTC)

== Image not displaying ==

Can anyone work out why this image isn't displaying?&lt;br&gt;
[[Educational Media Awareness Campaign/Physics/POTD 10]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:45, 11 June 2026 (UTC)

:Not sure, but it was an issue with the file itself and either way, it should be (and I have since done this) replaced with the SVG [[:File:Telescope-schematic.svg]]. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 13:59, 11 June 2026 (UTC)

== New nomination template(s) ==

I created {{tlx|Nomination}} when someone requests curator or custodian permissions, which often at least require mentorship. On the other hand, I might create {{tlx|Nomination 2}}, in which the latter does not have a section about mentorship (often used for bureaucrat or interface administrator nominations). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:29, 12 June 2026 (UTC)

== June 2026 Wikimedia Café meetups regarding the English Wikipedia Editor Reflections project ==

&lt;div class=&quot;border-box&quot; style=&quot;background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)&quot;&gt;
&lt;div class=&quot;box&quot; style=&quot;float:left; padding-top: 10px; padding-right: 10px; padding-left: 10px; padding-bottom: 10px;&quot;&gt;[[File:Wikimedia Café logo in plain SVG format.svg|60px|alt=The logo for the Wikimedia Café]]&lt;/div&gt;
Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities during the last weekend of June. Both sessions will focus on the [https://en.wikipedia.org/wiki/Wikipedia:Editor_reflections English Wikipedia Editor Reflections project]. The featured guest in the Café will be [https://en.wikipedia.org/wiki/User:Clovermoss User:Clovermoss]. Participants may attend either or both sessions.

#'''27 June 2026 15:00 UTC''' ([https://zonestamp.toolforge.org/1782572400 timestamp converter]), at a time friendly to the Americas, Africa, and Europe
#'''28 June 2026 03:00 UTC''' ([https://zonestamp.toolforge.org/1782615600 timestamp converter]), at a time friendly to Asia and the Pacific

Please see the Café page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]!
&lt;br /&gt;
[[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]&lt;/div&gt;
&lt;span style=&quot;white-space:nowrap;&quot;&gt;[[User:Pine|&lt;span style=&quot;color:#01796f; text-shadow:#00BFFF 0 0 1.0em&quot;&gt;↠Pine&lt;/span&gt;]] [[User talk:Pine|&lt;span style=&quot;color:DeepSkyBlue&quot;&gt;(&lt;b style=&quot;color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em&quot;&gt;✉&lt;/b&gt;)&lt;/span&gt;]]&lt;/span&gt; 04:00, 15 June 2026 (UTC)

== Mobile friendly main page ==

Hello, I have recently been using wikiversity on mobile and unlike wikipedia some images and boxes stick out instead of all having a set width which means you can scroll a little side to side, which makes the site feel a bit unfinished. Its just a suggestion but I think it will wake the user experience much better {{unsigned|AUBSTRAWBS}}
:{{Ping|AUBSTRAWBS}} I don't use a smartphone. Can you give me more details or even take some screenshots? You can upload them at [[:c:Category:English Wikiversity screenshots]]. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 13:30, 18 June 2026 (UTC)
::Hi i uploaded an image of the problem. Since some of the images are larger than the screen and not adjusted to fit they stick out and makes the page larger which lets you scroll right and have a big white rectangle on the side [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 14:03, 18 June 2026 (UTC)
:::Thanks. I agree that this is an issue, but it's a pretty minor-to-moderate one to me and I don't think I will be able to dedicate time to fix it myself. Showing it to others here is useful in case someone else wants to tinker with the CSS to resolve it. Thanks for bringing it to the community's attention. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 15:42, 18 June 2026 (UTC)
::::I do know CSS as I like to maintain a blog online so I could try and fix it but I don't know if I have the access to do that, would i need to be a curator/ custodian. Alternatively i could edit a sandbox version of the main page and then send it to someone. [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 20:00, 18 June 2026 (UTC)
:::::Oh great. There are a lot of draft versions of the main page like [[Wikiversity:Main Page/Draft version 0.2]], so you can make [[Wikiversity:Main Page/Sandbox]] if you want and edit there. If you can tinker it to your liking, I can edit the main page. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 20:14, 18 June 2026 (UTC)
::::::thank you, i'll check it out [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 22:16, 18 June 2026 (UTC)

== Main page titles ==

Currently, the title says &quot;Wikiversity:Main Page&quot;, but in my opinion, it's too basic. I would like to propose changing it with the following options (you may only pick one):
# Option 1: Set both [[MediaWiki:Mainpage-title]] and [[MediaWiki:Mainpage-title-loggedin]] to blank, giving the main page a portal-like design (as with English Wikipedia, English Wikibooks, etc.)
# Option 2: Modify [[MediaWiki:Mainpage-title]] to &lt;code&gt;Welcome to Wikiversity&lt;/code&gt; (for unregistered users), and [[MediaWiki:Mainpage-title-loggedin]] to &lt;code&gt;&lt;nowiki&gt;Welcome to Wikiversity, $1!&lt;/nowiki&gt;&lt;/code&gt;; the latter would display to me as &lt;code&gt;Welcome to Wikiversity, Codename Noreste!&lt;/code&gt;
Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:34, 18 June 2026 (UTC)
: Pinging @[[User:Jtneill|Jtneill]] and @[[User:Koavf|Koavf]] for input above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:03, 24 June 2026 (UTC)
:I'm afraid that I don't have strong feelings on this. Changing to either or staying with the status quo are all fine to me. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 16:10, 24 June 2026 (UTC)
: I like the option of being consistent with Wikipedia and Wikibooks -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:04, 25 June 2026 (UTC)
: {{done}}. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:34, 30 June 2026 (UTC)
: Thankyou - looks good. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:29, 2 July 2026 (UTC)

== Wiki x AI preconference day @ Wikimania ==
There will be a preconference day at Wikimania about [[meta:Artificial_intelligence/2026_Wiki_AI | Wiki AI]]. It will be mostly offline, but there will be at least one hybrid session for demos of community-developed AI tools and workflows. 

* If you've built something cool, that is a chance to show it off, list it on the gallery of tools in progress, and get feedback. 

* If you could ask the people shaping AI on the wikis (WMF, tool builders, model trainers, GLAM and policy folks) a question, what would it be? 

Cheers, &lt;span style=&quot;padding:0 2px 0 2px;background-color:white;color:#bbb;&quot;&gt;&amp;ndash;[[User:Sj|SJ]][[User Talk:Sj|&lt;span style=&quot;color:#ff9900;&quot;&gt;+&lt;/span&gt;]]&lt;/span&gt; 23:12, 20 June 2026 (UTC) and Alaexis&lt;br&gt;{{comment|1=Copied from https://en.wikiversity.org/w/index.php?title=Talk%3AMotivation_and_emotion%2FAssessment%2FUsing_generative_AI&amp;diff=2816357&amp;oldid=2807052}}

== RFC about AI-generated content in Wikimedia Commons ==

You are invited to participate in a [[c:Commons:Requests for comment/Policy update for AI content|request for comment on Wikimedia Commons about a policy update for AI content]]. This may affect files that are uploaded to Wikimedia Commons for use on this project. Thank you. [[m:User:Codename Noreste|Codename Noreste]] ([[m:User talk:Codename Noreste|discuss]]) 17:12, 23 June 2026 (UTC)
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== Deployment of Legal and Safety Contacts Link in the Footer of Your Wiki ==

&lt;section begin=&quot;Message&quot;/&gt;
'''Legal &amp; Safety Contacts'''

Hello community, the Wikimedia Foundation has provided a [[wmf:Special:MyLanguage/Legal:Wikimedia Foundation Legal and Safety Contact Information|single legal and safety contact page]], to be linked in the footer of your wiki, to ensure access to accurate legal information. This is a regulatory requirement. We have already rolled out links to English, German, Italian, Spanish and other wikis and we will deploy to your wiki soon. [[m:Special:MyLanguage/Wikimedia_Foundation_Legal_and_Safety_Contacts_FAQ|Please read more on the project page]] and leave any comments in this thread or on the [[m:Special:MyLanguage/Talk:Wikimedia Foundation Legal and Safety Contacts FAQ|talk page]].

&lt;section end=&quot;Message&quot;/&gt;

-- [[User:Sannita (WMF)|User:Sannita (WMF)]] ([[User talk:Sannita (WMF)|talk]]) 13:31, 25 June 2026 (UTC)
&lt;!-- Message sent by User:Sannita (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=User:Sannita_(WMF)/Mass_sending_test&amp;oldid=30731267 --&gt;

== Preparing manuscript for submission to the ''WikiJournal of Humanities'' ==

I am working on preparing an article in Wikipedia for a dual goal of submitting it for a featured article candidacy in Wikipedia and submitting it to the ''[[WikiJournal of Humanities]]''.  I have an open request for pre-submission peer review at [[en:Wikipedia:Wikipedia:Peer review/Rei Ayanami/archive2|Wikipedia:Peer review/Rei Ayanami/archive2]], and I am asking for someone experienced with submitting journals to WikiJournals.  The article is not ready for submission, and I would like to know where I can get assistance from users who submitted articles to the journal, but did not necessarily review them.  Furthermore, I said there that submitting to the ''WikiJournal of Humanities'' depends on whether the article attains featured article status in Wikipedia, as I would like to use the featured article as a manuscript for a journal article. [[User:Z. Patterson|Z. Patterson]] ([[User talk:Z. Patterson|discuss]] • [[Special:Contributions/Z. Patterson|contribs]]) 12:04, 30 June 2026 (UTC)

== I could probably save many kilobytes by compressing my LLM chat history ==

I could probably save many kilobytes by compressing my LLM chat history. Would that be ok? I like how my new &quot;method&quot; looks: [[User:ThinkingScience/All General AI Prompt History Archive]] very compressed and neat. I like how to find new ways to make people using LLMs not become &quot;secondary citizens&quot;. [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 05:05, 7 July 2026 (UTC)

:{{replyto|ThinkingScience}} I do not see why not. People should be able to look in your page's history for LLM chats before your compression. [[User:Z. Patterson|Z. Patterson]] ([[User talk:Z. Patterson|discuss]] • [[Special:Contributions/Z. Patterson|contribs]]) 02:34, 14 July 2026 (UTC)
::That's good. Then an admin if they have been instructed to compress...then they can delete the edit history perhaps and save space, then it's up to the admin and I don't need to worry about taking up too many resources. That's good! [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 07:50, 14 July 2026 (UTC)

== July 2026 Wikimedia Café meetups regarding Wikimedia governance and options for reform ==

&lt;div class=&quot;border-box&quot; style=&quot;background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)&quot;&gt;
&lt;div class=&quot;box&quot; style=&quot;float:left; padding-top: 10px; padding-right: 10px; padding-left: 10px; padding-bottom: 10px;&quot;&gt;[[File:Wikimedia Café logo in plain SVG format.svg|60px|alt=The logo for the Wikimedia Café]]&lt;/div&gt;
Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities in July. Both sessions will focus on Wikimedia governance, including possible follow-ups to the [https://meta.wikimedia.org/wiki/Movement_Charter Movement Charter] and options for reform. Participants may attend either or both Café sessions. 

This month, to deconflict the Café meetups from Wikimania, the meetups will be held one day later than usual.

#'''26 July 2026 15:00 UTC''' ([https://zonestamp.toolforge.org/1785078000 timestamp converter]), at a time friendly to the Americas, Africa, and Europe
#'''27 July 2026 03:00 UTC''' ([https://zonestamp.toolforge.org/1785121200 timestamp converter]), at a time friendly to Asia and the Pacific

Please see the Café page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]!
&lt;br /&gt;
[[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]&lt;/div&gt;
&lt;span style=&quot;white-space:nowrap;&quot;&gt;[[User:Pine|&lt;span style=&quot;color:#01796f; text-shadow:#00BFFF 0 0 1.0em&quot;&gt;↠Pine&lt;/span&gt;]] [[User talk:Pine|&lt;span style=&quot;color:DeepSkyBlue&quot;&gt;(&lt;b style=&quot;color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em&quot;&gt;✉&lt;/b&gt;)&lt;/span&gt;]]&lt;/span&gt; 03:51, 13 July 2026 (UTC)

:@[[User:Pine|Pine]], Some wikimedians who may contribute less popular, but hopefully productive, comments,  are those who are currently blocked on Meta. I myself was blocked infinitely on Meta several years ago, but was allowed to contInue contributing on other wikimedia projects, such as this one. There are many long term contributors, some who have contrIbuted tens of thousands of edits  to wikimedia projects who are are now blocked on specific projects. 
:Those who happen to be blocked on Meta have,  to all intents and purposes, been removed from the wikimedia movement.
:Ottawahitech [[Special:Contributions/&amp;#126;2026-40514-81|&amp;#126;2026-40514-81]] ([[User talk:&amp;#126;2026-40514-81|talk]]) 21:41, 18 July 2026 (UTC)
:*Hi Ottawahitech, thanks for the comment. Reviewing local bans on Meta is outside of my current scope, but if you have comments that you'd like to share here in the Colloquium relating to Wikiversity, I can consider the comments in the scope of the Café discussions. &lt;span style=&quot;white-space:nowrap;&quot;&gt;[[User:Pine|&lt;span style=&quot;color:#01796f; text-shadow:#00BFFF 0 0 1.0em&quot;&gt;↠Pine&lt;/span&gt;]] [[User talk:Pine|&lt;span style=&quot;color:DeepSkyBlue&quot;&gt;(&lt;b style=&quot;color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em&quot;&gt;✉&lt;/b&gt;)&lt;/span&gt;]]&lt;/span&gt; 20:05, 25 July 2026 (UTC)

== Request for comment (the future of Abstract Wikipedia) ==

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You are invited to voice your opinions in a [[:m:Requests for comment/The future of Abstract Wikipedia|request for comment about the future of Abstract Wikipedia]]. {{Int:Feedback-thanks-title}}
[[:m:User:Kowal2701|Kowal2701]] ([[:m:User talk:Kowal2701|talk]])  12:25, 24 July 2026 (UTC)
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== Draft inactivity policy ==

I created [[Wikiversity:Inactivity policy]] as a start. Any experienced Wikiversity user may feel free to expand it. This is also one-to-two step(s) towards opting out of the [[m:Admin activity review|AAR process]].

However, I made a bold change to reduce the response timeframe from one month to two weeks. In addition, should we reduce the inactivity timeframe to one year? For the latter, most projects use that timeframe and I suggested this for consistency. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:57, 4 June 2026 (UTC)

:I support those suggestions. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:55, 4 June 2026 (UTC)
: Juandev has posted some comments on the [[Wikiversity talk:Inactivity policy|talk page]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:30, 12 June 2026 (UTC)
: Thanks for creating this draft. I've made some changes (including moving back to a one month response timeframe) and moved it from draft to proposed policy.
: Based on the discussion on the talk page, I think it is close to ready (or ready) to be formally proposed as a policy (by adding it to the sitenotice) to allow wider discussion and hopefully adoption. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 06:45, 21 July 2026 (UTC)

== Proposed user group and/or possible policy changes ==

{{tracked|T430416|fixed}}
I want to discuss about user group and possible policy changes.

# First, interface administrators. I don't think we should allow interface administrators to remove their permission from their own account, since we have multiple active bureaucrats and we can ask them to remove the permission when done, or for them to add a temporary grant. This is according to the [[Wikiversity:IA|current IA policy]]. I also left [[Wikiversity talk:Interface administrators#My thoughts about this user group|my thoughts on the relevant talk page]].
# Second, curators. Given that curators have some sensitive custodian rights (such as &lt;code&gt;delete&lt;/code&gt; [but not &lt;code&gt;undelete&lt;/code&gt; or similar rights that allow viewing deleted content, unless the curatorship process is RFA-like] and &lt;code&gt;protect&lt;/code&gt;), it would probably make more sense only for bureaucrats to grant and remove it, on par with them granting (but not removing) custodian permissions.
# Third, about probationary custodians. [[Wikiversity:Probationary custodians]] is currently marked as historical, and the process might still exist on [[Wikiversity:Custodianship]]. Therefore, to maintain consistency with [[Wikiversity:Curatorship#How does one become a curator?]], I propose that we repeal the probationary custodianship process and change it more or less to align with the curatorship process, effectively making probationary custodians permanent ones. However, custodian mentors would still be retained.

Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:55, 5 June 2026 (UTC)

:#Yes, I agree.
:#Thats a good point, but I dont know. At least I dont think its a good idea that both groups i.e. crats and custodiants can do that, it may create chaos.
:#Another good point. It seems to me that the current situation is somewhat unclear and should be clarified. I understand the original status of [[Wikiversity:Probationary custodians|Probationary custodians]] as a historicall and invalid, but at the same time I consider myself a probationary custodian, because on the Wikiversity:Custodianship page in the ''[[Wikiversity:Custodianship#How does one become a custodian?|How does one become a custodian?]]'' section it says, I quote, ''&quot;II ...then you will be approved as a probationary custodian for a period of at least four weeks&quot;''.
:::Mentors should definitely be kept, but for certain applicants the probation and mentorship should be abolished. For example, if someone was an active custodian for 5 years, then loses their rights or gives them up for a year and then wants to resume their custodial activities, there is no reason for them to undergo a training period. It burdens both the mentors and the community with double voting. The only exception could be a situation where policies or tools for custodians change significantly during that year, or the candidate wants to.
:[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 06:08, 9 June 2026 (UTC)

== Towards an Ethics policy ==

In connection with the [[Wikiversity:Community Review/Removal of Wikidebates|discussion of Wikidebates]], I said that it would be good to establish a policy on ethics, or rather a boundary between ethical and unethical content, so that we don't have to discuss individual cases. In addition, today we also have some global policies that prohibit, for example, attacks on members of the Wikimedia movement or undermining other projects.

However, at the very beginning, I would start by collecting your opinions. What content or what research should not be allowed on Wikiversity? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 05:52, 9 June 2026 (UTC)

:One ethical issue that I think should be non-controversial is related to good faith in the learning modules. So, learning materials should not be hoaxes or encourage behavior or methods that don't work or that misrepresent the facts or the likelihood of something occurring, etc. and authors should also not plagiarize or misrepresent authorship, etc. That was quite a run-on, but I hope that others can tease out what I mean here. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:39, 9 June 2026 (UTC)
::I look at it from a practical perspective. We can give that to the policy, but I see the problem in that we are not able to check it except plagiarism.
::Plagiarism can be partially detected during patrolling. I see a new text, I put part of it in Google and I check if it is copied from the web. It is a problem with copying from books or other offline sources, but sometimes it happens that someone finds out that something is copied from somewhere and it can be deleted.
::The biggest issue we have here is that we are missing Wikipedia's control mechanism: references. Only some types of resources on Wikiversity require references. In-line references are not often used in courses, exercises, lectures, etc. We are thus deprived of one of the excellent control mechanisms and the only option is for the increase in the number of members with various qualifications to check it for their colleagues. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:59, 9 June 2026 (UTC)
:::Having a policy and enforcing that policy are indeed two different things. If we are only concerned with issues that we can definitively enforce, then that will definitely change this conversation. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 08:06, 9 June 2026 (UTC)
::::ok [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:55, 13 June 2026 (UTC)
:AI generated content should not be allowed as it is inherently plagiarism. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:14, 9 June 2026 (UTC)
::And if the user mention it was generated by an AI? Note that there is something called as public domain, that is the author wave its rights. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:53, 9 June 2026 (UTC)
:::Plagiarism isn’t copyright violation. Crediting the AI is not crediting the authors the AI stole from without credit. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 10:18, 9 June 2026 (UTC)
::::I see, now I understand your point. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:56, 13 June 2026 (UTC)

== New nomination template(s) ==

I created {{tlx|Nomination}} when someone requests curator or custodian permissions, which often at least require mentorship. On the other hand, I might create {{tlx|Nomination 2}}, in which the latter does not have a section about mentorship (often used for bureaucrat or interface administrator nominations). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:29, 12 June 2026 (UTC)

== June 2026 Wikimedia Café meetups regarding the English Wikipedia Editor Reflections project ==

&lt;div class=&quot;border-box&quot; style=&quot;background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)&quot;&gt;
&lt;div class=&quot;box&quot; style=&quot;float:left; padding-top: 10px; padding-right: 10px; padding-left: 10px; padding-bottom: 10px;&quot;&gt;[[File:Wikimedia Café logo in plain SVG format.svg|60px|alt=The logo for the Wikimedia Café]]&lt;/div&gt;
Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities during the last weekend of June. Both sessions will focus on the [https://en.wikipedia.org/wiki/Wikipedia:Editor_reflections English Wikipedia Editor Reflections project]. The featured guest in the Café will be [https://en.wikipedia.org/wiki/User:Clovermoss User:Clovermoss]. Participants may attend either or both sessions.

#'''27 June 2026 15:00 UTC''' ([https://zonestamp.toolforge.org/1782572400 timestamp converter]), at a time friendly to the Americas, Africa, and Europe
#'''28 June 2026 03:00 UTC''' ([https://zonestamp.toolforge.org/1782615600 timestamp converter]), at a time friendly to Asia and the Pacific

Please see the Café page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]!
&lt;br /&gt;
[[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]&lt;/div&gt;
&lt;span style=&quot;white-space:nowrap;&quot;&gt;[[User:Pine|&lt;span style=&quot;color:#01796f; text-shadow:#00BFFF 0 0 1.0em&quot;&gt;↠Pine&lt;/span&gt;]] [[User talk:Pine|&lt;span style=&quot;color:DeepSkyBlue&quot;&gt;(&lt;b style=&quot;color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em&quot;&gt;✉&lt;/b&gt;)&lt;/span&gt;]]&lt;/span&gt; 04:00, 15 June 2026 (UTC)

== Mobile friendly main page ==

Hello, I have recently been using wikiversity on mobile and unlike wikipedia some images and boxes stick out instead of all having a set width which means you can scroll a little side to side, which makes the site feel a bit unfinished. Its just a suggestion but I think it will wake the user experience much better {{unsigned|AUBSTRAWBS}}
:{{Ping|AUBSTRAWBS}} I don't use a smartphone. Can you give me more details or even take some screenshots? You can upload them at [[:c:Category:English Wikiversity screenshots]]. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 13:30, 18 June 2026 (UTC)
::Hi i uploaded an image of the problem. Since some of the images are larger than the screen and not adjusted to fit they stick out and makes the page larger which lets you scroll right and have a big white rectangle on the side [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 14:03, 18 June 2026 (UTC)
:::Thanks. I agree that this is an issue, but it's a pretty minor-to-moderate one to me and I don't think I will be able to dedicate time to fix it myself. Showing it to others here is useful in case someone else wants to tinker with the CSS to resolve it. Thanks for bringing it to the community's attention. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 15:42, 18 June 2026 (UTC)
::::I do know CSS as I like to maintain a blog online so I could try and fix it but I don't know if I have the access to do that, would i need to be a curator/ custodian. Alternatively i could edit a sandbox version of the main page and then send it to someone. [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 20:00, 18 June 2026 (UTC)
:::::Oh great. There are a lot of draft versions of the main page like [[Wikiversity:Main Page/Draft version 0.2]], so you can make [[Wikiversity:Main Page/Sandbox]] if you want and edit there. If you can tinker it to your liking, I can edit the main page. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 20:14, 18 June 2026 (UTC)
::::::thank you, i'll check it out [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 22:16, 18 June 2026 (UTC)

== Main page titles ==

Currently, the title says &quot;Wikiversity:Main Page&quot;, but in my opinion, it's too basic. I would like to propose changing it with the following options (you may only pick one):
# Option 1: Set both [[MediaWiki:Mainpage-title]] and [[MediaWiki:Mainpage-title-loggedin]] to blank, giving the main page a portal-like design (as with English Wikipedia, English Wikibooks, etc.)
# Option 2: Modify [[MediaWiki:Mainpage-title]] to &lt;code&gt;Welcome to Wikiversity&lt;/code&gt; (for unregistered users), and [[MediaWiki:Mainpage-title-loggedin]] to &lt;code&gt;&lt;nowiki&gt;Welcome to Wikiversity, $1!&lt;/nowiki&gt;&lt;/code&gt;; the latter would display to me as &lt;code&gt;Welcome to Wikiversity, Codename Noreste!&lt;/code&gt;
Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:34, 18 June 2026 (UTC)
: Pinging @[[User:Jtneill|Jtneill]] and @[[User:Koavf|Koavf]] for input above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:03, 24 June 2026 (UTC)
:I'm afraid that I don't have strong feelings on this. Changing to either or staying with the status quo are all fine to me. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 16:10, 24 June 2026 (UTC)
: I like the option of being consistent with Wikipedia and Wikibooks -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:04, 25 June 2026 (UTC)
: {{done}}. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:34, 30 June 2026 (UTC)
: Thankyou - looks good. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:29, 2 July 2026 (UTC)

== Wiki x AI preconference day @ Wikimania ==
There will be a preconference day at Wikimania about [[meta:Artificial_intelligence/2026_Wiki_AI | Wiki AI]]. It will be mostly offline, but there will be at least one hybrid session for demos of community-developed AI tools and workflows. 

* If you've built something cool, that is a chance to show it off, list it on the gallery of tools in progress, and get feedback. 

* If you could ask the people shaping AI on the wikis (WMF, tool builders, model trainers, GLAM and policy folks) a question, what would it be? 

Cheers, &lt;span style=&quot;padding:0 2px 0 2px;background-color:white;color:#bbb;&quot;&gt;&amp;ndash;[[User:Sj|SJ]][[User Talk:Sj|&lt;span style=&quot;color:#ff9900;&quot;&gt;+&lt;/span&gt;]]&lt;/span&gt; 23:12, 20 June 2026 (UTC) and Alaexis&lt;br&gt;{{comment|1=Copied from https://en.wikiversity.org/w/index.php?title=Talk%3AMotivation_and_emotion%2FAssessment%2FUsing_generative_AI&amp;diff=2816357&amp;oldid=2807052}}

== RFC about AI-generated content in Wikimedia Commons ==

You are invited to participate in a [[c:Commons:Requests for comment/Policy update for AI content|request for comment on Wikimedia Commons about a policy update for AI content]]. This may affect files that are uploaded to Wikimedia Commons for use on this project. Thank you. [[m:User:Codename Noreste|Codename Noreste]] ([[m:User talk:Codename Noreste|discuss]]) 17:12, 23 June 2026 (UTC)
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== Deployment of Legal and Safety Contacts Link in the Footer of Your Wiki ==

&lt;section begin=&quot;Message&quot;/&gt;
'''Legal &amp; Safety Contacts'''

Hello community, the Wikimedia Foundation has provided a [[wmf:Special:MyLanguage/Legal:Wikimedia Foundation Legal and Safety Contact Information|single legal and safety contact page]], to be linked in the footer of your wiki, to ensure access to accurate legal information. This is a regulatory requirement. We have already rolled out links to English, German, Italian, Spanish and other wikis and we will deploy to your wiki soon. [[m:Special:MyLanguage/Wikimedia_Foundation_Legal_and_Safety_Contacts_FAQ|Please read more on the project page]] and leave any comments in this thread or on the [[m:Special:MyLanguage/Talk:Wikimedia Foundation Legal and Safety Contacts FAQ|talk page]].

&lt;section end=&quot;Message&quot;/&gt;

-- [[User:Sannita (WMF)|User:Sannita (WMF)]] ([[User talk:Sannita (WMF)|talk]]) 13:31, 25 June 2026 (UTC)
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== Preparing manuscript for submission to the ''WikiJournal of Humanities'' ==

I am working on preparing an article in Wikipedia for a dual goal of submitting it for a featured article candidacy in Wikipedia and submitting it to the ''[[WikiJournal of Humanities]]''.  I have an open request for pre-submission peer review at [[en:Wikipedia:Wikipedia:Peer review/Rei Ayanami/archive2|Wikipedia:Peer review/Rei Ayanami/archive2]], and I am asking for someone experienced with submitting journals to WikiJournals.  The article is not ready for submission, and I would like to know where I can get assistance from users who submitted articles to the journal, but did not necessarily review them.  Furthermore, I said there that submitting to the ''WikiJournal of Humanities'' depends on whether the article attains featured article status in Wikipedia, as I would like to use the featured article as a manuscript for a journal article. [[User:Z. Patterson|Z. Patterson]] ([[User talk:Z. Patterson|discuss]] • [[Special:Contributions/Z. Patterson|contribs]]) 12:04, 30 June 2026 (UTC)

== I could probably save many kilobytes by compressing my LLM chat history ==

I could probably save many kilobytes by compressing my LLM chat history. Would that be ok? I like how my new &quot;method&quot; looks: [[User:ThinkingScience/All General AI Prompt History Archive]] very compressed and neat. I like how to find new ways to make people using LLMs not become &quot;secondary citizens&quot;. [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 05:05, 7 July 2026 (UTC)

:{{replyto|ThinkingScience}} I do not see why not. People should be able to look in your page's history for LLM chats before your compression. [[User:Z. Patterson|Z. Patterson]] ([[User talk:Z. Patterson|discuss]] • [[Special:Contributions/Z. Patterson|contribs]]) 02:34, 14 July 2026 (UTC)
::That's good. Then an admin if they have been instructed to compress...then they can delete the edit history perhaps and save space, then it's up to the admin and I don't need to worry about taking up too many resources. That's good! [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 07:50, 14 July 2026 (UTC)

== July 2026 Wikimedia Café meetups regarding Wikimedia governance and options for reform ==

&lt;div class=&quot;border-box&quot; style=&quot;background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)&quot;&gt;
&lt;div class=&quot;box&quot; style=&quot;float:left; padding-top: 10px; padding-right: 10px; padding-left: 10px; padding-bottom: 10px;&quot;&gt;[[File:Wikimedia Café logo in plain SVG format.svg|60px|alt=The logo for the Wikimedia Café]]&lt;/div&gt;
Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities in July. Both sessions will focus on Wikimedia governance, including possible follow-ups to the [https://meta.wikimedia.org/wiki/Movement_Charter Movement Charter] and options for reform. Participants may attend either or both Café sessions. 

This month, to deconflict the Café meetups from Wikimania, the meetups will be held one day later than usual.

#'''26 July 2026 15:00 UTC''' ([https://zonestamp.toolforge.org/1785078000 timestamp converter]), at a time friendly to the Americas, Africa, and Europe
#'''27 July 2026 03:00 UTC''' ([https://zonestamp.toolforge.org/1785121200 timestamp converter]), at a time friendly to Asia and the Pacific

Please see the Café page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]!
&lt;br /&gt;
[[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]&lt;/div&gt;
&lt;span style=&quot;white-space:nowrap;&quot;&gt;[[User:Pine|&lt;span style=&quot;color:#01796f; text-shadow:#00BFFF 0 0 1.0em&quot;&gt;↠Pine&lt;/span&gt;]] [[User talk:Pine|&lt;span style=&quot;color:DeepSkyBlue&quot;&gt;(&lt;b style=&quot;color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em&quot;&gt;✉&lt;/b&gt;)&lt;/span&gt;]]&lt;/span&gt; 03:51, 13 July 2026 (UTC)

:@[[User:Pine|Pine]], Some wikimedians who may contribute less popular, but hopefully productive, comments,  are those who are currently blocked on Meta. I myself was blocked infinitely on Meta several years ago, but was allowed to contInue contributing on other wikimedia projects, such as this one. There are many long term contributors, some who have contrIbuted tens of thousands of edits  to wikimedia projects who are are now blocked on specific projects. 
:Those who happen to be blocked on Meta have,  to all intents and purposes, been removed from the wikimedia movement.
:Ottawahitech [[Special:Contributions/&amp;#126;2026-40514-81|&amp;#126;2026-40514-81]] ([[User talk:&amp;#126;2026-40514-81|talk]]) 21:41, 18 July 2026 (UTC)
:*Hi Ottawahitech, thanks for the comment. Reviewing local bans on Meta is outside of my current scope, but if you have comments that you'd like to share here in the Colloquium relating to Wikiversity, I can consider the comments in the scope of the Café discussions. &lt;span style=&quot;white-space:nowrap;&quot;&gt;[[User:Pine|&lt;span style=&quot;color:#01796f; text-shadow:#00BFFF 0 0 1.0em&quot;&gt;↠Pine&lt;/span&gt;]] [[User talk:Pine|&lt;span style=&quot;color:DeepSkyBlue&quot;&gt;(&lt;b style=&quot;color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em&quot;&gt;✉&lt;/b&gt;)&lt;/span&gt;]]&lt;/span&gt; 20:05, 25 July 2026 (UTC)

== Request for comment (the future of Abstract Wikipedia) ==

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You are invited to voice your opinions in a [[:m:Requests for comment/The future of Abstract Wikipedia|request for comment about the future of Abstract Wikipedia]]. {{Int:Feedback-thanks-title}}
[[:m:User:Kowal2701|Kowal2701]] ([[:m:User talk:Kowal2701|talk]])  12:25, 24 July 2026 (UTC)
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    <revision>
      <id>2819789</id>
      <parentid>2819787</parentid>
      <timestamp>2026-07-30T04:49:30Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>/* June 2026 Wikimedia Café meetups regarding the English Wikipedia Editor Reflections project */ archive to [[Wikiversity:Colloquium/archives/June 2026#June 2026 Wikimedia Café meetups regarding the English Wikipedia Editor Reflections project]] ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]])</comment>
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      <text bytes="22992" sha1="574m1962xjzlikwbayfturbw9z4vi4a" xml:space="preserve">{{Wikiversity:Colloquium/Header}}
&lt;!-- MESSAGES GO BELOW --&gt;

== Draft inactivity policy ==

I created [[Wikiversity:Inactivity policy]] as a start. Any experienced Wikiversity user may feel free to expand it. This is also one-to-two step(s) towards opting out of the [[m:Admin activity review|AAR process]].

However, I made a bold change to reduce the response timeframe from one month to two weeks. In addition, should we reduce the inactivity timeframe to one year? For the latter, most projects use that timeframe and I suggested this for consistency. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:57, 4 June 2026 (UTC)

:I support those suggestions. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:55, 4 June 2026 (UTC)
: Juandev has posted some comments on the [[Wikiversity talk:Inactivity policy|talk page]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:30, 12 June 2026 (UTC)
: Thanks for creating this draft. I've made some changes (including moving back to a one month response timeframe) and moved it from draft to proposed policy.
: Based on the discussion on the talk page, I think it is close to ready (or ready) to be formally proposed as a policy (by adding it to the sitenotice) to allow wider discussion and hopefully adoption. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 06:45, 21 July 2026 (UTC)

== Proposed user group and/or possible policy changes ==

{{tracked|T430416|fixed}}
I want to discuss about user group and possible policy changes.

# First, interface administrators. I don't think we should allow interface administrators to remove their permission from their own account, since we have multiple active bureaucrats and we can ask them to remove the permission when done, or for them to add a temporary grant. This is according to the [[Wikiversity:IA|current IA policy]]. I also left [[Wikiversity talk:Interface administrators#My thoughts about this user group|my thoughts on the relevant talk page]].
# Second, curators. Given that curators have some sensitive custodian rights (such as &lt;code&gt;delete&lt;/code&gt; [but not &lt;code&gt;undelete&lt;/code&gt; or similar rights that allow viewing deleted content, unless the curatorship process is RFA-like] and &lt;code&gt;protect&lt;/code&gt;), it would probably make more sense only for bureaucrats to grant and remove it, on par with them granting (but not removing) custodian permissions.
# Third, about probationary custodians. [[Wikiversity:Probationary custodians]] is currently marked as historical, and the process might still exist on [[Wikiversity:Custodianship]]. Therefore, to maintain consistency with [[Wikiversity:Curatorship#How does one become a curator?]], I propose that we repeal the probationary custodianship process and change it more or less to align with the curatorship process, effectively making probationary custodians permanent ones. However, custodian mentors would still be retained.

Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:55, 5 June 2026 (UTC)

:#Yes, I agree.
:#Thats a good point, but I dont know. At least I dont think its a good idea that both groups i.e. crats and custodiants can do that, it may create chaos.
:#Another good point. It seems to me that the current situation is somewhat unclear and should be clarified. I understand the original status of [[Wikiversity:Probationary custodians|Probationary custodians]] as a historicall and invalid, but at the same time I consider myself a probationary custodian, because on the Wikiversity:Custodianship page in the ''[[Wikiversity:Custodianship#How does one become a custodian?|How does one become a custodian?]]'' section it says, I quote, ''&quot;II ...then you will be approved as a probationary custodian for a period of at least four weeks&quot;''.
:::Mentors should definitely be kept, but for certain applicants the probation and mentorship should be abolished. For example, if someone was an active custodian for 5 years, then loses their rights or gives them up for a year and then wants to resume their custodial activities, there is no reason for them to undergo a training period. It burdens both the mentors and the community with double voting. The only exception could be a situation where policies or tools for custodians change significantly during that year, or the candidate wants to.
:[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 06:08, 9 June 2026 (UTC)

== Towards an Ethics policy ==

In connection with the [[Wikiversity:Community Review/Removal of Wikidebates|discussion of Wikidebates]], I said that it would be good to establish a policy on ethics, or rather a boundary between ethical and unethical content, so that we don't have to discuss individual cases. In addition, today we also have some global policies that prohibit, for example, attacks on members of the Wikimedia movement or undermining other projects.

However, at the very beginning, I would start by collecting your opinions. What content or what research should not be allowed on Wikiversity? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 05:52, 9 June 2026 (UTC)

:One ethical issue that I think should be non-controversial is related to good faith in the learning modules. So, learning materials should not be hoaxes or encourage behavior or methods that don't work or that misrepresent the facts or the likelihood of something occurring, etc. and authors should also not plagiarize or misrepresent authorship, etc. That was quite a run-on, but I hope that others can tease out what I mean here. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:39, 9 June 2026 (UTC)
::I look at it from a practical perspective. We can give that to the policy, but I see the problem in that we are not able to check it except plagiarism.
::Plagiarism can be partially detected during patrolling. I see a new text, I put part of it in Google and I check if it is copied from the web. It is a problem with copying from books or other offline sources, but sometimes it happens that someone finds out that something is copied from somewhere and it can be deleted.
::The biggest issue we have here is that we are missing Wikipedia's control mechanism: references. Only some types of resources on Wikiversity require references. In-line references are not often used in courses, exercises, lectures, etc. We are thus deprived of one of the excellent control mechanisms and the only option is for the increase in the number of members with various qualifications to check it for their colleagues. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:59, 9 June 2026 (UTC)
:::Having a policy and enforcing that policy are indeed two different things. If we are only concerned with issues that we can definitively enforce, then that will definitely change this conversation. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 08:06, 9 June 2026 (UTC)
::::ok [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:55, 13 June 2026 (UTC)
:AI generated content should not be allowed as it is inherently plagiarism. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:14, 9 June 2026 (UTC)
::And if the user mention it was generated by an AI? Note that there is something called as public domain, that is the author wave its rights. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:53, 9 June 2026 (UTC)
:::Plagiarism isn’t copyright violation. Crediting the AI is not crediting the authors the AI stole from without credit. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 10:18, 9 June 2026 (UTC)
::::I see, now I understand your point. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:56, 13 June 2026 (UTC)

== New nomination template(s) ==

I created {{tlx|Nomination}} when someone requests curator or custodian permissions, which often at least require mentorship. On the other hand, I might create {{tlx|Nomination 2}}, in which the latter does not have a section about mentorship (often used for bureaucrat or interface administrator nominations). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:29, 12 June 2026 (UTC)

== Mobile friendly main page ==

Hello, I have recently been using wikiversity on mobile and unlike wikipedia some images and boxes stick out instead of all having a set width which means you can scroll a little side to side, which makes the site feel a bit unfinished. Its just a suggestion but I think it will wake the user experience much better {{unsigned|AUBSTRAWBS}}
:{{Ping|AUBSTRAWBS}} I don't use a smartphone. Can you give me more details or even take some screenshots? You can upload them at [[:c:Category:English Wikiversity screenshots]]. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 13:30, 18 June 2026 (UTC)
::Hi i uploaded an image of the problem. Since some of the images are larger than the screen and not adjusted to fit they stick out and makes the page larger which lets you scroll right and have a big white rectangle on the side [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 14:03, 18 June 2026 (UTC)
:::Thanks. I agree that this is an issue, but it's a pretty minor-to-moderate one to me and I don't think I will be able to dedicate time to fix it myself. Showing it to others here is useful in case someone else wants to tinker with the CSS to resolve it. Thanks for bringing it to the community's attention. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 15:42, 18 June 2026 (UTC)
::::I do know CSS as I like to maintain a blog online so I could try and fix it but I don't know if I have the access to do that, would i need to be a curator/ custodian. Alternatively i could edit a sandbox version of the main page and then send it to someone. [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 20:00, 18 June 2026 (UTC)
:::::Oh great. There are a lot of draft versions of the main page like [[Wikiversity:Main Page/Draft version 0.2]], so you can make [[Wikiversity:Main Page/Sandbox]] if you want and edit there. If you can tinker it to your liking, I can edit the main page. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 20:14, 18 June 2026 (UTC)
::::::thank you, i'll check it out [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 22:16, 18 June 2026 (UTC)

== Main page titles ==

Currently, the title says &quot;Wikiversity:Main Page&quot;, but in my opinion, it's too basic. I would like to propose changing it with the following options (you may only pick one):
# Option 1: Set both [[MediaWiki:Mainpage-title]] and [[MediaWiki:Mainpage-title-loggedin]] to blank, giving the main page a portal-like design (as with English Wikipedia, English Wikibooks, etc.)
# Option 2: Modify [[MediaWiki:Mainpage-title]] to &lt;code&gt;Welcome to Wikiversity&lt;/code&gt; (for unregistered users), and [[MediaWiki:Mainpage-title-loggedin]] to &lt;code&gt;&lt;nowiki&gt;Welcome to Wikiversity, $1!&lt;/nowiki&gt;&lt;/code&gt;; the latter would display to me as &lt;code&gt;Welcome to Wikiversity, Codename Noreste!&lt;/code&gt;
Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:34, 18 June 2026 (UTC)
: Pinging @[[User:Jtneill|Jtneill]] and @[[User:Koavf|Koavf]] for input above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:03, 24 June 2026 (UTC)
:I'm afraid that I don't have strong feelings on this. Changing to either or staying with the status quo are all fine to me. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 16:10, 24 June 2026 (UTC)
: I like the option of being consistent with Wikipedia and Wikibooks -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:04, 25 June 2026 (UTC)
: {{done}}. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:34, 30 June 2026 (UTC)
: Thankyou - looks good. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:29, 2 July 2026 (UTC)

== Wiki x AI preconference day @ Wikimania ==
There will be a preconference day at Wikimania about [[meta:Artificial_intelligence/2026_Wiki_AI | Wiki AI]]. It will be mostly offline, but there will be at least one hybrid session for demos of community-developed AI tools and workflows. 

* If you've built something cool, that is a chance to show it off, list it on the gallery of tools in progress, and get feedback. 

* If you could ask the people shaping AI on the wikis (WMF, tool builders, model trainers, GLAM and policy folks) a question, what would it be? 

Cheers, &lt;span style=&quot;padding:0 2px 0 2px;background-color:white;color:#bbb;&quot;&gt;&amp;ndash;[[User:Sj|SJ]][[User Talk:Sj|&lt;span style=&quot;color:#ff9900;&quot;&gt;+&lt;/span&gt;]]&lt;/span&gt; 23:12, 20 June 2026 (UTC) and Alaexis&lt;br&gt;{{comment|1=Copied from https://en.wikiversity.org/w/index.php?title=Talk%3AMotivation_and_emotion%2FAssessment%2FUsing_generative_AI&amp;diff=2816357&amp;oldid=2807052}}

== RFC about AI-generated content in Wikimedia Commons ==

You are invited to participate in a [[c:Commons:Requests for comment/Policy update for AI content|request for comment on Wikimedia Commons about a policy update for AI content]]. This may affect files that are uploaded to Wikimedia Commons for use on this project. Thank you. [[m:User:Codename Noreste|Codename Noreste]] ([[m:User talk:Codename Noreste|discuss]]) 17:12, 23 June 2026 (UTC)
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== Deployment of Legal and Safety Contacts Link in the Footer of Your Wiki ==

&lt;section begin=&quot;Message&quot;/&gt;
'''Legal &amp; Safety Contacts'''

Hello community, the Wikimedia Foundation has provided a [[wmf:Special:MyLanguage/Legal:Wikimedia Foundation Legal and Safety Contact Information|single legal and safety contact page]], to be linked in the footer of your wiki, to ensure access to accurate legal information. This is a regulatory requirement. We have already rolled out links to English, German, Italian, Spanish and other wikis and we will deploy to your wiki soon. [[m:Special:MyLanguage/Wikimedia_Foundation_Legal_and_Safety_Contacts_FAQ|Please read more on the project page]] and leave any comments in this thread or on the [[m:Special:MyLanguage/Talk:Wikimedia Foundation Legal and Safety Contacts FAQ|talk page]].

&lt;section end=&quot;Message&quot;/&gt;

-- [[User:Sannita (WMF)|User:Sannita (WMF)]] ([[User talk:Sannita (WMF)|talk]]) 13:31, 25 June 2026 (UTC)
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== Preparing manuscript for submission to the ''WikiJournal of Humanities'' ==

I am working on preparing an article in Wikipedia for a dual goal of submitting it for a featured article candidacy in Wikipedia and submitting it to the ''[[WikiJournal of Humanities]]''.  I have an open request for pre-submission peer review at [[en:Wikipedia:Wikipedia:Peer review/Rei Ayanami/archive2|Wikipedia:Peer review/Rei Ayanami/archive2]], and I am asking for someone experienced with submitting journals to WikiJournals.  The article is not ready for submission, and I would like to know where I can get assistance from users who submitted articles to the journal, but did not necessarily review them.  Furthermore, I said there that submitting to the ''WikiJournal of Humanities'' depends on whether the article attains featured article status in Wikipedia, as I would like to use the featured article as a manuscript for a journal article. [[User:Z. Patterson|Z. Patterson]] ([[User talk:Z. Patterson|discuss]] • [[Special:Contributions/Z. Patterson|contribs]]) 12:04, 30 June 2026 (UTC)

== I could probably save many kilobytes by compressing my LLM chat history ==

I could probably save many kilobytes by compressing my LLM chat history. Would that be ok? I like how my new &quot;method&quot; looks: [[User:ThinkingScience/All General AI Prompt History Archive]] very compressed and neat. I like how to find new ways to make people using LLMs not become &quot;secondary citizens&quot;. [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 05:05, 7 July 2026 (UTC)

:{{replyto|ThinkingScience}} I do not see why not. People should be able to look in your page's history for LLM chats before your compression. [[User:Z. Patterson|Z. Patterson]] ([[User talk:Z. Patterson|discuss]] • [[Special:Contributions/Z. Patterson|contribs]]) 02:34, 14 July 2026 (UTC)
::That's good. Then an admin if they have been instructed to compress...then they can delete the edit history perhaps and save space, then it's up to the admin and I don't need to worry about taking up too many resources. That's good! [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 07:50, 14 July 2026 (UTC)

== July 2026 Wikimedia Café meetups regarding Wikimedia governance and options for reform ==

&lt;div class=&quot;border-box&quot; style=&quot;background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)&quot;&gt;
&lt;div class=&quot;box&quot; style=&quot;float:left; padding-top: 10px; padding-right: 10px; padding-left: 10px; padding-bottom: 10px;&quot;&gt;[[File:Wikimedia Café logo in plain SVG format.svg|60px|alt=The logo for the Wikimedia Café]]&lt;/div&gt;
Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities in July. Both sessions will focus on Wikimedia governance, including possible follow-ups to the [https://meta.wikimedia.org/wiki/Movement_Charter Movement Charter] and options for reform. Participants may attend either or both Café sessions. 

This month, to deconflict the Café meetups from Wikimania, the meetups will be held one day later than usual.

#'''26 July 2026 15:00 UTC''' ([https://zonestamp.toolforge.org/1785078000 timestamp converter]), at a time friendly to the Americas, Africa, and Europe
#'''27 July 2026 03:00 UTC''' ([https://zonestamp.toolforge.org/1785121200 timestamp converter]), at a time friendly to Asia and the Pacific

Please see the Café page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]!
&lt;br /&gt;
[[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]&lt;/div&gt;
&lt;span style=&quot;white-space:nowrap;&quot;&gt;[[User:Pine|&lt;span style=&quot;color:#01796f; text-shadow:#00BFFF 0 0 1.0em&quot;&gt;↠Pine&lt;/span&gt;]] [[User talk:Pine|&lt;span style=&quot;color:DeepSkyBlue&quot;&gt;(&lt;b style=&quot;color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em&quot;&gt;✉&lt;/b&gt;)&lt;/span&gt;]]&lt;/span&gt; 03:51, 13 July 2026 (UTC)

:@[[User:Pine|Pine]], Some wikimedians who may contribute less popular, but hopefully productive, comments,  are those who are currently blocked on Meta. I myself was blocked infinitely on Meta several years ago, but was allowed to contInue contributing on other wikimedia projects, such as this one. There are many long term contributors, some who have contrIbuted tens of thousands of edits  to wikimedia projects who are are now blocked on specific projects. 
:Those who happen to be blocked on Meta have,  to all intents and purposes, been removed from the wikimedia movement.
:Ottawahitech [[Special:Contributions/&amp;#126;2026-40514-81|&amp;#126;2026-40514-81]] ([[User talk:&amp;#126;2026-40514-81|talk]]) 21:41, 18 July 2026 (UTC)
:*Hi Ottawahitech, thanks for the comment. Reviewing local bans on Meta is outside of my current scope, but if you have comments that you'd like to share here in the Colloquium relating to Wikiversity, I can consider the comments in the scope of the Café discussions. &lt;span style=&quot;white-space:nowrap;&quot;&gt;[[User:Pine|&lt;span style=&quot;color:#01796f; text-shadow:#00BFFF 0 0 1.0em&quot;&gt;↠Pine&lt;/span&gt;]] [[User talk:Pine|&lt;span style=&quot;color:DeepSkyBlue&quot;&gt;(&lt;b style=&quot;color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em&quot;&gt;✉&lt;/b&gt;)&lt;/span&gt;]]&lt;/span&gt; 20:05, 25 July 2026 (UTC)

== Request for comment (the future of Abstract Wikipedia) ==

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot; class=&quot;mw-content-ltr&quot;&gt;

You are invited to voice your opinions in a [[:m:Requests for comment/The future of Abstract Wikipedia|request for comment about the future of Abstract Wikipedia]]. {{Int:Feedback-thanks-title}}
[[:m:User:Kowal2701|Kowal2701]] ([[:m:User talk:Kowal2701|talk]])  12:25, 24 July 2026 (UTC)
&lt;/bdi&gt;
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      <comment>/* Mobile friendly main page */ archive to [[Wikiversity:Colloquium/archives/June 2026#Mobile friendly main page]] ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]])</comment>
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== Draft inactivity policy ==

I created [[Wikiversity:Inactivity policy]] as a start. Any experienced Wikiversity user may feel free to expand it. This is also one-to-two step(s) towards opting out of the [[m:Admin activity review|AAR process]].

However, I made a bold change to reduce the response timeframe from one month to two weeks. In addition, should we reduce the inactivity timeframe to one year? For the latter, most projects use that timeframe and I suggested this for consistency. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:57, 4 June 2026 (UTC)

:I support those suggestions. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:55, 4 June 2026 (UTC)
: Juandev has posted some comments on the [[Wikiversity talk:Inactivity policy|talk page]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:30, 12 June 2026 (UTC)
: Thanks for creating this draft. I've made some changes (including moving back to a one month response timeframe) and moved it from draft to proposed policy.
: Based on the discussion on the talk page, I think it is close to ready (or ready) to be formally proposed as a policy (by adding it to the sitenotice) to allow wider discussion and hopefully adoption. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 06:45, 21 July 2026 (UTC)

== Proposed user group and/or possible policy changes ==

{{tracked|T430416|fixed}}
I want to discuss about user group and possible policy changes.

# First, interface administrators. I don't think we should allow interface administrators to remove their permission from their own account, since we have multiple active bureaucrats and we can ask them to remove the permission when done, or for them to add a temporary grant. This is according to the [[Wikiversity:IA|current IA policy]]. I also left [[Wikiversity talk:Interface administrators#My thoughts about this user group|my thoughts on the relevant talk page]].
# Second, curators. Given that curators have some sensitive custodian rights (such as &lt;code&gt;delete&lt;/code&gt; [but not &lt;code&gt;undelete&lt;/code&gt; or similar rights that allow viewing deleted content, unless the curatorship process is RFA-like] and &lt;code&gt;protect&lt;/code&gt;), it would probably make more sense only for bureaucrats to grant and remove it, on par with them granting (but not removing) custodian permissions.
# Third, about probationary custodians. [[Wikiversity:Probationary custodians]] is currently marked as historical, and the process might still exist on [[Wikiversity:Custodianship]]. Therefore, to maintain consistency with [[Wikiversity:Curatorship#How does one become a curator?]], I propose that we repeal the probationary custodianship process and change it more or less to align with the curatorship process, effectively making probationary custodians permanent ones. However, custodian mentors would still be retained.

Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:55, 5 June 2026 (UTC)

:#Yes, I agree.
:#Thats a good point, but I dont know. At least I dont think its a good idea that both groups i.e. crats and custodiants can do that, it may create chaos.
:#Another good point. It seems to me that the current situation is somewhat unclear and should be clarified. I understand the original status of [[Wikiversity:Probationary custodians|Probationary custodians]] as a historicall and invalid, but at the same time I consider myself a probationary custodian, because on the Wikiversity:Custodianship page in the ''[[Wikiversity:Custodianship#How does one become a custodian?|How does one become a custodian?]]'' section it says, I quote, ''&quot;II ...then you will be approved as a probationary custodian for a period of at least four weeks&quot;''.
:::Mentors should definitely be kept, but for certain applicants the probation and mentorship should be abolished. For example, if someone was an active custodian for 5 years, then loses their rights or gives them up for a year and then wants to resume their custodial activities, there is no reason for them to undergo a training period. It burdens both the mentors and the community with double voting. The only exception could be a situation where policies or tools for custodians change significantly during that year, or the candidate wants to.
:[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 06:08, 9 June 2026 (UTC)

== Towards an Ethics policy ==

In connection with the [[Wikiversity:Community Review/Removal of Wikidebates|discussion of Wikidebates]], I said that it would be good to establish a policy on ethics, or rather a boundary between ethical and unethical content, so that we don't have to discuss individual cases. In addition, today we also have some global policies that prohibit, for example, attacks on members of the Wikimedia movement or undermining other projects.

However, at the very beginning, I would start by collecting your opinions. What content or what research should not be allowed on Wikiversity? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 05:52, 9 June 2026 (UTC)

:One ethical issue that I think should be non-controversial is related to good faith in the learning modules. So, learning materials should not be hoaxes or encourage behavior or methods that don't work or that misrepresent the facts or the likelihood of something occurring, etc. and authors should also not plagiarize or misrepresent authorship, etc. That was quite a run-on, but I hope that others can tease out what I mean here. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:39, 9 June 2026 (UTC)
::I look at it from a practical perspective. We can give that to the policy, but I see the problem in that we are not able to check it except plagiarism.
::Plagiarism can be partially detected during patrolling. I see a new text, I put part of it in Google and I check if it is copied from the web. It is a problem with copying from books or other offline sources, but sometimes it happens that someone finds out that something is copied from somewhere and it can be deleted.
::The biggest issue we have here is that we are missing Wikipedia's control mechanism: references. Only some types of resources on Wikiversity require references. In-line references are not often used in courses, exercises, lectures, etc. We are thus deprived of one of the excellent control mechanisms and the only option is for the increase in the number of members with various qualifications to check it for their colleagues. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:59, 9 June 2026 (UTC)
:::Having a policy and enforcing that policy are indeed two different things. If we are only concerned with issues that we can definitively enforce, then that will definitely change this conversation. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 08:06, 9 June 2026 (UTC)
::::ok [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:55, 13 June 2026 (UTC)
:AI generated content should not be allowed as it is inherently plagiarism. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:14, 9 June 2026 (UTC)
::And if the user mention it was generated by an AI? Note that there is something called as public domain, that is the author wave its rights. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:53, 9 June 2026 (UTC)
:::Plagiarism isn’t copyright violation. Crediting the AI is not crediting the authors the AI stole from without credit. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 10:18, 9 June 2026 (UTC)
::::I see, now I understand your point. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:56, 13 June 2026 (UTC)

== New nomination template(s) ==

I created {{tlx|Nomination}} when someone requests curator or custodian permissions, which often at least require mentorship. On the other hand, I might create {{tlx|Nomination 2}}, in which the latter does not have a section about mentorship (often used for bureaucrat or interface administrator nominations). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:29, 12 June 2026 (UTC)

== Main page titles ==

Currently, the title says &quot;Wikiversity:Main Page&quot;, but in my opinion, it's too basic. I would like to propose changing it with the following options (you may only pick one):
# Option 1: Set both [[MediaWiki:Mainpage-title]] and [[MediaWiki:Mainpage-title-loggedin]] to blank, giving the main page a portal-like design (as with English Wikipedia, English Wikibooks, etc.)
# Option 2: Modify [[MediaWiki:Mainpage-title]] to &lt;code&gt;Welcome to Wikiversity&lt;/code&gt; (for unregistered users), and [[MediaWiki:Mainpage-title-loggedin]] to &lt;code&gt;&lt;nowiki&gt;Welcome to Wikiversity, $1!&lt;/nowiki&gt;&lt;/code&gt;; the latter would display to me as &lt;code&gt;Welcome to Wikiversity, Codename Noreste!&lt;/code&gt;
Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:34, 18 June 2026 (UTC)
: Pinging @[[User:Jtneill|Jtneill]] and @[[User:Koavf|Koavf]] for input above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:03, 24 June 2026 (UTC)
:I'm afraid that I don't have strong feelings on this. Changing to either or staying with the status quo are all fine to me. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 16:10, 24 June 2026 (UTC)
: I like the option of being consistent with Wikipedia and Wikibooks -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:04, 25 June 2026 (UTC)
: {{done}}. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:34, 30 June 2026 (UTC)
: Thankyou - looks good. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:29, 2 July 2026 (UTC)

== Wiki x AI preconference day @ Wikimania ==
There will be a preconference day at Wikimania about [[meta:Artificial_intelligence/2026_Wiki_AI | Wiki AI]]. It will be mostly offline, but there will be at least one hybrid session for demos of community-developed AI tools and workflows. 

* If you've built something cool, that is a chance to show it off, list it on the gallery of tools in progress, and get feedback. 

* If you could ask the people shaping AI on the wikis (WMF, tool builders, model trainers, GLAM and policy folks) a question, what would it be? 

Cheers, &lt;span style=&quot;padding:0 2px 0 2px;background-color:white;color:#bbb;&quot;&gt;&amp;ndash;[[User:Sj|SJ]][[User Talk:Sj|&lt;span style=&quot;color:#ff9900;&quot;&gt;+&lt;/span&gt;]]&lt;/span&gt; 23:12, 20 June 2026 (UTC) and Alaexis&lt;br&gt;{{comment|1=Copied from https://en.wikiversity.org/w/index.php?title=Talk%3AMotivation_and_emotion%2FAssessment%2FUsing_generative_AI&amp;diff=2816357&amp;oldid=2807052}}

== RFC about AI-generated content in Wikimedia Commons ==

You are invited to participate in a [[c:Commons:Requests for comment/Policy update for AI content|request for comment on Wikimedia Commons about a policy update for AI content]]. This may affect files that are uploaded to Wikimedia Commons for use on this project. Thank you. [[m:User:Codename Noreste|Codename Noreste]] ([[m:User talk:Codename Noreste|discuss]]) 17:12, 23 June 2026 (UTC)
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== Deployment of Legal and Safety Contacts Link in the Footer of Your Wiki ==

&lt;section begin=&quot;Message&quot;/&gt;
'''Legal &amp; Safety Contacts'''

Hello community, the Wikimedia Foundation has provided a [[wmf:Special:MyLanguage/Legal:Wikimedia Foundation Legal and Safety Contact Information|single legal and safety contact page]], to be linked in the footer of your wiki, to ensure access to accurate legal information. This is a regulatory requirement. We have already rolled out links to English, German, Italian, Spanish and other wikis and we will deploy to your wiki soon. [[m:Special:MyLanguage/Wikimedia_Foundation_Legal_and_Safety_Contacts_FAQ|Please read more on the project page]] and leave any comments in this thread or on the [[m:Special:MyLanguage/Talk:Wikimedia Foundation Legal and Safety Contacts FAQ|talk page]].

&lt;section end=&quot;Message&quot;/&gt;

-- [[User:Sannita (WMF)|User:Sannita (WMF)]] ([[User talk:Sannita (WMF)|talk]]) 13:31, 25 June 2026 (UTC)
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== Preparing manuscript for submission to the ''WikiJournal of Humanities'' ==

I am working on preparing an article in Wikipedia for a dual goal of submitting it for a featured article candidacy in Wikipedia and submitting it to the ''[[WikiJournal of Humanities]]''.  I have an open request for pre-submission peer review at [[en:Wikipedia:Wikipedia:Peer review/Rei Ayanami/archive2|Wikipedia:Peer review/Rei Ayanami/archive2]], and I am asking for someone experienced with submitting journals to WikiJournals.  The article is not ready for submission, and I would like to know where I can get assistance from users who submitted articles to the journal, but did not necessarily review them.  Furthermore, I said there that submitting to the ''WikiJournal of Humanities'' depends on whether the article attains featured article status in Wikipedia, as I would like to use the featured article as a manuscript for a journal article. [[User:Z. Patterson|Z. Patterson]] ([[User talk:Z. Patterson|discuss]] • [[Special:Contributions/Z. Patterson|contribs]]) 12:04, 30 June 2026 (UTC)

== I could probably save many kilobytes by compressing my LLM chat history ==

I could probably save many kilobytes by compressing my LLM chat history. Would that be ok? I like how my new &quot;method&quot; looks: [[User:ThinkingScience/All General AI Prompt History Archive]] very compressed and neat. I like how to find new ways to make people using LLMs not become &quot;secondary citizens&quot;. [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 05:05, 7 July 2026 (UTC)

:{{replyto|ThinkingScience}} I do not see why not. People should be able to look in your page's history for LLM chats before your compression. [[User:Z. Patterson|Z. Patterson]] ([[User talk:Z. Patterson|discuss]] • [[Special:Contributions/Z. Patterson|contribs]]) 02:34, 14 July 2026 (UTC)
::That's good. Then an admin if they have been instructed to compress...then they can delete the edit history perhaps and save space, then it's up to the admin and I don't need to worry about taking up too many resources. That's good! [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 07:50, 14 July 2026 (UTC)

== July 2026 Wikimedia Café meetups regarding Wikimedia governance and options for reform ==

&lt;div class=&quot;border-box&quot; style=&quot;background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)&quot;&gt;
&lt;div class=&quot;box&quot; style=&quot;float:left; padding-top: 10px; padding-right: 10px; padding-left: 10px; padding-bottom: 10px;&quot;&gt;[[File:Wikimedia Café logo in plain SVG format.svg|60px|alt=The logo for the Wikimedia Café]]&lt;/div&gt;
Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities in July. Both sessions will focus on Wikimedia governance, including possible follow-ups to the [https://meta.wikimedia.org/wiki/Movement_Charter Movement Charter] and options for reform. Participants may attend either or both Café sessions. 

This month, to deconflict the Café meetups from Wikimania, the meetups will be held one day later than usual.

#'''26 July 2026 15:00 UTC''' ([https://zonestamp.toolforge.org/1785078000 timestamp converter]), at a time friendly to the Americas, Africa, and Europe
#'''27 July 2026 03:00 UTC''' ([https://zonestamp.toolforge.org/1785121200 timestamp converter]), at a time friendly to Asia and the Pacific

Please see the Café page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]!
&lt;br /&gt;
[[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]&lt;/div&gt;
&lt;span style=&quot;white-space:nowrap;&quot;&gt;[[User:Pine|&lt;span style=&quot;color:#01796f; text-shadow:#00BFFF 0 0 1.0em&quot;&gt;↠Pine&lt;/span&gt;]] [[User talk:Pine|&lt;span style=&quot;color:DeepSkyBlue&quot;&gt;(&lt;b style=&quot;color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em&quot;&gt;✉&lt;/b&gt;)&lt;/span&gt;]]&lt;/span&gt; 03:51, 13 July 2026 (UTC)

:@[[User:Pine|Pine]], Some wikimedians who may contribute less popular, but hopefully productive, comments,  are those who are currently blocked on Meta. I myself was blocked infinitely on Meta several years ago, but was allowed to contInue contributing on other wikimedia projects, such as this one. There are many long term contributors, some who have contrIbuted tens of thousands of edits  to wikimedia projects who are are now blocked on specific projects. 
:Those who happen to be blocked on Meta have,  to all intents and purposes, been removed from the wikimedia movement.
:Ottawahitech [[Special:Contributions/&amp;#126;2026-40514-81|&amp;#126;2026-40514-81]] ([[User talk:&amp;#126;2026-40514-81|talk]]) 21:41, 18 July 2026 (UTC)
:*Hi Ottawahitech, thanks for the comment. Reviewing local bans on Meta is outside of my current scope, but if you have comments that you'd like to share here in the Colloquium relating to Wikiversity, I can consider the comments in the scope of the Café discussions. &lt;span style=&quot;white-space:nowrap;&quot;&gt;[[User:Pine|&lt;span style=&quot;color:#01796f; text-shadow:#00BFFF 0 0 1.0em&quot;&gt;↠Pine&lt;/span&gt;]] [[User talk:Pine|&lt;span style=&quot;color:DeepSkyBlue&quot;&gt;(&lt;b style=&quot;color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em&quot;&gt;✉&lt;/b&gt;)&lt;/span&gt;]]&lt;/span&gt; 20:05, 25 July 2026 (UTC)

== Request for comment (the future of Abstract Wikipedia) ==

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot; class=&quot;mw-content-ltr&quot;&gt;

You are invited to voice your opinions in a [[:m:Requests for comment/The future of Abstract Wikipedia|request for comment about the future of Abstract Wikipedia]]. {{Int:Feedback-thanks-title}}
[[:m:User:Kowal2701|Kowal2701]] ([[:m:User talk:Kowal2701|talk]])  12:25, 24 July 2026 (UTC)
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== Draft inactivity policy ==

I created [[Wikiversity:Inactivity policy]] as a start. Any experienced Wikiversity user may feel free to expand it. This is also one-to-two step(s) towards opting out of the [[m:Admin activity review|AAR process]].

However, I made a bold change to reduce the response timeframe from one month to two weeks. In addition, should we reduce the inactivity timeframe to one year? For the latter, most projects use that timeframe and I suggested this for consistency. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:57, 4 June 2026 (UTC)

:I support those suggestions. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:55, 4 June 2026 (UTC)
: Juandev has posted some comments on the [[Wikiversity talk:Inactivity policy|talk page]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:30, 12 June 2026 (UTC)
: Thanks for creating this draft. I've made some changes (including moving back to a one month response timeframe) and moved it from draft to proposed policy.
: Based on the discussion on the talk page, I think it is close to ready (or ready) to be formally proposed as a policy (by adding it to the sitenotice) to allow wider discussion and hopefully adoption. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 06:45, 21 July 2026 (UTC)

== Proposed user group and/or possible policy changes ==

{{tracked|T430416|fixed}}
I want to discuss about user group and possible policy changes.

# First, interface administrators. I don't think we should allow interface administrators to remove their permission from their own account, since we have multiple active bureaucrats and we can ask them to remove the permission when done, or for them to add a temporary grant. This is according to the [[Wikiversity:IA|current IA policy]]. I also left [[Wikiversity talk:Interface administrators#My thoughts about this user group|my thoughts on the relevant talk page]].
# Second, curators. Given that curators have some sensitive custodian rights (such as &lt;code&gt;delete&lt;/code&gt; [but not &lt;code&gt;undelete&lt;/code&gt; or similar rights that allow viewing deleted content, unless the curatorship process is RFA-like] and &lt;code&gt;protect&lt;/code&gt;), it would probably make more sense only for bureaucrats to grant and remove it, on par with them granting (but not removing) custodian permissions.
# Third, about probationary custodians. [[Wikiversity:Probationary custodians]] is currently marked as historical, and the process might still exist on [[Wikiversity:Custodianship]]. Therefore, to maintain consistency with [[Wikiversity:Curatorship#How does one become a curator?]], I propose that we repeal the probationary custodianship process and change it more or less to align with the curatorship process, effectively making probationary custodians permanent ones. However, custodian mentors would still be retained.

Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:55, 5 June 2026 (UTC)

:#Yes, I agree.
:#Thats a good point, but I dont know. At least I dont think its a good idea that both groups i.e. crats and custodiants can do that, it may create chaos.
:#Another good point. It seems to me that the current situation is somewhat unclear and should be clarified. I understand the original status of [[Wikiversity:Probationary custodians|Probationary custodians]] as a historicall and invalid, but at the same time I consider myself a probationary custodian, because on the Wikiversity:Custodianship page in the ''[[Wikiversity:Custodianship#How does one become a custodian?|How does one become a custodian?]]'' section it says, I quote, ''&quot;II ...then you will be approved as a probationary custodian for a period of at least four weeks&quot;''.
:::Mentors should definitely be kept, but for certain applicants the probation and mentorship should be abolished. For example, if someone was an active custodian for 5 years, then loses their rights or gives them up for a year and then wants to resume their custodial activities, there is no reason for them to undergo a training period. It burdens both the mentors and the community with double voting. The only exception could be a situation where policies or tools for custodians change significantly during that year, or the candidate wants to.
:[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 06:08, 9 June 2026 (UTC)

== Towards an Ethics policy ==

In connection with the [[Wikiversity:Community Review/Removal of Wikidebates|discussion of Wikidebates]], I said that it would be good to establish a policy on ethics, or rather a boundary between ethical and unethical content, so that we don't have to discuss individual cases. In addition, today we also have some global policies that prohibit, for example, attacks on members of the Wikimedia movement or undermining other projects.

However, at the very beginning, I would start by collecting your opinions. What content or what research should not be allowed on Wikiversity? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 05:52, 9 June 2026 (UTC)

:One ethical issue that I think should be non-controversial is related to good faith in the learning modules. So, learning materials should not be hoaxes or encourage behavior or methods that don't work or that misrepresent the facts or the likelihood of something occurring, etc. and authors should also not plagiarize or misrepresent authorship, etc. That was quite a run-on, but I hope that others can tease out what I mean here. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:39, 9 June 2026 (UTC)
::I look at it from a practical perspective. We can give that to the policy, but I see the problem in that we are not able to check it except plagiarism.
::Plagiarism can be partially detected during patrolling. I see a new text, I put part of it in Google and I check if it is copied from the web. It is a problem with copying from books or other offline sources, but sometimes it happens that someone finds out that something is copied from somewhere and it can be deleted.
::The biggest issue we have here is that we are missing Wikipedia's control mechanism: references. Only some types of resources on Wikiversity require references. In-line references are not often used in courses, exercises, lectures, etc. We are thus deprived of one of the excellent control mechanisms and the only option is for the increase in the number of members with various qualifications to check it for their colleagues. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:59, 9 June 2026 (UTC)
:::Having a policy and enforcing that policy are indeed two different things. If we are only concerned with issues that we can definitively enforce, then that will definitely change this conversation. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 08:06, 9 June 2026 (UTC)
::::ok [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:55, 13 June 2026 (UTC)
:AI generated content should not be allowed as it is inherently plagiarism. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:14, 9 June 2026 (UTC)
::And if the user mention it was generated by an AI? Note that there is something called as public domain, that is the author wave its rights. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:53, 9 June 2026 (UTC)
:::Plagiarism isn’t copyright violation. Crediting the AI is not crediting the authors the AI stole from without credit. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 10:18, 9 June 2026 (UTC)
::::I see, now I understand your point. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:56, 13 June 2026 (UTC)

== New nomination template(s) ==

I created {{tlx|Nomination}} when someone requests curator or custodian permissions, which often at least require mentorship. On the other hand, I might create {{tlx|Nomination 2}}, in which the latter does not have a section about mentorship (often used for bureaucrat or interface administrator nominations). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:29, 12 June 2026 (UTC)

== Wiki x AI preconference day @ Wikimania ==
There will be a preconference day at Wikimania about [[meta:Artificial_intelligence/2026_Wiki_AI | Wiki AI]]. It will be mostly offline, but there will be at least one hybrid session for demos of community-developed AI tools and workflows. 

* If you've built something cool, that is a chance to show it off, list it on the gallery of tools in progress, and get feedback. 

* If you could ask the people shaping AI on the wikis (WMF, tool builders, model trainers, GLAM and policy folks) a question, what would it be? 

Cheers, &lt;span style=&quot;padding:0 2px 0 2px;background-color:white;color:#bbb;&quot;&gt;&amp;ndash;[[User:Sj|SJ]][[User Talk:Sj|&lt;span style=&quot;color:#ff9900;&quot;&gt;+&lt;/span&gt;]]&lt;/span&gt; 23:12, 20 June 2026 (UTC) and Alaexis&lt;br&gt;{{comment|1=Copied from https://en.wikiversity.org/w/index.php?title=Talk%3AMotivation_and_emotion%2FAssessment%2FUsing_generative_AI&amp;diff=2816357&amp;oldid=2807052}}

== RFC about AI-generated content in Wikimedia Commons ==

You are invited to participate in a [[c:Commons:Requests for comment/Policy update for AI content|request for comment on Wikimedia Commons about a policy update for AI content]]. This may affect files that are uploaded to Wikimedia Commons for use on this project. Thank you. [[m:User:Codename Noreste|Codename Noreste]] ([[m:User talk:Codename Noreste|discuss]]) 17:12, 23 June 2026 (UTC)
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== Deployment of Legal and Safety Contacts Link in the Footer of Your Wiki ==

&lt;section begin=&quot;Message&quot;/&gt;
'''Legal &amp; Safety Contacts'''

Hello community, the Wikimedia Foundation has provided a [[wmf:Special:MyLanguage/Legal:Wikimedia Foundation Legal and Safety Contact Information|single legal and safety contact page]], to be linked in the footer of your wiki, to ensure access to accurate legal information. This is a regulatory requirement. We have already rolled out links to English, German, Italian, Spanish and other wikis and we will deploy to your wiki soon. [[m:Special:MyLanguage/Wikimedia_Foundation_Legal_and_Safety_Contacts_FAQ|Please read more on the project page]] and leave any comments in this thread or on the [[m:Special:MyLanguage/Talk:Wikimedia Foundation Legal and Safety Contacts FAQ|talk page]].

&lt;section end=&quot;Message&quot;/&gt;

-- [[User:Sannita (WMF)|User:Sannita (WMF)]] ([[User talk:Sannita (WMF)|talk]]) 13:31, 25 June 2026 (UTC)
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== Preparing manuscript for submission to the ''WikiJournal of Humanities'' ==

I am working on preparing an article in Wikipedia for a dual goal of submitting it for a featured article candidacy in Wikipedia and submitting it to the ''[[WikiJournal of Humanities]]''.  I have an open request for pre-submission peer review at [[en:Wikipedia:Wikipedia:Peer review/Rei Ayanami/archive2|Wikipedia:Peer review/Rei Ayanami/archive2]], and I am asking for someone experienced with submitting journals to WikiJournals.  The article is not ready for submission, and I would like to know where I can get assistance from users who submitted articles to the journal, but did not necessarily review them.  Furthermore, I said there that submitting to the ''WikiJournal of Humanities'' depends on whether the article attains featured article status in Wikipedia, as I would like to use the featured article as a manuscript for a journal article. [[User:Z. Patterson|Z. Patterson]] ([[User talk:Z. Patterson|discuss]] • [[Special:Contributions/Z. Patterson|contribs]]) 12:04, 30 June 2026 (UTC)

== I could probably save many kilobytes by compressing my LLM chat history ==

I could probably save many kilobytes by compressing my LLM chat history. Would that be ok? I like how my new &quot;method&quot; looks: [[User:ThinkingScience/All General AI Prompt History Archive]] very compressed and neat. I like how to find new ways to make people using LLMs not become &quot;secondary citizens&quot;. [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 05:05, 7 July 2026 (UTC)

:{{replyto|ThinkingScience}} I do not see why not. People should be able to look in your page's history for LLM chats before your compression. [[User:Z. Patterson|Z. Patterson]] ([[User talk:Z. Patterson|discuss]] • [[Special:Contributions/Z. Patterson|contribs]]) 02:34, 14 July 2026 (UTC)
::That's good. Then an admin if they have been instructed to compress...then they can delete the edit history perhaps and save space, then it's up to the admin and I don't need to worry about taking up too many resources. That's good! [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 07:50, 14 July 2026 (UTC)

== July 2026 Wikimedia Café meetups regarding Wikimedia governance and options for reform ==

&lt;div class=&quot;border-box&quot; style=&quot;background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)&quot;&gt;
&lt;div class=&quot;box&quot; style=&quot;float:left; padding-top: 10px; padding-right: 10px; padding-left: 10px; padding-bottom: 10px;&quot;&gt;[[File:Wikimedia Café logo in plain SVG format.svg|60px|alt=The logo for the Wikimedia Café]]&lt;/div&gt;
Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities in July. Both sessions will focus on Wikimedia governance, including possible follow-ups to the [https://meta.wikimedia.org/wiki/Movement_Charter Movement Charter] and options for reform. Participants may attend either or both Café sessions. 

This month, to deconflict the Café meetups from Wikimania, the meetups will be held one day later than usual.

#'''26 July 2026 15:00 UTC''' ([https://zonestamp.toolforge.org/1785078000 timestamp converter]), at a time friendly to the Americas, Africa, and Europe
#'''27 July 2026 03:00 UTC''' ([https://zonestamp.toolforge.org/1785121200 timestamp converter]), at a time friendly to Asia and the Pacific

Please see the Café page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]!
&lt;br /&gt;
[[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]&lt;/div&gt;
&lt;span style=&quot;white-space:nowrap;&quot;&gt;[[User:Pine|&lt;span style=&quot;color:#01796f; text-shadow:#00BFFF 0 0 1.0em&quot;&gt;↠Pine&lt;/span&gt;]] [[User talk:Pine|&lt;span style=&quot;color:DeepSkyBlue&quot;&gt;(&lt;b style=&quot;color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em&quot;&gt;✉&lt;/b&gt;)&lt;/span&gt;]]&lt;/span&gt; 03:51, 13 July 2026 (UTC)

:@[[User:Pine|Pine]], Some wikimedians who may contribute less popular, but hopefully productive, comments,  are those who are currently blocked on Meta. I myself was blocked infinitely on Meta several years ago, but was allowed to contInue contributing on other wikimedia projects, such as this one. There are many long term contributors, some who have contrIbuted tens of thousands of edits  to wikimedia projects who are are now blocked on specific projects. 
:Those who happen to be blocked on Meta have,  to all intents and purposes, been removed from the wikimedia movement.
:Ottawahitech [[Special:Contributions/&amp;#126;2026-40514-81|&amp;#126;2026-40514-81]] ([[User talk:&amp;#126;2026-40514-81|talk]]) 21:41, 18 July 2026 (UTC)
:*Hi Ottawahitech, thanks for the comment. Reviewing local bans on Meta is outside of my current scope, but if you have comments that you'd like to share here in the Colloquium relating to Wikiversity, I can consider the comments in the scope of the Café discussions. &lt;span style=&quot;white-space:nowrap;&quot;&gt;[[User:Pine|&lt;span style=&quot;color:#01796f; text-shadow:#00BFFF 0 0 1.0em&quot;&gt;↠Pine&lt;/span&gt;]] [[User talk:Pine|&lt;span style=&quot;color:DeepSkyBlue&quot;&gt;(&lt;b style=&quot;color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em&quot;&gt;✉&lt;/b&gt;)&lt;/span&gt;]]&lt;/span&gt; 20:05, 25 July 2026 (UTC)

== Request for comment (the future of Abstract Wikipedia) ==

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot; class=&quot;mw-content-ltr&quot;&gt;

You are invited to voice your opinions in a [[:m:Requests for comment/The future of Abstract Wikipedia|request for comment about the future of Abstract Wikipedia]]. {{Int:Feedback-thanks-title}}
[[:m:User:Kowal2701|Kowal2701]] ([[:m:User talk:Kowal2701|talk]])  12:25, 24 July 2026 (UTC)
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      <id>2819795</id>
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        <id>10242</id>
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      <comment>/* Wiki x AI preconference day @ Wikimania */ archive to [[Wikiversity:Colloquium/archives/June 2026#Wiki x AI preconference day @ Wikimania]] ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]])</comment>
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== Draft inactivity policy ==

I created [[Wikiversity:Inactivity policy]] as a start. Any experienced Wikiversity user may feel free to expand it. This is also one-to-two step(s) towards opting out of the [[m:Admin activity review|AAR process]].

However, I made a bold change to reduce the response timeframe from one month to two weeks. In addition, should we reduce the inactivity timeframe to one year? For the latter, most projects use that timeframe and I suggested this for consistency. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:57, 4 June 2026 (UTC)

:I support those suggestions. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:55, 4 June 2026 (UTC)
: Juandev has posted some comments on the [[Wikiversity talk:Inactivity policy|talk page]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:30, 12 June 2026 (UTC)
: Thanks for creating this draft. I've made some changes (including moving back to a one month response timeframe) and moved it from draft to proposed policy.
: Based on the discussion on the talk page, I think it is close to ready (or ready) to be formally proposed as a policy (by adding it to the sitenotice) to allow wider discussion and hopefully adoption. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 06:45, 21 July 2026 (UTC)

== Proposed user group and/or possible policy changes ==

{{tracked|T430416|fixed}}
I want to discuss about user group and possible policy changes.

# First, interface administrators. I don't think we should allow interface administrators to remove their permission from their own account, since we have multiple active bureaucrats and we can ask them to remove the permission when done, or for them to add a temporary grant. This is according to the [[Wikiversity:IA|current IA policy]]. I also left [[Wikiversity talk:Interface administrators#My thoughts about this user group|my thoughts on the relevant talk page]].
# Second, curators. Given that curators have some sensitive custodian rights (such as &lt;code&gt;delete&lt;/code&gt; [but not &lt;code&gt;undelete&lt;/code&gt; or similar rights that allow viewing deleted content, unless the curatorship process is RFA-like] and &lt;code&gt;protect&lt;/code&gt;), it would probably make more sense only for bureaucrats to grant and remove it, on par with them granting (but not removing) custodian permissions.
# Third, about probationary custodians. [[Wikiversity:Probationary custodians]] is currently marked as historical, and the process might still exist on [[Wikiversity:Custodianship]]. Therefore, to maintain consistency with [[Wikiversity:Curatorship#How does one become a curator?]], I propose that we repeal the probationary custodianship process and change it more or less to align with the curatorship process, effectively making probationary custodians permanent ones. However, custodian mentors would still be retained.

Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:55, 5 June 2026 (UTC)

:#Yes, I agree.
:#Thats a good point, but I dont know. At least I dont think its a good idea that both groups i.e. crats and custodiants can do that, it may create chaos.
:#Another good point. It seems to me that the current situation is somewhat unclear and should be clarified. I understand the original status of [[Wikiversity:Probationary custodians|Probationary custodians]] as a historicall and invalid, but at the same time I consider myself a probationary custodian, because on the Wikiversity:Custodianship page in the ''[[Wikiversity:Custodianship#How does one become a custodian?|How does one become a custodian?]]'' section it says, I quote, ''&quot;II ...then you will be approved as a probationary custodian for a period of at least four weeks&quot;''.
:::Mentors should definitely be kept, but for certain applicants the probation and mentorship should be abolished. For example, if someone was an active custodian for 5 years, then loses their rights or gives them up for a year and then wants to resume their custodial activities, there is no reason for them to undergo a training period. It burdens both the mentors and the community with double voting. The only exception could be a situation where policies or tools for custodians change significantly during that year, or the candidate wants to.
:[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 06:08, 9 June 2026 (UTC)

== Towards an Ethics policy ==

In connection with the [[Wikiversity:Community Review/Removal of Wikidebates|discussion of Wikidebates]], I said that it would be good to establish a policy on ethics, or rather a boundary between ethical and unethical content, so that we don't have to discuss individual cases. In addition, today we also have some global policies that prohibit, for example, attacks on members of the Wikimedia movement or undermining other projects.

However, at the very beginning, I would start by collecting your opinions. What content or what research should not be allowed on Wikiversity? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 05:52, 9 June 2026 (UTC)

:One ethical issue that I think should be non-controversial is related to good faith in the learning modules. So, learning materials should not be hoaxes or encourage behavior or methods that don't work or that misrepresent the facts or the likelihood of something occurring, etc. and authors should also not plagiarize or misrepresent authorship, etc. That was quite a run-on, but I hope that others can tease out what I mean here. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:39, 9 June 2026 (UTC)
::I look at it from a practical perspective. We can give that to the policy, but I see the problem in that we are not able to check it except plagiarism.
::Plagiarism can be partially detected during patrolling. I see a new text, I put part of it in Google and I check if it is copied from the web. It is a problem with copying from books or other offline sources, but sometimes it happens that someone finds out that something is copied from somewhere and it can be deleted.
::The biggest issue we have here is that we are missing Wikipedia's control mechanism: references. Only some types of resources on Wikiversity require references. In-line references are not often used in courses, exercises, lectures, etc. We are thus deprived of one of the excellent control mechanisms and the only option is for the increase in the number of members with various qualifications to check it for their colleagues. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:59, 9 June 2026 (UTC)
:::Having a policy and enforcing that policy are indeed two different things. If we are only concerned with issues that we can definitively enforce, then that will definitely change this conversation. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 08:06, 9 June 2026 (UTC)
::::ok [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:55, 13 June 2026 (UTC)
:AI generated content should not be allowed as it is inherently plagiarism. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:14, 9 June 2026 (UTC)
::And if the user mention it was generated by an AI? Note that there is something called as public domain, that is the author wave its rights. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:53, 9 June 2026 (UTC)
:::Plagiarism isn’t copyright violation. Crediting the AI is not crediting the authors the AI stole from without credit. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 10:18, 9 June 2026 (UTC)
::::I see, now I understand your point. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:56, 13 June 2026 (UTC)

== New nomination template(s) ==

I created {{tlx|Nomination}} when someone requests curator or custodian permissions, which often at least require mentorship. On the other hand, I might create {{tlx|Nomination 2}}, in which the latter does not have a section about mentorship (often used for bureaucrat or interface administrator nominations). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:29, 12 June 2026 (UTC)

== RFC about AI-generated content in Wikimedia Commons ==

You are invited to participate in a [[c:Commons:Requests for comment/Policy update for AI content|request for comment on Wikimedia Commons about a policy update for AI content]]. This may affect files that are uploaded to Wikimedia Commons for use on this project. Thank you. [[m:User:Codename Noreste|Codename Noreste]] ([[m:User talk:Codename Noreste|discuss]]) 17:12, 23 June 2026 (UTC)
&lt;!-- Message sent by User:Codename Noreste@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&amp;oldid=30513860 --&gt;

== Deployment of Legal and Safety Contacts Link in the Footer of Your Wiki ==

&lt;section begin=&quot;Message&quot;/&gt;
'''Legal &amp; Safety Contacts'''

Hello community, the Wikimedia Foundation has provided a [[wmf:Special:MyLanguage/Legal:Wikimedia Foundation Legal and Safety Contact Information|single legal and safety contact page]], to be linked in the footer of your wiki, to ensure access to accurate legal information. This is a regulatory requirement. We have already rolled out links to English, German, Italian, Spanish and other wikis and we will deploy to your wiki soon. [[m:Special:MyLanguage/Wikimedia_Foundation_Legal_and_Safety_Contacts_FAQ|Please read more on the project page]] and leave any comments in this thread or on the [[m:Special:MyLanguage/Talk:Wikimedia Foundation Legal and Safety Contacts FAQ|talk page]].

&lt;section end=&quot;Message&quot;/&gt;

-- [[User:Sannita (WMF)|User:Sannita (WMF)]] ([[User talk:Sannita (WMF)|talk]]) 13:31, 25 June 2026 (UTC)
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== Preparing manuscript for submission to the ''WikiJournal of Humanities'' ==

I am working on preparing an article in Wikipedia for a dual goal of submitting it for a featured article candidacy in Wikipedia and submitting it to the ''[[WikiJournal of Humanities]]''.  I have an open request for pre-submission peer review at [[en:Wikipedia:Wikipedia:Peer review/Rei Ayanami/archive2|Wikipedia:Peer review/Rei Ayanami/archive2]], and I am asking for someone experienced with submitting journals to WikiJournals.  The article is not ready for submission, and I would like to know where I can get assistance from users who submitted articles to the journal, but did not necessarily review them.  Furthermore, I said there that submitting to the ''WikiJournal of Humanities'' depends on whether the article attains featured article status in Wikipedia, as I would like to use the featured article as a manuscript for a journal article. [[User:Z. Patterson|Z. Patterson]] ([[User talk:Z. Patterson|discuss]] • [[Special:Contributions/Z. Patterson|contribs]]) 12:04, 30 June 2026 (UTC)

== I could probably save many kilobytes by compressing my LLM chat history ==

I could probably save many kilobytes by compressing my LLM chat history. Would that be ok? I like how my new &quot;method&quot; looks: [[User:ThinkingScience/All General AI Prompt History Archive]] very compressed and neat. I like how to find new ways to make people using LLMs not become &quot;secondary citizens&quot;. [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 05:05, 7 July 2026 (UTC)

:{{replyto|ThinkingScience}} I do not see why not. People should be able to look in your page's history for LLM chats before your compression. [[User:Z. Patterson|Z. Patterson]] ([[User talk:Z. Patterson|discuss]] • [[Special:Contributions/Z. Patterson|contribs]]) 02:34, 14 July 2026 (UTC)
::That's good. Then an admin if they have been instructed to compress...then they can delete the edit history perhaps and save space, then it's up to the admin and I don't need to worry about taking up too many resources. That's good! [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 07:50, 14 July 2026 (UTC)

== July 2026 Wikimedia Café meetups regarding Wikimedia governance and options for reform ==

&lt;div class=&quot;border-box&quot; style=&quot;background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)&quot;&gt;
&lt;div class=&quot;box&quot; style=&quot;float:left; padding-top: 10px; padding-right: 10px; padding-left: 10px; padding-bottom: 10px;&quot;&gt;[[File:Wikimedia Café logo in plain SVG format.svg|60px|alt=The logo for the Wikimedia Café]]&lt;/div&gt;
Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities in July. Both sessions will focus on Wikimedia governance, including possible follow-ups to the [https://meta.wikimedia.org/wiki/Movement_Charter Movement Charter] and options for reform. Participants may attend either or both Café sessions. 

This month, to deconflict the Café meetups from Wikimania, the meetups will be held one day later than usual.

#'''26 July 2026 15:00 UTC''' ([https://zonestamp.toolforge.org/1785078000 timestamp converter]), at a time friendly to the Americas, Africa, and Europe
#'''27 July 2026 03:00 UTC''' ([https://zonestamp.toolforge.org/1785121200 timestamp converter]), at a time friendly to Asia and the Pacific

Please see the Café page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]!
&lt;br /&gt;
[[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]&lt;/div&gt;
&lt;span style=&quot;white-space:nowrap;&quot;&gt;[[User:Pine|&lt;span style=&quot;color:#01796f; text-shadow:#00BFFF 0 0 1.0em&quot;&gt;↠Pine&lt;/span&gt;]] [[User talk:Pine|&lt;span style=&quot;color:DeepSkyBlue&quot;&gt;(&lt;b style=&quot;color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em&quot;&gt;✉&lt;/b&gt;)&lt;/span&gt;]]&lt;/span&gt; 03:51, 13 July 2026 (UTC)

:@[[User:Pine|Pine]], Some wikimedians who may contribute less popular, but hopefully productive, comments,  are those who are currently blocked on Meta. I myself was blocked infinitely on Meta several years ago, but was allowed to contInue contributing on other wikimedia projects, such as this one. There are many long term contributors, some who have contrIbuted tens of thousands of edits  to wikimedia projects who are are now blocked on specific projects. 
:Those who happen to be blocked on Meta have,  to all intents and purposes, been removed from the wikimedia movement.
:Ottawahitech [[Special:Contributions/&amp;#126;2026-40514-81|&amp;#126;2026-40514-81]] ([[User talk:&amp;#126;2026-40514-81|talk]]) 21:41, 18 July 2026 (UTC)
:*Hi Ottawahitech, thanks for the comment. Reviewing local bans on Meta is outside of my current scope, but if you have comments that you'd like to share here in the Colloquium relating to Wikiversity, I can consider the comments in the scope of the Café discussions. &lt;span style=&quot;white-space:nowrap;&quot;&gt;[[User:Pine|&lt;span style=&quot;color:#01796f; text-shadow:#00BFFF 0 0 1.0em&quot;&gt;↠Pine&lt;/span&gt;]] [[User talk:Pine|&lt;span style=&quot;color:DeepSkyBlue&quot;&gt;(&lt;b style=&quot;color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em&quot;&gt;✉&lt;/b&gt;)&lt;/span&gt;]]&lt;/span&gt; 20:05, 25 July 2026 (UTC)

== Request for comment (the future of Abstract Wikipedia) ==

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot; class=&quot;mw-content-ltr&quot;&gt;

You are invited to voice your opinions in a [[:m:Requests for comment/The future of Abstract Wikipedia|request for comment about the future of Abstract Wikipedia]]. {{Int:Feedback-thanks-title}}
[[:m:User:Kowal2701|Kowal2701]] ([[:m:User talk:Kowal2701|talk]])  12:25, 24 July 2026 (UTC)
&lt;/bdi&gt;
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      <sha1>c8yuan3cqw4p25k11okl34gildc75ss</sha1>
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        <id>10242</id>
      </contributor>
      <comment>/* July 2026 Wikimedia Café meetups regarding Wikimedia governance and options for reform */ archive to [[Wikiversity:Colloquium/archives/July 2026#July 2026 Wikimedia Café meetups regarding Wikimedia governance and options for reform]] ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]])</comment>
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      <text bytes="14073" sha1="iqr8rz4jwmrewy1kda96ffbzkl4zc95" xml:space="preserve">{{Wikiversity:Colloquium/Header}}
&lt;!-- MESSAGES GO BELOW --&gt;

== Draft inactivity policy ==

I created [[Wikiversity:Inactivity policy]] as a start. Any experienced Wikiversity user may feel free to expand it. This is also one-to-two step(s) towards opting out of the [[m:Admin activity review|AAR process]].

However, I made a bold change to reduce the response timeframe from one month to two weeks. In addition, should we reduce the inactivity timeframe to one year? For the latter, most projects use that timeframe and I suggested this for consistency. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:57, 4 June 2026 (UTC)

:I support those suggestions. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:55, 4 June 2026 (UTC)
: Juandev has posted some comments on the [[Wikiversity talk:Inactivity policy|talk page]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:30, 12 June 2026 (UTC)
: Thanks for creating this draft. I've made some changes (including moving back to a one month response timeframe) and moved it from draft to proposed policy.
: Based on the discussion on the talk page, I think it is close to ready (or ready) to be formally proposed as a policy (by adding it to the sitenotice) to allow wider discussion and hopefully adoption. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 06:45, 21 July 2026 (UTC)

== Proposed user group and/or possible policy changes ==

{{tracked|T430416|fixed}}
I want to discuss about user group and possible policy changes.

# First, interface administrators. I don't think we should allow interface administrators to remove their permission from their own account, since we have multiple active bureaucrats and we can ask them to remove the permission when done, or for them to add a temporary grant. This is according to the [[Wikiversity:IA|current IA policy]]. I also left [[Wikiversity talk:Interface administrators#My thoughts about this user group|my thoughts on the relevant talk page]].
# Second, curators. Given that curators have some sensitive custodian rights (such as &lt;code&gt;delete&lt;/code&gt; [but not &lt;code&gt;undelete&lt;/code&gt; or similar rights that allow viewing deleted content, unless the curatorship process is RFA-like] and &lt;code&gt;protect&lt;/code&gt;), it would probably make more sense only for bureaucrats to grant and remove it, on par with them granting (but not removing) custodian permissions.
# Third, about probationary custodians. [[Wikiversity:Probationary custodians]] is currently marked as historical, and the process might still exist on [[Wikiversity:Custodianship]]. Therefore, to maintain consistency with [[Wikiversity:Curatorship#How does one become a curator?]], I propose that we repeal the probationary custodianship process and change it more or less to align with the curatorship process, effectively making probationary custodians permanent ones. However, custodian mentors would still be retained.

Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:55, 5 June 2026 (UTC)

:#Yes, I agree.
:#Thats a good point, but I dont know. At least I dont think its a good idea that both groups i.e. crats and custodiants can do that, it may create chaos.
:#Another good point. It seems to me that the current situation is somewhat unclear and should be clarified. I understand the original status of [[Wikiversity:Probationary custodians|Probationary custodians]] as a historicall and invalid, but at the same time I consider myself a probationary custodian, because on the Wikiversity:Custodianship page in the ''[[Wikiversity:Custodianship#How does one become a custodian?|How does one become a custodian?]]'' section it says, I quote, ''&quot;II ...then you will be approved as a probationary custodian for a period of at least four weeks&quot;''.
:::Mentors should definitely be kept, but for certain applicants the probation and mentorship should be abolished. For example, if someone was an active custodian for 5 years, then loses their rights or gives them up for a year and then wants to resume their custodial activities, there is no reason for them to undergo a training period. It burdens both the mentors and the community with double voting. The only exception could be a situation where policies or tools for custodians change significantly during that year, or the candidate wants to.
:[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 06:08, 9 June 2026 (UTC)

== Towards an Ethics policy ==

In connection with the [[Wikiversity:Community Review/Removal of Wikidebates|discussion of Wikidebates]], I said that it would be good to establish a policy on ethics, or rather a boundary between ethical and unethical content, so that we don't have to discuss individual cases. In addition, today we also have some global policies that prohibit, for example, attacks on members of the Wikimedia movement or undermining other projects.

However, at the very beginning, I would start by collecting your opinions. What content or what research should not be allowed on Wikiversity? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 05:52, 9 June 2026 (UTC)

:One ethical issue that I think should be non-controversial is related to good faith in the learning modules. So, learning materials should not be hoaxes or encourage behavior or methods that don't work or that misrepresent the facts or the likelihood of something occurring, etc. and authors should also not plagiarize or misrepresent authorship, etc. That was quite a run-on, but I hope that others can tease out what I mean here. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:39, 9 June 2026 (UTC)
::I look at it from a practical perspective. We can give that to the policy, but I see the problem in that we are not able to check it except plagiarism.
::Plagiarism can be partially detected during patrolling. I see a new text, I put part of it in Google and I check if it is copied from the web. It is a problem with copying from books or other offline sources, but sometimes it happens that someone finds out that something is copied from somewhere and it can be deleted.
::The biggest issue we have here is that we are missing Wikipedia's control mechanism: references. Only some types of resources on Wikiversity require references. In-line references are not often used in courses, exercises, lectures, etc. We are thus deprived of one of the excellent control mechanisms and the only option is for the increase in the number of members with various qualifications to check it for their colleagues. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:59, 9 June 2026 (UTC)
:::Having a policy and enforcing that policy are indeed two different things. If we are only concerned with issues that we can definitively enforce, then that will definitely change this conversation. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 08:06, 9 June 2026 (UTC)
::::ok [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:55, 13 June 2026 (UTC)
:AI generated content should not be allowed as it is inherently plagiarism. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:14, 9 June 2026 (UTC)
::And if the user mention it was generated by an AI? Note that there is something called as public domain, that is the author wave its rights. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:53, 9 June 2026 (UTC)
:::Plagiarism isn’t copyright violation. Crediting the AI is not crediting the authors the AI stole from without credit. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 10:18, 9 June 2026 (UTC)
::::I see, now I understand your point. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:56, 13 June 2026 (UTC)

== New nomination template(s) ==

I created {{tlx|Nomination}} when someone requests curator or custodian permissions, which often at least require mentorship. On the other hand, I might create {{tlx|Nomination 2}}, in which the latter does not have a section about mentorship (often used for bureaucrat or interface administrator nominations). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:29, 12 June 2026 (UTC)

== RFC about AI-generated content in Wikimedia Commons ==

You are invited to participate in a [[c:Commons:Requests for comment/Policy update for AI content|request for comment on Wikimedia Commons about a policy update for AI content]]. This may affect files that are uploaded to Wikimedia Commons for use on this project. Thank you. [[m:User:Codename Noreste|Codename Noreste]] ([[m:User talk:Codename Noreste|discuss]]) 17:12, 23 June 2026 (UTC)
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== Deployment of Legal and Safety Contacts Link in the Footer of Your Wiki ==

&lt;section begin=&quot;Message&quot;/&gt;
'''Legal &amp; Safety Contacts'''

Hello community, the Wikimedia Foundation has provided a [[wmf:Special:MyLanguage/Legal:Wikimedia Foundation Legal and Safety Contact Information|single legal and safety contact page]], to be linked in the footer of your wiki, to ensure access to accurate legal information. This is a regulatory requirement. We have already rolled out links to English, German, Italian, Spanish and other wikis and we will deploy to your wiki soon. [[m:Special:MyLanguage/Wikimedia_Foundation_Legal_and_Safety_Contacts_FAQ|Please read more on the project page]] and leave any comments in this thread or on the [[m:Special:MyLanguage/Talk:Wikimedia Foundation Legal and Safety Contacts FAQ|talk page]].

&lt;section end=&quot;Message&quot;/&gt;

-- [[User:Sannita (WMF)|User:Sannita (WMF)]] ([[User talk:Sannita (WMF)|talk]]) 13:31, 25 June 2026 (UTC)
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== Preparing manuscript for submission to the ''WikiJournal of Humanities'' ==

I am working on preparing an article in Wikipedia for a dual goal of submitting it for a featured article candidacy in Wikipedia and submitting it to the ''[[WikiJournal of Humanities]]''.  I have an open request for pre-submission peer review at [[en:Wikipedia:Wikipedia:Peer review/Rei Ayanami/archive2|Wikipedia:Peer review/Rei Ayanami/archive2]], and I am asking for someone experienced with submitting journals to WikiJournals.  The article is not ready for submission, and I would like to know where I can get assistance from users who submitted articles to the journal, but did not necessarily review them.  Furthermore, I said there that submitting to the ''WikiJournal of Humanities'' depends on whether the article attains featured article status in Wikipedia, as I would like to use the featured article as a manuscript for a journal article. [[User:Z. Patterson|Z. Patterson]] ([[User talk:Z. Patterson|discuss]] • [[Special:Contributions/Z. Patterson|contribs]]) 12:04, 30 June 2026 (UTC)

== I could probably save many kilobytes by compressing my LLM chat history ==

I could probably save many kilobytes by compressing my LLM chat history. Would that be ok? I like how my new &quot;method&quot; looks: [[User:ThinkingScience/All General AI Prompt History Archive]] very compressed and neat. I like how to find new ways to make people using LLMs not become &quot;secondary citizens&quot;. [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 05:05, 7 July 2026 (UTC)

:{{replyto|ThinkingScience}} I do not see why not. People should be able to look in your page's history for LLM chats before your compression. [[User:Z. Patterson|Z. Patterson]] ([[User talk:Z. Patterson|discuss]] • [[Special:Contributions/Z. Patterson|contribs]]) 02:34, 14 July 2026 (UTC)
::That's good. Then an admin if they have been instructed to compress...then they can delete the edit history perhaps and save space, then it's up to the admin and I don't need to worry about taking up too many resources. That's good! [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 07:50, 14 July 2026 (UTC)

== Request for comment (the future of Abstract Wikipedia) ==

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot; class=&quot;mw-content-ltr&quot;&gt;

You are invited to voice your opinions in a [[:m:Requests for comment/The future of Abstract Wikipedia|request for comment about the future of Abstract Wikipedia]]. {{Int:Feedback-thanks-title}}
[[:m:User:Kowal2701|Kowal2701]] ([[:m:User talk:Kowal2701|talk]])  12:25, 24 July 2026 (UTC)
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      <sha1>iqr8rz4jwmrewy1kda96ffbzkl4zc95</sha1>
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  </page>
  <page>
    <title>Help:Wikitext quick reference</title>
    <ns>12</ns>
    <id>1644</id>
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      <id>2819860</id>
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      <minor />
      <comment>/* Wikitext markup */</comment>
      <origin>2819860</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10650" sha1="f5g8ypjszddwlea93l57i088qfg260q" xml:space="preserve">==Wikitext markup==
&lt;noinclude&gt;{{main|Help:Editing}}&lt;/noinclude&gt;
{{for|further guidance|Wikiversity:Wikitext}}
Making your page look the way you want with [[w:wikitext|Wikitext]]

* You can see some more detailed examples at [[Help:Wiki markup examples]].
* If you want to try out things without danger of doing any harm, you can do so in the [[{{ns:4}}:Sandbox]].

=== Basic text formatting ===
{| class=&quot;wikitable&quot; style=&quot;width:100%;&quot;
 ! width=&quot;25%&quot; | Name
 ! width=&quot;37%&quot; | Typing
 ! width=&quot;37%&quot; | Displays
 |-
 | [[File:Button bold.png]] Bold
 | &lt;syntaxhighlight lang=&quot;wikitext&quot; inline=&quot;yes&quot; &gt;Make text '''bold'''&lt;/syntaxhighlight&gt;
 | Make text '''bold'''
 |-
 | [[File:Button italic.png]] Italic
 | &lt;syntaxhighlight lang=&quot;wikitext&quot; inline=&quot;yes&quot;&gt;Make text ''italic''&lt;/syntaxhighlight&gt;
 | Make text ''italic''
 |-
 | [[File:Button link.png]] Internal Link
 | &lt;syntaxhighlight lang=&quot;wikitext&quot; inline=&quot;yes&quot;&gt;Link to [[Help:Editing|a page]] or [[#Basic text formatting|a section of a page]].&lt;/syntaxhighlight&gt;
 | Link to [[Help:Editing|a page]] or [[#Basic text formatting|a section of a page]].
 |-
 | [[File:Button extlink.png]] External Link
 | &lt;syntaxhighlight lang=&quot;wikitext&quot; inline=&quot;yes&quot;&gt;Link to [http://en.wikiversity.org/wiki/Help:Editing a page] or [http://en.wikiversity.org/wiki/Help:Editing#Editing_basics a section of a page].&lt;/syntaxhighlight&gt;
 | Link to [http://en.wikiversity.org/wiki/Help:Editing a page] or [http://en.wikiversity.org/wiki/Help:Editing#Editing_basics a section of a page].
 |-
 | [[File:Button headline.png]] Level 2 Heading
 | &lt;syntaxhighlight lang=&quot;wikitext&quot; inline=&quot;yes&quot;&gt;== Section Heading ==&lt;/syntaxhighlight&gt;
 | {{font|font=Georgia;|size=1.5em|Section Heading}}
 |-
 | [[File:Button image.png]] Embed File
 | &lt;syntaxhighlight lang=&quot;wikitext&quot; inline=&quot;yes&quot;&gt;[[File:Button image.png]]&lt;/syntaxhighlight&gt;
 | [[File:Button image.png]]
 |-
 | [[File:Button media.png]] File link
 | &lt;syntaxhighlight lang=&quot;wikitext&quot; inline=&quot;yes&quot;&gt;[[Media:Button media.png]]&lt;/syntaxhighlight&gt;
 | [[Media:Button media.png]]
 |-
 | [[File:Button math.png]] Mathematical Formulas ([[m:Help:Displaying a formula|LaTeX]])
 | [[Help:Formula|&lt;nowiki&gt;&lt;math&gt;2^3&lt;/math&gt;&lt;/nowiki&gt;]]
 | &lt;math&gt;2^3&lt;/math&gt;
 |-
 | [[File:Nowiki_icon.png]] Ignore wiki markup
 | &lt;syntaxhighlight lang=&quot;wikitext&quot; inline=&quot;yes&quot;&gt;&lt;nowiki&gt;&lt;math&gt;2^3&lt;/math&gt;&amp;lt;/nowiki&amp;gt;&lt;/syntaxhighlight&gt;
 | &lt;nowiki&gt;&lt;math&gt;2^3&lt;/math&gt;&lt;/nowiki&gt;
 |-
 | [[File:Button sig.png]] Signature
 | &lt;syntaxhighlight lang=&quot;wikitext&quot; inline=&quot;yes&quot;&gt;Thanks guys --~~~~&lt;/syntaxhighlight&gt;
 | Thanks guys --[[User:Karl Wick|Karl Wick]] 07:46, 27 November 2005 (UTC)
 |-
 | [[File:Button hr.png]] Horizontal line
 | &lt;syntaxhighlight lang=&quot;wikitext&quot; inline=&quot;yes&quot;&gt;----&lt;/syntaxhighlight&gt;
 |
----
 |-
 | [[File:Button redirect.png]] Page Redirect
 | &lt;syntaxhighlight lang=&quot;wikitext&quot; inline=&quot;yes&quot;&gt;#redirect [[Help:Editing#Basic text formatting]]&lt;/syntaxhighlight&gt;
 | [[File:Redirect arrow without text.svg|#REDIRECT]] [[Help:Editing#Basic text formatting]]
 |}

For a list of HTML tags that are allowed, see [[Help:HTML in wikitext|HTML in wikitext]]. However, you should avoid HTML in favor of Wiki markup whenever possible.

===Just show what I typed===
A few different kinds of formatting will tell the Wiki to display things as you typed them.

{| class=&quot;wikitable&quot; style=&quot;width:100%;&quot;
 ! width=&quot;25%&quot; | Name
 ! width=&quot;37%&quot; | Typing
 ! width=&quot;37%&quot; | Displays
 |-
 | '''&lt;syntaxhighlight lang=&quot;wikitext&quot; inline=yes&gt;&lt;nowiki&gt;&lt;/syntaxhighlight&gt; tags'''
 |&lt;syntaxhighlight lang=&quot;wikitext&quot;&gt;
&lt;nowiki&gt;
The nowiki tag ignores [[Wiki]] ''markup''.
It reformats text by removing newlines    and multiple spaces.
It still interprets special characters: &amp;amp;rarr;
&lt;/nowiki&gt;
&lt;/syntaxhighlight&gt;
|
&lt;nowiki&gt;
The nowiki tag ignores [[Wiki]] ''markup''.
It reformats text by removing newlines    and multiple spaces.
It still interprets special characters: &amp;rarr;
&lt;/nowiki&gt;
 |-
 | '''&amp;lt;pre&amp;gt;'''
 |&lt;syntaxhighlight lang=&quot;wikitext&quot;&gt;
&lt;pre&gt;
The pre tag ignores [[Wiki]] and &lt;strong&gt;HTML&lt;/strong&gt; ''markup''.
It also doesn't     reformat text.
It still interprets special characters: &amp;rarr;
&lt;/pre&gt;
&lt;/syntaxhighlight&gt;
 |
&lt;pre&gt;
The pre tag ignores [[Wiki]] and &lt;strong&gt;HTML&lt;/strong&gt; ''markup''.
It also doesn't     reformat text.
It still interprets HTML special characters: &amp;rarr;
&lt;/pre&gt;
 |-
 | '''Leading spaces'''
 |
Leading spaces are another way to preserve formatting.

&lt;syntaxhighlight lang=&quot;wikitext&quot;&gt;
 Putting a space at the beginning of each line
 stops the text   from being reformatted. It still
 interprets [[Wiki]] ''markup'' and special
 characters: &amp;amp;rarr;
&lt;/syntaxhighlight&gt;

but it puts the text in a box.
 |
Leading spaces are another way to preserve formatting.

 Putting a space at the beginning of each line
 stops the text   from being reformatted. It still
 interprets [[Wiki]] ''markup'' and special
 characters: &amp;rarr;

but it puts the text in a box.
 |}

=== Organizing your writing ===
{| border=&quot;1&quot; cellpadding=&quot;2&quot; cellspacing=&quot;0&quot;
|-
!What it looks like
!What you type
{{Help:Headings}}
|-
| {{anchor|anchor=lists}}
* ''Unordered [[Help:List|list]]s'' are easy to do:
** Start every line with a star.
*** More stars indicate a deeper level.
*A newline
*in a list  
marks the end of the list.
*Of course you can start again.
|&lt;syntaxhighlight lang=&quot;wikitext&quot;&gt;
* ''Unordered lists'' are easy to do:
** Start every line with a star.
*** More stars indicate a deeper level.
*A newline
*in a list  
marks the end of the list.
*Of course you can start again.
&lt;/syntaxhighlight&gt;
|-
|
# Numbered lists are also good:
## Very organized
## Easy to follow
A newline marks the end of the list.
#New numbering starts with 1.

|&lt;syntaxhighlight lang=&quot;wikitext&quot;&gt;
# Numbered lists are also good:
## Very organized
## Easy to follow
A newline marks the end of the list.
#New numbering starts with 1.
&lt;/syntaxhighlight&gt;
|-
|
* You can even do mixed lists
*# and nest them
*#* or break lines&lt;br&gt;in lists.
|&lt;syntaxhighlight lang=&quot;wikitext&quot;&gt;
* You can even do mixed lists
*# and nest them
*#* or break lines&lt;br&gt;in lists.&lt;/syntaxhighlight&gt;
|-
| {{anchor|Description list|DL}}
Another kind of list is a '''description list''':
; word : description of the word
; longer phrase 
: phrase described at length, such that, even with a relatively large window width one can see that the wrapping of the text maintains the indentation of the description and thus a highlight of the described term.  This accentuates readability of the list.
|&lt;syntaxhighlight lang=&quot;wikitext&quot;&gt;
Another kind of list is a '''description list''':
; word : description of the word
; longer phrase 
: phrase described at length, such that, even with a
relatively large window width one can see that the 
wrapping of the text maintains the indentation of the 
description and thus a highlight of the described term.  
This accentuates readability of the list.&lt;/syntaxhighlight&gt;
|-
|
:A colon indents a line or paragraph.
A newline after that starts a new paragraph.
::This is often used for discussion on [[Help:Talk page|Talk page]]s.
|&lt;syntaxhighlight lang=&quot;wikitext&quot;&gt;
:A colon indents a line or paragraph.
A newline after that starts a new paragraph.
::This is often used for discussion on talk pages.
&lt;/syntaxhighlight&gt;
|-
|
You can make [[w:horizontal dividing line|horizontal dividing line]]s
to separate text.
----
But you should usually use sections instead,
so that they go in the table of contents.
|&lt;syntaxhighlight lang=&quot;wikitext&quot;&gt;
You can make horizontal dividing lines
to separate text.
----
But you should usually use sections instead,
so that they go in the table of contents.
&lt;/syntaxhighlight&gt;
|}

===Images, tables, video, and sounds===
This is a very quick introduction (please do not forget to [[Uploading files|upload your files]]). For more information, see:
* [[w:Wikipedia:Media help|Images and sound files]] for help with media files
* [[w:Help:Extended image syntax]] for how to arrange images on the page
* [[Help:Table]] for how to create a table

{| border=&quot;1&quot; cellpadding=&quot;2&quot; cellspacing=&quot;0&quot;
|-
!What it looks like
!What you type
|-
|
A picture, including alternate text:

[[File:Wikiversity_beta.png|50px|The logo for this Wiki]]

You can put the image in a frame with a caption:
[[File:Wikiversity_beta.png|thumb|The logo for this Wiki]]
|&lt;syntaxhighlight lang=&quot;wikitext&quot;&gt;
A picture, including alternate text:

[[File:Wikiversity_beta.png|50px|The logo for this Wiki]]

You can put the image in a frame with a caption:
[[File:Wikiversity_beta.png|frame|The logo for this Wiki]]
&lt;/syntaxhighlight&gt;
|-
|
A link to Wikiversity's page for the image:
[[:File:Wikiversity beta.png]]

Or a link directly to the image itself:
[[Media:Wikiversity beta.png]]
|&lt;syntaxhighlight lang=&quot;wikitext&quot;&gt;
A link to Wikiversity's page for the image:
[[:File:Wikiversity beta.png]]

Or a link directly to the image itself:
[[Media:Wikiversity beta.png]]
&lt;/syntaxhighlight&gt;
|-
|
Use '''media:''' links to link to sounds
or videos.

Sound File Example: [[media:Sound_Of_Fear.ogg|A sound file]]

pdf File Example: [[Media:Darkside-graphak1.pdf|A pdf file]]
|&lt;syntaxhighlight lang=&quot;wikitext&quot;&gt;
Use '''media:''' links to link to sounds
or videos.
Sound File Example: [[media:Sound_Of_Fear.ogg|A sound file]]
pdf File Example: [[Media:Darkside-graphak1.pdf|A pdf file]]
&lt;/syntaxhighlight&gt;
|-
|
{{center top}}
{| border=1 cellspacing=0 cellpadding=5
| This
| is
|- 
| a &lt;!-- much less intuitive than a standard html --&gt;
| '''table'''
|}
{{center bottom}}
|&lt;syntaxhighlight lang=&quot;wikitext&quot;&gt;
{{center top}}
{| border=1 cellspacing=0 cellpadding=5
| This
| is
|- 
| a
| '''table'''
|}
{{center bottom}}
&lt;/syntaxhighlight&gt;
|}

===Templates===
'''[[Help:Template|Templates]]''' are segments of Wiki markup that are meant to be copied automatically (&quot;transcluded&quot;) into a page.
You add them by putting the template's name in &lt;nowiki&gt;{{double braces}}&lt;/nowiki&gt;.

Some templates take ''parameters'', as well, which you separate with the pipe character.
{| border=&quot;1&quot; cellpadding=&quot;2&quot; cellspacing=&quot;0&quot;
|-
!What it looks like
!What you type
|-
|
{{Transclusion demo}}
|&lt;syntaxhighlight lang=&quot;wikitext&quot;&gt;
{{Transclusion demo}}
&lt;/syntaxhighlight&gt;
|-
|
This template takes two parameters, and creates
underlined text with a hover box:

{{H:title|This is the hover text|
Hover your mouse over this text}}

Go to [[template:H:title|this page]] to see the H:title template itself.

|&lt;syntaxhighlight lang=&quot;wikitext&quot;&gt;
This template takes two parameters, and creates
underlined text with a hover box:

{{H:title|This is the hover text|
Hover your mouse over this text}}

Go to [[template:H:title|this page]] to see the H:title template itself.
&lt;/syntaxhighlight&gt;
|}

[[Category:Help]]</text>
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    </revision>
  </page>
  <page>
    <title>Wikiversity:FAQ/Editing</title>
    <ns>4</ns>
    <id>1646</id>
    <revision>
      <id>2819857</id>
      <parentid>2809384</parentid>
      <timestamp>2026-07-30T11:38:06Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>Overview</comment>
      <origin>2819857</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3801" sha1="hqmpkgqgls4w1473swoizh8jlgyrmsp" xml:space="preserve">__TOC__ {{Editing help}}
==Overview==
You can just dive in and start typing the way you'd type an e-mail - '''[[Help:Be bold|be bold!]]'''. Come to this page, however, for a jumping-off place for more technical editing instructions. To experiment with editing, visit the [[Wikiversity:Sandbox|sandbox]].

This page is like a manual for editing [[Wikiversity]] and provides many examples. You may want to keep this page open in a separate browser window for reference while you edit. If you are new to [[wiki]], you may benefit from [[Introduction to Wikiversity]] before trying to use this page.

Additional help pages for editing are shown in the box on the right.
:''Wikipedia has additional help for editing such as the [[w:Wikipedia:Cheatsheet|Wikipedia Cheatsheet]]. There is an attempt to centralize help for all Wikimedia projects at [[m:Help:Editing|this page]].''

== Editing basics ==
[[file:mediawiki editthis.jpg|right|link={{fullurl:Project:Sandbox|action=edit}}]]
[[file:mediawiki editlink.jpg|right|link={{fullurl:Project:Sandbox|action=edit&amp;section=1}}]]

*'''[[Wikiversity:Introduction|Short introductory tutorial]]'''
;Start editing
:To start editing a [[w:MediaWiki|MediaWiki]] page, click on the &quot;'''Edit this page'''&quot; (or just &quot;'''edit'''&quot;) link at one of its edges. This will bring you to the '''edit page''': a page with a text box containing the ''[[w:wikitext|wikitext]]'': the editable source code from which the server produces the webpage. ''If you just want to experiment, please do so in the [[{{ns:4}}:Sandbox|sandbox]], not here''.

;Summarize your changes
:You should write a short [[Help:Edit summary|edit summary]] in the small field below the edit-box. You may use shorthand to describe your changes, as described in the [[{{ns:4}}:Edit summary legend|legend]].

;Preview before saving
:When you have finished, click [[Help:Show preview|preview]] to see how your changes will look -- '''before''' you make them permanent.  Repeat the edit/preview process until you are satisfied, then click &quot;Save&quot; and your changes will be immediately applied to the article. Small changes such as spelling corrections can be marked as &quot;minor&quot; before saving. Marking a significant change as a minor change is considered bad Wikiquette.

;Review previous changes to the page
:Yes, you can step back in time to see every single version of a page. Just click on the History tab at the top of the page. Here you can see who did the editing, the date of the change, their summary notes about the change. There are even tools to let you compare what's different between one version and another. Click on an editor's Talk URL and go to their talk page where you can engage them in discussion.

{{:Help:Wikitext quick reference}} &lt;!-- Edit this at [[Help:Wikitext quick reference]] --&gt;

{{:Help:Editing tips and tricks}} &lt;!-- Edit this at [[Help:Editing tips and tricks]] --&gt;

==References==
For help with adding a numbered reference list to a Wikiversity page, see [[w:Wikipedia:Footnotes|Wikipedia:Footnotes]]. If your reference list looks odd, go to the small window of your browser that shows the URL of the page and add '''?action=purg'''e after the article name. Hit return.

==See also==
*[[Help:Editing FAQ]]
*[[Help:Calculation]]
*[[Help:Editing toolbar]]
*[[Help:HTML in wikitext]]
*[[Help:Starting a new page]]
*[[Help:Variable]]
*[[w:UseModWiki|UseModWiki]] and [[w:Wikipedia:PHP script|Wikipedia:PHP script]].
*[[w:HTML element|HTML elements]].
*[[w:User:Pilaf/Live Preview|Live preview]] - a way to preview your edits without contacting the server.

==External links==
*[http://home.earthlink.net/~awinkelried/keyboard_shortcuts.html Special characters in HTML]
* [http://www.youtube.com/watch?v=lj7huccQV2c Creating a course page on Wikiversity]

[[Category:Wikiversity FAQ]]</text>
      <sha1>hqmpkgqgls4w1473swoizh8jlgyrmsp</sha1>
    </revision>
    <revision>
      <id>2819858</id>
      <parentid>2819857</parentid>
      <timestamp>2026-07-30T11:38:34Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <minor />
      <origin>2819858</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3801" sha1="eiy7zqro4zdoplb7s2app5rzq6q7cv6" xml:space="preserve">__TOC__
==Overview==
{{Editing help}}
You can just dive in and start typing the way you'd type an e-mail - '''[[Help:Be bold|be bold!]]'''. Come to this page, however, for a jumping-off place for more technical editing instructions. To experiment with editing, visit the [[Wikiversity:Sandbox|sandbox]].

This page is like a manual for editing [[Wikiversity]] and provides many examples. You may want to keep this page open in a separate browser window for reference while you edit. If you are new to [[wiki]], you may benefit from [[Introduction to Wikiversity]] before trying to use this page.

Additional help pages for editing are shown in the box on the right.
:''Wikipedia has additional help for editing such as the [[w:Wikipedia:Cheatsheet|Wikipedia Cheatsheet]]. There is an attempt to centralize help for all Wikimedia projects at [[m:Help:Editing|this page]].''

== Editing basics ==
[[file:mediawiki editthis.jpg|right|link={{fullurl:Project:Sandbox|action=edit}}]]
[[file:mediawiki editlink.jpg|right|link={{fullurl:Project:Sandbox|action=edit&amp;section=1}}]]

*'''[[Wikiversity:Introduction|Short introductory tutorial]]'''
;Start editing
:To start editing a [[w:MediaWiki|MediaWiki]] page, click on the &quot;'''Edit this page'''&quot; (or just &quot;'''edit'''&quot;) link at one of its edges. This will bring you to the '''edit page''': a page with a text box containing the ''[[w:wikitext|wikitext]]'': the editable source code from which the server produces the webpage. ''If you just want to experiment, please do so in the [[{{ns:4}}:Sandbox|sandbox]], not here''.

;Summarize your changes
:You should write a short [[Help:Edit summary|edit summary]] in the small field below the edit-box. You may use shorthand to describe your changes, as described in the [[{{ns:4}}:Edit summary legend|legend]].

;Preview before saving
:When you have finished, click [[Help:Show preview|preview]] to see how your changes will look -- '''before''' you make them permanent.  Repeat the edit/preview process until you are satisfied, then click &quot;Save&quot; and your changes will be immediately applied to the article. Small changes such as spelling corrections can be marked as &quot;minor&quot; before saving. Marking a significant change as a minor change is considered bad Wikiquette.

;Review previous changes to the page
:Yes, you can step back in time to see every single version of a page. Just click on the History tab at the top of the page. Here you can see who did the editing, the date of the change, their summary notes about the change. There are even tools to let you compare what's different between one version and another. Click on an editor's Talk URL and go to their talk page where you can engage them in discussion.

{{:Help:Wikitext quick reference}} &lt;!-- Edit this at [[Help:Wikitext quick reference]] --&gt;

{{:Help:Editing tips and tricks}} &lt;!-- Edit this at [[Help:Editing tips and tricks]] --&gt;

==References==
For help with adding a numbered reference list to a Wikiversity page, see [[w:Wikipedia:Footnotes|Wikipedia:Footnotes]]. If your reference list looks odd, go to the small window of your browser that shows the URL of the page and add '''?action=purg'''e after the article name. Hit return.

==See also==
*[[Help:Editing FAQ]]
*[[Help:Calculation]]
*[[Help:Editing toolbar]]
*[[Help:HTML in wikitext]]
*[[Help:Starting a new page]]
*[[Help:Variable]]
*[[w:UseModWiki|UseModWiki]] and [[w:Wikipedia:PHP script|Wikipedia:PHP script]].
*[[w:HTML element|HTML elements]].
*[[w:User:Pilaf/Live Preview|Live preview]] - a way to preview your edits without contacting the server.

==External links==
*[http://home.earthlink.net/~awinkelried/keyboard_shortcuts.html Special characters in HTML]
* [http://www.youtube.com/watch?v=lj7huccQV2c Creating a course page on Wikiversity]

[[Category:Wikiversity FAQ]]</text>
      <sha1>eiy7zqro4zdoplb7s2app5rzq6q7cv6</sha1>
    </revision>
    <revision>
      <id>2819859</id>
      <parentid>2819858</parentid>
      <timestamp>2026-07-30T11:40:04Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>Minor update</comment>
      <origin>2819859</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3782" sha1="hgl942q9y4mcb8p4jgz9novch2hm8e6" xml:space="preserve">__TOC__
==Overview==
{{Editing help}}
You can just dive in and start typing the way you'd type an e-mail - '''[[Help:Be bold|be bold!]]'''. Come to this page, however, for a jumping-off place for more technical editing instructions. To experiment with editing, visit the [[Wikiversity:Sandbox|sandbox]].

This page is like a manual for editing [[Wikiversity]] and provides many examples. You may want to keep this page open in a separate browser window for reference while you edit. If you are new to [[wiki]], you may benefit from [[Introduction to Wikiversity]] before trying to use this page.

Additional help pages for editing are shown in the box on the right.
:''Wikipedia has additional help for editing such as the [[w:Wikipedia:Cheatsheet|Wikipedia Cheatsheet]]. There is  centralized editing help information on [[m:Help:Editing|this page]].''

== Editing basics ==
[[file:mediawiki editthis.jpg|right|link={{fullurl:Project:Sandbox|action=edit}}]]
[[file:mediawiki editlink.jpg|right|link={{fullurl:Project:Sandbox|action=edit&amp;section=1}}]]

*'''[[Wikiversity:Introduction|Short introductory tutorial]]'''
;Start editing
:To start editing a [[w:MediaWiki|MediaWiki]] page, click on the &quot;'''Edit this page'''&quot; (or just &quot;'''edit'''&quot;) link at one of its edges. This will bring you to the '''edit page''': a page with a text box containing the ''[[w:wikitext|wikitext]]'': the editable source code from which the server produces the webpage. ''If you just want to experiment, please do so in the [[{{ns:4}}:Sandbox|sandbox]], not here''.

;Summarize your changes
:You should write a short [[Help:Edit summary|edit summary]] in the small field below the edit-box. You may use shorthand to describe your changes, as described in the [[{{ns:4}}:Edit summary legend|legend]].

;Preview before saving
:When you have finished, click [[Help:Show preview|preview]] to see how your changes will look -- '''before''' you make them permanent.  Repeat the edit/preview process until you are satisfied, then click &quot;Save&quot; and your changes will be immediately applied to the article. Small changes such as spelling corrections can be marked as &quot;minor&quot; before saving. Marking a significant change as a minor change is considered bad Wikiquette.

;Review previous changes to the page
:Yes, you can step back in time to see every single version of a page. Just click on the History tab at the top of the page. Here you can see who did the editing, the date of the change, their summary notes about the change. There are even tools to let you compare what's different between one version and another. Click on an editor's Talk URL and go to their talk page where you can engage them in discussion.

{{:Help:Wikitext quick reference}} &lt;!-- Edit this at [[Help:Wikitext quick reference]] --&gt;

{{:Help:Editing tips and tricks}} &lt;!-- Edit this at [[Help:Editing tips and tricks]] --&gt;

==References==
For help with adding a numbered reference list to a Wikiversity page, see [[w:Wikipedia:Footnotes|Wikipedia:Footnotes]]. If your reference list looks odd, go to the small window of your browser that shows the URL of the page and add '''?action=purg'''e after the article name. Hit return.

==See also==
*[[Help:Editing FAQ]]
*[[Help:Calculation]]
*[[Help:Editing toolbar]]
*[[Help:HTML in wikitext]]
*[[Help:Starting a new page]]
*[[Help:Variable]]
*[[w:UseModWiki|UseModWiki]] and [[w:Wikipedia:PHP script|Wikipedia:PHP script]].
*[[w:HTML element|HTML elements]].
*[[w:User:Pilaf/Live Preview|Live preview]] - a way to preview your edits without contacting the server.

==External links==
*[http://home.earthlink.net/~awinkelried/keyboard_shortcuts.html Special characters in HTML]
* [http://www.youtube.com/watch?v=lj7huccQV2c Creating a course page on Wikiversity]

[[Category:Wikiversity FAQ]]</text>
      <sha1>hgl942q9y4mcb8p4jgz9novch2hm8e6</sha1>
    </revision>
  </page>
  <page>
    <title>Wikiversity:FAQ/Editing/Edit summary</title>
    <ns>4</ns>
    <id>4558</id>
    <revision>
      <id>2819853</id>
      <parentid>2716296</parentid>
      <timestamp>2026-07-30T11:13:19Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>Revert</comment>
      <origin>2819853</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9409" sha1="7m3ippfvmz6q9dw816kxj9ftudgb7sa" xml:space="preserve">{{Editing help}}
When [[Help:editing|editing]] a [[learning resource]] with the wikicode editor, there is a small field labeled &quot;Summary&quot; under the main edit-box. It looks like this:

[[Image:Edit_Summary-2.png|900px|Edit summary text box]]

It is highly recommended that one fill in the edit summary field, as it makes it easier for you and your fellow contributors to understand what has changed, and is helpful when going through the history of the page.

==Writing edit summaries==

'''Always fill in the summary field'''. This is considered an important guideline. Even a short summary is better than no summary. An edit summary is even more important if you delete any text; otherwise, people may think you're being sneaky. Also, mentioning one change but not another one can be misleading to someone who finds the other one more important; add &quot;and misc.&quot; to cover the other change(s).

Accurate summaries help people decide whether it is worthwhile for them to check a change. We've found that summaries often pique the interest of contributors with expertise in the area. This may not be as necessary for &quot;minor changes&quot;, but &quot;fixed spelling&quot; would be nice even then.

The edit summary box can hold one line of 200 characters. If you attempt to type or paste more than this only the first 200 characters will be displayed - the rest will be disregarded. Also, for example attempting to add 10 new characters (at the end or in between) to a summary already containing 195 characters will result in the first 5 new characters being inserted and the second 5 being disregarded. 

In the case of a small addition to an article, it is highly recommended to copy the full text of this addition to the summary field, giving a maximum of information with a minimum of effort. Put ''ft'' in front, as an abbreviation of &quot;full text&quot; (see the [[#Abbreviations|Abbreviations section]] for other abbreviations). This way, readers of the summary will be unlikely to check the page itself as they already know the extent of the edit. These kinds of edits allow users to check Recent changes, Page history and User contributions (see below) very efficiently - this also reduces the load on the servers.

If the addition is more than 200 characters, so it does not fit fully in the edit summary box, you should write a short summary of the changes you have introduced into the article. For an addition of, say, 400 characters you can also save time by simply copying that into the summary field. The excess will fall off, and the first 200 characters will usually be acceptable as a crude &quot;summary&quot;.

Unfortunately you can copy only one line of text from the edit box into the edit summary box. The contents of further lines can be pasted at the end of the line. Thus, for example, a bulleted &quot;see also&quot; list is cumbersome to put in the edit summary box. One possible workaround for a new list is putting the list on one line, separated by the asterisks for the bullets, copying it to the edit summary box, and then, in the main edit box, putting the new lines before the asterisks.

In addition to a summary of the change itself, the summary field may also contain an explanation of the change; note that if the reason for an edit is not clear, it is more likely to be reverted, especially in the case that some text is deleted. To give a longer explanation, use the Talk page and put in the edit summary &quot;see Talk&quot;.

After saving the page, the summary can not be edited--another reason to avoid spelling errors.

In the case of important omissions or errors in the edit summary, you can make a [[Help:Dummy edit|dummy edit]] just to put the correction in the edit summary.

If you find it hard to remember to fill in the edit summary (it is an acquired habit), then try switching on the option which alerts you whenever you try to save an edit without an edit summary: Go to '''Preferences''' (top-right), choose the '''Edit''' tab (top-right), then click this box: [[Image:Edit summary prompt.png|300px]], then '''Save'''.

==Places where the edit summary appears==

The edit summary appears in black italics in the following places:

*[[Help:Page history|Page history]] - list of changes to the page you edited
*[[Help:User contributions|User contributions]] - list of all your edits
*[[Help:Watching pages|Watchlist]]'''&lt;nowiki&gt;*&lt;/nowiki&gt;''' - list of recent changes to watched pages ([[Help: Logging in|logged-in]] users only)
*[[Help:Diff|Diff page]] - shows the difference between two edits
*[[Help:Recent changes|Recent changes]] - list of all recent edits
*[[w:Wikipedia:IRC channels|Wikipedia IRC channels]] - real time list of all edits 
*[[Help:Related changes|Related changes]] - list of recent changes to pages linked to the page you edited
*List of new pages: shows the edit summary of the creation.

'''&lt;nowiki&gt;*&lt;/nowiki&gt;''' Use the enhanced watchlist to see all recent changes in the watched pages, not just the last change in each page.

== Abbreviations ==

See [[w:Wikipedia:Edit summary legend|Edit summary legend]] for a list of commonly used abbreviations in edit summaries.

==Searching==

The [[Help:Searching|Wikimedia search function]] can not search edit summaries, and they are not indexed by external search engines.

==File upload summary==

When uploading an image one can supply an upload summary. This serves multiple purposes:
*as second part of the automatically created edit summary of the upload log (the first part giving the file name)
*as text in the entry of the [[Help:Page history#Image_history|image history]]
*in the case that the file name of the image is new:
**as edit summary for the creation of the image page
**as wikitext for the editable part of the image page, which includes the following possibilities:
***briefly describe the image
***provide internal or external links
***call templates
***specify one or more categories the image is in

The capacity of the upload summary is one line of 250 characters; in the upload log the last part may fall off, because this can contain 255 characters, including &quot;uploaded &quot;''filename''&quot;&quot;.

Note that there is no preview function to check the code for the links, template calls and category tags, but of course, if needed one can edit the image page after uploading, to correct errors and also to extend the text.

See also [[Help:Image page]].

==Section editing==

When applying the [[Help:Section#Section editing|section editing]] feature the section title is automatically inserted as initial version or first part of the edit summary. Put more details after this text. In the case that you provide a long summary yourself you can delete the section title in order to stay within the limit of 200 characters. The automatic text appears by default in grey, with the manually typed text being black. If you create a new section before or after an existing section by clicking a section &quot;''edit''&quot; link, please delete the automatic edit summary to avoid confusion.

Prior to May 2004, the automatic part of the summary was surrounded by &quot;=&quot;-signs. The old style remains in edits from that time.

===Multiple sections in edit summary===
From Dec 2004, the section title that forms the automatic part of the edit summary is preceded by a right arrow which links to the section. For this purpose the section title appears within the marks /* and */ in the edit summary box. This code can be manually inserted, and by using the /* and */ syntax, it is possible to include links to multiple sections. This may be useful when editing a section which contains many sub-sections (to list changes made to each section), or when editing an existing section and also adding a new (sub-)section after it.

For example, the edit summary:
 /* Foo */ test /* Bar */ test

Is rendered as:

&lt;span class=&quot;comment&quot;&gt;&lt;span class=&quot;autocomment&quot;&gt;[[#Foo|&amp;rarr;]]Foo&lt;/span&gt; test &lt;span class=&quot;autocomment&quot;&gt;[[#Bar|&amp;rarr;]]Bar&lt;/span&gt; test&lt;/span&gt;

==&quot;Post a comment&quot; feature==

When starting a new thread on a Talk page, the &quot;Post a comment&quot; feature can be used. Click the plus sign next to the Edit link. A box labelled &quot;''Subject/headline''&quot; appears before the main editing box. Text typed here becomes both a new heading and the edit summary.

==Redirect==
When creating a redirect, an edit summary is automatically provided, with the text according to [[MediaWiki:Autoredircomment]], but only if no edit summary is supplied (as opposed to the automatic edit summary in section editing, which can be supplemented by the user).

==Rendering of wikitext; URLs==

Text in edit summaries renders internal links, including [[Help:Piped link|piped link]]s, and [[Help:Interwiki linking|interwiki]] links, even when enclosed within &lt;nowiki&gt;&lt;nowiki&gt; and &lt;/nowiki&gt;&lt;/nowiki&gt;. Therefore, copying wikitext in the edit summary box may be preferable to copying text from the preview, except when one wants to save space.

Other wikitext coding is not interpreted.

When copying an external link from the preview into the edit summary box then, depending on the operating system, the 
&quot;printable version&quot; is copied, i.e. how it is normally rendered, and in addition, between parentheses, the URL; hence the same information as in the wikitext, but in a different format, as well as a possible sequential number.

==See also==
*[[Template:Edit summary]]
*[[Wikiversity:Rollback]]


[[Category:Wikiversity FAQ|Editing, Summary]]

[[es:Ayuda:Resumen de edición]]</text>
      <sha1>7m3ippfvmz6q9dw816kxj9ftudgb7sa</sha1>
    </revision>
    <revision>
      <id>2819854</id>
      <parentid>2819853</parentid>
      <timestamp>2026-07-30T11:15:03Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <minor />
      <origin>2819854</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9407" sha1="33syx83tsnzscrnb7uqx61zxiizw1al" xml:space="preserve">When [[Help:editing|editing]] a [[learning resource]] with the wikicode editor, there is a small field labeled &quot;Summary&quot; under the main edit-box. It looks like this:

[[File:Edit_Summary-2.png|900px|Edit summary text box]]

It is highly recommended that one fill in the edit summary field, as it makes it easier for you and your fellow contributors to understand what has changed, and is helpful when going through the history of the page.

==Writing edit summaries==
{{Editing help}}
'''Always fill in the summary field'''. This is considered an important guideline. Even a short summary is better than no summary. An edit summary is even more important if you delete any text; otherwise, people may think you're being sneaky. Also, mentioning one change but not another one can be misleading to someone who finds the other one more important; add &quot;and misc.&quot; to cover the other change(s).

Accurate summaries help people decide whether it is worthwhile for them to check a change. We've found that summaries often pique the interest of contributors with expertise in the area. This may not be as necessary for &quot;minor changes&quot;, but &quot;fixed spelling&quot; would be nice even then.

The edit summary box can hold one line of 200 characters. If you attempt to type or paste more than this only the first 200 characters will be displayed - the rest will be disregarded. Also, for example attempting to add 10 new characters (at the end or in between) to a summary already containing 195 characters will result in the first 5 new characters being inserted and the second 5 being disregarded. 

In the case of a small addition to an article, it is highly recommended to copy the full text of this addition to the summary field, giving a maximum of information with a minimum of effort. Put ''ft'' in front, as an abbreviation of &quot;full text&quot; (see the [[#Abbreviations|Abbreviations section]] for other abbreviations). This way, readers of the summary will be unlikely to check the page itself as they already know the extent of the edit. These kinds of edits allow users to check Recent changes, Page history and User contributions (see below) very efficiently - this also reduces the load on the servers.

If the addition is more than 200 characters, so it does not fit fully in the edit summary box, you should write a short summary of the changes you have introduced into the article. For an addition of, say, 400 characters you can also save time by simply copying that into the summary field. The excess will fall off, and the first 200 characters will usually be acceptable as a crude &quot;summary&quot;.

Unfortunately you can copy only one line of text from the edit box into the edit summary box. The contents of further lines can be pasted at the end of the line. Thus, for example, a bulleted &quot;see also&quot; list is cumbersome to put in the edit summary box. One possible workaround for a new list is putting the list on one line, separated by the asterisks for the bullets, copying it to the edit summary box, and then, in the main edit box, putting the new lines before the asterisks.

In addition to a summary of the change itself, the summary field may also contain an explanation of the change; note that if the reason for an edit is not clear, it is more likely to be reverted, especially in the case that some text is deleted. To give a longer explanation, use the Talk page and put in the edit summary &quot;see Talk&quot;.

After saving the page, the summary can not be edited--another reason to avoid spelling errors.

In the case of important omissions or errors in the edit summary, you can make a [[Help:Dummy edit|dummy edit]] just to put the correction in the edit summary.

If you find it hard to remember to fill in the edit summary (it is an acquired habit), then try switching on the option which alerts you whenever you try to save an edit without an edit summary: Go to '''Preferences''' (top-right), choose the '''Edit''' tab (top-right), then click this box: [[Image:Edit summary prompt.png|300px]], then '''Save'''.

==Places where the edit summary appears==

The edit summary appears in black italics in the following places:

*[[Help:Page history|Page history]] - list of changes to the page you edited
*[[Help:User contributions|User contributions]] - list of all your edits
*[[Help:Watching pages|Watchlist]]'''&lt;nowiki&gt;*&lt;/nowiki&gt;''' - list of recent changes to watched pages ([[Help: Logging in|logged-in]] users only)
*[[Help:Diff|Diff page]] - shows the difference between two edits
*[[Help:Recent changes|Recent changes]] - list of all recent edits
*[[w:Wikipedia:IRC channels|Wikipedia IRC channels]] - real time list of all edits 
*[[Help:Related changes|Related changes]] - list of recent changes to pages linked to the page you edited
*List of new pages: shows the edit summary of the creation.

'''&lt;nowiki&gt;*&lt;/nowiki&gt;''' Use the enhanced watchlist to see all recent changes in the watched pages, not just the last change in each page.

== Abbreviations ==

See [[w:Wikipedia:Edit summary legend|Edit summary legend]] for a list of commonly used abbreviations in edit summaries.

==Searching==

The [[Help:Searching|Wikimedia search function]] can not search edit summaries, and they are not indexed by external search engines.

==File upload summary==

When uploading an image one can supply an upload summary. This serves multiple purposes:
*as second part of the automatically created edit summary of the upload log (the first part giving the file name)
*as text in the entry of the [[Help:Page history#Image_history|image history]]
*in the case that the file name of the image is new:
**as edit summary for the creation of the image page
**as wikitext for the editable part of the image page, which includes the following possibilities:
***briefly describe the image
***provide internal or external links
***call templates
***specify one or more categories the image is in

The capacity of the upload summary is one line of 250 characters; in the upload log the last part may fall off, because this can contain 255 characters, including &quot;uploaded &quot;''filename''&quot;&quot;.

Note that there is no preview function to check the code for the links, template calls and category tags, but of course, if needed one can edit the image page after uploading, to correct errors and also to extend the text.

See also [[Help:Image page]].

==Section editing==

When applying the [[Help:Section#Section editing|section editing]] feature the section title is automatically inserted as initial version or first part of the edit summary. Put more details after this text. In the case that you provide a long summary yourself you can delete the section title in order to stay within the limit of 200 characters. The automatic text appears by default in grey, with the manually typed text being black. If you create a new section before or after an existing section by clicking a section &quot;''edit''&quot; link, please delete the automatic edit summary to avoid confusion.

Prior to May 2004, the automatic part of the summary was surrounded by &quot;=&quot;-signs. The old style remains in edits from that time.

===Multiple sections in edit summary===
From Dec 2004, the section title that forms the automatic part of the edit summary is preceded by a right arrow which links to the section. For this purpose the section title appears within the marks /* and */ in the edit summary box. This code can be manually inserted, and by using the /* and */ syntax, it is possible to include links to multiple sections. This may be useful when editing a section which contains many sub-sections (to list changes made to each section), or when editing an existing section and also adding a new (sub-)section after it.

For example, the edit summary:
 /* Foo */ test /* Bar */ test

Is rendered as:

&lt;span class=&quot;comment&quot;&gt;&lt;span class=&quot;autocomment&quot;&gt;[[#Foo|&amp;rarr;]]Foo&lt;/span&gt; test &lt;span class=&quot;autocomment&quot;&gt;[[#Bar|&amp;rarr;]]Bar&lt;/span&gt; test&lt;/span&gt;

==&quot;Post a comment&quot; feature==

When starting a new thread on a Talk page, the &quot;Post a comment&quot; feature can be used. Click the plus sign next to the Edit link. A box labelled &quot;''Subject/headline''&quot; appears before the main editing box. Text typed here becomes both a new heading and the edit summary.

==Redirect==
When creating a redirect, an edit summary is automatically provided, with the text according to [[MediaWiki:Autoredircomment]], but only if no edit summary is supplied (as opposed to the automatic edit summary in section editing, which can be supplemented by the user).

==Rendering of wikitext; URLs==

Text in edit summaries renders internal links, including [[Help:Piped link|piped link]]s, and [[Help:Interwiki linking|interwiki]] links, even when enclosed within &lt;nowiki&gt;&lt;nowiki&gt; and &lt;/nowiki&gt;&lt;/nowiki&gt;. Therefore, copying wikitext in the edit summary box may be preferable to copying text from the preview, except when one wants to save space.

Other wikitext coding is not interpreted.

When copying an external link from the preview into the edit summary box then, depending on the operating system, the 
&quot;printable version&quot; is copied, i.e. how it is normally rendered, and in addition, between parentheses, the URL; hence the same information as in the wikitext, but in a different format, as well as a possible sequential number.

==See also==
*[[Template:Edit summary]]
*[[Wikiversity:Rollback]]


[[Category:Wikiversity FAQ|Editing, Summary]]

[[es:Ayuda:Resumen de edición]]</text>
      <sha1>33syx83tsnzscrnb7uqx61zxiizw1al</sha1>
    </revision>
    <revision>
      <id>2819855</id>
      <parentid>2819854</parentid>
      <timestamp>2026-07-30T11:28:46Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>Revise to shorten</comment>
      <origin>2819855</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9454" sha1="odts1c4huoiq9xjjqv3xgsnz9j3hkzu" xml:space="preserve">When [[Help:editing|editing]] a [[learning resource]] there is a small field labeled &quot;Summary&quot;.

It is highly recommended that contributors fill in the edit summary field, as it makes it easier for you and your fellow contributors to understand what has changed, and is helpful when going through the history of the page.

When using the wikicode editor (by clicking &quot;Edit source&quot;), the Summary field appears under the main edit-box. It looks like this:

[[File:Edit_Summary-2.png|900px|Edit summary text box]]

==Writing edit summaries==
{{Editing help}}
'''Always fill in the summary field'''. This is an important guideline. Even a short summary is better than no summary. An edit summary is even more important if you remove text; otherwise, people may think you're being sneaky. Also, mentioning one change but not another one can be misleading to someone who finds the other one more important; you could, for example, add &quot;and misc.&quot; to cover the other change(s).

Accurate summaries help people decide whether it is worthwhile for them to check a change. Summaries often pique the interest of contributors with interest and/or expertise in the area. This may not be as necessary for &quot;minor changes&quot;, but &quot;fixed spelling&quot; would be nice even then.

The edit summary box can hold one line of 200 characters. If you attempt to type or paste more than this only the first 200 characters will be displayed - the rest will be disregarded.&lt;!-- Excessive detail: Also, for example attempting to add 10 new characters (at the end or in between) to a summary already containing 195 characters will result in the first 5 new characters being inserted and the second 5 being disregarded. --&gt;

In the case of a small addition, it is highly recommended to copy the full text of this addition to the summary field, giving a maximum of information with a minimum of effort. You can put ''ft'' in front, as an abbreviation of &quot;full text&quot; (see the [[#Abbreviations|Abbreviations section]] for other abbreviations). This way, readers of the summary will be unlikely to check the page itself as they already know the extent of the edit. These kinds of edits allow users to check Recent changes, Page history and User contributions very efficiently.

If the addition is more than 200 characters, so it does not fit fully in the edit summary box, you should write a short summary of the changes you have introduced. For an addition of, say, 400 characters you can also save time by simply copying that into the summary field. The excess will fall off, and the first 200 characters will usually be acceptable as a crude &quot;summary&quot;.

Unfortunately you can copy only one line of text from the edit box into the edit summary box. The contents of further lines can be pasted at the end of the line. Thus, for example, a bulleted &quot;see also&quot; list is cumbersome to put in the edit summary box. One possible workaround for a new list is putting the list on one line, separated by the asterisks for the bullets, copying it to the edit summary box, and then, in the main edit box, putting the new lines before the asterisks.

In addition to a summary of the change itself, the summary field may also contain an explanation of the change; note that if the reason for an edit is not clear, it is more likely to be reverted, especially in the case that some text is deleted. To give a longer explanation, use the Talk page and put in the edit summary &quot;see Talk&quot;.

After saving the page, the summary can not be edited—another reason to avoid spelling errors.

In the case of important omissions or errors in the edit summary, you can make a [[Help:Dummy edit|dummy edit]] just to put the correction in the edit summary.

If you find it hard to remember to fill in the edit summary (it is an acquired habit), then try switching on the option which alerts you whenever you try to save an edit without an edit summary: Go to '''Preferences''' (top-right), choose the '''Edit''' tab (top-right), then click this box: [[Image:Edit summary prompt.png|300px]], then '''Save'''.

==Places where the edit summary appears==
The edit summary appears in black italics in the following places:
*[[Help:Page history|Page history]] - list of changes to the page you edited
*[[Help:User contributions|User contributions]] - list of all your edits
*[[Help:Watching pages|Watchlist]]'''&lt;nowiki&gt;*&lt;/nowiki&gt;''' - list of recent changes to watched pages ([[Help: Logging in|logged-in]] users only)
*[[Help:Diff|Diff page]] - shows the difference between two edits
*[[Help:Recent changes|Recent changes]] - list of all recent edits
*[[w:Wikipedia:IRC channels|Wikipedia IRC channels]] - real time list of all edits 
*[[Help:Related changes|Related changes]] - list of recent changes to pages linked to the page you edited
*List of new pages: shows the edit summary of the creation.

'''&lt;nowiki&gt;*&lt;/nowiki&gt;''' Use the enhanced watchlist to see all recent changes in the watched pages, not just the last change in each page.

== Abbreviations ==
See [[w:Wikipedia:Edit summary legend|Edit summary legend]] for a list of commonly used abbreviations in edit summaries.

==Searching==
The [[Help:Searching|Wikimedia search function]] can not search edit summaries, and they are not indexed by external search engines.

==File upload summary==
When uploading an image one can supply an upload summary. This serves multiple purposes:
*as second part of the automatically created edit summary of the upload log (the first part giving the file name)
*as text in the entry of the [[Help:Page history#Image_history|image history]]
*in the case that the file name of the image is new:
**as edit summary for the creation of the image page
**as wikitext for the editable part of the image page, which includes the following possibilities:
***briefly describe the image
***provide internal or external links
***call templates
***specify one or more categories the image is in

The capacity of the upload summary is one line of 250 characters; in the upload log the last part may fall off, because this can contain 255 characters, including &quot;uploaded &quot;''filename''&quot;&quot;.

Note that there is no preview function to check the code for the links, template calls and category tags, but of course, if needed one can edit the image page after uploading, to correct errors and also to extend the text.

See also [[Help:Image page]].

==Section editing==
When using the [[Help:Section#Section editing|section editing]] feature the section title is automatically inserted as initial version or first part of the edit summary. Put more details after this text. In the case that you provide a long summary yourself you can delete the section title in order to stay within the limit of 200 characters. The automatic text appears by default in grey, with the manually typed text being black. If you create a new section before or after an existing section by clicking a section &quot;''edit''&quot; link, please delete the automatic edit summary to avoid confusion.
&lt;!--
Prior to May 2004, the automatic part of the summary was surrounded by &quot;=&quot;-signs. The old style remains in edits from that time.
--&gt;
===Multiple sections in edit summary===
From Dec 2004, the section title that forms the automatic part of the edit summary is preceded by a right arrow which links to the section. For this purpose the section title appears within the marks /* and */ in the edit summary box. This code can be manually inserted, and by using the /* and */ syntax, it is possible to include links to multiple sections. This may be useful when editing a section which contains many sub-sections (to list changes made to each section), or when editing an existing section and also adding a new (sub-)section after it.

For example, the edit summary:
 /* Foo */ test /* Bar */ test

Is rendered as:

&lt;span class=&quot;comment&quot;&gt;&lt;span class=&quot;autocomment&quot;&gt;[[#Foo|&amp;rarr;]]Foo&lt;/span&gt; test &lt;span class=&quot;autocomment&quot;&gt;[[#Bar|&amp;rarr;]]Bar&lt;/span&gt; test&lt;/span&gt;

==&quot;Post a comment&quot; feature==
When starting a new thread on a Talk page, the &quot;Post a comment&quot; feature can be used. Click the plus sign next to the Edit link. A box labelled &quot;''Subject/headline''&quot; appears before the main editing box. Text typed here becomes both a new heading and the edit summary.

==Redirect==
When creating a redirect, an edit summary is automatically provided, with the text according to [[MediaWiki:Autoredircomment]], but only if no edit summary is supplied (as opposed to the automatic edit summary in section editing, which can be supplemented by the user).

==Rendering of wikitext; URLs==
Text in edit summaries renders internal links, including [[Help:Piped link|piped link]]s, and [[Help:Interwiki linking|interwiki]] links, even when enclosed within &lt;nowiki&gt;&lt;nowiki&gt; and &lt;/nowiki&gt;&lt;/nowiki&gt;. Therefore, copying wikitext in the edit summary box may be preferable to copying text from the preview, except when one wants to save space.

Other wikitext coding is not interpreted.
&lt;!-- Excessive detail

When copying an external link from the preview into the edit summary box then, depending on the operating system, the 
&quot;printable version&quot; is copied, i.e. how it is normally rendered, and in addition, between parentheses, the URL; hence the same information as in the wikitext, but in a different format, as well as a possible sequential number.
--&gt;
==See also==
*[[Template:Edit summary]]
*[[Wikiversity:Rollback]]


[[Category:Wikiversity FAQ|Editing, Summary]]

[[es:Ayuda:Resumen de edición]]</text>
      <sha1>odts1c4huoiq9xjjqv3xgsnz9j3hkzu</sha1>
    </revision>
  </page>
  <page>
    <title>Business finance</title>
    <ns>0</ns>
    <id>9569</id>
    <revision>
      <id>2819802</id>
      <parentid>2486619</parentid>
      <timestamp>2026-07-30T05:12:09Z</timestamp>
      <contributor>
        <username>~2026-42034-08</username>
        <id>3104165</id>
      </contributor>
      <comment>/* If you are going to start a business you need to keep track of your finances. */</comment>
      <origin>2819802</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17570" sha1="7qbffx82hdb072k43rroo757mldpnil" xml:space="preserve">{{Cleanup|Wikifying}}

== If you are going to start a business you need to keep track of your finances. ==
MONEY is the life blood of the economy and BUSINESS is the way in which people and capital are organized within the economy to create money and productivity. In essence, business is like the '''heart''' of the economy while money is the '''blood'''. 

Over the years a variety of laws [https://www.sec.gov/about/laws.shtml] have been introduced to regulate financial aspects of business. Keeping track of finances in business helps to reduce fraud and waste. Keeping track of finances also allows for maximum efficiency to be attained. It is also the law. 

This course is intended to help give you an idea of how business finances work. This is a wiki, so you are encouraged to add and edit the course. This course will probably not lead you towards earning a degree in business, finance, accounting, or business finance; however, it will hopefully give you a better idea of how business finance works, and maybe, just maybe, if you wanted to start your own business and use sound financial practices, you could.

The Time Value of Money

== Balance sheet structure ==

The following balance sheet structure is just an example. It does not show all possible kinds of assets, equity and liabilities, but it shows the most usual ones. Because it shows [[Goodwill]] it could be a [[/Consolidated account/|consolidated]] balance sheet. Monetary values are not shown, summary (total) rows are missing as well.

 '''Balance Sheet of XYZ, Ltd. as on 31 December 2005'''

 '''ASSETS'''

 '''Current Assets'''
 Cash and cash equivalents
     Marketable Securities
  [[Accounts receivable]]
 [[Inventory|Inventories]]
   Prepaid Expenses
Investments held for trading
 Other current assets

 '''Non-Current Assets (Fixed Assets)'''
 Property, plant and equipment 
 Accumulated Depreciation
 [[Goodwill]] 
 [[Intangible asset|Other intangible fixed assets]]
 [[Associate company|Investments in]]
 [[Deferred tax]] assets

 '''and EQUITY'''

  '''Current liabilities'''
 [[Accounts payable]]
 Current income tax liabilities
 Current portion of bank loans payable
 Short-term [[/Provision|provisions]]
 Other current liabilities

'''Long term  Liabilities (Fixed Liabilities)'''
 [[Loan|Bank loans]]
 [[Security (finance)|Issued debt securities]]
 [[Deferred tax]] liability
 [[/Provision|Provisions]]
   [[Minority interest]]


'''Capital and reserves''' 
 [[Share capital]]
 Capital [[Reserve (Accounting)|reserves]]
 Revaluation [[Reserve (Accounting)|reserve]]
 Translation [[Reserve (Accounting)|reserve]]
 [[Retained earnings]]


'''Analogy'''

Knowing is not the same as understanding, so it is helpful to present an analogy.

    * Think of an investment as a water reservoir. The value of the investment is the volume of water in it. The shareholder equity
      (net equity) on the balance sheet measures this value.
    * Streams empty into the reservoir, adding more water. These inflows are measured by Revenues on the Income statement.
    * Streams run out of the reservoir, depleting it. These outflow are measured by Expenses on the Income statement.
          o The difference between these inflows and outflows is the net income, also shown in the Income statement.

    * When a neighbor joins in the investment as a partner, he digs a canal from his own reservoir so it drains into the reservoir.
      This additional water is measured by an increase in the share capital.
    * If you ask a neighbor to add to the reservoir, it is considered as liability, thus reducing net equity but increasing assets.
      These are both shown on the Balance Sheet.

While measuring the volume of water in the reservoir at a point in time (balance sheet) is relatively easy, keeping track of the streams' volumes at every second of the year is difficult. Accountants may choose to ignore some streams: they may not know some streams exist: water may be evaporating, and unmeasurable. As a result the income statement is easily wrong. Regardless, the net sum of inflows less outflows should equal the difference in the reservoir (beginning vs. ending). The statement of changes in shareholder equity attempts this reconciliation.

None of the financial statements measures your own personal share of the reservoir when you have partners. It is up to the individual investor to measure, not the business totals, but his share. The methods to use 'equity per share' is shown at shareholders' equity.

==Users of Financial Statements==

Financial statements are used by a diverse group of parties, both inside and outside a business. Generally, these users are:

'''1. Internal Users:''' are owners, managers, employees and other parties who are directly connected with a company.

    * '''Owners and managers''' require financial statements to make important business decisions
         that affect its continued operations. Financial analyses are then performed on these 
         statements to provide management with a more detailed understanding of the figures. 
         These statements are also used as part of management's report to its stockholders,
         as it form part of its Annual Report.

    * '''Employees''' also need these reports in making collective bargaining agreements (CBA) with
         the management, in the case of labor unions or for individuals in discussing their
         compensation, promotion and rankings. 

'''2. External Users:''' are potential investors, banks, government agencies and other parties who are outside the business but need financial information about the business for a diverse number of reasons.

    * '''Prospective investors''' make use of financial statements to assess the viability of
         investing in a business. Financial analysis is often used by investors and is prepared 
         by professionals (Financial Analysts), thus providing them with the basis in making
         investment decisions.

    * '''Financial institutions''' (banks and other lending companies) use them to decide whether to
         grant a company with fresh working capital or extend debt securities (such as a long-term
         bank loan or debentures) to finance expansion and other significant expenditures.

    * '''Government entities (Tax Authorities)''' need financial statements to ascertain the
         propriety and accuracy of taxes and other duties declared and paid by a company.

    * '''Media and the general public''' are also interested in financial statements for a variety 
         of reasons.

==Government financial statements==

The rules for the recording, measurement and presentation of government financial statements may be different from those required for business and even for non-profit organizations. They may use either accrual accounting, or cash accounting, or a combination of the two. A complete set of Chart of Accounts is also used that is substantially different from the Chart of a profit-oriented business.

===Audit and Legal Implications===

Although the legal statutes may differ from country to country, an audit of financial statements is required. These are usually performed by independent accountants or auditing firms. Results of the audit are summarized in an audit report that either provide an unqualified opinion on the financial statements or qualifications as to its fairness and accuracy. The audit opinion on the financial statements is usually included in the Annual Report.

There has been much legal debate over who an auditor is liable to. Since audit reports tend to be addressed to the current shareholders, it is commonly thought that they owe a legal duty of care to them. But this may not be the case as determined by common law precedent. In Canada, auditors are liable only to investors using a prospectus to buy shares in the primary market. In the UK, they have been held liable to potential investors when the auditor was aware of the potential investor and how they would use the information in the financial statements. Nowadays auditors tend to include in their report liability restricting language, discouraging anyone other than the addressees of their report from relying on it. Liability is an important issue: in the UK, for example, auditors have unlimited liability.

In the United States, especially in the post-Enron era there has been substantial concern about the accuracy of financial statements. Corporate officers (the CEO and CFO) are personally liable for attesting that financial statements &quot;do not contain any untrue statement of a material fact or omit to state a material fact necessary to make the statements made, in light of the circumstances under which such statements were made, not misleading with respect to the period covered by th[e] report&quot;. Making or certifiing misleading financial statements exposes the people involved to substantial civil and criminal liability. For example Bernie Ebbers (former CEO of WorldCom) was sentenced to 25 years in federal prison for allowing WorldCom's revenues to be overstated by $11 billion over five years.

===Standards and Regulations===

Different countries have developed their own accounting principles over time, making international comparisons of companies difficult. To ensure uniformity and comparability between financial statements prepared by different companies, a set of guidelines and rules are used. Commonly referred to as Generally Accepted Accounting Principles (GAAP), these set of guidelines provide the basis in the preparation of financial statements.

Recently there has been a push towards standardizing accounting rules made by the International Accounting Standards Board (&quot;IASB&quot;). IASB develops International Financial Reporting Standards that have been adopted by Australia, Canada and the European Union (for publicly quoted companies only), are under consideration in South Africa and other countries. The United States Federal Accounting Standards Board has made a commitment to converge the US GAAP and IFRS over time.

===Inclusion in Annual Reports===

To entice new investors, most public companies assemble their financial statements on fine paper with pleasing graphics and photos in an annual report to shareholders, attempting to capture the excitement and culture of the organization in a &quot;marketing brochure&quot; of sorts. Usually the company's chief executive officer will write a letter to shareholders, describing management's performance and the company's financial highlights.

In the United States, prior to the advent of the internet, the annual report is considered the most effective way for corporations to communicate with individual shareholders. Blue chip companies went to great expense to produce and mail out attractive annual reports to every shareholder. The annual report was often prepared in the style of a coffee table book

In formal bookkeeping and accounting, a balance sheet is a statement of the book value of a business or other organization or person at a particular date, at the end of a period such as a &quot;fiscal year,&quot; as distinct from an income statement, also known as a profit and loss account (P&amp;L), which records revenue and expenses over a specified period of time.

A balance sheet is often described as a &quot;snapshot&quot; of the company's financial condition on a given date. Of the four basic financial statements, the balance sheet is the only statement which applies to a single point in time, instead of a period of time.

A simple business operating entirely in cash could measure its profits by simply withdrawing the entire bank balance at the end of the period, plus any cash in hand. However, real businesses are not paid immediately; they build up inventories of goods to sell and they acquire buildings and equipment. In other words: businesses have assets and so they could not, even if they wanted to, immediately turn these into cash at the end of each period. Real businesses also owe money to suppliers and to tax authorities, and the proprietors do not withdraw all their original capital and profits at the end of each period. In other words businesses also have liabilities.

A modern balance sheet usually has three parts: assets, liabilities and shareholders' equity. The main categories of assets are usually listed first and are followed by the liabilities. The difference between the assets and the liabilities is known as the 'net assets' or the 'net worth' of the company.

The net assets shown by the balance sheet equals the third part of the balance sheet, which is known as the shareholders' equity. This balance is not a coincidence. Records of the values of each account in the balance sheet are maintained using a system of accounting known as double-entry bookkeeping.

====Some Future Value Definitions====

• '''Future Value (FV):''' The amount an investment is worth after one or more periods.

• '''Simple Interest:''' Interest earned only on the original principal amount invested.

• '''Compound Interest:''' Interest earned on
both the initial principal and the interest
reinvested from prior periods.

• '''Compounding:''' The process of
accumulating interest on an investment
over time to earn more interest.

==Financial Statements==
Financial statements (or financial reports) are formal records of a business' financial activities. These statements provide an overview of a business' profitability and financial condition in both short and long term. There are four basic financial statements:

'''1. Balance Sheet:''' also referred to as statement of financial condition, reports on a company's assets, liabilities and net equity as of a given point in time.

'''2. Income Statement:''' also referred to as Profit or loss statement, reports on a company's results of operations over a period of time.

'''3. Cash Flow Statement:''' reports on a company's cash flow activities, particularly its operating, investing and financing activities.

'''4. Statement of changes in shareholder equity:''' reconciles the difference between net equity at two different points in time.

Because these statements are often complex, an extensive set of Notes to the Financial Statements and management discussion and analysis is usually included. The notes will typically describe each item on the Balance sheet, Income statement and Cash flow statement in further details. Notes to Financial Statements are considered as an integral part of the Financial Statements.

==Constructing a Balance Sheet==
'''Case Study'''

'''1.1'''&lt;br&gt;
A new business starts up as a limited company called Sunrise Ltd by raising $10,000 from the owners i.e. share holders. The money is put in to a new bank account. What would the assets, liabilities and equity be?
 '''Assets:'''
 Bank Balance		10,000

 '''Equity &amp; Liabilities:'''
 Share Capital		10,000

'''1.2'''&lt;br&gt;
They then use 6,000 of its bank account to buy a delivery van. Assets and liabilities after this transaction:
 '''Assets:'''
 Bank Balance		 4,000
 Delivery Van		 6,000

 '''Equity &amp; Liabilities:'''
 Share Capital		10,000

'''1.3'''&lt;br&gt;
Sunrise Ltd then buys some inventory at 3,000 on credit. Assets and liabilities after this transaction:
 '''Assets:'''
 Bank Balance		 4,000
 Delivery Van		 6,000
 Inventory		 3,000

 '''Liabilities:'''
 Accounts Payable	 3,000	(to be paid to creditors)

 '''Equity:'''
 Share Capital		10,000

Total assets must always equal total liabilities (and equity). It is inevitable as the liabilities (and equity) are providing the funds that we are spending on these assets.

'''1.4'''&lt;br&gt;
Shortly afterwards, after selling 1,000 of inventory for 2,500, payment of 2,600 of the accounts payable and the purchase of 2,200 of machinery financed by a 2,200 bank loan, the assets and liabilities change to the following: 

 '''Sunrise Ltd.'''
 '''Balance Sheet'''
 As of December 31, 2005
 -----------------------------------

 '''Fixed Assets'''
 Delivery Van		 6,000
 Machinery		 2,200
 -----------------------------------
 '''Total fixed assets			 8,200'''

 '''Current Assets'''
 Bank Balance		 1,400
 Inventory		 2,000
 Accounts Receivable	 2,500
 -----------------------------------
 Total 			 5,900

 Accounts Payable	        400
 -----------------------------------
 '''Net current assets 			 5,500'''
 '''Total assets                        13,700'''

 '''Long-Term Liabilities'''
 Loans Repayable		 2,200
 -----------------------------------
 '''Total Long Term Liabilities		 2,200'''
 -----------------------------------
 '''NET ASSETS				11,500'''
 -----------------------------------

 '''Shareholders' Equity'''
 Share Capital		10,000
 Retained profits	 1,500
 -----------------------------------
 '''TOTAL SHAREHOLDERS' EQUITY		11,500'''
 -----------------------------------

Points to note:&lt;br&gt;
* Must be headed with the name of the reporting entity (e.g. Sunrise Ltd) and the date.
* The van has not been depreciated and there are no other trading expenses
* The terms 'Current Liability' and 'Long-Term Liability' are the traditional names possibly used by sole traders or partnerships. Limited companies may use the phrases 'Liabilities: Amounts falling due within 1 year' and 'Liabilities: Amounts falling due after 1 year'.
* The Total Equity may also be called the 'Net Worth'.
* The Net Worth is in principle what the company is worth, it shows the monetary amount that would effectively be left, if all assets were sold and all liabilities paid off.


==See also==
*[[Topic:Finance]]
[http://www.trintech.com/ Financial Governance]
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{{shortcut|WV:HELP}}
{{Help header}}

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{| style=&quot;vertical-align:top; margin:0em 0em 1em 0em; border:1px solid Black; background-color:White; {{text color default}};width:100%&quot; cellpadding=&quot;10&quot;
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'''Wikiversity''' is a [[w:Wikimedia Foundation|Wikimedia Foundation]] project devoted to learning resources, learning projects, and research for use in all levels, types, and styles of education from pre-school to university, including professional training and informal learning. We invite teachers, students, and researchers to join us in creating [[w:Open educational resources|open educational resources]] and collaborative learning communities. 

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Having trouble finding out what you need to know? If it's not covered by the '''[[Wikiversity:FAQ|FAQ]]''', try asking at the '''[[Wikiversity:Colloquium|Colloquium]]'''.
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=== MediaWiki ===
The function of [[w:MediaWiki|MediaWiki]] (the software of Wikiversity) is described at the '''[[m:Help:Contents|MediaWiki User's Guide]]''' in Meta-wiki. &lt;!--The MediaWiki-handbook in Meta-wiki is not kept here in order to avoid duplicate maintenance work (the same applies to all help pages not specific to Wikiversity).--&gt;

If you suspect a software bug, ask for feedback at the [[WV:C|Colloquium]], and then report it using [[Phabricator:|'''Wikimedia Phabricator''']]. This is the only way to get MediaWiki developers to notice your bug reports.

=== Other help page collections ===
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&lt;small&gt;&lt;p style=&quot;text-align:left;&quot;&gt;'''Archived versions:''' [[Help:Contents/Old|2007 Help:Contents]] | [[Help:Contents/Old/2008|2008 Help:Contents]]&lt;/p&gt;&lt;/small&gt;

__NOTOC__
__NOEDITSECTION__

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[[Category:Help| ]]
[[Category:Web_Technology]]
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      <contributor>
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      <comment>Update</comment>
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      <text bytes="1538" sha1="oah7rsq4m71jyfaqw0ipjwuxjnjtver" xml:space="preserve">==Editing tip - Using edit summaries==
{| style=&quot;float: center;background: inherit;&quot;
|-
| [[Image:Crystal Clear app ktip.svg|48px|left]]
| Thank-you for your contributions to Wikiversity!

I would like to offer an [[Wikiversity:Etiquette|wikiquette]] editing tip. Could you try to provide [[Help:Edit summary|edit summaries]] more often? The edit summary box appears at the bottom of pages you are editing. It looks something like this: [[File:Edit Summary.png]].

Filling in this edit summary box each time you edit doesn't take very long, and provides other users with a quick summary of the changes you made. This can save others a lot of time in working out what has changed.

If you find it hard to remember to fill in the edit summary (it is an acquired habit), then try switching on the option which alerts you whenever you try to save an edit without an edit summary: Go to '''Preferences''' (top-right), choose the '''Edit''' tab (top-right), click this box: [[File:Edit summary prompt.png|300px]], then '''Save'''.

To learn more, see [[Wikiversity:FAQ/Editing/Edit summary|edit summary]] (FAQ).

I hope this helps empower you to keep up the great work! ~~&lt;includeonly&gt;&lt;/includeonly&gt;~~
|}&lt;noinclude&gt;

==Notes==
* This is a template to add to user talk pages, to help mentor/encourage them to get into the desirable wiki editing practice of providing edit summaries.
* Include on a talk page by using &lt;code&gt;&lt;nowiki&gt;{{subst:Edit summary}}&lt;/nowiki&gt;&lt;/code&gt;
[[Category:User talk templates]]
[[Category:Wikiversity tips]]
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      <comment>/* Open access psychology journals */ International Journal of Psychological Research</comment>
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      <text bytes="3801" sha1="tdoiid7udkif7ddlyj3g2ei5m671t9w" xml:space="preserve">==All psychology [[journals]]==
* [http://lamp.infosys.deakin.edu.au/era/?page=jfordet15&amp;selfor=17 ERA FOR Code 17: Psychology]
* [[w:List of psychology journals|List of psychology journals]] (Wikipedia)

==[[Open access]] psychology [[journals]]==
This section lists open psychology journals and their rankings.
* [https://www.scimagojr.com/journalrank.php?category=3201&amp;openaccess=true scimagojr]
* [https://www.doaj.org/search?source=%7B%22query%22%3A%7B%22filtered%22%3A%7B%22filter%22%3A%7B%22bool%22%3A%7B%22must%22%3A%5B%7B%22term%22%3A%7B%22index.classification.exact%22%3A%22Psychology%22%7D%7D%2C%7B%22term%22%3A%7B%22_type%22%3A%22journal%22%7D%7D%5D%7D%7D%2C%22query%22%3A%7B%22match_all%22%3A%7B%7D%7D%7D%7D%7D DOAJ]

{| class=&quot;wikitable sortable&quot;
 ! Journal !! Notes
 |-
 | [http://www.ac-psych.org/ Advances in Cognitive Psychology]
 |
 |-
 | [http://www.discourseunit.com/arcp.htm Annual Review of Critical Psychology]
 |
 |-
 | [http://www.apcj.org/ Applied Psychology In Criminal Justice]
 | 
 |-
 | [http://www.athleticinsight.com/ Athletic Insight]
 | 
 |-
 | [http://www.auseinet.com/journal The Australian e-Journal for the Advancement of Mental Health]
 | 
 |-
 | [http://www.newcastle.edu.au/group/ajedp/ Australian Journal of Educational &amp; Developmental Psychology]
 | [http://lamp.infosys.deakin.edu.au/era/?page=fordet&amp;selfor=1701 ERA Rank B]. Papers accepted for publication become the copyright of the publisher. Green: Author pre- and post-prints can be made available.
 |-
 | [http://www.behavior-analyst-today.com/ The Behavior Analyst Today]
 |
 |-
 | [http://www.uic.edu/htbin/cgiwrap/bin/ojs/index.php?journal=bsi&amp;page=index Behavior and Social Issues]
 |
 |-
 | [http://cpl.revues.org/ Current psychology letters]
 |
 |-
 | [http://www.uiowa.edu/~grpproc/crisp/crisp.html Current Research In Social Psychology]
 |
 |-
 | [http://goertzel.org/dynapsyc/dynacon.html Dynamical Psychology: An International, Interdisciplinary Journal of Complex Mental Processes]
 |
 |-
 | [http://ojs.lib.swin.edu.au/index.php/ejap E-journal of Applied Psychology]
 | ERA Rank C&lt;br&gt;Uses [[Open Journal Systems]]. Authors give journal a license to publish. The journal uses [Attribution-NonCommercial-NoDerivs 2.1 Australia  cc-nd 2.1 Australia licensing]. 
 |-
 | [http://www.internationaljournalofwellbeing.org/index.php/ijow/index International Journal of Wellbeing]
 | 
 
|-
|[https://revistas.usb.edu.co/index.php/IJPR International Journal of Psychological Research]
|Could suit student research projects
|-
 | [http://seab.envmed.rochester.edu/jaba/ Journal of Applied Behavior Analysis]
 |
 |-
 | [http://seab.envmed.rochester.edu/jeab/ Journal of Experimental Analysis of Behavior]
 |
 |-
 | [http://www.brains-minds-media.org/ Journal of New Media in Neural and Cognitive Science and Education]
 |
 |-
 | [http://www.psychologie-aktuell.com/index.php?id=200 Psychological Test and Assessment Modeling]
 | ERA Rank C&lt;br&gt;Articles are available at no-cost electronically&lt;br&gt;Licensing is unclear
 |-
 | [http://hrcak.srce.hr/psihologijske-teme?lang=en Psychological Topics]
 | ERA Rank: None
 |-
 | [http://www.sgcp.org.uk/coachingpsy/publications.cfm Special Group in Coaching Psychology]
 |
 |-
 | [http://bentham.org/open/topsyj The Open Psychology Journal]
 | ERA Rank: None&lt;br&gt;Authors who publish in Bentham OPEN Journals retain copyright to their work. Articles are licensed under the terms of the [http://creativecommons.org/licenses/by-nc/3.0/ Creative Commons Attribution non-commercial License]. [http://www.bentham.org/open/topsyj/MSandI.htm Author is required to pay an article processing fee of several hundred $US.]
|}

See also: [[w:List of open-access journals|List of open-access journals]] (Wikipedia)

[[Category:Open journals]]
[[Category:Psychology/Journals]]</text>
      <sha1>tdoiid7udkif7ddlyj3g2ei5m671t9w</sha1>
    </revision>
  </page>
  <page>
    <title>Walsh permutation; inversions</title>
    <ns>0</ns>
    <id>134630</id>
    <revision>
      <id>2819769</id>
      <parentid>2773704</parentid>
      <timestamp>2026-07-29T21:06:04Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <comment>/* Bit permutations */</comment>
      <origin>2819769</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="41115" sha1="46ntt3t99tgha9zec8nead2w697fa4s" xml:space="preserve">{| align=&quot;right&quot; style=&quot;text-align:right; border:1px solid lightgray&quot; cellspacing=&quot;10&quot;
| [[Walsh permutation]] {{rdrup}}
|-
| [[Inversion (discrete mathematics)]] {{rdrup}}
|}

[[File:Construction site ahead (17184445992).jpg|thumb|The vectors need to be corrected. See [[Walsh permutation#Notation warning]]. Done only for section [[#Nimber multiplication|Nimber multiplication]].]]

{|
|style=&quot;vertical-align:top;&quot;|
&lt;!---▶▶▶---&gt;{| style=&quot;border: 1px solid #cccccc&quot;
! bgcolor=&quot;#dddddd&quot; colspan=&quot;2&quot;|Abbreviations
|-
|
|-
|'''CV'''
|Compression vector
|-
|'''CM'''
|Compression matrix
|-
|'''Σ'''
|Number of ones in the compression matrix
|-
|'''IS'''
|Inversion set
|-
|'''#'''
|Index number of the finite permutation
|-
|'''IN'''
|Inversion number
|}&lt;!---◀◀◀---&gt;
|style=&quot;vertical-align:top;&quot;|__TOC__
|}

&lt;br style=&quot;clear: both;&quot;&gt;

==Stripes==
Among the ''n''-bit Walsh permutations are 2&lt;sup&gt;''n''&lt;/sup&gt; with striped inversion sets.&lt;br&gt;
As seen below they correspond to Walsh functions, i.e. the rows of the [[w:Walsh matrix|Walsh matrix]] of order 2&lt;sup&gt;''n''&lt;/sup&gt;.&lt;br&gt;
These permutations under [[w:Function composition|composition]] form the group Z&lt;sub&gt;2&lt;/sub&gt;&lt;sup&gt;''n''&lt;/sup&gt;, just like the row numbers under bitwise XOR.

=== 3-bit ===

An extract of the [[3-bit Walsh permutation; table|'''table''']] of [[3-bit Walsh permutation]]s:
{| class=&quot;wikitable sortable&quot; style=&quot;height:495px; text-align:center;&quot;
|-style=&quot;height:55px&quot;
!
!style=&quot;width:55px&quot;|CV
!style=&quot;width:55px&quot; class=&quot;unsortable&quot;|&amp;nbsp;&amp;nbsp;CM&amp;nbsp;&amp;nbsp;
!D
!Σ
!#
!Permutation&lt;br&gt;F
!IN
!class=&quot;unsortable&quot;|IS
!style=&quot;width:55px&quot; class=&quot;unsortable&quot;|PM
!class=&quot;unsortable&quot;|FPC
!Cy
|-   
|0 || {{font|color=#aaa|1 2 4}} || {{BIN1start}}{{BIN0}}{{BIN0;}}&lt;br&gt;{{BIN0}}{{BIN1}}{{BIN0;}}&lt;br&gt;{{BIN0}}{{BIN0}}{{BIN1end}} || + || 3 || align=&quot;right&quot;|  0 || {{font|color=#aaa|0 1 2 3 &amp;nbsp; 4 5 6 7}}&lt;br&gt;{{font|color=#adf|0  0  0  0 &amp;nbsp; 0  0  0  0}} || {{font|color=#adf|&amp;#8199;0}} ||{{Inversions8|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0}}  || {{Row8,0start}}{{Row8,1}}{{Row8,2}}{{Row8,3}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,4}}{{Row8,5}}{{Row8,6}}{{Row8,7end}} || [[File:Walsh permutation 124 Fano.svg|53px]] || align=&quot;left&quot;|{{sms|1 }}{{font|color=#c07|&amp;nbsp;&amp;#8199;0}}
|-   
|style=&quot;background:lightgray;{{text color default}};&quot;| 1 || 3 5 1  || {{BIN1start}}{{BIN1}}{{BIN1;}}&lt;br&gt;{{BIN1}}{{BIN0}}{{BIN0;}}&lt;br&gt;{{BIN0}}{{BIN1}}{{BIN0end}} || + || 5 || align=&quot;right&quot;|17576 || {{font|color=#aaa|0}}  7 1 6  &amp;nbsp;  2  {{font|color=#aaa|5}}  3 4 &lt;br&gt;{{font|color=#adf|0  0  1  1 &amp;nbsp; 2  2  3  3}} || {{font|color=#adf|12}} ||{{Inversions8|0|0|1|0|1|0|0|1|0|1|0|1|0|1|0|0|1|0|1|0|1|0|1|0|1|0|1|0}}  || {{Row8,0start}}{{Row8,7}}{{Row8,1}}{{Row8,6}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,2}}{{Row8,5}}{{Row8,3}}{{Row8,4end}} || || align=&quot;left&quot;|{{sms|3 }}{{font|color=#c07|&amp;nbsp;&amp;#8199;8}}&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;span style=&quot;font-size:60%&quot;&gt;(4+2)&lt;/span&gt;&gt;
|-   
|style=&quot;background:lightgray;{{text color default}};&quot;| 2 || 3 6 2  || {{BIN1start}}{{BIN0}}{{BIN0;}}&lt;br&gt;{{BIN1}}{{BIN1}}{{BIN1;}}&lt;br&gt;{{BIN0}}{{BIN1}}{{BIN0end}} || − || 5 || align=&quot;right&quot;|16854 || {{font|color=#aaa|0 1}}  7 6  &amp;nbsp;  2 3 5 4 &lt;br&gt;{{font|color=#adf|0  0  0  1 &amp;nbsp; 2  2  2  3}} || {{font|color=#adf|10}} ||{{Inversions8|0|0|0|0|0|1|0|0|1|1|0|0|1|1|0|0|0|1|1|0|0|0|0|1|1|0|0|1}}  || {{Row8,0start}}{{Row8,1}}{{Row8,7}}{{Row8,6}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,2}}{{Row8,3}}{{Row8,5}}{{Row8,4end}} || || align=&quot;left&quot;|{{sms|4 }}{{font|color=#c07|&amp;nbsp;&amp;#8199;9}}&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;span style=&quot;font-size:60%&quot;&gt;(3+3)&lt;/span&gt;&gt;
|-   
|3 || {{font|color=#aaa|1}}  7 3  || {{BIN1start}}{{BIN1}}{{BIN1;}}&lt;br&gt;{{BIN0}}{{BIN1}}{{BIN1;}}&lt;br&gt;{{BIN0}}{{BIN1}}{{BIN0end}} || − || 6 || align=&quot;right&quot;|22622 || {{font|color=#aaa|0}}  7 6 1  &amp;nbsp;  2  {{font|color=#aaa|5}}  4 3 &lt;br&gt;{{font|color=#adf|0  0  1  2 &amp;nbsp; 2  2  3  4}} || {{font|color=#adf|14}} ||{{Inversions8|0|0|1|0|1|1|0|1|1|0|0|1|1|0|0|0|1|1|0|0|1|0|1|1|0|0|1|1}}  || {{Row8,0start}}{{Row8,7}}{{Row8,6}}{{Row8,1}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,2}}{{Row8,5}}{{Row8,4}}{{Row8,3end}} || || align=&quot;left&quot;|{{sms|4 }}{{font|color=#c07|&amp;nbsp;&amp;#8199;9}}&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;span style=&quot;font-size:60%&quot;&gt;(3+3)&lt;/span&gt;&gt;
|-   
|style=&quot;background:lightgray;{{text color default}};&quot;| 4 || 5 6  {{font|color=#aaa|4}} || {{BIN1start}}{{BIN0}}{{BIN0;}}&lt;br&gt;{{BIN0}}{{BIN1}}{{BIN0;}}&lt;br&gt;{{BIN1}}{{BIN1}}{{BIN1end}} || + || 5 || align=&quot;right&quot;|16680 || {{font|color=#aaa|0 1 2 3}} &amp;nbsp;  7 6 5 4 &lt;br&gt;{{font|color=#adf|0  0  0  0 &amp;nbsp; 0  1  2  3}} || {{font|color=#adf|&amp;#8199;6}} ||{{Inversions8|0|0|0|0|0|0|0|0|0|0|0|0|0|0|1|0|0|0|0|1|1|0|0|0|0|1|1|1}}  || {{Row8,0start}}{{Row8,1}}{{Row8,2}}{{Row8,3}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,7}}{{Row8,6}}{{Row8,5}}{{Row8,4end}} || || align=&quot;left&quot;|{{sms|2 }}{{font|color=#c07|&amp;nbsp;&amp;#8199;3}}&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;span style=&quot;font-size:60%&quot;&gt;(2+2)&lt;/span&gt;&gt;
|-   
|5 || 7 1 5  || {{BIN1start}}{{BIN1}}{{BIN1;}}&lt;br&gt;{{BIN1}}{{BIN0}}{{BIN0;}}&lt;br&gt;{{BIN1}}{{BIN0}}{{BIN1end}} || − || 6 || align=&quot;right&quot;|22736 || {{font|color=#aaa|0}}  7 1 6  &amp;nbsp;  5 2 4 3 &lt;br&gt;{{font|color=#adf|0  0  1  1 &amp;nbsp; 2  3  3  4}} || {{font|color=#adf|14}} ||{{Inversions8|0|0|1|0|1|0|0|1|0|1|0|1|0|1|1|0|1|0|1|1|0|0|1|0|1|1|0|1}}  || {{Row8,0start}}{{Row8,7}}{{Row8,1}}{{Row8,6}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,5}}{{Row8,2}}{{Row8,4}}{{Row8,3end}} || || align=&quot;left&quot;|{{sms|6 }}{{font|color=#c07|&amp;nbsp;14}}&lt;sub&gt;b&lt;/sub&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;span style=&quot;font-size:60%&quot;&gt;(7)&lt;/span&gt;&gt;
|-   
|6 || 7  {{font|color=#aaa|2}}  6  || {{BIN1start}}{{BIN0}}{{BIN0;}}&lt;br&gt;{{BIN1}}{{BIN1}}{{BIN1;}}&lt;br&gt;{{BIN1}}{{BIN0}}{{BIN1end}} || + || 6 || align=&quot;right&quot;|23454 || {{font|color=#aaa|0 1}}  7 6  &amp;nbsp;  5 4 2 3 &lt;br&gt;{{font|color=#adf|0  0  0  1 &amp;nbsp; 2  3  4  4}} || {{font|color=#adf|14}} ||{{Inversions8|0|0|0|0|0|1|0|0|1|1|0|0|1|1|1|0|0|1|1|1|1|0|0|1|1|1|1|0}}  || {{Row8,0start}}{{Row8,1}}{{Row8,7}}{{Row8,6}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,5}}{{Row8,4}}{{Row8,2}}{{Row8,3end}} || [[File:Walsh permutation 165 Fano red.svg|53px]] || align=&quot;left&quot;|{{sms|3 }}{{font|color=#c07|&amp;nbsp;&amp;#8199;8}}&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;span style=&quot;font-size:60%&quot;&gt;(4+2)&lt;/span&gt;&gt;
|-   
|style=&quot;background:lightgray;{{text color default}};&quot;| 7 || 5 3 7  || {{BIN1start}}{{BIN1}}{{BIN1;}}&lt;br&gt;{{BIN0}}{{BIN1}}{{BIN1;}}&lt;br&gt;{{BIN1}}{{BIN0}}{{BIN1end}} || + || 7 || align=&quot;right&quot;|17702 || {{font|color=#aaa|0}}  7 6 1  &amp;nbsp;  5 2 3 4 &lt;br&gt;{{font|color=#adf|0  0  1  2 &amp;nbsp; 2  3  3  3}} || {{font|color=#adf|14}} ||{{Inversions8|0|0|1|0|1|1|0|1|1|0|0|1|1|0|1|0|1|1|0|1|0|0|1|1|0|1|0|0}}  || {{Row8,0start}}{{Row8,7}}{{Row8,6}}{{Row8,1}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,5}}{{Row8,2}}{{Row8,3}}{{Row8,4end}} || [[File:Walsh permutation 357 Fano red.svg|53px]] || align=&quot;left&quot;|{{sms|5 }}{{font|color=#c07|&amp;nbsp;14}}&lt;sub&gt;a&lt;/sub&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;span style=&quot;font-size:60%&quot;&gt;(7)&lt;/span&gt;&gt;
|}
{|
|Sorting by '''#''' gives the permutation (0,4,2,1,7,3,5,6) in the left column, which is ''reflection'' [[w:Function composition|∘]] [[w:Circular shift|''right shift'']] ∘ ''wp''(6,5,7) (with the compression matrix || {{Bin0start}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1end}} || ).
|}
Sorting by '''Permutation''' gives the (0,4,2,6, 1,5,3,7) in the left column, which is the bit-reversal permutation ''wp''(4,2,1).

=== 4-bit ===

[[File:Variadic logical XOR.svg|thumb|right|Binary Walsh matrix of order 16]]

{| class=&quot;wikitable sortable&quot; style=&quot;text-align:right;&quot;
|-
!
!#
!class=&quot;unsortable&quot;|IS
!CV
!class=&quot;unsortable&quot;|CM
!Σ
!Permutation&lt;br&gt;F
!IN
|-
|0
|0
|[[File:16 choose 2 array.svg|36px]]
|{{font|face=Courier|{{nowrap|'''( 1, 2, 4, 8)'''}}}}
|&lt;div style=&quot;white-space: nowrap&quot;&gt;{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin0}}{{Bin1end}}&lt;/div&gt;
|4
|{{font|face=Courier|{{nowrap|'''( 0, 1, 2, 3,&amp;nbsp; 4, 5, 6, 7,&amp;nbsp; 8, 9,10,11,&amp;nbsp;12,13,14,15)'''}}&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 0,&amp;nbsp; 0, 0, 0, 0,&amp;nbsp; 0, 0, 0, 0,&amp;nbsp; 0, 0, 0, 0)'''}}}}
|{{font|color=#adf|0}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|1
|9804424107176
|[[File:Inversion set 16; Walsh (15, 1, 2, 4).svg|36px]]
|{{font|face=Courier|'''( 3, 5, 9, 1)'''}}
|{{Bin1start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0end}}
|7
|{{font|face=Courier|'''( 0,15, 1,14,&amp;nbsp; 2,13, 3,12,&amp;nbsp; 4,11, 5,10,&amp;nbsp; 6, 9, 7, 8)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 1,&amp;nbsp; 2, 2, 3, 3,&amp;nbsp; 4, 4, 5, 5,&amp;nbsp; 6, 6, 7, 7)'''}}}}
|{{font|color=#adf|56}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|2
|9717242186454
|[[File:Inversion set 16; Walsh (1, 15, 2, 4).svg|36px]]
|{{font|face=Courier|'''( 3, 6,10, 2)'''}}
|{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0end}}
|7
|{{font|face=Courier|'''( 0, 1,15,14,&amp;nbsp; 2, 3,13,12,&amp;nbsp; 4, 5,11,10,&amp;nbsp; 6, 7, 9, 8)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 1,&amp;nbsp; 2, 2, 2, 3,&amp;nbsp; 4, 4, 4, 5,&amp;nbsp; 6, 6, 6, 7)'''}}}}
|{{font|color=#adf|52}}
|-
|3
|11112138397022
|[[File:Inversion set 16; wp( 1, 7,11, 3).svg|36px]]
|{{font|face=Courier|'''( 1, 7,11, 3)'''}}
|{{Bin1start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0end}}
|9
|{{font|face=Courier|'''( 0,15,14, 1,&amp;nbsp; 2,13,12, 3,&amp;nbsp; 4,11,10, 5,&amp;nbsp; 6, 9, 8, 7)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 2,&amp;nbsp; 2, 2, 3, 4,&amp;nbsp; 4, 4, 5, 6,&amp;nbsp; 6, 6, 7, 8)'''}}}}
|{{font|color=#adf|60}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|4
|9710017245480
|[[File:Inversion set 16; wp( 5, 6,12, 4).svg|36px]]
|{{font|face=Courier|'''( 5, 6,12, 4)'''}}
|{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0end}}
|7
|{{font|face=Courier|'''( 0, 1, 2, 3,&amp;nbsp;15,14,13,12,&amp;nbsp; 4, 5, 6, 7,&amp;nbsp;11,10, 9, 8)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 0,&amp;nbsp; 0, 1, 2, 3,&amp;nbsp; 4, 4, 4, 4,&amp;nbsp; 4, 5, 6, 7)'''}}}}
|{{font|color=#adf|44}}
|-
|5
|11118325501136
|[[File:Inversion set 16; Walsh (15, 1, 13, 4).svg|36px]]
|{{font|face=Courier|'''( 7, 1,13, 5)'''}}
|{{Bin1start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0end}}
|9
|{{font|face=Courier|'''( 0,15, 1,14,&amp;nbsp;13, 2,12, 3,&amp;nbsp; 4,11, 5,10,&amp;nbsp; 9, 6, 8, 7)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 1,&amp;nbsp; 2, 3, 3, 4,&amp;nbsp; 4, 4, 5, 5,&amp;nbsp; 6, 7, 7, 8)'''}}}}
|{{font|color=#adf|60}}
|-
|6
|11205500164254
|[[File:Inversion set 16; Walsh (1, 15, 13, 4).svg|36px]]
|{{font|face=Courier|'''( 7, 2,14, 6)'''}}
|{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0end}}
|9
|{{font|face=Courier|'''( 0, 1,15,14,&amp;nbsp;13,12, 2, 3,&amp;nbsp; 4, 5,11,10,&amp;nbsp; 9, 8, 6, 7)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 1,&amp;nbsp; 2, 3, 4, 4,&amp;nbsp; 4, 4, 4, 5,&amp;nbsp; 6, 7, 8, 8)'''}}}}
|{{font|color=#adf|60}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|7
|9810691044902
|[[File:Inversion set 16; Walsh (15, 14, 13, 4).svg|36px]]
|{{font|face=Courier|'''( 5, 3,15, 7)'''}}
|{{Bin1start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0end}}
|11
|{{font|face=Courier|'''( 0,15,14, 1,&amp;nbsp;13, 2, 3,12,&amp;nbsp; 4,11,10, 5,&amp;nbsp; 9, 6, 7, 8)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 2,&amp;nbsp; 2, 3, 3, 3,&amp;nbsp; 4, 4, 5, 6,&amp;nbsp; 6, 7, 7, 7)'''}}}}
|{{font|color=#adf|60}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|8
|9709968804480
|[[File:Inversion set 16; Walsh (1, 2, 4, 15).svg|36px]]
|{{font|face=Courier|'''( 9,10,12, 8)'''}}
|{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin1end}}
|7
|{{font|face=Courier|'''( 0, 1, 2, 3,&amp;nbsp; 4, 5, 6, 7,&amp;nbsp;15,14,13,12,&amp;nbsp;11,10, 9, 8)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 0,&amp;nbsp; 0, 0, 0, 0,&amp;nbsp;  0, 1, 2, 3,&amp;nbsp; 4, 5, 6, 7)'''}}}}
|{{font|color=#adf|28}}
|-
|9
|11118365775656
|[[File:Inversion set 16; Walsh (15, 1, 2, 11).svg|36px]]
|{{font|face=Courier|'''(11,13, 1, 9)'''}}
|{{Bin1start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin0}}{{Bin1end}}
|9
|{{font|face=Courier|'''( 0,15, 1,14,&amp;nbsp; 2,13, 3,12,&amp;nbsp;11, 4,10, 5,&amp;nbsp; 9, 6, 8, 7)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 1,&amp;nbsp; 2, 2, 3, 3,&amp;nbsp; 4, 5, 5, 6,&amp;nbsp; 6, 7, 7, 8)'''}}}}
|{{font|color=#adf|60}}
|-
|10
|11205547694934
|[[File:Inversion set 16; Walsh (1, 15, 2, 11).svg|36px]]
|{{font|face=Courier|'''(11,14, 2,10)'''}}
|{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin0}}{{Bin1end}}
|9
|{{font|face=Courier|'''( 0, 1,15,14,&amp;nbsp; 2, 3,13,12,&amp;nbsp;11,10, 4, 5,&amp;nbsp; 9, 8, 6, 7)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 1,&amp;nbsp; 2, 2, 2, 3,&amp;nbsp; 4, 5, 6, 6,&amp;nbsp; 6, 7, 8, 8)'''}}}}
|{{font|color=#adf|60}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|11
|9810651495902
|[[File:Inversion set 16; Walsh (15, 14, 2, 11).svg|36px]]
|{{font|face=Courier|'''( 9,15, 3,11)'''}}
|{{Bin1start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin0}}{{Bin1end}}
|11
|{{font|face=Courier|'''( 0,15,14, 1,&amp;nbsp; 2,13,12, 3,&amp;nbsp;11, 4, 5,10,&amp;nbsp; 9, 6, 7, 8)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 2,&amp;nbsp; 2, 2, 3, 4,&amp;nbsp; 4, 5, 5, 5,&amp;nbsp; 6, 7, 7, 7)'''}}}}
|{{font|color=#adf|60}}
|-
|12
|11212772635560
|[[File:Inversion set 16; Walsh (1, 2, 15, 11).svg|36px]]
|{{font|face=Courier|'''(13,14, 4,12)'''}}
|{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin0}}{{Bin1end}}
|9
|{{font|face=Courier|'''( 0, 1, 2, 3,&amp;nbsp;15,14,13,12,&amp;nbsp;11,10, 9, 8,&amp;nbsp; 4, 5, 6, 7)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 0,&amp;nbsp;  0, 1, 2, 3,&amp;nbsp; 4, 5, 6, 7,&amp;nbsp; 8, 8, 8, 8)'''}}}}
|{{font|color=#adf|60}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|13
|9804464392016
|[[File:Inversion set 16; Walsh (15, 1, 13, 11).svg|36px]]
|{{font|face=Courier|'''(15, 9, 5,13)'''}}
|{{Bin1start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin0}}{{Bin1end}}
|11
|{{font|face=Courier|'''( 0,15, 1,14,&amp;nbsp;13, 2,12, 3,&amp;nbsp;11, 4,10, 5,&amp;nbsp; 6, 9, 7, 8)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 1,&amp;nbsp; 2, 3, 3, 4,&amp;nbsp; 4, 5, 5, 6,&amp;nbsp; 6, 6, 7, 7)'''}}}}
|{{font|color=#adf|60}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|14
|9717289730334
|[[File:Inversion set 16; Walsh (1, 15, 13, 11).svg|36px]]
|{{font|face=Courier|'''(15,10, 6,14)'''}}
|{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin0}}{{Bin1end}}
|11
|{{font|face=Courier|'''( 0, 1,15,14,&amp;nbsp;13,12, 2, 3,&amp;nbsp;11,10, 4, 5,&amp;nbsp; 6, 7, 9, 8)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 1,&amp;nbsp; 2, 3, 4, 4,&amp;nbsp; 4, 5, 6, 6,&amp;nbsp; 6, 6, 6, 7)'''}}}}
|{{font|color=#adf|60}}
|-
|15
|11112098838182
|[[File:Inversion set 16; Walsh (15, 14, 13, 11).svg|36px]]
|{{font|face=Courier|'''(13,11, 7,15)'''}}
|{{Bin1start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin0}}{{Bin1end}}
|13
|{{font|face=Courier|'''( 0,15,14, 1,&amp;nbsp;13, 2, 3,12,&amp;nbsp;11, 4, 5,10,&amp;nbsp; 6, 9, 8, 7)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 2,&amp;nbsp; 2, 3, 3, 3,&amp;nbsp; 4, 5, 5, 5,&amp;nbsp; 6, 6, 7, 8)'''}}}}
|{{font|color=#adf|60}}
|}

{|
|Sorting by '''#''' gives the (0, 8,4,2,14, 1,13,11,7, 15,3,5,9, 6,10,12) in the left column, which is ''reflection'' [[w:Function composition|∘]] [[w:Circular shift|''right shift'']] ∘ ''wp''(12,10, 9, 7) (with the compression matrix || {{Bin0start}}{{Bin0}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin0}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin0end}} || ).
|}
Sorting by '''Permutation''' gives the (0,8,4,12, 2,10,6,14, 1,9,5,13, 3,11,7,15) in the left column, which is the [[w:Bit-reversal permutation|bit-reversal permutation]] ''wp''( 8, 4, 2, 1).

{| class=&quot;collapsible collapsed&quot; style=&quot;width: 100%; border: 1px solid #cccccc; background:white;{{text color default}};&quot;
! bgcolor=&quot;#f0f0f0&quot;|{{font|color=#999999|Matrices}}
|-
|
 Compression vectors:
     1     2     4     8
     3     5     9     1
     3     6    10     2
     1     7    11     3
     5     6    12     4
     7     1    13     5
     7     2    14     6
     5     3    15     7
     9    10    12     8
    11    13     1     9
    11    14     2    10
     9    15     3    11
    13    14     4    12
    15     9     5    13
    15    10     6    14
    13    11     7    15
 &amp;nbsp;
 Permutations:
     0     1     2     3     4     5     6     7     8     9    10    11    12    13    14    15
     0    15     1    14     2    13     3    12     4    11     5    10     6     9     7     8
     0     1    15    14     2     3    13    12     4     5    11    10     6     7     9     8
     0    15    14     1     2    13    12     3     4    11    10     5     6     9     8     7
     0     1     2     3    15    14    13    12     4     5     6     7    11    10     9     8
     0    15     1    14    13     2    12     3     4    11     5    10     9     6     8     7
     0     1    15    14    13    12     2     3     4     5    11    10     9     8     6     7
     0    15    14     1    13     2     3    12     4    11    10     5     9     6     7     8
     0     1     2     3     4     5     6     7    15    14    13    12    11    10     9     8
     0    15     1    14     2    13     3    12    11     4    10     5     9     6     8     7
     0     1    15    14     2     3    13    12    11    10     4     5     9     8     6     7
     0    15    14     1     2    13    12     3    11     4     5    10     9     6     7     8
     0     1     2     3    15    14    13    12    11    10     9     8     4     5     6     7
     0    15     1    14    13     2    12     3    11     4    10     5     6     9     7     8
     0     1    15    14    13    12     2     3    11    10     4     5     6     7     9     8
     0    15    14     1    13     2     3    12    11     4     5    10     6     9     8     7
 &amp;nbsp;
 Little-endian factorial numbers:
     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0
     0     0     1     1     2     2     3     3     4     4     5     5     6     6     7     7
     0     0     0     1     2     2     2     3     4     4     4     5     6     6     6     7
     0     0     1     2     2     2     3     4     4     4     5     6     6     6     7     8
     0     0     0     0     0     1     2     3     4     4     4     4     4     5     6     7
     0     0     1     1     2     3     3     4     4     4     5     5     6     7     7     8
     0     0     0     1     2     3     4     4     4     4     4     5     6     7     8     8
     0     0     1     2     2     3     3     3     4     4     5     6     6     7     7     7
     0     0     0     0     0     0     0     0     0     1     2     3     4     5     6     7
     0     0     1     1     2     2     3     3     4     5     5     6     6     7     7     8
     0     0     0     1     2     2     2     3     4     5     6     6     6     7     8     8
     0     0     1     2     2     2     3     4     4     5     5     5     6     7     7     7
     0     0     0     0     0     1     2     3     4     5     6     7     8     8     8     8
     0     0     1     1     2     3     3     4     4     5     5     6     6     6     7     7
     0     0     0     1     2     3     4     4     4     5     6     6     6     6     6     7
     0     0     1     2     2     3     3     3     4     5     5     5     6     6     7     8
|}

=== Compression matrices ===

[[File:Compression matrices of Walsh permutations with striped inversion sets.svg|left|800px]]
&lt;br style=&quot;clear: both;&quot;&gt;
The CM of the ''n''-th permutation is a matrix ''B'' with horizontal stripes for the binary digits of ''n'' &lt;small&gt;(gray lines in the image)&lt;/small&gt;, XORed with a modified unity matrix ''U'' &lt;small&gt;(black dots in the image)&lt;/small&gt;.&lt;br&gt;
''U'' is modified by including an empty row where ''n'' has its lowest binary digit ({{oeis|A001511}}). It pushes the other rows of the unity matrix down, and the last row &quot;out of the matrix&quot;.


==Bended stripes==

Among the ''n''-bit Walsh permutations are 2&lt;sup&gt;''n-1''&lt;/sup&gt; whose inversion sets show bended stripes.&lt;br&gt;
They correspond to Walsh functions, i.e. the rows of the [[w:Walsh matrix|Walsh matrix]] of order 2&lt;sup&gt;''n-1''&lt;/sup&gt;.&lt;br&gt;
These permutations under composition form the group Z&lt;sub&gt;2&lt;/sub&gt;&lt;sup&gt;''n''-1&lt;/sup&gt;, just like the row numbers under bitwise XOR.

=== 3-bit ===

{| class=&quot;wikitable sortable&quot; style=&quot;height:275px; text-align:center;&quot;
|-style=&quot;height:55px&quot;
!
!style=&quot;width:55px&quot;|CV
!style=&quot;width:55px&quot; class=&quot;unsortable&quot;|&amp;nbsp;&amp;nbsp;CM&amp;nbsp;&amp;nbsp;
!D
!#
!Permutation&lt;br&gt;F
!IN
!class=&quot;unsortable&quot;|IS
!style=&quot;width:55px&quot; class=&quot;unsortable&quot;|PM
!class=&quot;unsortable&quot;|FPC
!Cy
|-   
|0 || {{font|color=#aaa|1 2 4}} || {{BIN1start}}{{BIN0}}{{BIN0;}}&lt;br&gt;{{BIN0}}{{BIN1}}{{BIN0;}}&lt;br&gt;{{BIN0}}{{BIN0}}{{BIN1end}} || + || align=&quot;right&quot;|  0 || {{font|color=#aaa|0 1 2 3 &amp;nbsp; 4 5 6 7}}&lt;br&gt;{{font|color=#adf|0  0  0  0 &amp;nbsp; 0  0  0  0}} || {{font|color=#adf|&amp;#8199;0}} ||{{Inversions8|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0}}  || {{Row8,0start}}{{Row8,1}}{{Row8,2}}{{Row8,3}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,4}}{{Row8,5}}{{Row8,6}}{{Row8,7end}} || [[File:Walsh permutation 124 Fano.svg|53px]] || align=&quot;left&quot;|{{sms|1 }}{{font|color=#c07|&amp;nbsp;&amp;#8199;0}}
|-   
|style=&quot;background:lightgray;{{text color default}};&quot;|1 ||  4 7 1  || {{BIN0start}}{{BIN1}}{{BIN1;}}&lt;br&gt;{{BIN0}}{{BIN1}}{{BIN0;}}&lt;br&gt;{{BIN1}}{{BIN1}}{{BIN0end}} || − || align=&quot;right&quot;| 3776 || {{font|color=#aaa|0}}  6  {{font|color=#aaa|2}}  4  &amp;nbsp;  3  {{font|color=#aaa|5}}  1  {{font|color=#aaa|7}}&lt;br&gt;{{font|color=#adf|0  0  1  1 &amp;nbsp; 2  1  5  0}} || {{font|color=#adf|10}} ||{{Inversions8|0|0|1|0|1|0|0|1|0|1|0|1|0|0|0|0|1|1|1|1|1|0|0|0|0|0|0|0}}  || {{Row8,0start}}{{Row8,6}}{{Row8,2}}{{Row8,4}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,3}}{{Row8,5}}{{Row8,1}}{{Row8,7end}} || || align=&quot;left&quot;|{{sms|2 }}{{font|color=#c07|&amp;nbsp;&amp;#8199;3}}&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;span style=&quot;font-size:60%&quot;&gt;(2+2)&lt;/span&gt;&gt;
|-   
|style=&quot;background:lightgray;{{text color default}};&quot;|2 ||  7 4 2  || {{BIN1start}}{{BIN0}}{{BIN0;}}&lt;br&gt;{{BIN1}}{{BIN0}}{{BIN1;}}&lt;br&gt;{{BIN1}}{{BIN1}}{{BIN0end}} || − || align=&quot;right&quot;|  414 || {{font|color=#aaa|0 1}}  5 4  &amp;nbsp;  3 2  {{font|color=#aaa|6 7}}&lt;br&gt;{{font|color=#adf|0  0  0  1 &amp;nbsp; 2  3  0  0}} || {{font|color=#adf|&amp;#8199;6}} ||{{Inversions8|0|0|0|0|0|1|0|0|1|1|0|0|1|1|1|0|0|0|0|0|0|0|0|0|0|0|0|0}}  || {{Row8,0start}}{{Row8,1}}{{Row8,5}}{{Row8,4}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,3}}{{Row8,2}}{{Row8,6}}{{Row8,7end}} || || align=&quot;left&quot;|{{sms|2 }}{{font|color=#c07|&amp;nbsp;&amp;#8199;3}}&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;span style=&quot;font-size:60%&quot;&gt;(2+2)&lt;/span&gt;&gt;
|-   
|3 ||  2 1 7  || {{BIN0start}}{{BIN1}}{{BIN1;}}&lt;br&gt;{{BIN1}}{{BIN0}}{{BIN1;}}&lt;br&gt;{{BIN0}}{{BIN0}}{{BIN1end}} || − || align=&quot;right&quot;| 4142 || {{font|color=#aaa|0}}  6 5  {{font|color=#aaa|3 &amp;nbsp; 4}}  2 1  {{font|color=#aaa|7}}&lt;br&gt;{{font|color=#adf|0  0  1  2 &amp;nbsp; 2  4  5  0}} || {{font|color=#adf|14}} ||{{Inversions8|0|0|1|0|1|1|0|1|1|0|0|1|1|1|1|0|1|1|1|1|1|0|0|0|0|0|0|0}}  || {{Row8,0start}}{{Row8,6}}{{Row8,5}}{{Row8,3}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,4}}{{Row8,2}}{{Row8,1}}{{Row8,7end}} || || align=&quot;left&quot;|{{sms|2 }}{{font|color=#c07|&amp;nbsp;&amp;#8199;3}}&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;span style=&quot;font-size:60%&quot;&gt;(2+2)&lt;/span&gt;&gt;
|}

Sorting by '''#''' or '''Permutation''' gives (0,2,1,3) in the left column, which is the bit-reversal permutation ''wp''(2,1).&lt;br&gt;
Sorting by '''CV''' gives (0,3,1,2) in the left column, which is ''wp''(3,1).


=== 4-bit ===

{| class=&quot;wikitable sortable&quot; style=&quot;text-align:right;&quot;
|-
!
!#
!class=&quot;unsortable&quot;|IS
!CV
!class=&quot;unsortable&quot;|CM
!Permutation&lt;br&gt;F
!IN
|-
|0
|0
|[[File:16 choose 2 array.svg|36px]]
|{{font|face=Courier|{{nowrap|'''( 1, 2, 4, 8)'''}}}}
|&lt;div style=&quot;white-space: nowrap&quot;&gt;{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin0}}{{Bin1end}}&lt;/div&gt;
|{{font|face=Courier|{{nowrap|'''( 0, 1, 2, 3,&amp;nbsp; 4, 5, 6, 7,&amp;nbsp; 8, 9,10,11,&amp;nbsp;12,13,14,15)'''}}&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 0,&amp;nbsp; 0, 0, 0, 0,&amp;nbsp; 0, 0, 0, 0,&amp;nbsp; 0, 0, 0, 0)'''}}}}
|{{font|color=#adf|0}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|1
|1144441325096
|[[File:Inversion set 16; Walsh (14, 2, 4, 7).svg|36px]]
|{{font|face=Courier|'''( 8,11,13, 1)'''}}
|{{Bin0start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin0end}}
|{{font|face=Courier|'''( 0,14, 2,12,&amp;nbsp; 4,10, 6, 8,&amp;nbsp; 7, 9, 5,11,&amp;nbsp; 3,13, 1,15)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 1,&amp;nbsp; 2, 2, 3, 3,&amp;nbsp; 4, 3, 7, 2,&amp;nbsp;10, 1,13, 0)'''}}}}
|{{font|color=#adf|52}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|2
|73376328534
|[[File:Inversion set 16; Walsh (1, 13, 4, 7).svg|36px]]
|{{font|face=Courier|'''(11, 8,14, 2)'''}}
|{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin0end}}
|{{font|face=Courier|'''( 0, 1,13,12,&amp;nbsp; 4, 5, 9, 8,&amp;nbsp; 7, 6,10,11,&amp;nbsp; 3, 2,14,15)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 1,&amp;nbsp; 2, 2, 2, 3,&amp;nbsp; 4, 5, 2, 2,&amp;nbsp;10,11, 0, 0)'''}}}}
|{{font|color=#adf|44}}
|-
|3
|1209107634782
|[[File:Inversion set 16; Walsh (14, 13, 4, 8).svg|36px]]
|{{font|face=Courier|'''( 2, 1, 7,11)'''}}
|{{Bin0start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin0}}{{Bin1end}}
|{{font|face=Courier|'''( 0,14,13, 3,&amp;nbsp; 4,10, 9, 7,&amp;nbsp; 8, 6, 5,11,&amp;nbsp;12, 2, 1,15)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 2,&amp;nbsp; 2, 2, 3, 4,&amp;nbsp; 4, 6, 7, 2,&amp;nbsp; 2,12,13, 0)'''}}}}
|{{font|color=#adf|60}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|4
|303182760
|[[File:Inversion set 16; Walsh (1, 2, 11, 7).svg|36px]]
|{{font|face=Courier|'''(13,14, 8, 4)'''}}
|{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin0}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin0end}}
|{{font|face=Courier|'''( 0, 1, 2, 3,&amp;nbsp;11,10, 9, 8,&amp;nbsp; 7, 6, 5, 4,&amp;nbsp;12,13,14,15)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 0,&amp;nbsp; 0, 1, 2, 3,&amp;nbsp; 4, 5, 6, 7,&amp;nbsp; 0, 0, 0, 0)'''}}}}
|{{font|color=#adf|28}}
|-
|5
|1144707321296
|[[File:Inversion set 16; Walsh (14, 2, 11, 8).svg|36px]]
|{{font|face=Courier|'''( 4, 7, 1,13)'''}}
|{{Bin0start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin0}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin0}}{{Bin1end}}
|{{font|face=Courier|'''( 0,14, 2,12,&amp;nbsp;11, 5, 9, 7,&amp;nbsp; 8, 6,10, 4,&amp;nbsp; 3,13, 1,15)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 1,&amp;nbsp; 2, 3, 3, 4,&amp;nbsp; 4, 6, 3, 9,&amp;nbsp;10, 1,13, 0)'''}}}}
|{{font|color=#adf|60}}
|-
|6
|73677162654
|[[File:Inversion set 16; Walsh (1, 13, 11, 8).svg|36px]]
|{{font|face=Courier|'''( 7, 4, 2,14)'''}}
|{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin0}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin0}}{{Bin1end}}
|{{font|face=Courier|'''( 0, 1,13,12,&amp;nbsp;11,10, 6, 7,&amp;nbsp; 8, 9, 5, 4,&amp;nbsp; 3, 2,14,15)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 1,&amp;nbsp; 2, 3, 4, 4,&amp;nbsp; 4, 4, 8, 9,&amp;nbsp;10,11, 0, 0)'''}}}}
|{{font|color=#adf|60}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|7
|1209371443622
|[[File:Inversion set 16; Walsh (14, 13, 11, 7).svg|36px]]
|{{font|face=Courier|'''(14,13,11, 7)'''}}
|{{Bin0start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin0}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin0end}}
|{{font|face=Courier|'''( 0,14,13, 3,&amp;nbsp;11, 5, 6, 8,&amp;nbsp; 7, 9,10, 4,&amp;nbsp;12, 2, 1,15)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 2,&amp;nbsp; 2, 3, 3, 3,&amp;nbsp; 4, 3, 3, 9,&amp;nbsp; 2,12,13, 0)'''}}}}
|{{font|color=#adf|60}}
|}

Sorting by '''#''' or '''Permutation''' gives (0,4,2,6, 1,5,3,7) in the left column, which is the bit-reversal permutation ''wp''(4,2,1).&lt;br&gt;
Sorting by '''CV''' gives (0,3,5,6, 1,2,4,7) in the left column, which is ''wp''(7,1,2).

===Compression matrices===

[[File:Compression matrices of Walsh permutations with bendedly striped inversion sets.svg|800px]]
&lt;br style=&quot;clear: both;&quot;&gt;
The CM are matrices with an even number of horizontal stripes &lt;small&gt;(gray lines in the image)&lt;/small&gt; XORed with the unity matrix &lt;small&gt;(black dots in the image)&lt;/small&gt;.&lt;br&gt;
The order by the ''evil numbers'' ({{oeis|A001969}}) of the matrices, used in the image, corresponds to the lexicographical order of the compression vectors.

==Bit permutations==
{| align=&quot;right&quot;
|colspan=&quot;2&quot;|
{| align=right style=&quot;text-align=right; border:1px solid lightgray&quot; cellspacing=&quot;10&quot;
|[[Walsh permutation]]; [[Walsh permutation; bit permutation|bit permutation]] {{Rdrup}}
|}
|-
|style=&quot;vertical-align:top;&quot;|[[File:Inversion set 16; Walsh (2, 8, 1, 4).svg|thumb|Inversion set of ''wp''( 4, 1, 8, 2)&lt;br&gt;All columns are symmetrical.]]
|style=&quot;vertical-align:top;&quot;|[[File:Inversion set 16; Walsh (14, 13, 11, 7).svg|thumb|Inversion set of ''wp''(14,13,11, 7)&lt;br&gt;Here the symmetry is easily seen.]]
|-
|colspan=&quot;2&quot;|
{| align=right
|-
|style=&quot;vertical-align:top;&quot;|[[File:Symmetric group 4; cycle graph; details.svg|thumb|center|500px|[[w:Cycle graph (algebra)|Cycle graph]] of the bit permutations]]
|style=&quot;vertical-align:top;&quot;|[[File:Symmetric group 4; permutation list (0-based).svg|thumb|center|Permutations of 4 elements&lt;br&gt;{{Rdr}} [[Symmetric group S4|Symmetric group S&lt;sub&gt;4&lt;/sub&gt;]]]]
|}
|}


{| class=&quot;wikitable sortable&quot; style=&quot;text-align:right;&quot;
|-
!
!#
!class=&quot;unsortable&quot;|IS
!CV
|-
|style=&quot;background:white;{{text color default}};&quot;|&lt;big&gt;'''0'''&lt;/big&gt;
|0
|[[File:16 choose 2 array.svg|36px]]
|{{font|face=Courier|'''( 1, 2, 4, 8)'''}}
|-
|style=&quot;background:#33eb2a;{{text color default}};&quot;|1
|87181920722
|[[File:Inversion set 16; Walsh (2, 1, 4, 8).svg|36px]]
|{{font|face=Courier|'''( 2, 1, 4, 8)'''}}
|-
|style=&quot;background:#33eb2a;{{text color default}};&quot;|2
|13412045088
|[[File:Inversion set 16; wp( 1, 4, 2, 8).svg|36px]]
|{{font|face=Courier|'''( 1, 4, 2, 8)'''}}
|-
|style=&quot;background:white;{{text color default}};&quot;|3
|101069338632
|[[File:Inversion set 16; Walsh (2, 4, 1, 8).svg|36px]]
|{{font|face=Courier|'''( 4, 1, 2, 8)'''}}
|-
|style=&quot;background:white;{{text color default}};&quot;|4
|262496507810
|[[File:Inversion set 16; Walsh (4, 1, 2, 8).svg|36px]]
|{{font|face=Courier|'''( 2, 4, 1, 8)'''}}
|-
|style=&quot;background:#33eb2a;{{text color default}};&quot;|5
|269202530354
|[[File:Inversion set 16; Walsh (4, 2, 1, 8).svg|36px]]
|{{font|face=Courier|'''( 4, 2, 1, 8)'''}}
|-
|style=&quot;background:#33eb2a;{{text color default}};&quot;|6
|175795200
|[[File:Inversion set 16; Walsh (1, 2, 8, 4).svg|36px]]
|{{font|face=Courier|'''( 1, 2, 8, 4)'''}}
|-
|style=&quot;background:white;{{text color default}};&quot;|&lt;big&gt;'''7'''&lt;/big&gt;
|87357715922
|[[File:Inversion set 16; Walsh (2, 1, 8, 4).svg|36px]]
|{{font|face=Courier|'''( 2, 1, 8, 4)'''}}
|-
|style=&quot;background:white;{{text color default}};&quot;|8
|13588646400
|[[File:Inversion set 16; Walsh (1, 4, 8, 2).svg|36px]]
|{{font|face=Courier|'''( 1, 8, 2, 4)'''}}
|-
|style=&quot;background:#ffbb00;{{text color default}};&quot;|9
|101249608320
|[[File:Inversion set 16; Walsh (2, 4, 8, 1).svg|36px]]
|{{font|face=Courier|'''( 8, 1, 2, 4)'''}}
|-
|style=&quot;background:#ffbb00;{{text color default}};&quot;|10
|262679639762
|[[File:Inversion set 16; Walsh (4, 1, 8, 2).svg|36px]]
|{{font|face=Courier|'''( 2, 8, 1, 4)'''}}
|-
|style=&quot;background:white;{{text color default}};&quot;|11
|269386065362
|[[File:Inversion set 16; Walsh (4, 2, 8, 1).svg|36px]]
|{{font|face=Courier|'''( 8, 2, 1, 4)'''}}
|-
|style=&quot;background:white;{{text color default}};&quot;|12
|40237747488
|[[File:Inversion set 16; Walsh (1, 8, 2, 4).svg|36px]]
|{{font|face=Courier|'''( 1, 4, 8, 2)'''}}
|-
|style=&quot;background:#ffbb00;{{text color default}};&quot;|13
|127895041032
|[[File:Inversion set 16; Walsh (2, 8, 1, 4).svg|36px]]
|{{font|face=Courier|'''( 4, 1, 8, 2)'''}}
|-
|style=&quot;background:#33eb2a;{{text color default}};&quot;|14
|40325645088
|[[File:Inversion set 16; wp( 1, 8, 4, 2).svg|36px]]
|{{font|face=Courier|'''( 1, 8, 4, 2)'''}}
|-
|style=&quot;background:white;{{text color default}};&quot;|15
|127986607008
|[[File:Inversion set 16; Walsh (2, 8, 4, 1).svg|36px]]
|{{font|face=Courier|'''( 8, 1, 4, 2)'''}}
|-
|style=&quot;background:white;{{text color default}};&quot;|&lt;big&gt;'''16'''&lt;/big&gt;
|303216964872
|[[File:Inversion set 16; wp( 4, 8, 1, 2).svg|36px]]
|{{font|face=Courier|'''( 4, 8, 1, 2)'''}}
|-
|style=&quot;background:#ffbb00;{{text color default}};&quot;|17
|303260913672
|[[File:Inversion set 16; Walsh (4, 8, 2, 1).svg|36px]]
|{{font|face=Courier|'''( 8, 4, 1, 2)'''}}
|-
|style=&quot;background:#ffbb00;{{text color default}};&quot;|18
|613140355490
|[[File:Inversion set 16; Walsh (8, 1, 2, 4).svg|36px]]
|{{font|face=Courier|'''( 2, 4, 8, 1)'''}}
|-
|style=&quot;background:white;{{text color default}};&quot;|19
|619846378034
|[[File:Inversion set 16; Walsh (8, 2, 1, 4).svg|36px]]
|{{font|face=Courier|'''( 4, 2, 8, 1)'''}}
|-
|style=&quot;background:white;{{text color default}};&quot;|20
|613228253090
|[[File:Inversion set 16; Walsh (8, 1, 4, 2).svg|36px]]
|{{font|face=Courier|'''( 2, 8, 4, 1)'''}}
|-
|style=&quot;background:#33eb2a;{{text color default}};&quot;|21
|619934678690
|[[File:Inversion set 16; Walsh (8, 2, 4, 1).svg|36px]]
|{{font|face=Courier|'''( 8, 2, 4, 1)'''}}
|-
|style=&quot;background:#ffbb00;{{text color default}};&quot;|22
|633259229234
|[[File:Inversion set 16; wp( 4, 8, 2, 1).svg|36px]]
|{{font|face=Courier|'''( 4, 8, 2, 1)'''}}
|-
|style=&quot;background:white;{{text color default}};&quot;|&lt;big&gt;'''23'''&lt;/big&gt;
|633303178034
|[[File:Inversion set 16; Walsh (8, 4, 2, 1).svg|36px]]
|{{font|face=Courier|'''( 8, 4, 2, 1)'''}}
|}

{| class=&quot;collapsible collapsed&quot; style=&quot;width: 100%; border: 1px solid #cccccc; background:white;{{text color default}};&quot;
! bgcolor=&quot;#f0f0f0&quot;|{{font|color=#999999|F}}
|-
|
 Little-endian factorial numbers:
     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0
     0     0     1     0     0     0     1     0     0     0     1     0     0     0     1     0
     0     0     0     0     2     2     0     0     0     0     0     0     2     2     0     0
     0     0     0     0     3     2     1     0     0     0     0     0     3     2     1     0
     0     0     1     0     2     0     3     0     0     0     1     0     2     0     3     0
     0     0     1     0     3     1     3     0     0     0     1     0     3     1     3     0
     0     0     0     0     0     0     0     0     4     4     4     4     0     0     0     0
     0     0     1     0     0     0     1     0     4     4     5     4     0     0     1     0
     0     0     0     0     0     0     0     0     6     6     4     4     2     2     0     0
     0     0     0     0     0     0     0     0     7     6     5     4     3     2     1     0
     0     0     1     0     0     0     1     0     6     4     7     4     2     0     3     0
     0     0     1     0     0     0     1     0     7     5     7     4     3     1     3     0
     0     0     0     0     2     2     0     0     4     4     0     0     6     6     0     0
     0     0     0     0     3     2     1     0     4     4     0     0     7     6     1     0
     0     0     0     0     2     2     0     0     6     6     2     2     6     6     0     0
     0     0     0     0     2     2     0     0     7     6     3     2     7     6     1     0
     0     0     0     0     3     2     1     0     6     4     2     0     9     6     3     0
     0     0     0     0     3     2     1     0     7     5     3     1     9     6     3     0
     0     0     1     0     2     0     3     0     4     0     5     0     6     0     7     0
     0     0     1     0     3     1     3     0     4     0     5     0     7     1     7     0
     0     0     1     0     2     0     3     0     6     2     7     2     6     0     7     0
     0     0     1     0     2     0     3     0     7     3     7     2     7     1     7     0
     0     0     1     0     3     1     3     0     6     2     5     0     9     3     7     0
     0     0     1     0     3     1     3     0     7     3     6     1     9     3     7     0
 &amp;nbsp;
 Inversion numbers:
     0
     4
     8
    12
    12
    16
    16
    20
    24
    28
    28
    32
    24
    28
    32
    36
    36
    40
    28
    32
    36
    40
    40
    44
|}

==Nimber multiplication==
{| align=&quot;right&quot; style=&quot;text-align:right; border:1px solid lightgray&quot; cellspacing=&quot;10&quot;
| [[Walsh permutation]]; [[Walsh permutation; nimber multiplication|nimber multiplication]] {{rdrup}}
|}

Images: {{commons catlink|Inversion set of a 4-bit Walsh permutation; nimber multiplication}}

{{clear}}

{| class=&quot;wikitable sortable&quot; style=&quot;text-align:right&quot;
|-
! #
! CV
! Permutation
! F
! IN
|- style=&quot;font-family:Courier&quot;
|              0||( 1, 2, 4, 8)||(0, '''1''', 2, 3, &amp;nbsp; 4, 5, 6, 7, &amp;nbsp; 8, 9,10,11, &amp;nbsp;12,13,14,15) || (0, 0, 0, 0, &amp;nbsp; 0, 0, 0, 0, &amp;nbsp; 0, 0, 0, 0, &amp;nbsp; 0, 0, 0, '''0''') ||  0
|- style=&quot;font-family:Courier&quot;
| 13827898032492||( 2, 3, 8,12)||(0, '''2''', 3, 1, &amp;nbsp; 8,10,11, 9, &amp;nbsp;12,14,15,13, &amp;nbsp; 4, 6, 7, 5) || (0, 0, 0, 2, &amp;nbsp; 0, 0, 0, 2, &amp;nbsp; 0, 0, 0, 2, &amp;nbsp; 8, 8, 8,'''10''') || 40
|- style=&quot;font-family:Courier&quot;
|  7001306732168||( 3, 1,12, 4)||(0, '''3''', 1, 2, &amp;nbsp;12,15,13,14, &amp;nbsp; 4, 7, 5, 6, &amp;nbsp; 8,11, 9,10) || (0, 0, 1, 1, &amp;nbsp; 0, 0, 1, 1, &amp;nbsp; 4, 4, 5, 5, &amp;nbsp; 4, 4, 5, '''5''') || 40
|- style=&quot;font-family:Courier&quot;
| 19217812814448||( 4, 8, 6,11)||(0, '''4''', 8,12, &amp;nbsp; 6, 2,14,10, &amp;nbsp;11,15, 3, 7, &amp;nbsp;13, 9, 5, 1) || (0, 0, 0, 0, &amp;nbsp; 2, 4, 0, 2, &amp;nbsp; 2, 0, 8, 6, &amp;nbsp; 2, 6,10,'''14''') || 56
|- style=&quot;font-family:Courier&quot;
|  1630614229512||( 5,10, 2, 3)||(0, '''5''',10,15, &amp;nbsp; 2, 7, 8,13, &amp;nbsp; 3, 6, 9,12, &amp;nbsp; 1, 4,11,14) || (0, 0, 0, 0, &amp;nbsp; 3, 2, 2, 1, &amp;nbsp; 6, 5, 3, 2, &amp;nbsp;11, 9, 3, '''1''') || 48
|- style=&quot;font-family:Courier&quot;
| 15433122573720||( 6,11,14, 7)||(0, '''6''',11,13, &amp;nbsp;14, 8, 5, 3, &amp;nbsp; 7, 1,12,10, &amp;nbsp; 9,15, 2, 4) || (0, 0, 0, 0, &amp;nbsp; 0, 3, 5, 6, &amp;nbsp; 4, 8, 2, 4, &amp;nbsp; 5, 0,12,'''11''') || 60
|- style=&quot;font-family:Courier&quot;
|  5564988000144||( 7, 9,10,15)||(0, '''7''', 9,14, &amp;nbsp;10,13, 3, 4, &amp;nbsp;15, 8, 6, 1, &amp;nbsp; 5, 2,12,11) || (0, 0, 0, 0, &amp;nbsp; 1, 1, 5, 5, &amp;nbsp; 0, 5, 7,10, &amp;nbsp; 8,11, 3, '''4''') || 60
|- style=&quot;font-family:Courier&quot;
| 17445432818676||( 8,12,11,13)||(0, '''8''',12, 4, &amp;nbsp;11, 3, 7,15, &amp;nbsp;13, 5, 1, 9, &amp;nbsp; 6,14,10, 2) || (0, 0, 0, 2, &amp;nbsp; 1, 4, 3, 0, &amp;nbsp; 1, 6, 9, 4, &amp;nbsp; 7, 1, 5,'''13''') || 56
|- style=&quot;font-family:Courier&quot;
|  3551688986892||( 9,14,15, 5)||(0, '''9''',14, 7, &amp;nbsp;15, 6, 1, 8, &amp;nbsp; 5,12,11, 2, &amp;nbsp;10, 3, 4,13) || (0, 0, 0, 2, &amp;nbsp; 0, 4, 5, 3, &amp;nbsp; 6, 2, 3, 9, &amp;nbsp; 4,10,10, '''2''') || 60
|- style=&quot;font-family:Courier&quot;
| 10678227659604||(10,15, 3, 1)||(0,'''10''',15, 5, &amp;nbsp; 3, 9,12, 6, &amp;nbsp; 1,11,14, 4, &amp;nbsp; 2, 8,13, 7) || (0, 0, 0, 2, &amp;nbsp; 3, 2, 1, 4, &amp;nbsp; 7, 2, 1, 8, &amp;nbsp;10, 6, 2, '''8''') || 56
|- style=&quot;font-family:Courier&quot;
| 10169232103620||(11,13, 7, 9)||(0,'''11''',13, 6, &amp;nbsp; 7,12,10, 1, &amp;nbsp; 9, 2, 4,15, &amp;nbsp;14, 5, 3, 8) || (0, 0, 0, 2, &amp;nbsp; 2, 1, 3, 6, &amp;nbsp; 4, 7, 7, 0, &amp;nbsp; 1, 9,11, '''7''') || 60
|- style=&quot;font-family:Courier&quot;
| 15737148505928||(12, 4,13, 6)||(0,'''12''', 4, 8, &amp;nbsp;13, 1, 9, 5, &amp;nbsp; 6,10, 2,14, &amp;nbsp;11, 7,15, 3) || (0, 0, 1, 1, &amp;nbsp; 0, 4, 2, 4, &amp;nbsp; 4, 2, 8, 0, &amp;nbsp; 3, 7, 0,'''12''') || 48
|- style=&quot;font-family:Courier&quot;
|  5084423155256||(13, 6, 9,14)||(0,'''13''', 6,11, &amp;nbsp; 9, 4,15, 2, &amp;nbsp;14, 3, 8, 5, &amp;nbsp; 7,10, 1,12) || (0, 0, 1, 1, &amp;nbsp; 2, 4, 0, 6, &amp;nbsp; 1, 7, 5, 7, &amp;nbsp; 6, 4,13, '''3''') || 60
|- style=&quot;font-family:Courier&quot;
| 12454407149960||(14, 7, 5,10)||(0,'''14''', 7, 9, &amp;nbsp; 5,11, 2,12, &amp;nbsp;10, 4,13, 3, &amp;nbsp;15, 1, 8, 6) || (0, 0, 1, 1, &amp;nbsp; 3, 1, 5, 1, &amp;nbsp; 3, 7, 1, 9, &amp;nbsp; 0,12, 7, '''9''') || 60
|- style=&quot;font-family:Courier&quot;
|  8566683323240||(15, 5, 1, 2)||(0,'''15''', 5,10, &amp;nbsp; 1,14, 4,11, &amp;nbsp; 2,13, 7, 8, &amp;nbsp; 3,12, 6, 9) || (0, 0, 1, 1, &amp;nbsp; 3, 1, 4, 2, &amp;nbsp; 6, 2, 5, 5, &amp;nbsp; 9, 3, 8, '''6''') || 56
|}

[[Category:Walsh permutation]]</text>
      <sha1>46ntt3t99tgha9zec8nead2w697fa4s</sha1>
    </revision>
    <revision>
      <id>2819770</id>
      <parentid>2819769</parentid>
      <timestamp>2026-07-29T21:06:51Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <comment>/* Bit permutations */</comment>
      <origin>2819770</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="41078" sha1="3i3qwtzi76pup6xekkqm9gu83i9tv64" xml:space="preserve">{| align=&quot;right&quot; style=&quot;text-align:right; border:1px solid lightgray&quot; cellspacing=&quot;10&quot;
| [[Walsh permutation]] {{rdrup}}
|-
| [[Inversion (discrete mathematics)]] {{rdrup}}
|}

[[File:Construction site ahead (17184445992).jpg|thumb|The vectors need to be corrected. See [[Walsh permutation#Notation warning]]. Done only for section [[#Nimber multiplication|Nimber multiplication]].]]

{|
|style=&quot;vertical-align:top;&quot;|
&lt;!---▶▶▶---&gt;{| style=&quot;border: 1px solid #cccccc&quot;
! bgcolor=&quot;#dddddd&quot; colspan=&quot;2&quot;|Abbreviations
|-
|
|-
|'''CV'''
|Compression vector
|-
|'''CM'''
|Compression matrix
|-
|'''Σ'''
|Number of ones in the compression matrix
|-
|'''IS'''
|Inversion set
|-
|'''#'''
|Index number of the finite permutation
|-
|'''IN'''
|Inversion number
|}&lt;!---◀◀◀---&gt;
|style=&quot;vertical-align:top;&quot;|__TOC__
|}

&lt;br style=&quot;clear: both;&quot;&gt;

==Stripes==
Among the ''n''-bit Walsh permutations are 2&lt;sup&gt;''n''&lt;/sup&gt; with striped inversion sets.&lt;br&gt;
As seen below they correspond to Walsh functions, i.e. the rows of the [[w:Walsh matrix|Walsh matrix]] of order 2&lt;sup&gt;''n''&lt;/sup&gt;.&lt;br&gt;
These permutations under [[w:Function composition|composition]] form the group Z&lt;sub&gt;2&lt;/sub&gt;&lt;sup&gt;''n''&lt;/sup&gt;, just like the row numbers under bitwise XOR.

=== 3-bit ===

An extract of the [[3-bit Walsh permutation; table|'''table''']] of [[3-bit Walsh permutation]]s:
{| class=&quot;wikitable sortable&quot; style=&quot;height:495px; text-align:center;&quot;
|-style=&quot;height:55px&quot;
!
!style=&quot;width:55px&quot;|CV
!style=&quot;width:55px&quot; class=&quot;unsortable&quot;|&amp;nbsp;&amp;nbsp;CM&amp;nbsp;&amp;nbsp;
!D
!Σ
!#
!Permutation&lt;br&gt;F
!IN
!class=&quot;unsortable&quot;|IS
!style=&quot;width:55px&quot; class=&quot;unsortable&quot;|PM
!class=&quot;unsortable&quot;|FPC
!Cy
|-   
|0 || {{font|color=#aaa|1 2 4}} || {{BIN1start}}{{BIN0}}{{BIN0;}}&lt;br&gt;{{BIN0}}{{BIN1}}{{BIN0;}}&lt;br&gt;{{BIN0}}{{BIN0}}{{BIN1end}} || + || 3 || align=&quot;right&quot;|  0 || {{font|color=#aaa|0 1 2 3 &amp;nbsp; 4 5 6 7}}&lt;br&gt;{{font|color=#adf|0  0  0  0 &amp;nbsp; 0  0  0  0}} || {{font|color=#adf|&amp;#8199;0}} ||{{Inversions8|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0}}  || {{Row8,0start}}{{Row8,1}}{{Row8,2}}{{Row8,3}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,4}}{{Row8,5}}{{Row8,6}}{{Row8,7end}} || [[File:Walsh permutation 124 Fano.svg|53px]] || align=&quot;left&quot;|{{sms|1 }}{{font|color=#c07|&amp;nbsp;&amp;#8199;0}}
|-   
|style=&quot;background:lightgray;{{text color default}};&quot;| 1 || 3 5 1  || {{BIN1start}}{{BIN1}}{{BIN1;}}&lt;br&gt;{{BIN1}}{{BIN0}}{{BIN0;}}&lt;br&gt;{{BIN0}}{{BIN1}}{{BIN0end}} || + || 5 || align=&quot;right&quot;|17576 || {{font|color=#aaa|0}}  7 1 6  &amp;nbsp;  2  {{font|color=#aaa|5}}  3 4 &lt;br&gt;{{font|color=#adf|0  0  1  1 &amp;nbsp; 2  2  3  3}} || {{font|color=#adf|12}} ||{{Inversions8|0|0|1|0|1|0|0|1|0|1|0|1|0|1|0|0|1|0|1|0|1|0|1|0|1|0|1|0}}  || {{Row8,0start}}{{Row8,7}}{{Row8,1}}{{Row8,6}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,2}}{{Row8,5}}{{Row8,3}}{{Row8,4end}} || || align=&quot;left&quot;|{{sms|3 }}{{font|color=#c07|&amp;nbsp;&amp;#8199;8}}&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;span style=&quot;font-size:60%&quot;&gt;(4+2)&lt;/span&gt;&gt;
|-   
|style=&quot;background:lightgray;{{text color default}};&quot;| 2 || 3 6 2  || {{BIN1start}}{{BIN0}}{{BIN0;}}&lt;br&gt;{{BIN1}}{{BIN1}}{{BIN1;}}&lt;br&gt;{{BIN0}}{{BIN1}}{{BIN0end}} || − || 5 || align=&quot;right&quot;|16854 || {{font|color=#aaa|0 1}}  7 6  &amp;nbsp;  2 3 5 4 &lt;br&gt;{{font|color=#adf|0  0  0  1 &amp;nbsp; 2  2  2  3}} || {{font|color=#adf|10}} ||{{Inversions8|0|0|0|0|0|1|0|0|1|1|0|0|1|1|0|0|0|1|1|0|0|0|0|1|1|0|0|1}}  || {{Row8,0start}}{{Row8,1}}{{Row8,7}}{{Row8,6}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,2}}{{Row8,3}}{{Row8,5}}{{Row8,4end}} || || align=&quot;left&quot;|{{sms|4 }}{{font|color=#c07|&amp;nbsp;&amp;#8199;9}}&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;span style=&quot;font-size:60%&quot;&gt;(3+3)&lt;/span&gt;&gt;
|-   
|3 || {{font|color=#aaa|1}}  7 3  || {{BIN1start}}{{BIN1}}{{BIN1;}}&lt;br&gt;{{BIN0}}{{BIN1}}{{BIN1;}}&lt;br&gt;{{BIN0}}{{BIN1}}{{BIN0end}} || − || 6 || align=&quot;right&quot;|22622 || {{font|color=#aaa|0}}  7 6 1  &amp;nbsp;  2  {{font|color=#aaa|5}}  4 3 &lt;br&gt;{{font|color=#adf|0  0  1  2 &amp;nbsp; 2  2  3  4}} || {{font|color=#adf|14}} ||{{Inversions8|0|0|1|0|1|1|0|1|1|0|0|1|1|0|0|0|1|1|0|0|1|0|1|1|0|0|1|1}}  || {{Row8,0start}}{{Row8,7}}{{Row8,6}}{{Row8,1}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,2}}{{Row8,5}}{{Row8,4}}{{Row8,3end}} || || align=&quot;left&quot;|{{sms|4 }}{{font|color=#c07|&amp;nbsp;&amp;#8199;9}}&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;span style=&quot;font-size:60%&quot;&gt;(3+3)&lt;/span&gt;&gt;
|-   
|style=&quot;background:lightgray;{{text color default}};&quot;| 4 || 5 6  {{font|color=#aaa|4}} || {{BIN1start}}{{BIN0}}{{BIN0;}}&lt;br&gt;{{BIN0}}{{BIN1}}{{BIN0;}}&lt;br&gt;{{BIN1}}{{BIN1}}{{BIN1end}} || + || 5 || align=&quot;right&quot;|16680 || {{font|color=#aaa|0 1 2 3}} &amp;nbsp;  7 6 5 4 &lt;br&gt;{{font|color=#adf|0  0  0  0 &amp;nbsp; 0  1  2  3}} || {{font|color=#adf|&amp;#8199;6}} ||{{Inversions8|0|0|0|0|0|0|0|0|0|0|0|0|0|0|1|0|0|0|0|1|1|0|0|0|0|1|1|1}}  || {{Row8,0start}}{{Row8,1}}{{Row8,2}}{{Row8,3}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,7}}{{Row8,6}}{{Row8,5}}{{Row8,4end}} || || align=&quot;left&quot;|{{sms|2 }}{{font|color=#c07|&amp;nbsp;&amp;#8199;3}}&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;span style=&quot;font-size:60%&quot;&gt;(2+2)&lt;/span&gt;&gt;
|-   
|5 || 7 1 5  || {{BIN1start}}{{BIN1}}{{BIN1;}}&lt;br&gt;{{BIN1}}{{BIN0}}{{BIN0;}}&lt;br&gt;{{BIN1}}{{BIN0}}{{BIN1end}} || − || 6 || align=&quot;right&quot;|22736 || {{font|color=#aaa|0}}  7 1 6  &amp;nbsp;  5 2 4 3 &lt;br&gt;{{font|color=#adf|0  0  1  1 &amp;nbsp; 2  3  3  4}} || {{font|color=#adf|14}} ||{{Inversions8|0|0|1|0|1|0|0|1|0|1|0|1|0|1|1|0|1|0|1|1|0|0|1|0|1|1|0|1}}  || {{Row8,0start}}{{Row8,7}}{{Row8,1}}{{Row8,6}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,5}}{{Row8,2}}{{Row8,4}}{{Row8,3end}} || || align=&quot;left&quot;|{{sms|6 }}{{font|color=#c07|&amp;nbsp;14}}&lt;sub&gt;b&lt;/sub&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;span style=&quot;font-size:60%&quot;&gt;(7)&lt;/span&gt;&gt;
|-   
|6 || 7  {{font|color=#aaa|2}}  6  || {{BIN1start}}{{BIN0}}{{BIN0;}}&lt;br&gt;{{BIN1}}{{BIN1}}{{BIN1;}}&lt;br&gt;{{BIN1}}{{BIN0}}{{BIN1end}} || + || 6 || align=&quot;right&quot;|23454 || {{font|color=#aaa|0 1}}  7 6  &amp;nbsp;  5 4 2 3 &lt;br&gt;{{font|color=#adf|0  0  0  1 &amp;nbsp; 2  3  4  4}} || {{font|color=#adf|14}} ||{{Inversions8|0|0|0|0|0|1|0|0|1|1|0|0|1|1|1|0|0|1|1|1|1|0|0|1|1|1|1|0}}  || {{Row8,0start}}{{Row8,1}}{{Row8,7}}{{Row8,6}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,5}}{{Row8,4}}{{Row8,2}}{{Row8,3end}} || [[File:Walsh permutation 165 Fano red.svg|53px]] || align=&quot;left&quot;|{{sms|3 }}{{font|color=#c07|&amp;nbsp;&amp;#8199;8}}&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;span style=&quot;font-size:60%&quot;&gt;(4+2)&lt;/span&gt;&gt;
|-   
|style=&quot;background:lightgray;{{text color default}};&quot;| 7 || 5 3 7  || {{BIN1start}}{{BIN1}}{{BIN1;}}&lt;br&gt;{{BIN0}}{{BIN1}}{{BIN1;}}&lt;br&gt;{{BIN1}}{{BIN0}}{{BIN1end}} || + || 7 || align=&quot;right&quot;|17702 || {{font|color=#aaa|0}}  7 6 1  &amp;nbsp;  5 2 3 4 &lt;br&gt;{{font|color=#adf|0  0  1  2 &amp;nbsp; 2  3  3  3}} || {{font|color=#adf|14}} ||{{Inversions8|0|0|1|0|1|1|0|1|1|0|0|1|1|0|1|0|1|1|0|1|0|0|1|1|0|1|0|0}}  || {{Row8,0start}}{{Row8,7}}{{Row8,6}}{{Row8,1}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,5}}{{Row8,2}}{{Row8,3}}{{Row8,4end}} || [[File:Walsh permutation 357 Fano red.svg|53px]] || align=&quot;left&quot;|{{sms|5 }}{{font|color=#c07|&amp;nbsp;14}}&lt;sub&gt;a&lt;/sub&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;span style=&quot;font-size:60%&quot;&gt;(7)&lt;/span&gt;&gt;
|}
{|
|Sorting by '''#''' gives the permutation (0,4,2,1,7,3,5,6) in the left column, which is ''reflection'' [[w:Function composition|∘]] [[w:Circular shift|''right shift'']] ∘ ''wp''(6,5,7) (with the compression matrix || {{Bin0start}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1end}} || ).
|}
Sorting by '''Permutation''' gives the (0,4,2,6, 1,5,3,7) in the left column, which is the bit-reversal permutation ''wp''(4,2,1).

=== 4-bit ===

[[File:Variadic logical XOR.svg|thumb|right|Binary Walsh matrix of order 16]]

{| class=&quot;wikitable sortable&quot; style=&quot;text-align:right;&quot;
|-
!
!#
!class=&quot;unsortable&quot;|IS
!CV
!class=&quot;unsortable&quot;|CM
!Σ
!Permutation&lt;br&gt;F
!IN
|-
|0
|0
|[[File:16 choose 2 array.svg|36px]]
|{{font|face=Courier|{{nowrap|'''( 1, 2, 4, 8)'''}}}}
|&lt;div style=&quot;white-space: nowrap&quot;&gt;{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin0}}{{Bin1end}}&lt;/div&gt;
|4
|{{font|face=Courier|{{nowrap|'''( 0, 1, 2, 3,&amp;nbsp; 4, 5, 6, 7,&amp;nbsp; 8, 9,10,11,&amp;nbsp;12,13,14,15)'''}}&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 0,&amp;nbsp; 0, 0, 0, 0,&amp;nbsp; 0, 0, 0, 0,&amp;nbsp; 0, 0, 0, 0)'''}}}}
|{{font|color=#adf|0}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|1
|9804424107176
|[[File:Inversion set 16; Walsh (15, 1, 2, 4).svg|36px]]
|{{font|face=Courier|'''( 3, 5, 9, 1)'''}}
|{{Bin1start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0end}}
|7
|{{font|face=Courier|'''( 0,15, 1,14,&amp;nbsp; 2,13, 3,12,&amp;nbsp; 4,11, 5,10,&amp;nbsp; 6, 9, 7, 8)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 1,&amp;nbsp; 2, 2, 3, 3,&amp;nbsp; 4, 4, 5, 5,&amp;nbsp; 6, 6, 7, 7)'''}}}}
|{{font|color=#adf|56}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|2
|9717242186454
|[[File:Inversion set 16; Walsh (1, 15, 2, 4).svg|36px]]
|{{font|face=Courier|'''( 3, 6,10, 2)'''}}
|{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0end}}
|7
|{{font|face=Courier|'''( 0, 1,15,14,&amp;nbsp; 2, 3,13,12,&amp;nbsp; 4, 5,11,10,&amp;nbsp; 6, 7, 9, 8)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 1,&amp;nbsp; 2, 2, 2, 3,&amp;nbsp; 4, 4, 4, 5,&amp;nbsp; 6, 6, 6, 7)'''}}}}
|{{font|color=#adf|52}}
|-
|3
|11112138397022
|[[File:Inversion set 16; wp( 1, 7,11, 3).svg|36px]]
|{{font|face=Courier|'''( 1, 7,11, 3)'''}}
|{{Bin1start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0end}}
|9
|{{font|face=Courier|'''( 0,15,14, 1,&amp;nbsp; 2,13,12, 3,&amp;nbsp; 4,11,10, 5,&amp;nbsp; 6, 9, 8, 7)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 2,&amp;nbsp; 2, 2, 3, 4,&amp;nbsp; 4, 4, 5, 6,&amp;nbsp; 6, 6, 7, 8)'''}}}}
|{{font|color=#adf|60}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|4
|9710017245480
|[[File:Inversion set 16; wp( 5, 6,12, 4).svg|36px]]
|{{font|face=Courier|'''( 5, 6,12, 4)'''}}
|{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0end}}
|7
|{{font|face=Courier|'''( 0, 1, 2, 3,&amp;nbsp;15,14,13,12,&amp;nbsp; 4, 5, 6, 7,&amp;nbsp;11,10, 9, 8)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 0,&amp;nbsp; 0, 1, 2, 3,&amp;nbsp; 4, 4, 4, 4,&amp;nbsp; 4, 5, 6, 7)'''}}}}
|{{font|color=#adf|44}}
|-
|5
|11118325501136
|[[File:Inversion set 16; Walsh (15, 1, 13, 4).svg|36px]]
|{{font|face=Courier|'''( 7, 1,13, 5)'''}}
|{{Bin1start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0end}}
|9
|{{font|face=Courier|'''( 0,15, 1,14,&amp;nbsp;13, 2,12, 3,&amp;nbsp; 4,11, 5,10,&amp;nbsp; 9, 6, 8, 7)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 1,&amp;nbsp; 2, 3, 3, 4,&amp;nbsp; 4, 4, 5, 5,&amp;nbsp; 6, 7, 7, 8)'''}}}}
|{{font|color=#adf|60}}
|-
|6
|11205500164254
|[[File:Inversion set 16; Walsh (1, 15, 13, 4).svg|36px]]
|{{font|face=Courier|'''( 7, 2,14, 6)'''}}
|{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0end}}
|9
|{{font|face=Courier|'''( 0, 1,15,14,&amp;nbsp;13,12, 2, 3,&amp;nbsp; 4, 5,11,10,&amp;nbsp; 9, 8, 6, 7)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 1,&amp;nbsp; 2, 3, 4, 4,&amp;nbsp; 4, 4, 4, 5,&amp;nbsp; 6, 7, 8, 8)'''}}}}
|{{font|color=#adf|60}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|7
|9810691044902
|[[File:Inversion set 16; Walsh (15, 14, 13, 4).svg|36px]]
|{{font|face=Courier|'''( 5, 3,15, 7)'''}}
|{{Bin1start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0end}}
|11
|{{font|face=Courier|'''( 0,15,14, 1,&amp;nbsp;13, 2, 3,12,&amp;nbsp; 4,11,10, 5,&amp;nbsp; 9, 6, 7, 8)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 2,&amp;nbsp; 2, 3, 3, 3,&amp;nbsp; 4, 4, 5, 6,&amp;nbsp; 6, 7, 7, 7)'''}}}}
|{{font|color=#adf|60}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|8
|9709968804480
|[[File:Inversion set 16; Walsh (1, 2, 4, 15).svg|36px]]
|{{font|face=Courier|'''( 9,10,12, 8)'''}}
|{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin1end}}
|7
|{{font|face=Courier|'''( 0, 1, 2, 3,&amp;nbsp; 4, 5, 6, 7,&amp;nbsp;15,14,13,12,&amp;nbsp;11,10, 9, 8)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 0,&amp;nbsp; 0, 0, 0, 0,&amp;nbsp;  0, 1, 2, 3,&amp;nbsp; 4, 5, 6, 7)'''}}}}
|{{font|color=#adf|28}}
|-
|9
|11118365775656
|[[File:Inversion set 16; Walsh (15, 1, 2, 11).svg|36px]]
|{{font|face=Courier|'''(11,13, 1, 9)'''}}
|{{Bin1start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin0}}{{Bin1end}}
|9
|{{font|face=Courier|'''( 0,15, 1,14,&amp;nbsp; 2,13, 3,12,&amp;nbsp;11, 4,10, 5,&amp;nbsp; 9, 6, 8, 7)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 1,&amp;nbsp; 2, 2, 3, 3,&amp;nbsp; 4, 5, 5, 6,&amp;nbsp; 6, 7, 7, 8)'''}}}}
|{{font|color=#adf|60}}
|-
|10
|11205547694934
|[[File:Inversion set 16; Walsh (1, 15, 2, 11).svg|36px]]
|{{font|face=Courier|'''(11,14, 2,10)'''}}
|{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin0}}{{Bin1end}}
|9
|{{font|face=Courier|'''( 0, 1,15,14,&amp;nbsp; 2, 3,13,12,&amp;nbsp;11,10, 4, 5,&amp;nbsp; 9, 8, 6, 7)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 1,&amp;nbsp; 2, 2, 2, 3,&amp;nbsp; 4, 5, 6, 6,&amp;nbsp; 6, 7, 8, 8)'''}}}}
|{{font|color=#adf|60}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|11
|9810651495902
|[[File:Inversion set 16; Walsh (15, 14, 2, 11).svg|36px]]
|{{font|face=Courier|'''( 9,15, 3,11)'''}}
|{{Bin1start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin0}}{{Bin1end}}
|11
|{{font|face=Courier|'''( 0,15,14, 1,&amp;nbsp; 2,13,12, 3,&amp;nbsp;11, 4, 5,10,&amp;nbsp; 9, 6, 7, 8)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 2,&amp;nbsp; 2, 2, 3, 4,&amp;nbsp; 4, 5, 5, 5,&amp;nbsp; 6, 7, 7, 7)'''}}}}
|{{font|color=#adf|60}}
|-
|12
|11212772635560
|[[File:Inversion set 16; Walsh (1, 2, 15, 11).svg|36px]]
|{{font|face=Courier|'''(13,14, 4,12)'''}}
|{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin0}}{{Bin1end}}
|9
|{{font|face=Courier|'''( 0, 1, 2, 3,&amp;nbsp;15,14,13,12,&amp;nbsp;11,10, 9, 8,&amp;nbsp; 4, 5, 6, 7)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 0,&amp;nbsp;  0, 1, 2, 3,&amp;nbsp; 4, 5, 6, 7,&amp;nbsp; 8, 8, 8, 8)'''}}}}
|{{font|color=#adf|60}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|13
|9804464392016
|[[File:Inversion set 16; Walsh (15, 1, 13, 11).svg|36px]]
|{{font|face=Courier|'''(15, 9, 5,13)'''}}
|{{Bin1start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin0}}{{Bin1end}}
|11
|{{font|face=Courier|'''( 0,15, 1,14,&amp;nbsp;13, 2,12, 3,&amp;nbsp;11, 4,10, 5,&amp;nbsp; 6, 9, 7, 8)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 1,&amp;nbsp; 2, 3, 3, 4,&amp;nbsp; 4, 5, 5, 6,&amp;nbsp; 6, 6, 7, 7)'''}}}}
|{{font|color=#adf|60}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|14
|9717289730334
|[[File:Inversion set 16; Walsh (1, 15, 13, 11).svg|36px]]
|{{font|face=Courier|'''(15,10, 6,14)'''}}
|{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin0}}{{Bin1end}}
|11
|{{font|face=Courier|'''( 0, 1,15,14,&amp;nbsp;13,12, 2, 3,&amp;nbsp;11,10, 4, 5,&amp;nbsp; 6, 7, 9, 8)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 1,&amp;nbsp; 2, 3, 4, 4,&amp;nbsp; 4, 5, 6, 6,&amp;nbsp; 6, 6, 6, 7)'''}}}}
|{{font|color=#adf|60}}
|-
|15
|11112098838182
|[[File:Inversion set 16; Walsh (15, 14, 13, 11).svg|36px]]
|{{font|face=Courier|'''(13,11, 7,15)'''}}
|{{Bin1start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin0}}{{Bin1end}}
|13
|{{font|face=Courier|'''( 0,15,14, 1,&amp;nbsp;13, 2, 3,12,&amp;nbsp;11, 4, 5,10,&amp;nbsp; 6, 9, 8, 7)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 2,&amp;nbsp; 2, 3, 3, 3,&amp;nbsp; 4, 5, 5, 5,&amp;nbsp; 6, 6, 7, 8)'''}}}}
|{{font|color=#adf|60}}
|}

{|
|Sorting by '''#''' gives the (0, 8,4,2,14, 1,13,11,7, 15,3,5,9, 6,10,12) in the left column, which is ''reflection'' [[w:Function composition|∘]] [[w:Circular shift|''right shift'']] ∘ ''wp''(12,10, 9, 7) (with the compression matrix || {{Bin0start}}{{Bin0}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin0}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin0end}} || ).
|}
Sorting by '''Permutation''' gives the (0,8,4,12, 2,10,6,14, 1,9,5,13, 3,11,7,15) in the left column, which is the [[w:Bit-reversal permutation|bit-reversal permutation]] ''wp''( 8, 4, 2, 1).

{| class=&quot;collapsible collapsed&quot; style=&quot;width: 100%; border: 1px solid #cccccc; background:white;{{text color default}};&quot;
! bgcolor=&quot;#f0f0f0&quot;|{{font|color=#999999|Matrices}}
|-
|
 Compression vectors:
     1     2     4     8
     3     5     9     1
     3     6    10     2
     1     7    11     3
     5     6    12     4
     7     1    13     5
     7     2    14     6
     5     3    15     7
     9    10    12     8
    11    13     1     9
    11    14     2    10
     9    15     3    11
    13    14     4    12
    15     9     5    13
    15    10     6    14
    13    11     7    15
 &amp;nbsp;
 Permutations:
     0     1     2     3     4     5     6     7     8     9    10    11    12    13    14    15
     0    15     1    14     2    13     3    12     4    11     5    10     6     9     7     8
     0     1    15    14     2     3    13    12     4     5    11    10     6     7     9     8
     0    15    14     1     2    13    12     3     4    11    10     5     6     9     8     7
     0     1     2     3    15    14    13    12     4     5     6     7    11    10     9     8
     0    15     1    14    13     2    12     3     4    11     5    10     9     6     8     7
     0     1    15    14    13    12     2     3     4     5    11    10     9     8     6     7
     0    15    14     1    13     2     3    12     4    11    10     5     9     6     7     8
     0     1     2     3     4     5     6     7    15    14    13    12    11    10     9     8
     0    15     1    14     2    13     3    12    11     4    10     5     9     6     8     7
     0     1    15    14     2     3    13    12    11    10     4     5     9     8     6     7
     0    15    14     1     2    13    12     3    11     4     5    10     9     6     7     8
     0     1     2     3    15    14    13    12    11    10     9     8     4     5     6     7
     0    15     1    14    13     2    12     3    11     4    10     5     6     9     7     8
     0     1    15    14    13    12     2     3    11    10     4     5     6     7     9     8
     0    15    14     1    13     2     3    12    11     4     5    10     6     9     8     7
 &amp;nbsp;
 Little-endian factorial numbers:
     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0
     0     0     1     1     2     2     3     3     4     4     5     5     6     6     7     7
     0     0     0     1     2     2     2     3     4     4     4     5     6     6     6     7
     0     0     1     2     2     2     3     4     4     4     5     6     6     6     7     8
     0     0     0     0     0     1     2     3     4     4     4     4     4     5     6     7
     0     0     1     1     2     3     3     4     4     4     5     5     6     7     7     8
     0     0     0     1     2     3     4     4     4     4     4     5     6     7     8     8
     0     0     1     2     2     3     3     3     4     4     5     6     6     7     7     7
     0     0     0     0     0     0     0     0     0     1     2     3     4     5     6     7
     0     0     1     1     2     2     3     3     4     5     5     6     6     7     7     8
     0     0     0     1     2     2     2     3     4     5     6     6     6     7     8     8
     0     0     1     2     2     2     3     4     4     5     5     5     6     7     7     7
     0     0     0     0     0     1     2     3     4     5     6     7     8     8     8     8
     0     0     1     1     2     3     3     4     4     5     5     6     6     6     7     7
     0     0     0     1     2     3     4     4     4     5     6     6     6     6     6     7
     0     0     1     2     2     3     3     3     4     5     5     5     6     6     7     8
|}

=== Compression matrices ===

[[File:Compression matrices of Walsh permutations with striped inversion sets.svg|left|800px]]
&lt;br style=&quot;clear: both;&quot;&gt;
The CM of the ''n''-th permutation is a matrix ''B'' with horizontal stripes for the binary digits of ''n'' &lt;small&gt;(gray lines in the image)&lt;/small&gt;, XORed with a modified unity matrix ''U'' &lt;small&gt;(black dots in the image)&lt;/small&gt;.&lt;br&gt;
''U'' is modified by including an empty row where ''n'' has its lowest binary digit ({{oeis|A001511}}). It pushes the other rows of the unity matrix down, and the last row &quot;out of the matrix&quot;.


==Bended stripes==

Among the ''n''-bit Walsh permutations are 2&lt;sup&gt;''n-1''&lt;/sup&gt; whose inversion sets show bended stripes.&lt;br&gt;
They correspond to Walsh functions, i.e. the rows of the [[w:Walsh matrix|Walsh matrix]] of order 2&lt;sup&gt;''n-1''&lt;/sup&gt;.&lt;br&gt;
These permutations under composition form the group Z&lt;sub&gt;2&lt;/sub&gt;&lt;sup&gt;''n''-1&lt;/sup&gt;, just like the row numbers under bitwise XOR.

=== 3-bit ===

{| class=&quot;wikitable sortable&quot; style=&quot;height:275px; text-align:center;&quot;
|-style=&quot;height:55px&quot;
!
!style=&quot;width:55px&quot;|CV
!style=&quot;width:55px&quot; class=&quot;unsortable&quot;|&amp;nbsp;&amp;nbsp;CM&amp;nbsp;&amp;nbsp;
!D
!#
!Permutation&lt;br&gt;F
!IN
!class=&quot;unsortable&quot;|IS
!style=&quot;width:55px&quot; class=&quot;unsortable&quot;|PM
!class=&quot;unsortable&quot;|FPC
!Cy
|-   
|0 || {{font|color=#aaa|1 2 4}} || {{BIN1start}}{{BIN0}}{{BIN0;}}&lt;br&gt;{{BIN0}}{{BIN1}}{{BIN0;}}&lt;br&gt;{{BIN0}}{{BIN0}}{{BIN1end}} || + || align=&quot;right&quot;|  0 || {{font|color=#aaa|0 1 2 3 &amp;nbsp; 4 5 6 7}}&lt;br&gt;{{font|color=#adf|0  0  0  0 &amp;nbsp; 0  0  0  0}} || {{font|color=#adf|&amp;#8199;0}} ||{{Inversions8|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0}}  || {{Row8,0start}}{{Row8,1}}{{Row8,2}}{{Row8,3}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,4}}{{Row8,5}}{{Row8,6}}{{Row8,7end}} || [[File:Walsh permutation 124 Fano.svg|53px]] || align=&quot;left&quot;|{{sms|1 }}{{font|color=#c07|&amp;nbsp;&amp;#8199;0}}
|-   
|style=&quot;background:lightgray;{{text color default}};&quot;|1 ||  4 7 1  || {{BIN0start}}{{BIN1}}{{BIN1;}}&lt;br&gt;{{BIN0}}{{BIN1}}{{BIN0;}}&lt;br&gt;{{BIN1}}{{BIN1}}{{BIN0end}} || − || align=&quot;right&quot;| 3776 || {{font|color=#aaa|0}}  6  {{font|color=#aaa|2}}  4  &amp;nbsp;  3  {{font|color=#aaa|5}}  1  {{font|color=#aaa|7}}&lt;br&gt;{{font|color=#adf|0  0  1  1 &amp;nbsp; 2  1  5  0}} || {{font|color=#adf|10}} ||{{Inversions8|0|0|1|0|1|0|0|1|0|1|0|1|0|0|0|0|1|1|1|1|1|0|0|0|0|0|0|0}}  || {{Row8,0start}}{{Row8,6}}{{Row8,2}}{{Row8,4}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,3}}{{Row8,5}}{{Row8,1}}{{Row8,7end}} || || align=&quot;left&quot;|{{sms|2 }}{{font|color=#c07|&amp;nbsp;&amp;#8199;3}}&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;span style=&quot;font-size:60%&quot;&gt;(2+2)&lt;/span&gt;&gt;
|-   
|style=&quot;background:lightgray;{{text color default}};&quot;|2 ||  7 4 2  || {{BIN1start}}{{BIN0}}{{BIN0;}}&lt;br&gt;{{BIN1}}{{BIN0}}{{BIN1;}}&lt;br&gt;{{BIN1}}{{BIN1}}{{BIN0end}} || − || align=&quot;right&quot;|  414 || {{font|color=#aaa|0 1}}  5 4  &amp;nbsp;  3 2  {{font|color=#aaa|6 7}}&lt;br&gt;{{font|color=#adf|0  0  0  1 &amp;nbsp; 2  3  0  0}} || {{font|color=#adf|&amp;#8199;6}} ||{{Inversions8|0|0|0|0|0|1|0|0|1|1|0|0|1|1|1|0|0|0|0|0|0|0|0|0|0|0|0|0}}  || {{Row8,0start}}{{Row8,1}}{{Row8,5}}{{Row8,4}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,3}}{{Row8,2}}{{Row8,6}}{{Row8,7end}} || || align=&quot;left&quot;|{{sms|2 }}{{font|color=#c07|&amp;nbsp;&amp;#8199;3}}&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;span style=&quot;font-size:60%&quot;&gt;(2+2)&lt;/span&gt;&gt;
|-   
|3 ||  2 1 7  || {{BIN0start}}{{BIN1}}{{BIN1;}}&lt;br&gt;{{BIN1}}{{BIN0}}{{BIN1;}}&lt;br&gt;{{BIN0}}{{BIN0}}{{BIN1end}} || − || align=&quot;right&quot;| 4142 || {{font|color=#aaa|0}}  6 5  {{font|color=#aaa|3 &amp;nbsp; 4}}  2 1  {{font|color=#aaa|7}}&lt;br&gt;{{font|color=#adf|0  0  1  2 &amp;nbsp; 2  4  5  0}} || {{font|color=#adf|14}} ||{{Inversions8|0|0|1|0|1|1|0|1|1|0|0|1|1|1|1|0|1|1|1|1|1|0|0|0|0|0|0|0}}  || {{Row8,0start}}{{Row8,6}}{{Row8,5}}{{Row8,3}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,4}}{{Row8,2}}{{Row8,1}}{{Row8,7end}} || || align=&quot;left&quot;|{{sms|2 }}{{font|color=#c07|&amp;nbsp;&amp;#8199;3}}&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;span style=&quot;font-size:60%&quot;&gt;(2+2)&lt;/span&gt;&gt;
|}

Sorting by '''#''' or '''Permutation''' gives (0,2,1,3) in the left column, which is the bit-reversal permutation ''wp''(2,1).&lt;br&gt;
Sorting by '''CV''' gives (0,3,1,2) in the left column, which is ''wp''(3,1).


=== 4-bit ===

{| class=&quot;wikitable sortable&quot; style=&quot;text-align:right;&quot;
|-
!
!#
!class=&quot;unsortable&quot;|IS
!CV
!class=&quot;unsortable&quot;|CM
!Permutation&lt;br&gt;F
!IN
|-
|0
|0
|[[File:16 choose 2 array.svg|36px]]
|{{font|face=Courier|{{nowrap|'''( 1, 2, 4, 8)'''}}}}
|&lt;div style=&quot;white-space: nowrap&quot;&gt;{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin0}}{{Bin1end}}&lt;/div&gt;
|{{font|face=Courier|{{nowrap|'''( 0, 1, 2, 3,&amp;nbsp; 4, 5, 6, 7,&amp;nbsp; 8, 9,10,11,&amp;nbsp;12,13,14,15)'''}}&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 0,&amp;nbsp; 0, 0, 0, 0,&amp;nbsp; 0, 0, 0, 0,&amp;nbsp; 0, 0, 0, 0)'''}}}}
|{{font|color=#adf|0}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|1
|1144441325096
|[[File:Inversion set 16; Walsh (14, 2, 4, 7).svg|36px]]
|{{font|face=Courier|'''( 8,11,13, 1)'''}}
|{{Bin0start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin0end}}
|{{font|face=Courier|'''( 0,14, 2,12,&amp;nbsp; 4,10, 6, 8,&amp;nbsp; 7, 9, 5,11,&amp;nbsp; 3,13, 1,15)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 1,&amp;nbsp; 2, 2, 3, 3,&amp;nbsp; 4, 3, 7, 2,&amp;nbsp;10, 1,13, 0)'''}}}}
|{{font|color=#adf|52}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|2
|73376328534
|[[File:Inversion set 16; Walsh (1, 13, 4, 7).svg|36px]]
|{{font|face=Courier|'''(11, 8,14, 2)'''}}
|{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin0end}}
|{{font|face=Courier|'''( 0, 1,13,12,&amp;nbsp; 4, 5, 9, 8,&amp;nbsp; 7, 6,10,11,&amp;nbsp; 3, 2,14,15)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 1,&amp;nbsp; 2, 2, 2, 3,&amp;nbsp; 4, 5, 2, 2,&amp;nbsp;10,11, 0, 0)'''}}}}
|{{font|color=#adf|44}}
|-
|3
|1209107634782
|[[File:Inversion set 16; Walsh (14, 13, 4, 8).svg|36px]]
|{{font|face=Courier|'''( 2, 1, 7,11)'''}}
|{{Bin0start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin0}}{{Bin1end}}
|{{font|face=Courier|'''( 0,14,13, 3,&amp;nbsp; 4,10, 9, 7,&amp;nbsp; 8, 6, 5,11,&amp;nbsp;12, 2, 1,15)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 2,&amp;nbsp; 2, 2, 3, 4,&amp;nbsp; 4, 6, 7, 2,&amp;nbsp; 2,12,13, 0)'''}}}}
|{{font|color=#adf|60}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|4
|303182760
|[[File:Inversion set 16; Walsh (1, 2, 11, 7).svg|36px]]
|{{font|face=Courier|'''(13,14, 8, 4)'''}}
|{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin0}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin0end}}
|{{font|face=Courier|'''( 0, 1, 2, 3,&amp;nbsp;11,10, 9, 8,&amp;nbsp; 7, 6, 5, 4,&amp;nbsp;12,13,14,15)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 0,&amp;nbsp; 0, 1, 2, 3,&amp;nbsp; 4, 5, 6, 7,&amp;nbsp; 0, 0, 0, 0)'''}}}}
|{{font|color=#adf|28}}
|-
|5
|1144707321296
|[[File:Inversion set 16; Walsh (14, 2, 11, 8).svg|36px]]
|{{font|face=Courier|'''( 4, 7, 1,13)'''}}
|{{Bin0start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin0}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin0}}{{Bin1end}}
|{{font|face=Courier|'''( 0,14, 2,12,&amp;nbsp;11, 5, 9, 7,&amp;nbsp; 8, 6,10, 4,&amp;nbsp; 3,13, 1,15)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 1,&amp;nbsp; 2, 3, 3, 4,&amp;nbsp; 4, 6, 3, 9,&amp;nbsp;10, 1,13, 0)'''}}}}
|{{font|color=#adf|60}}
|-
|6
|73677162654
|[[File:Inversion set 16; Walsh (1, 13, 11, 8).svg|36px]]
|{{font|face=Courier|'''( 7, 4, 2,14)'''}}
|{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin0}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin0}}{{Bin1end}}
|{{font|face=Courier|'''( 0, 1,13,12,&amp;nbsp;11,10, 6, 7,&amp;nbsp; 8, 9, 5, 4,&amp;nbsp; 3, 2,14,15)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 1,&amp;nbsp; 2, 3, 4, 4,&amp;nbsp; 4, 4, 8, 9,&amp;nbsp;10,11, 0, 0)'''}}}}
|{{font|color=#adf|60}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|7
|1209371443622
|[[File:Inversion set 16; Walsh (14, 13, 11, 7).svg|36px]]
|{{font|face=Courier|'''(14,13,11, 7)'''}}
|{{Bin0start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin0}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin0end}}
|{{font|face=Courier|'''( 0,14,13, 3,&amp;nbsp;11, 5, 6, 8,&amp;nbsp; 7, 9,10, 4,&amp;nbsp;12, 2, 1,15)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 2,&amp;nbsp; 2, 3, 3, 3,&amp;nbsp; 4, 3, 3, 9,&amp;nbsp; 2,12,13, 0)'''}}}}
|{{font|color=#adf|60}}
|}

Sorting by '''#''' or '''Permutation''' gives (0,4,2,6, 1,5,3,7) in the left column, which is the bit-reversal permutation ''wp''(4,2,1).&lt;br&gt;
Sorting by '''CV''' gives (0,3,5,6, 1,2,4,7) in the left column, which is ''wp''(7,1,2).

===Compression matrices===

[[File:Compression matrices of Walsh permutations with bendedly striped inversion sets.svg|800px]]
&lt;br style=&quot;clear: both;&quot;&gt;
The CM are matrices with an even number of horizontal stripes &lt;small&gt;(gray lines in the image)&lt;/small&gt; XORed with the unity matrix &lt;small&gt;(black dots in the image)&lt;/small&gt;.&lt;br&gt;
The order by the ''evil numbers'' ({{oeis|A001969}}) of the matrices, used in the image, corresponds to the lexicographical order of the compression vectors.

==Bit permutations==
{| align=&quot;right&quot;
|colspan=&quot;2&quot;|
{| align=right style=&quot;text-align=right; border:1px solid lightgray&quot; cellspacing=&quot;10&quot;
|[[Walsh permutation]]; [[Walsh permutation; bit permutation|bit permutation]] {{Rdrup}}
|}
|-
|style=&quot;vertical-align:top;&quot;|[[File:Inversion set 16; Walsh (2, 8, 1, 4).svg|thumb|Inversion set of ''wp''( 4, 1, 8, 2)&lt;br&gt;All columns are symmetrical.]]
|style=&quot;vertical-align:top;&quot;|[[File:Inversion set 16; Walsh (14, 13, 11, 7).svg|thumb|Inversion set of ''wp''(14,13,11, 7)]]
|-
|colspan=&quot;2&quot;|
{| align=right
|-
|style=&quot;vertical-align:top;&quot;|[[File:Symmetric group 4; cycle graph; details.svg|thumb|center|500px|[[w:Cycle graph (algebra)|Cycle graph]] of the bit permutations]]
|style=&quot;vertical-align:top;&quot;|[[File:Symmetric group 4; permutation list (0-based).svg|thumb|center|Permutations of 4 elements&lt;br&gt;{{Rdr}} [[Symmetric group S4|Symmetric group S&lt;sub&gt;4&lt;/sub&gt;]]]]
|}
|}


{| class=&quot;wikitable sortable&quot; style=&quot;text-align:right;&quot;
|-
!
!#
!class=&quot;unsortable&quot;|IS
!CV
|-
|style=&quot;background:white;{{text color default}};&quot;|&lt;big&gt;'''0'''&lt;/big&gt;
|0
|[[File:16 choose 2 array.svg|36px]]
|{{font|face=Courier|'''( 1, 2, 4, 8)'''}}
|-
|style=&quot;background:#33eb2a;{{text color default}};&quot;|1
|87181920722
|[[File:Inversion set 16; Walsh (2, 1, 4, 8).svg|36px]]
|{{font|face=Courier|'''( 2, 1, 4, 8)'''}}
|-
|style=&quot;background:#33eb2a;{{text color default}};&quot;|2
|13412045088
|[[File:Inversion set 16; wp( 1, 4, 2, 8).svg|36px]]
|{{font|face=Courier|'''( 1, 4, 2, 8)'''}}
|-
|style=&quot;background:white;{{text color default}};&quot;|3
|101069338632
|[[File:Inversion set 16; Walsh (2, 4, 1, 8).svg|36px]]
|{{font|face=Courier|'''( 4, 1, 2, 8)'''}}
|-
|style=&quot;background:white;{{text color default}};&quot;|4
|262496507810
|[[File:Inversion set 16; Walsh (4, 1, 2, 8).svg|36px]]
|{{font|face=Courier|'''( 2, 4, 1, 8)'''}}
|-
|style=&quot;background:#33eb2a;{{text color default}};&quot;|5
|269202530354
|[[File:Inversion set 16; Walsh (4, 2, 1, 8).svg|36px]]
|{{font|face=Courier|'''( 4, 2, 1, 8)'''}}
|-
|style=&quot;background:#33eb2a;{{text color default}};&quot;|6
|175795200
|[[File:Inversion set 16; Walsh (1, 2, 8, 4).svg|36px]]
|{{font|face=Courier|'''( 1, 2, 8, 4)'''}}
|-
|style=&quot;background:white;{{text color default}};&quot;|&lt;big&gt;'''7'''&lt;/big&gt;
|87357715922
|[[File:Inversion set 16; Walsh (2, 1, 8, 4).svg|36px]]
|{{font|face=Courier|'''( 2, 1, 8, 4)'''}}
|-
|style=&quot;background:white;{{text color default}};&quot;|8
|13588646400
|[[File:Inversion set 16; Walsh (1, 4, 8, 2).svg|36px]]
|{{font|face=Courier|'''( 1, 8, 2, 4)'''}}
|-
|style=&quot;background:#ffbb00;{{text color default}};&quot;|9
|101249608320
|[[File:Inversion set 16; Walsh (2, 4, 8, 1).svg|36px]]
|{{font|face=Courier|'''( 8, 1, 2, 4)'''}}
|-
|style=&quot;background:#ffbb00;{{text color default}};&quot;|10
|262679639762
|[[File:Inversion set 16; Walsh (4, 1, 8, 2).svg|36px]]
|{{font|face=Courier|'''( 2, 8, 1, 4)'''}}
|-
|style=&quot;background:white;{{text color default}};&quot;|11
|269386065362
|[[File:Inversion set 16; Walsh (4, 2, 8, 1).svg|36px]]
|{{font|face=Courier|'''( 8, 2, 1, 4)'''}}
|-
|style=&quot;background:white;{{text color default}};&quot;|12
|40237747488
|[[File:Inversion set 16; Walsh (1, 8, 2, 4).svg|36px]]
|{{font|face=Courier|'''( 1, 4, 8, 2)'''}}
|-
|style=&quot;background:#ffbb00;{{text color default}};&quot;|13
|127895041032
|[[File:Inversion set 16; Walsh (2, 8, 1, 4).svg|36px]]
|{{font|face=Courier|'''( 4, 1, 8, 2)'''}}
|-
|style=&quot;background:#33eb2a;{{text color default}};&quot;|14
|40325645088
|[[File:Inversion set 16; wp( 1, 8, 4, 2).svg|36px]]
|{{font|face=Courier|'''( 1, 8, 4, 2)'''}}
|-
|style=&quot;background:white;{{text color default}};&quot;|15
|127986607008
|[[File:Inversion set 16; Walsh (2, 8, 4, 1).svg|36px]]
|{{font|face=Courier|'''( 8, 1, 4, 2)'''}}
|-
|style=&quot;background:white;{{text color default}};&quot;|&lt;big&gt;'''16'''&lt;/big&gt;
|303216964872
|[[File:Inversion set 16; wp( 4, 8, 1, 2).svg|36px]]
|{{font|face=Courier|'''( 4, 8, 1, 2)'''}}
|-
|style=&quot;background:#ffbb00;{{text color default}};&quot;|17
|303260913672
|[[File:Inversion set 16; Walsh (4, 8, 2, 1).svg|36px]]
|{{font|face=Courier|'''( 8, 4, 1, 2)'''}}
|-
|style=&quot;background:#ffbb00;{{text color default}};&quot;|18
|613140355490
|[[File:Inversion set 16; Walsh (8, 1, 2, 4).svg|36px]]
|{{font|face=Courier|'''( 2, 4, 8, 1)'''}}
|-
|style=&quot;background:white;{{text color default}};&quot;|19
|619846378034
|[[File:Inversion set 16; Walsh (8, 2, 1, 4).svg|36px]]
|{{font|face=Courier|'''( 4, 2, 8, 1)'''}}
|-
|style=&quot;background:white;{{text color default}};&quot;|20
|613228253090
|[[File:Inversion set 16; Walsh (8, 1, 4, 2).svg|36px]]
|{{font|face=Courier|'''( 2, 8, 4, 1)'''}}
|-
|style=&quot;background:#33eb2a;{{text color default}};&quot;|21
|619934678690
|[[File:Inversion set 16; Walsh (8, 2, 4, 1).svg|36px]]
|{{font|face=Courier|'''( 8, 2, 4, 1)'''}}
|-
|style=&quot;background:#ffbb00;{{text color default}};&quot;|22
|633259229234
|[[File:Inversion set 16; wp( 4, 8, 2, 1).svg|36px]]
|{{font|face=Courier|'''( 4, 8, 2, 1)'''}}
|-
|style=&quot;background:white;{{text color default}};&quot;|&lt;big&gt;'''23'''&lt;/big&gt;
|633303178034
|[[File:Inversion set 16; Walsh (8, 4, 2, 1).svg|36px]]
|{{font|face=Courier|'''( 8, 4, 2, 1)'''}}
|}

{| class=&quot;collapsible collapsed&quot; style=&quot;width: 100%; border: 1px solid #cccccc; background:white;{{text color default}};&quot;
! bgcolor=&quot;#f0f0f0&quot;|{{font|color=#999999|F}}
|-
|
 Little-endian factorial numbers:
     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0
     0     0     1     0     0     0     1     0     0     0     1     0     0     0     1     0
     0     0     0     0     2     2     0     0     0     0     0     0     2     2     0     0
     0     0     0     0     3     2     1     0     0     0     0     0     3     2     1     0
     0     0     1     0     2     0     3     0     0     0     1     0     2     0     3     0
     0     0     1     0     3     1     3     0     0     0     1     0     3     1     3     0
     0     0     0     0     0     0     0     0     4     4     4     4     0     0     0     0
     0     0     1     0     0     0     1     0     4     4     5     4     0     0     1     0
     0     0     0     0     0     0     0     0     6     6     4     4     2     2     0     0
     0     0     0     0     0     0     0     0     7     6     5     4     3     2     1     0
     0     0     1     0     0     0     1     0     6     4     7     4     2     0     3     0
     0     0     1     0     0     0     1     0     7     5     7     4     3     1     3     0
     0     0     0     0     2     2     0     0     4     4     0     0     6     6     0     0
     0     0     0     0     3     2     1     0     4     4     0     0     7     6     1     0
     0     0     0     0     2     2     0     0     6     6     2     2     6     6     0     0
     0     0     0     0     2     2     0     0     7     6     3     2     7     6     1     0
     0     0     0     0     3     2     1     0     6     4     2     0     9     6     3     0
     0     0     0     0     3     2     1     0     7     5     3     1     9     6     3     0
     0     0     1     0     2     0     3     0     4     0     5     0     6     0     7     0
     0     0     1     0     3     1     3     0     4     0     5     0     7     1     7     0
     0     0     1     0     2     0     3     0     6     2     7     2     6     0     7     0
     0     0     1     0     2     0     3     0     7     3     7     2     7     1     7     0
     0     0     1     0     3     1     3     0     6     2     5     0     9     3     7     0
     0     0     1     0     3     1     3     0     7     3     6     1     9     3     7     0
 &amp;nbsp;
 Inversion numbers:
     0
     4
     8
    12
    12
    16
    16
    20
    24
    28
    28
    32
    24
    28
    32
    36
    36
    40
    28
    32
    36
    40
    40
    44
|}

==Nimber multiplication==
{| align=&quot;right&quot; style=&quot;text-align:right; border:1px solid lightgray&quot; cellspacing=&quot;10&quot;
| [[Walsh permutation]]; [[Walsh permutation; nimber multiplication|nimber multiplication]] {{rdrup}}
|}

Images: {{commons catlink|Inversion set of a 4-bit Walsh permutation; nimber multiplication}}

{{clear}}

{| class=&quot;wikitable sortable&quot; style=&quot;text-align:right&quot;
|-
! #
! CV
! Permutation
! F
! IN
|- style=&quot;font-family:Courier&quot;
|              0||( 1, 2, 4, 8)||(0, '''1''', 2, 3, &amp;nbsp; 4, 5, 6, 7, &amp;nbsp; 8, 9,10,11, &amp;nbsp;12,13,14,15) || (0, 0, 0, 0, &amp;nbsp; 0, 0, 0, 0, &amp;nbsp; 0, 0, 0, 0, &amp;nbsp; 0, 0, 0, '''0''') ||  0
|- style=&quot;font-family:Courier&quot;
| 13827898032492||( 2, 3, 8,12)||(0, '''2''', 3, 1, &amp;nbsp; 8,10,11, 9, &amp;nbsp;12,14,15,13, &amp;nbsp; 4, 6, 7, 5) || (0, 0, 0, 2, &amp;nbsp; 0, 0, 0, 2, &amp;nbsp; 0, 0, 0, 2, &amp;nbsp; 8, 8, 8,'''10''') || 40
|- style=&quot;font-family:Courier&quot;
|  7001306732168||( 3, 1,12, 4)||(0, '''3''', 1, 2, &amp;nbsp;12,15,13,14, &amp;nbsp; 4, 7, 5, 6, &amp;nbsp; 8,11, 9,10) || (0, 0, 1, 1, &amp;nbsp; 0, 0, 1, 1, &amp;nbsp; 4, 4, 5, 5, &amp;nbsp; 4, 4, 5, '''5''') || 40
|- style=&quot;font-family:Courier&quot;
| 19217812814448||( 4, 8, 6,11)||(0, '''4''', 8,12, &amp;nbsp; 6, 2,14,10, &amp;nbsp;11,15, 3, 7, &amp;nbsp;13, 9, 5, 1) || (0, 0, 0, 0, &amp;nbsp; 2, 4, 0, 2, &amp;nbsp; 2, 0, 8, 6, &amp;nbsp; 2, 6,10,'''14''') || 56
|- style=&quot;font-family:Courier&quot;
|  1630614229512||( 5,10, 2, 3)||(0, '''5''',10,15, &amp;nbsp; 2, 7, 8,13, &amp;nbsp; 3, 6, 9,12, &amp;nbsp; 1, 4,11,14) || (0, 0, 0, 0, &amp;nbsp; 3, 2, 2, 1, &amp;nbsp; 6, 5, 3, 2, &amp;nbsp;11, 9, 3, '''1''') || 48
|- style=&quot;font-family:Courier&quot;
| 15433122573720||( 6,11,14, 7)||(0, '''6''',11,13, &amp;nbsp;14, 8, 5, 3, &amp;nbsp; 7, 1,12,10, &amp;nbsp; 9,15, 2, 4) || (0, 0, 0, 0, &amp;nbsp; 0, 3, 5, 6, &amp;nbsp; 4, 8, 2, 4, &amp;nbsp; 5, 0,12,'''11''') || 60
|- style=&quot;font-family:Courier&quot;
|  5564988000144||( 7, 9,10,15)||(0, '''7''', 9,14, &amp;nbsp;10,13, 3, 4, &amp;nbsp;15, 8, 6, 1, &amp;nbsp; 5, 2,12,11) || (0, 0, 0, 0, &amp;nbsp; 1, 1, 5, 5, &amp;nbsp; 0, 5, 7,10, &amp;nbsp; 8,11, 3, '''4''') || 60
|- style=&quot;font-family:Courier&quot;
| 17445432818676||( 8,12,11,13)||(0, '''8''',12, 4, &amp;nbsp;11, 3, 7,15, &amp;nbsp;13, 5, 1, 9, &amp;nbsp; 6,14,10, 2) || (0, 0, 0, 2, &amp;nbsp; 1, 4, 3, 0, &amp;nbsp; 1, 6, 9, 4, &amp;nbsp; 7, 1, 5,'''13''') || 56
|- style=&quot;font-family:Courier&quot;
|  3551688986892||( 9,14,15, 5)||(0, '''9''',14, 7, &amp;nbsp;15, 6, 1, 8, &amp;nbsp; 5,12,11, 2, &amp;nbsp;10, 3, 4,13) || (0, 0, 0, 2, &amp;nbsp; 0, 4, 5, 3, &amp;nbsp; 6, 2, 3, 9, &amp;nbsp; 4,10,10, '''2''') || 60
|- style=&quot;font-family:Courier&quot;
| 10678227659604||(10,15, 3, 1)||(0,'''10''',15, 5, &amp;nbsp; 3, 9,12, 6, &amp;nbsp; 1,11,14, 4, &amp;nbsp; 2, 8,13, 7) || (0, 0, 0, 2, &amp;nbsp; 3, 2, 1, 4, &amp;nbsp; 7, 2, 1, 8, &amp;nbsp;10, 6, 2, '''8''') || 56
|- style=&quot;font-family:Courier&quot;
| 10169232103620||(11,13, 7, 9)||(0,'''11''',13, 6, &amp;nbsp; 7,12,10, 1, &amp;nbsp; 9, 2, 4,15, &amp;nbsp;14, 5, 3, 8) || (0, 0, 0, 2, &amp;nbsp; 2, 1, 3, 6, &amp;nbsp; 4, 7, 7, 0, &amp;nbsp; 1, 9,11, '''7''') || 60
|- style=&quot;font-family:Courier&quot;
| 15737148505928||(12, 4,13, 6)||(0,'''12''', 4, 8, &amp;nbsp;13, 1, 9, 5, &amp;nbsp; 6,10, 2,14, &amp;nbsp;11, 7,15, 3) || (0, 0, 1, 1, &amp;nbsp; 0, 4, 2, 4, &amp;nbsp; 4, 2, 8, 0, &amp;nbsp; 3, 7, 0,'''12''') || 48
|- style=&quot;font-family:Courier&quot;
|  5084423155256||(13, 6, 9,14)||(0,'''13''', 6,11, &amp;nbsp; 9, 4,15, 2, &amp;nbsp;14, 3, 8, 5, &amp;nbsp; 7,10, 1,12) || (0, 0, 1, 1, &amp;nbsp; 2, 4, 0, 6, &amp;nbsp; 1, 7, 5, 7, &amp;nbsp; 6, 4,13, '''3''') || 60
|- style=&quot;font-family:Courier&quot;
| 12454407149960||(14, 7, 5,10)||(0,'''14''', 7, 9, &amp;nbsp; 5,11, 2,12, &amp;nbsp;10, 4,13, 3, &amp;nbsp;15, 1, 8, 6) || (0, 0, 1, 1, &amp;nbsp; 3, 1, 5, 1, &amp;nbsp; 3, 7, 1, 9, &amp;nbsp; 0,12, 7, '''9''') || 60
|- style=&quot;font-family:Courier&quot;
|  8566683323240||(15, 5, 1, 2)||(0,'''15''', 5,10, &amp;nbsp; 1,14, 4,11, &amp;nbsp; 2,13, 7, 8, &amp;nbsp; 3,12, 6, 9) || (0, 0, 1, 1, &amp;nbsp; 3, 1, 4, 2, &amp;nbsp; 6, 2, 5, 5, &amp;nbsp; 9, 3, 8, '''6''') || 56
|}

[[Category:Walsh permutation]]</text>
      <sha1>3i3qwtzi76pup6xekkqm9gu83i9tv64</sha1>
    </revision>
    <revision>
      <id>2819771</id>
      <parentid>2819770</parentid>
      <timestamp>2026-07-29T21:08:44Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <origin>2819771</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="41074" sha1="askto5be4rektvonn9w4htek8eu84qi" xml:space="preserve">{| align=&quot;right&quot; style=&quot;text-align:right; border:1px solid lightgray&quot; cellspacing=&quot;10&quot;
| [[Walsh permutation]] {{rdrup}}
|-
| [[Inversion (discrete mathematics)]] {{rdrup}}
|}

[[File:Construction site ahead (17184445992).jpg|thumb|The vectors need to be corrected. See [[Walsh permutation#Notation warning]]. Done only for section [[#Nimber multiplication|Nimber multiplication]].]]

{|
|style=&quot;vertical-align:top;&quot;|
&lt;!---▶▶▶---&gt;{| style=&quot;border: 1px solid #cccccc&quot;
! bgcolor=&quot;#dddddd&quot; colspan=&quot;2&quot;|Abbreviations
|-
|
|-
|'''CV'''
|Compression vector
|-
|'''CM'''
|Compression matrix
|-
|'''Σ'''
|Number of ones in the compression matrix
|-
|'''IS'''
|Inversion set
|-
|'''#'''
|Index number of the finite permutation
|-
|'''IN'''
|Inversion number
|}&lt;!---◀◀◀---&gt;
|style=&quot;vertical-align:top;&quot;|__TOC__
|}

&lt;br style=&quot;clear: both;&quot;&gt;

==Stripes==
Among the ''n''-bit Walsh permutations are 2&lt;sup&gt;''n''&lt;/sup&gt; with striped inversion sets.&lt;br&gt;
As seen below they correspond to Walsh functions, i.e. the rows of the [[w:Walsh matrix|Walsh matrix]] of order 2&lt;sup&gt;''n''&lt;/sup&gt;.&lt;br&gt;
These permutations under [[w:Function composition|composition]] form the group Z&lt;sub&gt;2&lt;/sub&gt;&lt;sup&gt;''n''&lt;/sup&gt;, just like the row numbers under bitwise XOR.

=== 3-bit ===

An extract of the [[3-bit Walsh permutation; table|'''table''']] of [[3-bit Walsh permutation]]s:
{| class=&quot;wikitable sortable&quot; style=&quot;height:495px; text-align:center;&quot;
|-style=&quot;height:55px&quot;
!
!style=&quot;width:55px&quot;|CV
!style=&quot;width:55px&quot; class=&quot;unsortable&quot;|&amp;nbsp;&amp;nbsp;CM&amp;nbsp;&amp;nbsp;
!D
!Σ
!#
!Permutation&lt;br&gt;F
!IN
!class=&quot;unsortable&quot;|IS
!style=&quot;width:55px&quot; class=&quot;unsortable&quot;|PM
!class=&quot;unsortable&quot;|FPC
!Cy
|-   
|0 || {{font|color=#aaa|1 2 4}} || {{BIN1start}}{{BIN0}}{{BIN0;}}&lt;br&gt;{{BIN0}}{{BIN1}}{{BIN0;}}&lt;br&gt;{{BIN0}}{{BIN0}}{{BIN1end}} || + || 3 || align=&quot;right&quot;|  0 || {{font|color=#aaa|0 1 2 3 &amp;nbsp; 4 5 6 7}}&lt;br&gt;{{font|color=#adf|0  0  0  0 &amp;nbsp; 0  0  0  0}} || {{font|color=#adf|&amp;#8199;0}} ||{{Inversions8|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0}}  || {{Row8,0start}}{{Row8,1}}{{Row8,2}}{{Row8,3}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,4}}{{Row8,5}}{{Row8,6}}{{Row8,7end}} || [[File:Walsh permutation 124 Fano.svg|53px]] || align=&quot;left&quot;|{{sms|1 }}{{font|color=#c07|&amp;nbsp;&amp;#8199;0}}
|-   
|style=&quot;background:lightgray;{{text color default}};&quot;| 1 || 3 5 1  || {{BIN1start}}{{BIN1}}{{BIN1;}}&lt;br&gt;{{BIN1}}{{BIN0}}{{BIN0;}}&lt;br&gt;{{BIN0}}{{BIN1}}{{BIN0end}} || + || 5 || align=&quot;right&quot;|17576 || {{font|color=#aaa|0}}  7 1 6  &amp;nbsp;  2  {{font|color=#aaa|5}}  3 4 &lt;br&gt;{{font|color=#adf|0  0  1  1 &amp;nbsp; 2  2  3  3}} || {{font|color=#adf|12}} ||{{Inversions8|0|0|1|0|1|0|0|1|0|1|0|1|0|1|0|0|1|0|1|0|1|0|1|0|1|0|1|0}}  || {{Row8,0start}}{{Row8,7}}{{Row8,1}}{{Row8,6}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,2}}{{Row8,5}}{{Row8,3}}{{Row8,4end}} || || align=&quot;left&quot;|{{sms|3 }}{{font|color=#c07|&amp;nbsp;&amp;#8199;8}}&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;span style=&quot;font-size:60%&quot;&gt;(4+2)&lt;/span&gt;&gt;
|-   
|style=&quot;background:lightgray;{{text color default}};&quot;| 2 || 3 6 2  || {{BIN1start}}{{BIN0}}{{BIN0;}}&lt;br&gt;{{BIN1}}{{BIN1}}{{BIN1;}}&lt;br&gt;{{BIN0}}{{BIN1}}{{BIN0end}} || − || 5 || align=&quot;right&quot;|16854 || {{font|color=#aaa|0 1}}  7 6  &amp;nbsp;  2 3 5 4 &lt;br&gt;{{font|color=#adf|0  0  0  1 &amp;nbsp; 2  2  2  3}} || {{font|color=#adf|10}} ||{{Inversions8|0|0|0|0|0|1|0|0|1|1|0|0|1|1|0|0|0|1|1|0|0|0|0|1|1|0|0|1}}  || {{Row8,0start}}{{Row8,1}}{{Row8,7}}{{Row8,6}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,2}}{{Row8,3}}{{Row8,5}}{{Row8,4end}} || || align=&quot;left&quot;|{{sms|4 }}{{font|color=#c07|&amp;nbsp;&amp;#8199;9}}&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;span style=&quot;font-size:60%&quot;&gt;(3+3)&lt;/span&gt;&gt;
|-   
|3 || {{font|color=#aaa|1}}  7 3  || {{BIN1start}}{{BIN1}}{{BIN1;}}&lt;br&gt;{{BIN0}}{{BIN1}}{{BIN1;}}&lt;br&gt;{{BIN0}}{{BIN1}}{{BIN0end}} || − || 6 || align=&quot;right&quot;|22622 || {{font|color=#aaa|0}}  7 6 1  &amp;nbsp;  2  {{font|color=#aaa|5}}  4 3 &lt;br&gt;{{font|color=#adf|0  0  1  2 &amp;nbsp; 2  2  3  4}} || {{font|color=#adf|14}} ||{{Inversions8|0|0|1|0|1|1|0|1|1|0|0|1|1|0|0|0|1|1|0|0|1|0|1|1|0|0|1|1}}  || {{Row8,0start}}{{Row8,7}}{{Row8,6}}{{Row8,1}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,2}}{{Row8,5}}{{Row8,4}}{{Row8,3end}} || || align=&quot;left&quot;|{{sms|4 }}{{font|color=#c07|&amp;nbsp;&amp;#8199;9}}&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;span style=&quot;font-size:60%&quot;&gt;(3+3)&lt;/span&gt;&gt;
|-   
|style=&quot;background:lightgray;{{text color default}};&quot;| 4 || 5 6  {{font|color=#aaa|4}} || {{BIN1start}}{{BIN0}}{{BIN0;}}&lt;br&gt;{{BIN0}}{{BIN1}}{{BIN0;}}&lt;br&gt;{{BIN1}}{{BIN1}}{{BIN1end}} || + || 5 || align=&quot;right&quot;|16680 || {{font|color=#aaa|0 1 2 3}} &amp;nbsp;  7 6 5 4 &lt;br&gt;{{font|color=#adf|0  0  0  0 &amp;nbsp; 0  1  2  3}} || {{font|color=#adf|&amp;#8199;6}} ||{{Inversions8|0|0|0|0|0|0|0|0|0|0|0|0|0|0|1|0|0|0|0|1|1|0|0|0|0|1|1|1}}  || {{Row8,0start}}{{Row8,1}}{{Row8,2}}{{Row8,3}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,7}}{{Row8,6}}{{Row8,5}}{{Row8,4end}} || || align=&quot;left&quot;|{{sms|2 }}{{font|color=#c07|&amp;nbsp;&amp;#8199;3}}&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;span style=&quot;font-size:60%&quot;&gt;(2+2)&lt;/span&gt;&gt;
|-   
|5 || 7 1 5  || {{BIN1start}}{{BIN1}}{{BIN1;}}&lt;br&gt;{{BIN1}}{{BIN0}}{{BIN0;}}&lt;br&gt;{{BIN1}}{{BIN0}}{{BIN1end}} || − || 6 || align=&quot;right&quot;|22736 || {{font|color=#aaa|0}}  7 1 6  &amp;nbsp;  5 2 4 3 &lt;br&gt;{{font|color=#adf|0  0  1  1 &amp;nbsp; 2  3  3  4}} || {{font|color=#adf|14}} ||{{Inversions8|0|0|1|0|1|0|0|1|0|1|0|1|0|1|1|0|1|0|1|1|0|0|1|0|1|1|0|1}}  || {{Row8,0start}}{{Row8,7}}{{Row8,1}}{{Row8,6}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,5}}{{Row8,2}}{{Row8,4}}{{Row8,3end}} || || align=&quot;left&quot;|{{sms|6 }}{{font|color=#c07|&amp;nbsp;14}}&lt;sub&gt;b&lt;/sub&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;span style=&quot;font-size:60%&quot;&gt;(7)&lt;/span&gt;&gt;
|-   
|6 || 7  {{font|color=#aaa|2}}  6  || {{BIN1start}}{{BIN0}}{{BIN0;}}&lt;br&gt;{{BIN1}}{{BIN1}}{{BIN1;}}&lt;br&gt;{{BIN1}}{{BIN0}}{{BIN1end}} || + || 6 || align=&quot;right&quot;|23454 || {{font|color=#aaa|0 1}}  7 6  &amp;nbsp;  5 4 2 3 &lt;br&gt;{{font|color=#adf|0  0  0  1 &amp;nbsp; 2  3  4  4}} || {{font|color=#adf|14}} ||{{Inversions8|0|0|0|0|0|1|0|0|1|1|0|0|1|1|1|0|0|1|1|1|1|0|0|1|1|1|1|0}}  || {{Row8,0start}}{{Row8,1}}{{Row8,7}}{{Row8,6}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,5}}{{Row8,4}}{{Row8,2}}{{Row8,3end}} || [[File:Walsh permutation 165 Fano red.svg|53px]] || align=&quot;left&quot;|{{sms|3 }}{{font|color=#c07|&amp;nbsp;&amp;#8199;8}}&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;span style=&quot;font-size:60%&quot;&gt;(4+2)&lt;/span&gt;&gt;
|-   
|style=&quot;background:lightgray;{{text color default}};&quot;| 7 || 5 3 7  || {{BIN1start}}{{BIN1}}{{BIN1;}}&lt;br&gt;{{BIN0}}{{BIN1}}{{BIN1;}}&lt;br&gt;{{BIN1}}{{BIN0}}{{BIN1end}} || + || 7 || align=&quot;right&quot;|17702 || {{font|color=#aaa|0}}  7 6 1  &amp;nbsp;  5 2 3 4 &lt;br&gt;{{font|color=#adf|0  0  1  2 &amp;nbsp; 2  3  3  3}} || {{font|color=#adf|14}} ||{{Inversions8|0|0|1|0|1|1|0|1|1|0|0|1|1|0|1|0|1|1|0|1|0|0|1|1|0|1|0|0}}  || {{Row8,0start}}{{Row8,7}}{{Row8,6}}{{Row8,1}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,5}}{{Row8,2}}{{Row8,3}}{{Row8,4end}} || [[File:Walsh permutation 357 Fano red.svg|53px]] || align=&quot;left&quot;|{{sms|5 }}{{font|color=#c07|&amp;nbsp;14}}&lt;sub&gt;a&lt;/sub&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;span style=&quot;font-size:60%&quot;&gt;(7)&lt;/span&gt;&gt;
|}
{|
|Sorting by '''#''' gives the permutation (0,4,2,1,7,3,5,6) in the left column, which is ''reflection'' [[w:Function composition|∘]] [[w:Circular shift|''right shift'']] ∘ ''wp''(6,5,7) (with the compression matrix || {{Bin0start}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1end}} || ).
|}
Sorting by '''Permutation''' gives the (0,4,2,6, 1,5,3,7) in the left column, which is the bit-reversal permutation ''wp''(4,2,1).

=== 4-bit ===

[[File:Variadic logical XOR.svg|thumb|right|Binary Walsh matrix of order 16]]

{| class=&quot;wikitable sortable&quot; style=&quot;text-align:right;&quot;
|-
!
!#
!class=&quot;unsortable&quot;|IS
!CV
!class=&quot;unsortable&quot;|CM
!Σ
!Permutation&lt;br&gt;F
!IN
|-
|0
|0
|[[File:16 choose 2 array.svg|36px]]
|{{font|face=Courier|{{nowrap|'''( 1, 2, 4, 8)'''}}}}
|&lt;div style=&quot;white-space: nowrap&quot;&gt;{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin0}}{{Bin1end}}&lt;/div&gt;
|4
|{{font|face=Courier|{{nowrap|'''( 0, 1, 2, 3,&amp;nbsp; 4, 5, 6, 7,&amp;nbsp; 8, 9,10,11,&amp;nbsp;12,13,14,15)'''}}&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 0,&amp;nbsp; 0, 0, 0, 0,&amp;nbsp; 0, 0, 0, 0,&amp;nbsp; 0, 0, 0, 0)'''}}}}
|{{font|color=#adf|0}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|1
|9804424107176
|[[File:Inversion set 16; Walsh (15, 1, 2, 4).svg|36px]]
|{{font|face=Courier|'''( 3, 5, 9, 1)'''}}
|{{Bin1start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0end}}
|7
|{{font|face=Courier|'''( 0,15, 1,14,&amp;nbsp; 2,13, 3,12,&amp;nbsp; 4,11, 5,10,&amp;nbsp; 6, 9, 7, 8)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 1,&amp;nbsp; 2, 2, 3, 3,&amp;nbsp; 4, 4, 5, 5,&amp;nbsp; 6, 6, 7, 7)'''}}}}
|{{font|color=#adf|56}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|2
|9717242186454
|[[File:Inversion set 16; Walsh (1, 15, 2, 4).svg|36px]]
|{{font|face=Courier|'''( 3, 6,10, 2)'''}}
|{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0end}}
|7
|{{font|face=Courier|'''( 0, 1,15,14,&amp;nbsp; 2, 3,13,12,&amp;nbsp; 4, 5,11,10,&amp;nbsp; 6, 7, 9, 8)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 1,&amp;nbsp; 2, 2, 2, 3,&amp;nbsp; 4, 4, 4, 5,&amp;nbsp; 6, 6, 6, 7)'''}}}}
|{{font|color=#adf|52}}
|-
|3
|11112138397022
|[[File:Inversion set 16; wp( 1, 7,11, 3).svg|36px]]
|{{font|face=Courier|'''( 1, 7,11, 3)'''}}
|{{Bin1start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0end}}
|9
|{{font|face=Courier|'''( 0,15,14, 1,&amp;nbsp; 2,13,12, 3,&amp;nbsp; 4,11,10, 5,&amp;nbsp; 6, 9, 8, 7)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 2,&amp;nbsp; 2, 2, 3, 4,&amp;nbsp; 4, 4, 5, 6,&amp;nbsp; 6, 6, 7, 8)'''}}}}
|{{font|color=#adf|60}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|4
|9710017245480
|[[File:Inversion set 16; wp( 5, 6,12, 4).svg|36px]]
|{{font|face=Courier|'''( 5, 6,12, 4)'''}}
|{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0end}}
|7
|{{font|face=Courier|'''( 0, 1, 2, 3,&amp;nbsp;15,14,13,12,&amp;nbsp; 4, 5, 6, 7,&amp;nbsp;11,10, 9, 8)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 0,&amp;nbsp; 0, 1, 2, 3,&amp;nbsp; 4, 4, 4, 4,&amp;nbsp; 4, 5, 6, 7)'''}}}}
|{{font|color=#adf|44}}
|-
|5
|11118325501136
|[[File:Inversion set 16; Walsh (15, 1, 13, 4).svg|36px]]
|{{font|face=Courier|'''( 7, 1,13, 5)'''}}
|{{Bin1start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0end}}
|9
|{{font|face=Courier|'''( 0,15, 1,14,&amp;nbsp;13, 2,12, 3,&amp;nbsp; 4,11, 5,10,&amp;nbsp; 9, 6, 8, 7)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 1,&amp;nbsp; 2, 3, 3, 4,&amp;nbsp; 4, 4, 5, 5,&amp;nbsp; 6, 7, 7, 8)'''}}}}
|{{font|color=#adf|60}}
|-
|6
|11205500164254
|[[File:Inversion set 16; Walsh (1, 15, 13, 4).svg|36px]]
|{{font|face=Courier|'''( 7, 2,14, 6)'''}}
|{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0end}}
|9
|{{font|face=Courier|'''( 0, 1,15,14,&amp;nbsp;13,12, 2, 3,&amp;nbsp; 4, 5,11,10,&amp;nbsp; 9, 8, 6, 7)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 1,&amp;nbsp; 2, 3, 4, 4,&amp;nbsp; 4, 4, 4, 5,&amp;nbsp; 6, 7, 8, 8)'''}}}}
|{{font|color=#adf|60}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|7
|9810691044902
|[[File:Inversion set 16; Walsh (15, 14, 13, 4).svg|36px]]
|{{font|face=Courier|'''( 5, 3,15, 7)'''}}
|{{Bin1start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0end}}
|11
|{{font|face=Courier|'''( 0,15,14, 1,&amp;nbsp;13, 2, 3,12,&amp;nbsp; 4,11,10, 5,&amp;nbsp; 9, 6, 7, 8)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 2,&amp;nbsp; 2, 3, 3, 3,&amp;nbsp; 4, 4, 5, 6,&amp;nbsp; 6, 7, 7, 7)'''}}}}
|{{font|color=#adf|60}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|8
|9709968804480
|[[File:Inversion set 16; Walsh (1, 2, 4, 15).svg|36px]]
|{{font|face=Courier|'''( 9,10,12, 8)'''}}
|{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin1end}}
|7
|{{font|face=Courier|'''( 0, 1, 2, 3,&amp;nbsp; 4, 5, 6, 7,&amp;nbsp;15,14,13,12,&amp;nbsp;11,10, 9, 8)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 0,&amp;nbsp; 0, 0, 0, 0,&amp;nbsp;  0, 1, 2, 3,&amp;nbsp; 4, 5, 6, 7)'''}}}}
|{{font|color=#adf|28}}
|-
|9
|11118365775656
|[[File:Inversion set 16; Walsh (15, 1, 2, 11).svg|36px]]
|{{font|face=Courier|'''(11,13, 1, 9)'''}}
|{{Bin1start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin0}}{{Bin1end}}
|9
|{{font|face=Courier|'''( 0,15, 1,14,&amp;nbsp; 2,13, 3,12,&amp;nbsp;11, 4,10, 5,&amp;nbsp; 9, 6, 8, 7)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 1,&amp;nbsp; 2, 2, 3, 3,&amp;nbsp; 4, 5, 5, 6,&amp;nbsp; 6, 7, 7, 8)'''}}}}
|{{font|color=#adf|60}}
|-
|10
|11205547694934
|[[File:Inversion set 16; Walsh (1, 15, 2, 11).svg|36px]]
|{{font|face=Courier|'''(11,14, 2,10)'''}}
|{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin0}}{{Bin1end}}
|9
|{{font|face=Courier|'''( 0, 1,15,14,&amp;nbsp; 2, 3,13,12,&amp;nbsp;11,10, 4, 5,&amp;nbsp; 9, 8, 6, 7)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 1,&amp;nbsp; 2, 2, 2, 3,&amp;nbsp; 4, 5, 6, 6,&amp;nbsp; 6, 7, 8, 8)'''}}}}
|{{font|color=#adf|60}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|11
|9810651495902
|[[File:Inversion set 16; Walsh (15, 14, 2, 11).svg|36px]]
|{{font|face=Courier|'''( 9,15, 3,11)'''}}
|{{Bin1start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin0}}{{Bin1end}}
|11
|{{font|face=Courier|'''( 0,15,14, 1,&amp;nbsp; 2,13,12, 3,&amp;nbsp;11, 4, 5,10,&amp;nbsp; 9, 6, 7, 8)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 2,&amp;nbsp; 2, 2, 3, 4,&amp;nbsp; 4, 5, 5, 5,&amp;nbsp; 6, 7, 7, 7)'''}}}}
|{{font|color=#adf|60}}
|-
|12
|11212772635560
|[[File:Inversion set 16; Walsh (1, 2, 15, 11).svg|36px]]
|{{font|face=Courier|'''(13,14, 4,12)'''}}
|{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin0}}{{Bin1end}}
|9
|{{font|face=Courier|'''( 0, 1, 2, 3,&amp;nbsp;15,14,13,12,&amp;nbsp;11,10, 9, 8,&amp;nbsp; 4, 5, 6, 7)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 0,&amp;nbsp;  0, 1, 2, 3,&amp;nbsp; 4, 5, 6, 7,&amp;nbsp; 8, 8, 8, 8)'''}}}}
|{{font|color=#adf|60}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|13
|9804464392016
|[[File:Inversion set 16; Walsh (15, 1, 13, 11).svg|36px]]
|{{font|face=Courier|'''(15, 9, 5,13)'''}}
|{{Bin1start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin0}}{{Bin1end}}
|11
|{{font|face=Courier|'''( 0,15, 1,14,&amp;nbsp;13, 2,12, 3,&amp;nbsp;11, 4,10, 5,&amp;nbsp; 6, 9, 7, 8)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 1,&amp;nbsp; 2, 3, 3, 4,&amp;nbsp; 4, 5, 5, 6,&amp;nbsp; 6, 6, 7, 7)'''}}}}
|{{font|color=#adf|60}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|14
|9717289730334
|[[File:Inversion set 16; Walsh (1, 15, 13, 11).svg|36px]]
|{{font|face=Courier|'''(15,10, 6,14)'''}}
|{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin0}}{{Bin1end}}
|11
|{{font|face=Courier|'''( 0, 1,15,14,&amp;nbsp;13,12, 2, 3,&amp;nbsp;11,10, 4, 5,&amp;nbsp; 6, 7, 9, 8)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 1,&amp;nbsp; 2, 3, 4, 4,&amp;nbsp; 4, 5, 6, 6,&amp;nbsp; 6, 6, 6, 7)'''}}}}
|{{font|color=#adf|60}}
|-
|15
|11112098838182
|[[File:Inversion set 16; Walsh (15, 14, 13, 11).svg|36px]]
|{{font|face=Courier|'''(13,11, 7,15)'''}}
|{{Bin1start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin0}}{{Bin1end}}
|13
|{{font|face=Courier|'''( 0,15,14, 1,&amp;nbsp;13, 2, 3,12,&amp;nbsp;11, 4, 5,10,&amp;nbsp; 6, 9, 8, 7)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 2,&amp;nbsp; 2, 3, 3, 3,&amp;nbsp; 4, 5, 5, 5,&amp;nbsp; 6, 6, 7, 8)'''}}}}
|{{font|color=#adf|60}}
|}

{|
|Sorting by '''#''' gives the (0, 8,4,2,14, 1,13,11,7, 15,3,5,9, 6,10,12) in the left column, which is ''reflection'' [[w:Function composition|∘]] [[w:Circular shift|''right shift'']] ∘ ''wp''(12,10, 9, 7) (with the compression matrix || {{Bin0start}}{{Bin0}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin0}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin0end}} || ).
|}
Sorting by '''Permutation''' gives the (0,8,4,12, 2,10,6,14, 1,9,5,13, 3,11,7,15) in the left column, which is the [[w:Bit-reversal permutation|bit-reversal permutation]] ''wp''( 8, 4, 2, 1).

{| class=&quot;collapsible collapsed&quot; style=&quot;width: 100%; border: 1px solid #cccccc; background:white;{{text color default}};&quot;
! bgcolor=&quot;#f0f0f0&quot;|{{font|color=#999999|Matrices}}
|-
|
 Compression vectors:
     1     2     4     8
     3     5     9     1
     3     6    10     2
     1     7    11     3
     5     6    12     4
     7     1    13     5
     7     2    14     6
     5     3    15     7
     9    10    12     8
    11    13     1     9
    11    14     2    10
     9    15     3    11
    13    14     4    12
    15     9     5    13
    15    10     6    14
    13    11     7    15
 &amp;nbsp;
 Permutations:
     0     1     2     3     4     5     6     7     8     9    10    11    12    13    14    15
     0    15     1    14     2    13     3    12     4    11     5    10     6     9     7     8
     0     1    15    14     2     3    13    12     4     5    11    10     6     7     9     8
     0    15    14     1     2    13    12     3     4    11    10     5     6     9     8     7
     0     1     2     3    15    14    13    12     4     5     6     7    11    10     9     8
     0    15     1    14    13     2    12     3     4    11     5    10     9     6     8     7
     0     1    15    14    13    12     2     3     4     5    11    10     9     8     6     7
     0    15    14     1    13     2     3    12     4    11    10     5     9     6     7     8
     0     1     2     3     4     5     6     7    15    14    13    12    11    10     9     8
     0    15     1    14     2    13     3    12    11     4    10     5     9     6     8     7
     0     1    15    14     2     3    13    12    11    10     4     5     9     8     6     7
     0    15    14     1     2    13    12     3    11     4     5    10     9     6     7     8
     0     1     2     3    15    14    13    12    11    10     9     8     4     5     6     7
     0    15     1    14    13     2    12     3    11     4    10     5     6     9     7     8
     0     1    15    14    13    12     2     3    11    10     4     5     6     7     9     8
     0    15    14     1    13     2     3    12    11     4     5    10     6     9     8     7
 &amp;nbsp;
 Little-endian factorial numbers:
     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0
     0     0     1     1     2     2     3     3     4     4     5     5     6     6     7     7
     0     0     0     1     2     2     2     3     4     4     4     5     6     6     6     7
     0     0     1     2     2     2     3     4     4     4     5     6     6     6     7     8
     0     0     0     0     0     1     2     3     4     4     4     4     4     5     6     7
     0     0     1     1     2     3     3     4     4     4     5     5     6     7     7     8
     0     0     0     1     2     3     4     4     4     4     4     5     6     7     8     8
     0     0     1     2     2     3     3     3     4     4     5     6     6     7     7     7
     0     0     0     0     0     0     0     0     0     1     2     3     4     5     6     7
     0     0     1     1     2     2     3     3     4     5     5     6     6     7     7     8
     0     0     0     1     2     2     2     3     4     5     6     6     6     7     8     8
     0     0     1     2     2     2     3     4     4     5     5     5     6     7     7     7
     0     0     0     0     0     1     2     3     4     5     6     7     8     8     8     8
     0     0     1     1     2     3     3     4     4     5     5     6     6     6     7     7
     0     0     0     1     2     3     4     4     4     5     6     6     6     6     6     7
     0     0     1     2     2     3     3     3     4     5     5     5     6     6     7     8
|}

=== Compression matrices ===

[[File:Compression matrices of Walsh permutations with striped inversion sets.svg|left|800px]]
&lt;br style=&quot;clear: both;&quot;&gt;
The CM of the ''n''-th permutation is a matrix ''B'' with vertical stripes for the binary digits of ''n'' &lt;small&gt;(gray lines in the image)&lt;/small&gt;, XORed with a modified unity matrix ''U'' &lt;small&gt;(black dots in the image)&lt;/small&gt;.&lt;br&gt;
''U'' is modified by including an empty row where ''n'' has its lowest binary digit ({{oeis|A001511}}). It pushes the other rows of the unity matrix down, and the last row &quot;out of the matrix&quot;.


==Bended stripes==

Among the ''n''-bit Walsh permutations are 2&lt;sup&gt;''n-1''&lt;/sup&gt; whose inversion sets show bended stripes.&lt;br&gt;
They correspond to Walsh functions, i.e. the rows of the [[w:Walsh matrix|Walsh matrix]] of order 2&lt;sup&gt;''n-1''&lt;/sup&gt;.&lt;br&gt;
These permutations under composition form the group Z&lt;sub&gt;2&lt;/sub&gt;&lt;sup&gt;''n''-1&lt;/sup&gt;, just like the row numbers under bitwise XOR.

=== 3-bit ===

{| class=&quot;wikitable sortable&quot; style=&quot;height:275px; text-align:center;&quot;
|-style=&quot;height:55px&quot;
!
!style=&quot;width:55px&quot;|CV
!style=&quot;width:55px&quot; class=&quot;unsortable&quot;|&amp;nbsp;&amp;nbsp;CM&amp;nbsp;&amp;nbsp;
!D
!#
!Permutation&lt;br&gt;F
!IN
!class=&quot;unsortable&quot;|IS
!style=&quot;width:55px&quot; class=&quot;unsortable&quot;|PM
!class=&quot;unsortable&quot;|FPC
!Cy
|-   
|0 || {{font|color=#aaa|1 2 4}} || {{BIN1start}}{{BIN0}}{{BIN0;}}&lt;br&gt;{{BIN0}}{{BIN1}}{{BIN0;}}&lt;br&gt;{{BIN0}}{{BIN0}}{{BIN1end}} || + || align=&quot;right&quot;|  0 || {{font|color=#aaa|0 1 2 3 &amp;nbsp; 4 5 6 7}}&lt;br&gt;{{font|color=#adf|0  0  0  0 &amp;nbsp; 0  0  0  0}} || {{font|color=#adf|&amp;#8199;0}} ||{{Inversions8|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0}}  || {{Row8,0start}}{{Row8,1}}{{Row8,2}}{{Row8,3}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,4}}{{Row8,5}}{{Row8,6}}{{Row8,7end}} || [[File:Walsh permutation 124 Fano.svg|53px]] || align=&quot;left&quot;|{{sms|1 }}{{font|color=#c07|&amp;nbsp;&amp;#8199;0}}
|-   
|style=&quot;background:lightgray;{{text color default}};&quot;|1 ||  4 7 1  || {{BIN0start}}{{BIN1}}{{BIN1;}}&lt;br&gt;{{BIN0}}{{BIN1}}{{BIN0;}}&lt;br&gt;{{BIN1}}{{BIN1}}{{BIN0end}} || − || align=&quot;right&quot;| 3776 || {{font|color=#aaa|0}}  6  {{font|color=#aaa|2}}  4  &amp;nbsp;  3  {{font|color=#aaa|5}}  1  {{font|color=#aaa|7}}&lt;br&gt;{{font|color=#adf|0  0  1  1 &amp;nbsp; 2  1  5  0}} || {{font|color=#adf|10}} ||{{Inversions8|0|0|1|0|1|0|0|1|0|1|0|1|0|0|0|0|1|1|1|1|1|0|0|0|0|0|0|0}}  || {{Row8,0start}}{{Row8,6}}{{Row8,2}}{{Row8,4}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,3}}{{Row8,5}}{{Row8,1}}{{Row8,7end}} || || align=&quot;left&quot;|{{sms|2 }}{{font|color=#c07|&amp;nbsp;&amp;#8199;3}}&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;span style=&quot;font-size:60%&quot;&gt;(2+2)&lt;/span&gt;&gt;
|-   
|style=&quot;background:lightgray;{{text color default}};&quot;|2 ||  7 4 2  || {{BIN1start}}{{BIN0}}{{BIN0;}}&lt;br&gt;{{BIN1}}{{BIN0}}{{BIN1;}}&lt;br&gt;{{BIN1}}{{BIN1}}{{BIN0end}} || − || align=&quot;right&quot;|  414 || {{font|color=#aaa|0 1}}  5 4  &amp;nbsp;  3 2  {{font|color=#aaa|6 7}}&lt;br&gt;{{font|color=#adf|0  0  0  1 &amp;nbsp; 2  3  0  0}} || {{font|color=#adf|&amp;#8199;6}} ||{{Inversions8|0|0|0|0|0|1|0|0|1|1|0|0|1|1|1|0|0|0|0|0|0|0|0|0|0|0|0|0}}  || {{Row8,0start}}{{Row8,1}}{{Row8,5}}{{Row8,4}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,3}}{{Row8,2}}{{Row8,6}}{{Row8,7end}} || || align=&quot;left&quot;|{{sms|2 }}{{font|color=#c07|&amp;nbsp;&amp;#8199;3}}&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;span style=&quot;font-size:60%&quot;&gt;(2+2)&lt;/span&gt;&gt;
|-   
|3 ||  2 1 7  || {{BIN0start}}{{BIN1}}{{BIN1;}}&lt;br&gt;{{BIN1}}{{BIN0}}{{BIN1;}}&lt;br&gt;{{BIN0}}{{BIN0}}{{BIN1end}} || − || align=&quot;right&quot;| 4142 || {{font|color=#aaa|0}}  6 5  {{font|color=#aaa|3 &amp;nbsp; 4}}  2 1  {{font|color=#aaa|7}}&lt;br&gt;{{font|color=#adf|0  0  1  2 &amp;nbsp; 2  4  5  0}} || {{font|color=#adf|14}} ||{{Inversions8|0|0|1|0|1|1|0|1|1|0|0|1|1|1|1|0|1|1|1|1|1|0|0|0|0|0|0|0}}  || {{Row8,0start}}{{Row8,6}}{{Row8,5}}{{Row8,3}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,4}}{{Row8,2}}{{Row8,1}}{{Row8,7end}} || || align=&quot;left&quot;|{{sms|2 }}{{font|color=#c07|&amp;nbsp;&amp;#8199;3}}&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;span style=&quot;font-size:60%&quot;&gt;(2+2)&lt;/span&gt;&gt;
|}

Sorting by '''#''' or '''Permutation''' gives (0,2,1,3) in the left column, which is the bit-reversal permutation ''wp''(2,1).&lt;br&gt;
Sorting by '''CV''' gives (0,3,1,2) in the left column, which is ''wp''(3,1).


=== 4-bit ===

{| class=&quot;wikitable sortable&quot; style=&quot;text-align:right;&quot;
|-
!
!#
!class=&quot;unsortable&quot;|IS
!CV
!class=&quot;unsortable&quot;|CM
!Permutation&lt;br&gt;F
!IN
|-
|0
|0
|[[File:16 choose 2 array.svg|36px]]
|{{font|face=Courier|{{nowrap|'''( 1, 2, 4, 8)'''}}}}
|&lt;div style=&quot;white-space: nowrap&quot;&gt;{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin0}}{{Bin1end}}&lt;/div&gt;
|{{font|face=Courier|{{nowrap|'''( 0, 1, 2, 3,&amp;nbsp; 4, 5, 6, 7,&amp;nbsp; 8, 9,10,11,&amp;nbsp;12,13,14,15)'''}}&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 0,&amp;nbsp; 0, 0, 0, 0,&amp;nbsp; 0, 0, 0, 0,&amp;nbsp; 0, 0, 0, 0)'''}}}}
|{{font|color=#adf|0}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|1
|1144441325096
|[[File:Inversion set 16; Walsh (14, 2, 4, 7).svg|36px]]
|{{font|face=Courier|'''( 8,11,13, 1)'''}}
|{{Bin0start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin0end}}
|{{font|face=Courier|'''( 0,14, 2,12,&amp;nbsp; 4,10, 6, 8,&amp;nbsp; 7, 9, 5,11,&amp;nbsp; 3,13, 1,15)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 1,&amp;nbsp; 2, 2, 3, 3,&amp;nbsp; 4, 3, 7, 2,&amp;nbsp;10, 1,13, 0)'''}}}}
|{{font|color=#adf|52}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|2
|73376328534
|[[File:Inversion set 16; Walsh (1, 13, 4, 7).svg|36px]]
|{{font|face=Courier|'''(11, 8,14, 2)'''}}
|{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin0end}}
|{{font|face=Courier|'''( 0, 1,13,12,&amp;nbsp; 4, 5, 9, 8,&amp;nbsp; 7, 6,10,11,&amp;nbsp; 3, 2,14,15)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 1,&amp;nbsp; 2, 2, 2, 3,&amp;nbsp; 4, 5, 2, 2,&amp;nbsp;10,11, 0, 0)'''}}}}
|{{font|color=#adf|44}}
|-
|3
|1209107634782
|[[File:Inversion set 16; Walsh (14, 13, 4, 8).svg|36px]]
|{{font|face=Courier|'''( 2, 1, 7,11)'''}}
|{{Bin0start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin0}}{{Bin1end}}
|{{font|face=Courier|'''( 0,14,13, 3,&amp;nbsp; 4,10, 9, 7,&amp;nbsp; 8, 6, 5,11,&amp;nbsp;12, 2, 1,15)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 2,&amp;nbsp; 2, 2, 3, 4,&amp;nbsp; 4, 6, 7, 2,&amp;nbsp; 2,12,13, 0)'''}}}}
|{{font|color=#adf|60}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|4
|303182760
|[[File:Inversion set 16; Walsh (1, 2, 11, 7).svg|36px]]
|{{font|face=Courier|'''(13,14, 8, 4)'''}}
|{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin0}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin0end}}
|{{font|face=Courier|'''( 0, 1, 2, 3,&amp;nbsp;11,10, 9, 8,&amp;nbsp; 7, 6, 5, 4,&amp;nbsp;12,13,14,15)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 0,&amp;nbsp; 0, 1, 2, 3,&amp;nbsp; 4, 5, 6, 7,&amp;nbsp; 0, 0, 0, 0)'''}}}}
|{{font|color=#adf|28}}
|-
|5
|1144707321296
|[[File:Inversion set 16; Walsh (14, 2, 11, 8).svg|36px]]
|{{font|face=Courier|'''( 4, 7, 1,13)'''}}
|{{Bin0start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin0}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin0}}{{Bin1end}}
|{{font|face=Courier|'''( 0,14, 2,12,&amp;nbsp;11, 5, 9, 7,&amp;nbsp; 8, 6,10, 4,&amp;nbsp; 3,13, 1,15)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 1,&amp;nbsp; 2, 3, 3, 4,&amp;nbsp; 4, 6, 3, 9,&amp;nbsp;10, 1,13, 0)'''}}}}
|{{font|color=#adf|60}}
|-
|6
|73677162654
|[[File:Inversion set 16; Walsh (1, 13, 11, 8).svg|36px]]
|{{font|face=Courier|'''( 7, 4, 2,14)'''}}
|{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin0}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin0}}{{Bin1end}}
|{{font|face=Courier|'''( 0, 1,13,12,&amp;nbsp;11,10, 6, 7,&amp;nbsp; 8, 9, 5, 4,&amp;nbsp; 3, 2,14,15)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 1,&amp;nbsp; 2, 3, 4, 4,&amp;nbsp; 4, 4, 8, 9,&amp;nbsp;10,11, 0, 0)'''}}}}
|{{font|color=#adf|60}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|7
|1209371443622
|[[File:Inversion set 16; Walsh (14, 13, 11, 7).svg|36px]]
|{{font|face=Courier|'''(14,13,11, 7)'''}}
|{{Bin0start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin0}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin0end}}
|{{font|face=Courier|'''( 0,14,13, 3,&amp;nbsp;11, 5, 6, 8,&amp;nbsp; 7, 9,10, 4,&amp;nbsp;12, 2, 1,15)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 2,&amp;nbsp; 2, 3, 3, 3,&amp;nbsp; 4, 3, 3, 9,&amp;nbsp; 2,12,13, 0)'''}}}}
|{{font|color=#adf|60}}
|}

Sorting by '''#''' or '''Permutation''' gives (0,4,2,6, 1,5,3,7) in the left column, which is the bit-reversal permutation ''wp''(4,2,1).&lt;br&gt;
Sorting by '''CV''' gives (0,3,5,6, 1,2,4,7) in the left column, which is ''wp''(7,1,2).

===Compression matrices===

[[File:Compression matrices of Walsh permutations with bendedly striped inversion sets.svg|800px]]
&lt;br style=&quot;clear: both;&quot;&gt;
The CM are matrices with an even number of vertical stripes &lt;small&gt;(gray lines in the image)&lt;/small&gt; XORed with the unity matrix &lt;small&gt;(black dots in the image)&lt;/small&gt;.&lt;br&gt;
The order by the ''evil numbers'' ({{oeis|A001969}}) of the matrices, used in the image, corresponds to the lexicographical order of the compression vectors.

==Bit permutations==
{| align=&quot;right&quot;
|colspan=&quot;2&quot;|
{| align=right style=&quot;text-align=right; border:1px solid lightgray&quot; cellspacing=&quot;10&quot;
|[[Walsh permutation]]; [[Walsh permutation; bit permutation|bit permutation]] {{Rdrup}}
|}
|-
|style=&quot;vertical-align:top;&quot;|[[File:Inversion set 16; Walsh (2, 8, 1, 4).svg|thumb|Inversion set of ''wp''( 4, 1, 8, 2)&lt;br&gt;All columns are symmetrical.]]
|style=&quot;vertical-align:top;&quot;|[[File:Inversion set 16; Walsh (14, 13, 11, 7).svg|thumb|Inversion set of ''wp''(14,13,11, 7)]]
|-
|colspan=&quot;2&quot;|
{| align=right
|-
|style=&quot;vertical-align:top;&quot;|[[File:Symmetric group 4; cycle graph; details.svg|thumb|center|500px|[[w:Cycle graph (algebra)|Cycle graph]] of the bit permutations]]
|style=&quot;vertical-align:top;&quot;|[[File:Symmetric group 4; permutation list (0-based).svg|thumb|center|Permutations of 4 elements&lt;br&gt;{{Rdr}} [[Symmetric group S4|Symmetric group S&lt;sub&gt;4&lt;/sub&gt;]]]]
|}
|}


{| class=&quot;wikitable sortable&quot; style=&quot;text-align:right;&quot;
|-
!
!#
!class=&quot;unsortable&quot;|IS
!CV
|-
|style=&quot;background:white;{{text color default}};&quot;|&lt;big&gt;'''0'''&lt;/big&gt;
|0
|[[File:16 choose 2 array.svg|36px]]
|{{font|face=Courier|'''( 1, 2, 4, 8)'''}}
|-
|style=&quot;background:#33eb2a;{{text color default}};&quot;|1
|87181920722
|[[File:Inversion set 16; Walsh (2, 1, 4, 8).svg|36px]]
|{{font|face=Courier|'''( 2, 1, 4, 8)'''}}
|-
|style=&quot;background:#33eb2a;{{text color default}};&quot;|2
|13412045088
|[[File:Inversion set 16; wp( 1, 4, 2, 8).svg|36px]]
|{{font|face=Courier|'''( 1, 4, 2, 8)'''}}
|-
|style=&quot;background:white;{{text color default}};&quot;|3
|101069338632
|[[File:Inversion set 16; Walsh (2, 4, 1, 8).svg|36px]]
|{{font|face=Courier|'''( 4, 1, 2, 8)'''}}
|-
|style=&quot;background:white;{{text color default}};&quot;|4
|262496507810
|[[File:Inversion set 16; Walsh (4, 1, 2, 8).svg|36px]]
|{{font|face=Courier|'''( 2, 4, 1, 8)'''}}
|-
|style=&quot;background:#33eb2a;{{text color default}};&quot;|5
|269202530354
|[[File:Inversion set 16; Walsh (4, 2, 1, 8).svg|36px]]
|{{font|face=Courier|'''( 4, 2, 1, 8)'''}}
|-
|style=&quot;background:#33eb2a;{{text color default}};&quot;|6
|175795200
|[[File:Inversion set 16; Walsh (1, 2, 8, 4).svg|36px]]
|{{font|face=Courier|'''( 1, 2, 8, 4)'''}}
|-
|style=&quot;background:white;{{text color default}};&quot;|&lt;big&gt;'''7'''&lt;/big&gt;
|87357715922
|[[File:Inversion set 16; Walsh (2, 1, 8, 4).svg|36px]]
|{{font|face=Courier|'''( 2, 1, 8, 4)'''}}
|-
|style=&quot;background:white;{{text color default}};&quot;|8
|13588646400
|[[File:Inversion set 16; Walsh (1, 4, 8, 2).svg|36px]]
|{{font|face=Courier|'''( 1, 8, 2, 4)'''}}
|-
|style=&quot;background:#ffbb00;{{text color default}};&quot;|9
|101249608320
|[[File:Inversion set 16; Walsh (2, 4, 8, 1).svg|36px]]
|{{font|face=Courier|'''( 8, 1, 2, 4)'''}}
|-
|style=&quot;background:#ffbb00;{{text color default}};&quot;|10
|262679639762
|[[File:Inversion set 16; Walsh (4, 1, 8, 2).svg|36px]]
|{{font|face=Courier|'''( 2, 8, 1, 4)'''}}
|-
|style=&quot;background:white;{{text color default}};&quot;|11
|269386065362
|[[File:Inversion set 16; Walsh (4, 2, 8, 1).svg|36px]]
|{{font|face=Courier|'''( 8, 2, 1, 4)'''}}
|-
|style=&quot;background:white;{{text color default}};&quot;|12
|40237747488
|[[File:Inversion set 16; Walsh (1, 8, 2, 4).svg|36px]]
|{{font|face=Courier|'''( 1, 4, 8, 2)'''}}
|-
|style=&quot;background:#ffbb00;{{text color default}};&quot;|13
|127895041032
|[[File:Inversion set 16; Walsh (2, 8, 1, 4).svg|36px]]
|{{font|face=Courier|'''( 4, 1, 8, 2)'''}}
|-
|style=&quot;background:#33eb2a;{{text color default}};&quot;|14
|40325645088
|[[File:Inversion set 16; wp( 1, 8, 4, 2).svg|36px]]
|{{font|face=Courier|'''( 1, 8, 4, 2)'''}}
|-
|style=&quot;background:white;{{text color default}};&quot;|15
|127986607008
|[[File:Inversion set 16; Walsh (2, 8, 4, 1).svg|36px]]
|{{font|face=Courier|'''( 8, 1, 4, 2)'''}}
|-
|style=&quot;background:white;{{text color default}};&quot;|&lt;big&gt;'''16'''&lt;/big&gt;
|303216964872
|[[File:Inversion set 16; wp( 4, 8, 1, 2).svg|36px]]
|{{font|face=Courier|'''( 4, 8, 1, 2)'''}}
|-
|style=&quot;background:#ffbb00;{{text color default}};&quot;|17
|303260913672
|[[File:Inversion set 16; Walsh (4, 8, 2, 1).svg|36px]]
|{{font|face=Courier|'''( 8, 4, 1, 2)'''}}
|-
|style=&quot;background:#ffbb00;{{text color default}};&quot;|18
|613140355490
|[[File:Inversion set 16; Walsh (8, 1, 2, 4).svg|36px]]
|{{font|face=Courier|'''( 2, 4, 8, 1)'''}}
|-
|style=&quot;background:white;{{text color default}};&quot;|19
|619846378034
|[[File:Inversion set 16; Walsh (8, 2, 1, 4).svg|36px]]
|{{font|face=Courier|'''( 4, 2, 8, 1)'''}}
|-
|style=&quot;background:white;{{text color default}};&quot;|20
|613228253090
|[[File:Inversion set 16; Walsh (8, 1, 4, 2).svg|36px]]
|{{font|face=Courier|'''( 2, 8, 4, 1)'''}}
|-
|style=&quot;background:#33eb2a;{{text color default}};&quot;|21
|619934678690
|[[File:Inversion set 16; Walsh (8, 2, 4, 1).svg|36px]]
|{{font|face=Courier|'''( 8, 2, 4, 1)'''}}
|-
|style=&quot;background:#ffbb00;{{text color default}};&quot;|22
|633259229234
|[[File:Inversion set 16; wp( 4, 8, 2, 1).svg|36px]]
|{{font|face=Courier|'''( 4, 8, 2, 1)'''}}
|-
|style=&quot;background:white;{{text color default}};&quot;|&lt;big&gt;'''23'''&lt;/big&gt;
|633303178034
|[[File:Inversion set 16; Walsh (8, 4, 2, 1).svg|36px]]
|{{font|face=Courier|'''( 8, 4, 2, 1)'''}}
|}

{| class=&quot;collapsible collapsed&quot; style=&quot;width: 100%; border: 1px solid #cccccc; background:white;{{text color default}};&quot;
! bgcolor=&quot;#f0f0f0&quot;|{{font|color=#999999|F}}
|-
|
 Little-endian factorial numbers:
     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0
     0     0     1     0     0     0     1     0     0     0     1     0     0     0     1     0
     0     0     0     0     2     2     0     0     0     0     0     0     2     2     0     0
     0     0     0     0     3     2     1     0     0     0     0     0     3     2     1     0
     0     0     1     0     2     0     3     0     0     0     1     0     2     0     3     0
     0     0     1     0     3     1     3     0     0     0     1     0     3     1     3     0
     0     0     0     0     0     0     0     0     4     4     4     4     0     0     0     0
     0     0     1     0     0     0     1     0     4     4     5     4     0     0     1     0
     0     0     0     0     0     0     0     0     6     6     4     4     2     2     0     0
     0     0     0     0     0     0     0     0     7     6     5     4     3     2     1     0
     0     0     1     0     0     0     1     0     6     4     7     4     2     0     3     0
     0     0     1     0     0     0     1     0     7     5     7     4     3     1     3     0
     0     0     0     0     2     2     0     0     4     4     0     0     6     6     0     0
     0     0     0     0     3     2     1     0     4     4     0     0     7     6     1     0
     0     0     0     0     2     2     0     0     6     6     2     2     6     6     0     0
     0     0     0     0     2     2     0     0     7     6     3     2     7     6     1     0
     0     0     0     0     3     2     1     0     6     4     2     0     9     6     3     0
     0     0     0     0     3     2     1     0     7     5     3     1     9     6     3     0
     0     0     1     0     2     0     3     0     4     0     5     0     6     0     7     0
     0     0     1     0     3     1     3     0     4     0     5     0     7     1     7     0
     0     0     1     0     2     0     3     0     6     2     7     2     6     0     7     0
     0     0     1     0     2     0     3     0     7     3     7     2     7     1     7     0
     0     0     1     0     3     1     3     0     6     2     5     0     9     3     7     0
     0     0     1     0     3     1     3     0     7     3     6     1     9     3     7     0
 &amp;nbsp;
 Inversion numbers:
     0
     4
     8
    12
    12
    16
    16
    20
    24
    28
    28
    32
    24
    28
    32
    36
    36
    40
    28
    32
    36
    40
    40
    44
|}

==Nimber multiplication==
{| align=&quot;right&quot; style=&quot;text-align:right; border:1px solid lightgray&quot; cellspacing=&quot;10&quot;
| [[Walsh permutation]]; [[Walsh permutation; nimber multiplication|nimber multiplication]] {{rdrup}}
|}

Images: {{commons catlink|Inversion set of a 4-bit Walsh permutation; nimber multiplication}}

{{clear}}

{| class=&quot;wikitable sortable&quot; style=&quot;text-align:right&quot;
|-
! #
! CV
! Permutation
! F
! IN
|- style=&quot;font-family:Courier&quot;
|              0||( 1, 2, 4, 8)||(0, '''1''', 2, 3, &amp;nbsp; 4, 5, 6, 7, &amp;nbsp; 8, 9,10,11, &amp;nbsp;12,13,14,15) || (0, 0, 0, 0, &amp;nbsp; 0, 0, 0, 0, &amp;nbsp; 0, 0, 0, 0, &amp;nbsp; 0, 0, 0, '''0''') ||  0
|- style=&quot;font-family:Courier&quot;
| 13827898032492||( 2, 3, 8,12)||(0, '''2''', 3, 1, &amp;nbsp; 8,10,11, 9, &amp;nbsp;12,14,15,13, &amp;nbsp; 4, 6, 7, 5) || (0, 0, 0, 2, &amp;nbsp; 0, 0, 0, 2, &amp;nbsp; 0, 0, 0, 2, &amp;nbsp; 8, 8, 8,'''10''') || 40
|- style=&quot;font-family:Courier&quot;
|  7001306732168||( 3, 1,12, 4)||(0, '''3''', 1, 2, &amp;nbsp;12,15,13,14, &amp;nbsp; 4, 7, 5, 6, &amp;nbsp; 8,11, 9,10) || (0, 0, 1, 1, &amp;nbsp; 0, 0, 1, 1, &amp;nbsp; 4, 4, 5, 5, &amp;nbsp; 4, 4, 5, '''5''') || 40
|- style=&quot;font-family:Courier&quot;
| 19217812814448||( 4, 8, 6,11)||(0, '''4''', 8,12, &amp;nbsp; 6, 2,14,10, &amp;nbsp;11,15, 3, 7, &amp;nbsp;13, 9, 5, 1) || (0, 0, 0, 0, &amp;nbsp; 2, 4, 0, 2, &amp;nbsp; 2, 0, 8, 6, &amp;nbsp; 2, 6,10,'''14''') || 56
|- style=&quot;font-family:Courier&quot;
|  1630614229512||( 5,10, 2, 3)||(0, '''5''',10,15, &amp;nbsp; 2, 7, 8,13, &amp;nbsp; 3, 6, 9,12, &amp;nbsp; 1, 4,11,14) || (0, 0, 0, 0, &amp;nbsp; 3, 2, 2, 1, &amp;nbsp; 6, 5, 3, 2, &amp;nbsp;11, 9, 3, '''1''') || 48
|- style=&quot;font-family:Courier&quot;
| 15433122573720||( 6,11,14, 7)||(0, '''6''',11,13, &amp;nbsp;14, 8, 5, 3, &amp;nbsp; 7, 1,12,10, &amp;nbsp; 9,15, 2, 4) || (0, 0, 0, 0, &amp;nbsp; 0, 3, 5, 6, &amp;nbsp; 4, 8, 2, 4, &amp;nbsp; 5, 0,12,'''11''') || 60
|- style=&quot;font-family:Courier&quot;
|  5564988000144||( 7, 9,10,15)||(0, '''7''', 9,14, &amp;nbsp;10,13, 3, 4, &amp;nbsp;15, 8, 6, 1, &amp;nbsp; 5, 2,12,11) || (0, 0, 0, 0, &amp;nbsp; 1, 1, 5, 5, &amp;nbsp; 0, 5, 7,10, &amp;nbsp; 8,11, 3, '''4''') || 60
|- style=&quot;font-family:Courier&quot;
| 17445432818676||( 8,12,11,13)||(0, '''8''',12, 4, &amp;nbsp;11, 3, 7,15, &amp;nbsp;13, 5, 1, 9, &amp;nbsp; 6,14,10, 2) || (0, 0, 0, 2, &amp;nbsp; 1, 4, 3, 0, &amp;nbsp; 1, 6, 9, 4, &amp;nbsp; 7, 1, 5,'''13''') || 56
|- style=&quot;font-family:Courier&quot;
|  3551688986892||( 9,14,15, 5)||(0, '''9''',14, 7, &amp;nbsp;15, 6, 1, 8, &amp;nbsp; 5,12,11, 2, &amp;nbsp;10, 3, 4,13) || (0, 0, 0, 2, &amp;nbsp; 0, 4, 5, 3, &amp;nbsp; 6, 2, 3, 9, &amp;nbsp; 4,10,10, '''2''') || 60
|- style=&quot;font-family:Courier&quot;
| 10678227659604||(10,15, 3, 1)||(0,'''10''',15, 5, &amp;nbsp; 3, 9,12, 6, &amp;nbsp; 1,11,14, 4, &amp;nbsp; 2, 8,13, 7) || (0, 0, 0, 2, &amp;nbsp; 3, 2, 1, 4, &amp;nbsp; 7, 2, 1, 8, &amp;nbsp;10, 6, 2, '''8''') || 56
|- style=&quot;font-family:Courier&quot;
| 10169232103620||(11,13, 7, 9)||(0,'''11''',13, 6, &amp;nbsp; 7,12,10, 1, &amp;nbsp; 9, 2, 4,15, &amp;nbsp;14, 5, 3, 8) || (0, 0, 0, 2, &amp;nbsp; 2, 1, 3, 6, &amp;nbsp; 4, 7, 7, 0, &amp;nbsp; 1, 9,11, '''7''') || 60
|- style=&quot;font-family:Courier&quot;
| 15737148505928||(12, 4,13, 6)||(0,'''12''', 4, 8, &amp;nbsp;13, 1, 9, 5, &amp;nbsp; 6,10, 2,14, &amp;nbsp;11, 7,15, 3) || (0, 0, 1, 1, &amp;nbsp; 0, 4, 2, 4, &amp;nbsp; 4, 2, 8, 0, &amp;nbsp; 3, 7, 0,'''12''') || 48
|- style=&quot;font-family:Courier&quot;
|  5084423155256||(13, 6, 9,14)||(0,'''13''', 6,11, &amp;nbsp; 9, 4,15, 2, &amp;nbsp;14, 3, 8, 5, &amp;nbsp; 7,10, 1,12) || (0, 0, 1, 1, &amp;nbsp; 2, 4, 0, 6, &amp;nbsp; 1, 7, 5, 7, &amp;nbsp; 6, 4,13, '''3''') || 60
|- style=&quot;font-family:Courier&quot;
| 12454407149960||(14, 7, 5,10)||(0,'''14''', 7, 9, &amp;nbsp; 5,11, 2,12, &amp;nbsp;10, 4,13, 3, &amp;nbsp;15, 1, 8, 6) || (0, 0, 1, 1, &amp;nbsp; 3, 1, 5, 1, &amp;nbsp; 3, 7, 1, 9, &amp;nbsp; 0,12, 7, '''9''') || 60
|- style=&quot;font-family:Courier&quot;
|  8566683323240||(15, 5, 1, 2)||(0,'''15''', 5,10, &amp;nbsp; 1,14, 4,11, &amp;nbsp; 2,13, 7, 8, &amp;nbsp; 3,12, 6, 9) || (0, 0, 1, 1, &amp;nbsp; 3, 1, 4, 2, &amp;nbsp; 6, 2, 5, 5, &amp;nbsp; 9, 3, 8, '''6''') || 56
|}

[[Category:Walsh permutation]]</text>
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    <revision>
      <id>2819772</id>
      <parentid>2819771</parentid>
      <timestamp>2026-07-29T21:14:11Z</timestamp>
      <contributor>
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        <id>137431</id>
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      <text bytes="41083" sha1="g14fy67rcktq2dd3kcils64hfvbk4hm" xml:space="preserve">{| align=&quot;right&quot; style=&quot;text-align:right; border:1px solid lightgray&quot; cellspacing=&quot;10&quot;
| [[Walsh permutation]] {{rdrup}}
|-
| [[Inversion (discrete mathematics)]] {{rdrup}}
|}

[[File:Construction site ahead (17184445992).jpg|thumb|The vectors need to be corrected. See [[Walsh permutation#Notation warning]]. Done only for section [[#Nimber multiplication|Nimber multiplication]].]]

{|
|style=&quot;vertical-align:top;&quot;|
&lt;!---▶▶▶---&gt;{| style=&quot;border: 1px solid #cccccc&quot;
! bgcolor=&quot;#dddddd&quot; colspan=&quot;2&quot;|Abbreviations
|-
|
|-
|'''CV'''
|Compression vector
|-
|'''CM'''
|Compression matrix
|-
|'''Σ'''
|Number of ones in the compression matrix
|-
|'''IS'''
|Inversion set
|-
|'''#'''
|Index number of the finite permutation
|-
|'''IN'''
|Inversion number
|}&lt;!---◀◀◀---&gt;
|style=&quot;vertical-align:top;&quot;|__TOC__
|}

&lt;br style=&quot;clear: both;&quot;&gt;

==Stripes==
Among the ''n''-bit Walsh permutations are 2&lt;sup&gt;''n''&lt;/sup&gt; with striped inversion sets.&lt;br&gt;
As seen below they correspond to Walsh functions, i.e. the rows of the [[w:Walsh matrix|Walsh matrix]] of order 2&lt;sup&gt;''n''&lt;/sup&gt;.&lt;br&gt;
These permutations under [[w:Function composition|composition]] form the group Z&lt;sub&gt;2&lt;/sub&gt;&lt;sup&gt;''n''&lt;/sup&gt;, just like the row numbers under bitwise XOR.

=== 3-bit ===

An extract of the [[3-bit Walsh permutation; table|'''table''']] of [[3-bit Walsh permutation]]s:
{| class=&quot;wikitable sortable&quot; style=&quot;height:495px; text-align:center;&quot;
|-style=&quot;height:55px&quot;
!
!style=&quot;width:55px&quot;|CV
!style=&quot;width:55px&quot; class=&quot;unsortable&quot;|&amp;nbsp;&amp;nbsp;CM&amp;nbsp;&amp;nbsp;
!D
!Σ
!#
!Permutation&lt;br&gt;F
!IN
!class=&quot;unsortable&quot;|IS
!style=&quot;width:55px&quot; class=&quot;unsortable&quot;|PM
!class=&quot;unsortable&quot;|FPC
!Cy
|-   
|0 || {{font|color=#aaa|1 2 4}} || {{BIN1start}}{{BIN0}}{{BIN0;}}&lt;br&gt;{{BIN0}}{{BIN1}}{{BIN0;}}&lt;br&gt;{{BIN0}}{{BIN0}}{{BIN1end}} || + || 3 || align=&quot;right&quot;|  0 || {{font|color=#aaa|0 1 2 3 &amp;nbsp; 4 5 6 7}}&lt;br&gt;{{font|color=#adf|0  0  0  0 &amp;nbsp; 0  0  0  0}} || {{font|color=#adf|&amp;#8199;0}} ||{{Inversions8|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0}}  || {{Row8,0start}}{{Row8,1}}{{Row8,2}}{{Row8,3}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,4}}{{Row8,5}}{{Row8,6}}{{Row8,7end}} || [[File:Walsh permutation 124 Fano.svg|53px]] || align=&quot;left&quot;|{{sms|1 }}{{font|color=#c07|&amp;nbsp;&amp;#8199;0}}
|-   
|style=&quot;background:lightgray;{{text color default}};&quot;| 1 || 3 5 1  || {{BIN1start}}{{BIN1}}{{BIN1;}}&lt;br&gt;{{BIN1}}{{BIN0}}{{BIN0;}}&lt;br&gt;{{BIN0}}{{BIN1}}{{BIN0end}} || + || 5 || align=&quot;right&quot;|17576 || {{font|color=#aaa|0}}  7 1 6  &amp;nbsp;  2  {{font|color=#aaa|5}}  3 4 &lt;br&gt;{{font|color=#adf|0  0  1  1 &amp;nbsp; 2  2  3  3}} || {{font|color=#adf|12}} ||{{Inversions8|0|0|1|0|1|0|0|1|0|1|0|1|0|1|0|0|1|0|1|0|1|0|1|0|1|0|1|0}}  || {{Row8,0start}}{{Row8,7}}{{Row8,1}}{{Row8,6}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,2}}{{Row8,5}}{{Row8,3}}{{Row8,4end}} || || align=&quot;left&quot;|{{sms|3 }}{{font|color=#c07|&amp;nbsp;&amp;#8199;8}}&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;span style=&quot;font-size:60%&quot;&gt;(4+2)&lt;/span&gt;&gt;
|-   
|style=&quot;background:lightgray;{{text color default}};&quot;| 2 || 3 6 2  || {{BIN1start}}{{BIN0}}{{BIN0;}}&lt;br&gt;{{BIN1}}{{BIN1}}{{BIN1;}}&lt;br&gt;{{BIN0}}{{BIN1}}{{BIN0end}} || − || 5 || align=&quot;right&quot;|16854 || {{font|color=#aaa|0 1}}  7 6  &amp;nbsp;  2 3 5 4 &lt;br&gt;{{font|color=#adf|0  0  0  1 &amp;nbsp; 2  2  2  3}} || {{font|color=#adf|10}} ||{{Inversions8|0|0|0|0|0|1|0|0|1|1|0|0|1|1|0|0|0|1|1|0|0|0|0|1|1|0|0|1}}  || {{Row8,0start}}{{Row8,1}}{{Row8,7}}{{Row8,6}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,2}}{{Row8,3}}{{Row8,5}}{{Row8,4end}} || || align=&quot;left&quot;|{{sms|4 }}{{font|color=#c07|&amp;nbsp;&amp;#8199;9}}&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;span style=&quot;font-size:60%&quot;&gt;(3+3)&lt;/span&gt;&gt;
|-   
|3 || {{font|color=#aaa|1}}  7 3  || {{BIN1start}}{{BIN1}}{{BIN1;}}&lt;br&gt;{{BIN0}}{{BIN1}}{{BIN1;}}&lt;br&gt;{{BIN0}}{{BIN1}}{{BIN0end}} || − || 6 || align=&quot;right&quot;|22622 || {{font|color=#aaa|0}}  7 6 1  &amp;nbsp;  2  {{font|color=#aaa|5}}  4 3 &lt;br&gt;{{font|color=#adf|0  0  1  2 &amp;nbsp; 2  2  3  4}} || {{font|color=#adf|14}} ||{{Inversions8|0|0|1|0|1|1|0|1|1|0|0|1|1|0|0|0|1|1|0|0|1|0|1|1|0|0|1|1}}  || {{Row8,0start}}{{Row8,7}}{{Row8,6}}{{Row8,1}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,2}}{{Row8,5}}{{Row8,4}}{{Row8,3end}} || || align=&quot;left&quot;|{{sms|4 }}{{font|color=#c07|&amp;nbsp;&amp;#8199;9}}&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;span style=&quot;font-size:60%&quot;&gt;(3+3)&lt;/span&gt;&gt;
|-   
|style=&quot;background:lightgray;{{text color default}};&quot;| 4 || 5 6  {{font|color=#aaa|4}} || {{BIN1start}}{{BIN0}}{{BIN0;}}&lt;br&gt;{{BIN0}}{{BIN1}}{{BIN0;}}&lt;br&gt;{{BIN1}}{{BIN1}}{{BIN1end}} || + || 5 || align=&quot;right&quot;|16680 || {{font|color=#aaa|0 1 2 3}} &amp;nbsp;  7 6 5 4 &lt;br&gt;{{font|color=#adf|0  0  0  0 &amp;nbsp; 0  1  2  3}} || {{font|color=#adf|&amp;#8199;6}} ||{{Inversions8|0|0|0|0|0|0|0|0|0|0|0|0|0|0|1|0|0|0|0|1|1|0|0|0|0|1|1|1}}  || {{Row8,0start}}{{Row8,1}}{{Row8,2}}{{Row8,3}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,7}}{{Row8,6}}{{Row8,5}}{{Row8,4end}} || || align=&quot;left&quot;|{{sms|2 }}{{font|color=#c07|&amp;nbsp;&amp;#8199;3}}&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;span style=&quot;font-size:60%&quot;&gt;(2+2)&lt;/span&gt;&gt;
|-   
|5 || 7 1 5  || {{BIN1start}}{{BIN1}}{{BIN1;}}&lt;br&gt;{{BIN1}}{{BIN0}}{{BIN0;}}&lt;br&gt;{{BIN1}}{{BIN0}}{{BIN1end}} || − || 6 || align=&quot;right&quot;|22736 || {{font|color=#aaa|0}}  7 1 6  &amp;nbsp;  5 2 4 3 &lt;br&gt;{{font|color=#adf|0  0  1  1 &amp;nbsp; 2  3  3  4}} || {{font|color=#adf|14}} ||{{Inversions8|0|0|1|0|1|0|0|1|0|1|0|1|0|1|1|0|1|0|1|1|0|0|1|0|1|1|0|1}}  || {{Row8,0start}}{{Row8,7}}{{Row8,1}}{{Row8,6}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,5}}{{Row8,2}}{{Row8,4}}{{Row8,3end}} || || align=&quot;left&quot;|{{sms|6 }}{{font|color=#c07|&amp;nbsp;14}}&lt;sub&gt;b&lt;/sub&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;span style=&quot;font-size:60%&quot;&gt;(7)&lt;/span&gt;&gt;
|-   
|6 || 7  {{font|color=#aaa|2}}  6  || {{BIN1start}}{{BIN0}}{{BIN0;}}&lt;br&gt;{{BIN1}}{{BIN1}}{{BIN1;}}&lt;br&gt;{{BIN1}}{{BIN0}}{{BIN1end}} || + || 6 || align=&quot;right&quot;|23454 || {{font|color=#aaa|0 1}}  7 6  &amp;nbsp;  5 4 2 3 &lt;br&gt;{{font|color=#adf|0  0  0  1 &amp;nbsp; 2  3  4  4}} || {{font|color=#adf|14}} ||{{Inversions8|0|0|0|0|0|1|0|0|1|1|0|0|1|1|1|0|0|1|1|1|1|0|0|1|1|1|1|0}}  || {{Row8,0start}}{{Row8,1}}{{Row8,7}}{{Row8,6}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,5}}{{Row8,4}}{{Row8,2}}{{Row8,3end}} || [[File:Walsh permutation 165 Fano red.svg|53px]] || align=&quot;left&quot;|{{sms|3 }}{{font|color=#c07|&amp;nbsp;&amp;#8199;8}}&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;span style=&quot;font-size:60%&quot;&gt;(4+2)&lt;/span&gt;&gt;
|-   
|style=&quot;background:lightgray;{{text color default}};&quot;| 7 || 5 3 7  || {{BIN1start}}{{BIN1}}{{BIN1;}}&lt;br&gt;{{BIN0}}{{BIN1}}{{BIN1;}}&lt;br&gt;{{BIN1}}{{BIN0}}{{BIN1end}} || + || 7 || align=&quot;right&quot;|17702 || {{font|color=#aaa|0}}  7 6 1  &amp;nbsp;  5 2 3 4 &lt;br&gt;{{font|color=#adf|0  0  1  2 &amp;nbsp; 2  3  3  3}} || {{font|color=#adf|14}} ||{{Inversions8|0|0|1|0|1|1|0|1|1|0|0|1|1|0|1|0|1|1|0|1|0|0|1|1|0|1|0|0}}  || {{Row8,0start}}{{Row8,7}}{{Row8,6}}{{Row8,1}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,5}}{{Row8,2}}{{Row8,3}}{{Row8,4end}} || [[File:Walsh permutation 357 Fano red.svg|53px]] || align=&quot;left&quot;|{{sms|5 }}{{font|color=#c07|&amp;nbsp;14}}&lt;sub&gt;a&lt;/sub&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;span style=&quot;font-size:60%&quot;&gt;(7)&lt;/span&gt;&gt;
|}
{|
|Sorting by '''#''' gives the permutation (0,4,2,1,7,3,5,6) in the left column, which is ''reflection'' [[w:Function composition|∘]] [[w:Circular shift|''right shift'']] ∘ ''wp''(6,5,7) (with the compression matrix || {{Bin0start}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1end}} || ).
|}
Sorting by '''Permutation''' gives the (0,4,2,6, 1,5,3,7) in the left column, which is the bit-reversal permutation ''wp''(4,2,1).

=== 4-bit ===

[[File:Variadic logical XOR.svg|thumb|right|Binary Walsh matrix of order 16]]

{| class=&quot;wikitable sortable&quot; style=&quot;text-align:right;&quot;
|-
!
!#
!class=&quot;unsortable&quot;|IS
!CV
!class=&quot;unsortable&quot;|CM
!Σ
!Permutation&lt;br&gt;F
!IN
|-
|0
|0
|[[File:16 choose 2 array.svg|36px]]
|{{font|face=Courier|{{nowrap|'''( 1, 2, 4, 8)'''}}}}
|&lt;div style=&quot;white-space: nowrap&quot;&gt;{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin0}}{{Bin1end}}&lt;/div&gt;
|4
|{{font|face=Courier|{{nowrap|'''( 0, 1, 2, 3,&amp;nbsp; 4, 5, 6, 7,&amp;nbsp; 8, 9,10,11,&amp;nbsp;12,13,14,15)'''}}&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 0,&amp;nbsp; 0, 0, 0, 0,&amp;nbsp; 0, 0, 0, 0,&amp;nbsp; 0, 0, 0, 0)'''}}}}
|{{font|color=#adf|0}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|1
|9804424107176
|[[File:Inversion set 16; Walsh (15, 1, 2, 4).svg|36px]]
|{{font|face=Courier|'''( 3, 5, 9, 1)'''}}
|{{Bin1start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0end}}
|7
|{{font|face=Courier|'''( 0,15, 1,14,&amp;nbsp; 2,13, 3,12,&amp;nbsp; 4,11, 5,10,&amp;nbsp; 6, 9, 7, 8)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 1,&amp;nbsp; 2, 2, 3, 3,&amp;nbsp; 4, 4, 5, 5,&amp;nbsp; 6, 6, 7, 7)'''}}}}
|{{font|color=#adf|56}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|2
|9717242186454
|[[File:Inversion set 16; Walsh (1, 15, 2, 4).svg|36px]]
|{{font|face=Courier|'''( 3, 6,10, 2)'''}}
|{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0end}}
|7
|{{font|face=Courier|'''( 0, 1,15,14,&amp;nbsp; 2, 3,13,12,&amp;nbsp; 4, 5,11,10,&amp;nbsp; 6, 7, 9, 8)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 1,&amp;nbsp; 2, 2, 2, 3,&amp;nbsp; 4, 4, 4, 5,&amp;nbsp; 6, 6, 6, 7)'''}}}}
|{{font|color=#adf|52}}
|-
|3
|11112138397022
|[[File:Inversion set 16; wp( 1, 7,11, 3).svg|36px]]
|{{font|face=Courier|'''( 1, 7,11, 3)'''}}
|{{Bin1start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0end}}
|9
|{{font|face=Courier|'''( 0,15,14, 1,&amp;nbsp; 2,13,12, 3,&amp;nbsp; 4,11,10, 5,&amp;nbsp; 6, 9, 8, 7)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 2,&amp;nbsp; 2, 2, 3, 4,&amp;nbsp; 4, 4, 5, 6,&amp;nbsp; 6, 6, 7, 8)'''}}}}
|{{font|color=#adf|60}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|4
|9710017245480
|[[File:Inversion set 16; wp( 5, 6,12, 4).svg|36px]]
|{{font|face=Courier|'''( 5, 6,12, 4)'''}}
|{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0end}}
|7
|{{font|face=Courier|'''( 0, 1, 2, 3,&amp;nbsp;15,14,13,12,&amp;nbsp; 4, 5, 6, 7,&amp;nbsp;11,10, 9, 8)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 0,&amp;nbsp; 0, 1, 2, 3,&amp;nbsp; 4, 4, 4, 4,&amp;nbsp; 4, 5, 6, 7)'''}}}}
|{{font|color=#adf|44}}
|-
|5
|11118325501136
|[[File:Inversion set 16; Walsh (15, 1, 13, 4).svg|36px]]
|{{font|face=Courier|'''( 7, 1,13, 5)'''}}
|{{Bin1start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0end}}
|9
|{{font|face=Courier|'''( 0,15, 1,14,&amp;nbsp;13, 2,12, 3,&amp;nbsp; 4,11, 5,10,&amp;nbsp; 9, 6, 8, 7)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 1,&amp;nbsp; 2, 3, 3, 4,&amp;nbsp; 4, 4, 5, 5,&amp;nbsp; 6, 7, 7, 8)'''}}}}
|{{font|color=#adf|60}}
|-
|6
|11205500164254
|[[File:Inversion set 16; Walsh (1, 15, 13, 4).svg|36px]]
|{{font|face=Courier|'''( 7, 2,14, 6)'''}}
|{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0end}}
|9
|{{font|face=Courier|'''( 0, 1,15,14,&amp;nbsp;13,12, 2, 3,&amp;nbsp; 4, 5,11,10,&amp;nbsp; 9, 8, 6, 7)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 1,&amp;nbsp; 2, 3, 4, 4,&amp;nbsp; 4, 4, 4, 5,&amp;nbsp; 6, 7, 8, 8)'''}}}}
|{{font|color=#adf|60}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|7
|9810691044902
|[[File:Inversion set 16; Walsh (15, 14, 13, 4).svg|36px]]
|{{font|face=Courier|'''( 5, 3,15, 7)'''}}
|{{Bin1start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0end}}
|11
|{{font|face=Courier|'''( 0,15,14, 1,&amp;nbsp;13, 2, 3,12,&amp;nbsp; 4,11,10, 5,&amp;nbsp; 9, 6, 7, 8)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 2,&amp;nbsp; 2, 3, 3, 3,&amp;nbsp; 4, 4, 5, 6,&amp;nbsp; 6, 7, 7, 7)'''}}}}
|{{font|color=#adf|60}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|8
|9709968804480
|[[File:Inversion set 16; Walsh (1, 2, 4, 15).svg|36px]]
|{{font|face=Courier|'''( 9,10,12, 8)'''}}
|{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin1end}}
|7
|{{font|face=Courier|'''( 0, 1, 2, 3,&amp;nbsp; 4, 5, 6, 7,&amp;nbsp;15,14,13,12,&amp;nbsp;11,10, 9, 8)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 0,&amp;nbsp; 0, 0, 0, 0,&amp;nbsp;  0, 1, 2, 3,&amp;nbsp; 4, 5, 6, 7)'''}}}}
|{{font|color=#adf|28}}
|-
|9
|11118365775656
|[[File:Inversion set 16; Walsh (15, 1, 2, 11).svg|36px]]
|{{font|face=Courier|'''(11,13, 1, 9)'''}}
|{{Bin1start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin0}}{{Bin1end}}
|9
|{{font|face=Courier|'''( 0,15, 1,14,&amp;nbsp; 2,13, 3,12,&amp;nbsp;11, 4,10, 5,&amp;nbsp; 9, 6, 8, 7)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 1,&amp;nbsp; 2, 2, 3, 3,&amp;nbsp; 4, 5, 5, 6,&amp;nbsp; 6, 7, 7, 8)'''}}}}
|{{font|color=#adf|60}}
|-
|10
|11205547694934
|[[File:Inversion set 16; Walsh (1, 15, 2, 11).svg|36px]]
|{{font|face=Courier|'''(11,14, 2,10)'''}}
|{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin0}}{{Bin1end}}
|9
|{{font|face=Courier|'''( 0, 1,15,14,&amp;nbsp; 2, 3,13,12,&amp;nbsp;11,10, 4, 5,&amp;nbsp; 9, 8, 6, 7)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 1,&amp;nbsp; 2, 2, 2, 3,&amp;nbsp; 4, 5, 6, 6,&amp;nbsp; 6, 7, 8, 8)'''}}}}
|{{font|color=#adf|60}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|11
|9810651495902
|[[File:Inversion set 16; Walsh (15, 14, 2, 11).svg|36px]]
|{{font|face=Courier|'''( 9,15, 3,11)'''}}
|{{Bin1start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin0}}{{Bin1end}}
|11
|{{font|face=Courier|'''( 0,15,14, 1,&amp;nbsp; 2,13,12, 3,&amp;nbsp;11, 4, 5,10,&amp;nbsp; 9, 6, 7, 8)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 2,&amp;nbsp; 2, 2, 3, 4,&amp;nbsp; 4, 5, 5, 5,&amp;nbsp; 6, 7, 7, 7)'''}}}}
|{{font|color=#adf|60}}
|-
|12
|11212772635560
|[[File:Inversion set 16; Walsh (1, 2, 15, 11).svg|36px]]
|{{font|face=Courier|'''(13,14, 4,12)'''}}
|{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin0}}{{Bin1end}}
|9
|{{font|face=Courier|'''( 0, 1, 2, 3,&amp;nbsp;15,14,13,12,&amp;nbsp;11,10, 9, 8,&amp;nbsp; 4, 5, 6, 7)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 0,&amp;nbsp;  0, 1, 2, 3,&amp;nbsp; 4, 5, 6, 7,&amp;nbsp; 8, 8, 8, 8)'''}}}}
|{{font|color=#adf|60}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|13
|9804464392016
|[[File:Inversion set 16; Walsh (15, 1, 13, 11).svg|36px]]
|{{font|face=Courier|'''(15, 9, 5,13)'''}}
|{{Bin1start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin0}}{{Bin1end}}
|11
|{{font|face=Courier|'''( 0,15, 1,14,&amp;nbsp;13, 2,12, 3,&amp;nbsp;11, 4,10, 5,&amp;nbsp; 6, 9, 7, 8)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 1,&amp;nbsp; 2, 3, 3, 4,&amp;nbsp; 4, 5, 5, 6,&amp;nbsp; 6, 6, 7, 7)'''}}}}
|{{font|color=#adf|60}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|14
|9717289730334
|[[File:Inversion set 16; Walsh (1, 15, 13, 11).svg|36px]]
|{{font|face=Courier|'''(15,10, 6,14)'''}}
|{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin0}}{{Bin1end}}
|11
|{{font|face=Courier|'''( 0, 1,15,14,&amp;nbsp;13,12, 2, 3,&amp;nbsp;11,10, 4, 5,&amp;nbsp; 6, 7, 9, 8)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 1,&amp;nbsp; 2, 3, 4, 4,&amp;nbsp; 4, 5, 6, 6,&amp;nbsp; 6, 6, 6, 7)'''}}}}
|{{font|color=#adf|60}}
|-
|15
|11112098838182
|[[File:Inversion set 16; Walsh (15, 14, 13, 11).svg|36px]]
|{{font|face=Courier|'''(13,11, 7,15)'''}}
|{{Bin1start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin0}}{{Bin1end}}
|13
|{{font|face=Courier|'''( 0,15,14, 1,&amp;nbsp;13, 2, 3,12,&amp;nbsp;11, 4, 5,10,&amp;nbsp; 6, 9, 8, 7)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 2,&amp;nbsp; 2, 3, 3, 3,&amp;nbsp; 4, 5, 5, 5,&amp;nbsp; 6, 6, 7, 8)'''}}}}
|{{font|color=#adf|60}}
|}

{|
|Sorting by '''#''' gives the (0, 8,4,2,14, 1,13,11,7, 15,3,5,9, 6,10,12) in the left column, which is ''reflection'' [[w:Function composition|∘]] [[w:Circular shift|''right shift'']] ∘ ''wp''(12,10, 9, 7) (with the compression matrix || {{Bin0start}}{{Bin0}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin0}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin0end}} || ).
|}
Sorting by '''Permutation''' gives the (0,8,4,12, 2,10,6,14, 1,9,5,13, 3,11,7,15) in the left column, which is the [[w:Bit-reversal permutation|bit-reversal permutation]] ''wp''( 8, 4, 2, 1).

{| class=&quot;collapsible collapsed&quot; style=&quot;width: 100%; border: 1px solid #cccccc; background:white;{{text color default}};&quot;
! bgcolor=&quot;#f0f0f0&quot;|{{font|color=#999999|Matrices}}
|-
|
 Compression vectors:
     1     2     4     8
     3     5     9     1
     3     6    10     2
     1     7    11     3
     5     6    12     4
     7     1    13     5
     7     2    14     6
     5     3    15     7
     9    10    12     8
    11    13     1     9
    11    14     2    10
     9    15     3    11
    13    14     4    12
    15     9     5    13
    15    10     6    14
    13    11     7    15
 &amp;nbsp;
 Permutations:
     0     1     2     3     4     5     6     7     8     9    10    11    12    13    14    15
     0    15     1    14     2    13     3    12     4    11     5    10     6     9     7     8
     0     1    15    14     2     3    13    12     4     5    11    10     6     7     9     8
     0    15    14     1     2    13    12     3     4    11    10     5     6     9     8     7
     0     1     2     3    15    14    13    12     4     5     6     7    11    10     9     8
     0    15     1    14    13     2    12     3     4    11     5    10     9     6     8     7
     0     1    15    14    13    12     2     3     4     5    11    10     9     8     6     7
     0    15    14     1    13     2     3    12     4    11    10     5     9     6     7     8
     0     1     2     3     4     5     6     7    15    14    13    12    11    10     9     8
     0    15     1    14     2    13     3    12    11     4    10     5     9     6     8     7
     0     1    15    14     2     3    13    12    11    10     4     5     9     8     6     7
     0    15    14     1     2    13    12     3    11     4     5    10     9     6     7     8
     0     1     2     3    15    14    13    12    11    10     9     8     4     5     6     7
     0    15     1    14    13     2    12     3    11     4    10     5     6     9     7     8
     0     1    15    14    13    12     2     3    11    10     4     5     6     7     9     8
     0    15    14     1    13     2     3    12    11     4     5    10     6     9     8     7
 &amp;nbsp;
 Little-endian factorial numbers:
     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0
     0     0     1     1     2     2     3     3     4     4     5     5     6     6     7     7
     0     0     0     1     2     2     2     3     4     4     4     5     6     6     6     7
     0     0     1     2     2     2     3     4     4     4     5     6     6     6     7     8
     0     0     0     0     0     1     2     3     4     4     4     4     4     5     6     7
     0     0     1     1     2     3     3     4     4     4     5     5     6     7     7     8
     0     0     0     1     2     3     4     4     4     4     4     5     6     7     8     8
     0     0     1     2     2     3     3     3     4     4     5     6     6     7     7     7
     0     0     0     0     0     0     0     0     0     1     2     3     4     5     6     7
     0     0     1     1     2     2     3     3     4     5     5     6     6     7     7     8
     0     0     0     1     2     2     2     3     4     5     6     6     6     7     8     8
     0     0     1     2     2     2     3     4     4     5     5     5     6     7     7     7
     0     0     0     0     0     1     2     3     4     5     6     7     8     8     8     8
     0     0     1     1     2     3     3     4     4     5     5     6     6     6     7     7
     0     0     0     1     2     3     4     4     4     5     6     6     6     6     6     7
     0     0     1     2     2     3     3     3     4     5     5     5     6     6     7     8
|}

=== Compression matrices ===

[[File:Compression matrices of Walsh permutations with striped inversion sets.svg|left|800px]]
&lt;br style=&quot;clear: both;&quot;&gt;
The CM of the ''n''-th permutation is a matrix ''B'' with vertical stripes for the binary digits of ''n'' &lt;small&gt;(gray lines in the image)&lt;/small&gt;, XORed with a modified unity matrix ''U'' &lt;small&gt;(black dots in the image)&lt;/small&gt;.&lt;br&gt;
''U'' is modified by including an column row where ''n'' has its lowest binary digit ({{oeis|A001511}}). It pushes the other rows of the unity matrix to the right, and the last row &quot;out of the matrix&quot;.


==Bended stripes==

Among the ''n''-bit Walsh permutations are 2&lt;sup&gt;''n-1''&lt;/sup&gt; whose inversion sets show bended stripes.&lt;br&gt;
They correspond to Walsh functions, i.e. the rows of the [[w:Walsh matrix|Walsh matrix]] of order 2&lt;sup&gt;''n-1''&lt;/sup&gt;.&lt;br&gt;
These permutations under composition form the group Z&lt;sub&gt;2&lt;/sub&gt;&lt;sup&gt;''n''-1&lt;/sup&gt;, just like the row numbers under bitwise XOR.

=== 3-bit ===

{| class=&quot;wikitable sortable&quot; style=&quot;height:275px; text-align:center;&quot;
|-style=&quot;height:55px&quot;
!
!style=&quot;width:55px&quot;|CV
!style=&quot;width:55px&quot; class=&quot;unsortable&quot;|&amp;nbsp;&amp;nbsp;CM&amp;nbsp;&amp;nbsp;
!D
!#
!Permutation&lt;br&gt;F
!IN
!class=&quot;unsortable&quot;|IS
!style=&quot;width:55px&quot; class=&quot;unsortable&quot;|PM
!class=&quot;unsortable&quot;|FPC
!Cy
|-   
|0 || {{font|color=#aaa|1 2 4}} || {{BIN1start}}{{BIN0}}{{BIN0;}}&lt;br&gt;{{BIN0}}{{BIN1}}{{BIN0;}}&lt;br&gt;{{BIN0}}{{BIN0}}{{BIN1end}} || + || align=&quot;right&quot;|  0 || {{font|color=#aaa|0 1 2 3 &amp;nbsp; 4 5 6 7}}&lt;br&gt;{{font|color=#adf|0  0  0  0 &amp;nbsp; 0  0  0  0}} || {{font|color=#adf|&amp;#8199;0}} ||{{Inversions8|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0}}  || {{Row8,0start}}{{Row8,1}}{{Row8,2}}{{Row8,3}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,4}}{{Row8,5}}{{Row8,6}}{{Row8,7end}} || [[File:Walsh permutation 124 Fano.svg|53px]] || align=&quot;left&quot;|{{sms|1 }}{{font|color=#c07|&amp;nbsp;&amp;#8199;0}}
|-   
|style=&quot;background:lightgray;{{text color default}};&quot;|1 ||  4 7 1  || {{BIN0start}}{{BIN1}}{{BIN1;}}&lt;br&gt;{{BIN0}}{{BIN1}}{{BIN0;}}&lt;br&gt;{{BIN1}}{{BIN1}}{{BIN0end}} || − || align=&quot;right&quot;| 3776 || {{font|color=#aaa|0}}  6  {{font|color=#aaa|2}}  4  &amp;nbsp;  3  {{font|color=#aaa|5}}  1  {{font|color=#aaa|7}}&lt;br&gt;{{font|color=#adf|0  0  1  1 &amp;nbsp; 2  1  5  0}} || {{font|color=#adf|10}} ||{{Inversions8|0|0|1|0|1|0|0|1|0|1|0|1|0|0|0|0|1|1|1|1|1|0|0|0|0|0|0|0}}  || {{Row8,0start}}{{Row8,6}}{{Row8,2}}{{Row8,4}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,3}}{{Row8,5}}{{Row8,1}}{{Row8,7end}} || || align=&quot;left&quot;|{{sms|2 }}{{font|color=#c07|&amp;nbsp;&amp;#8199;3}}&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;span style=&quot;font-size:60%&quot;&gt;(2+2)&lt;/span&gt;&gt;
|-   
|style=&quot;background:lightgray;{{text color default}};&quot;|2 ||  7 4 2  || {{BIN1start}}{{BIN0}}{{BIN0;}}&lt;br&gt;{{BIN1}}{{BIN0}}{{BIN1;}}&lt;br&gt;{{BIN1}}{{BIN1}}{{BIN0end}} || − || align=&quot;right&quot;|  414 || {{font|color=#aaa|0 1}}  5 4  &amp;nbsp;  3 2  {{font|color=#aaa|6 7}}&lt;br&gt;{{font|color=#adf|0  0  0  1 &amp;nbsp; 2  3  0  0}} || {{font|color=#adf|&amp;#8199;6}} ||{{Inversions8|0|0|0|0|0|1|0|0|1|1|0|0|1|1|1|0|0|0|0|0|0|0|0|0|0|0|0|0}}  || {{Row8,0start}}{{Row8,1}}{{Row8,5}}{{Row8,4}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,3}}{{Row8,2}}{{Row8,6}}{{Row8,7end}} || || align=&quot;left&quot;|{{sms|2 }}{{font|color=#c07|&amp;nbsp;&amp;#8199;3}}&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;span style=&quot;font-size:60%&quot;&gt;(2+2)&lt;/span&gt;&gt;
|-   
|3 ||  2 1 7  || {{BIN0start}}{{BIN1}}{{BIN1;}}&lt;br&gt;{{BIN1}}{{BIN0}}{{BIN1;}}&lt;br&gt;{{BIN0}}{{BIN0}}{{BIN1end}} || − || align=&quot;right&quot;| 4142 || {{font|color=#aaa|0}}  6 5  {{font|color=#aaa|3 &amp;nbsp; 4}}  2 1  {{font|color=#aaa|7}}&lt;br&gt;{{font|color=#adf|0  0  1  2 &amp;nbsp; 2  4  5  0}} || {{font|color=#adf|14}} ||{{Inversions8|0|0|1|0|1|1|0|1|1|0|0|1|1|1|1|0|1|1|1|1|1|0|0|0|0|0|0|0}}  || {{Row8,0start}}{{Row8,6}}{{Row8,5}}{{Row8,3}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,4}}{{Row8,2}}{{Row8,1}}{{Row8,7end}} || || align=&quot;left&quot;|{{sms|2 }}{{font|color=#c07|&amp;nbsp;&amp;#8199;3}}&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;span style=&quot;font-size:60%&quot;&gt;(2+2)&lt;/span&gt;&gt;
|}

Sorting by '''#''' or '''Permutation''' gives (0,2,1,3) in the left column, which is the bit-reversal permutation ''wp''(2,1).&lt;br&gt;
Sorting by '''CV''' gives (0,3,1,2) in the left column, which is ''wp''(3,1).


=== 4-bit ===

{| class=&quot;wikitable sortable&quot; style=&quot;text-align:right;&quot;
|-
!
!#
!class=&quot;unsortable&quot;|IS
!CV
!class=&quot;unsortable&quot;|CM
!Permutation&lt;br&gt;F
!IN
|-
|0
|0
|[[File:16 choose 2 array.svg|36px]]
|{{font|face=Courier|{{nowrap|'''( 1, 2, 4, 8)'''}}}}
|&lt;div style=&quot;white-space: nowrap&quot;&gt;{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin0}}{{Bin1end}}&lt;/div&gt;
|{{font|face=Courier|{{nowrap|'''( 0, 1, 2, 3,&amp;nbsp; 4, 5, 6, 7,&amp;nbsp; 8, 9,10,11,&amp;nbsp;12,13,14,15)'''}}&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 0,&amp;nbsp; 0, 0, 0, 0,&amp;nbsp; 0, 0, 0, 0,&amp;nbsp; 0, 0, 0, 0)'''}}}}
|{{font|color=#adf|0}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|1
|1144441325096
|[[File:Inversion set 16; Walsh (14, 2, 4, 7).svg|36px]]
|{{font|face=Courier|'''( 8,11,13, 1)'''}}
|{{Bin0start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin0end}}
|{{font|face=Courier|'''( 0,14, 2,12,&amp;nbsp; 4,10, 6, 8,&amp;nbsp; 7, 9, 5,11,&amp;nbsp; 3,13, 1,15)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 1,&amp;nbsp; 2, 2, 3, 3,&amp;nbsp; 4, 3, 7, 2,&amp;nbsp;10, 1,13, 0)'''}}}}
|{{font|color=#adf|52}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|2
|73376328534
|[[File:Inversion set 16; Walsh (1, 13, 4, 7).svg|36px]]
|{{font|face=Courier|'''(11, 8,14, 2)'''}}
|{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin0end}}
|{{font|face=Courier|'''( 0, 1,13,12,&amp;nbsp; 4, 5, 9, 8,&amp;nbsp; 7, 6,10,11,&amp;nbsp; 3, 2,14,15)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 1,&amp;nbsp; 2, 2, 2, 3,&amp;nbsp; 4, 5, 2, 2,&amp;nbsp;10,11, 0, 0)'''}}}}
|{{font|color=#adf|44}}
|-
|3
|1209107634782
|[[File:Inversion set 16; Walsh (14, 13, 4, 8).svg|36px]]
|{{font|face=Courier|'''( 2, 1, 7,11)'''}}
|{{Bin0start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin0}}{{Bin1end}}
|{{font|face=Courier|'''( 0,14,13, 3,&amp;nbsp; 4,10, 9, 7,&amp;nbsp; 8, 6, 5,11,&amp;nbsp;12, 2, 1,15)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 2,&amp;nbsp; 2, 2, 3, 4,&amp;nbsp; 4, 6, 7, 2,&amp;nbsp; 2,12,13, 0)'''}}}}
|{{font|color=#adf|60}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|4
|303182760
|[[File:Inversion set 16; Walsh (1, 2, 11, 7).svg|36px]]
|{{font|face=Courier|'''(13,14, 8, 4)'''}}
|{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin0}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin0end}}
|{{font|face=Courier|'''( 0, 1, 2, 3,&amp;nbsp;11,10, 9, 8,&amp;nbsp; 7, 6, 5, 4,&amp;nbsp;12,13,14,15)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 0,&amp;nbsp; 0, 1, 2, 3,&amp;nbsp; 4, 5, 6, 7,&amp;nbsp; 0, 0, 0, 0)'''}}}}
|{{font|color=#adf|28}}
|-
|5
|1144707321296
|[[File:Inversion set 16; Walsh (14, 2, 11, 8).svg|36px]]
|{{font|face=Courier|'''( 4, 7, 1,13)'''}}
|{{Bin0start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin0}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin0}}{{Bin1end}}
|{{font|face=Courier|'''( 0,14, 2,12,&amp;nbsp;11, 5, 9, 7,&amp;nbsp; 8, 6,10, 4,&amp;nbsp; 3,13, 1,15)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 1,&amp;nbsp; 2, 3, 3, 4,&amp;nbsp; 4, 6, 3, 9,&amp;nbsp;10, 1,13, 0)'''}}}}
|{{font|color=#adf|60}}
|-
|6
|73677162654
|[[File:Inversion set 16; Walsh (1, 13, 11, 8).svg|36px]]
|{{font|face=Courier|'''( 7, 4, 2,14)'''}}
|{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin0}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin0}}{{Bin1end}}
|{{font|face=Courier|'''( 0, 1,13,12,&amp;nbsp;11,10, 6, 7,&amp;nbsp; 8, 9, 5, 4,&amp;nbsp; 3, 2,14,15)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 1,&amp;nbsp; 2, 3, 4, 4,&amp;nbsp; 4, 4, 8, 9,&amp;nbsp;10,11, 0, 0)'''}}}}
|{{font|color=#adf|60}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|7
|1209371443622
|[[File:Inversion set 16; Walsh (14, 13, 11, 7).svg|36px]]
|{{font|face=Courier|'''(14,13,11, 7)'''}}
|{{Bin0start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin0}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin0end}}
|{{font|face=Courier|'''( 0,14,13, 3,&amp;nbsp;11, 5, 6, 8,&amp;nbsp; 7, 9,10, 4,&amp;nbsp;12, 2, 1,15)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 2,&amp;nbsp; 2, 3, 3, 3,&amp;nbsp; 4, 3, 3, 9,&amp;nbsp; 2,12,13, 0)'''}}}}
|{{font|color=#adf|60}}
|}

Sorting by '''#''' or '''Permutation''' gives (0,4,2,6, 1,5,3,7) in the left column, which is the bit-reversal permutation ''wp''(4,2,1).&lt;br&gt;
Sorting by '''CV''' gives (0,3,5,6, 1,2,4,7) in the left column, which is ''wp''(7,1,2).

===Compression matrices===

[[File:Compression matrices of Walsh permutations with bendedly striped inversion sets.svg|800px]]
&lt;br style=&quot;clear: both;&quot;&gt;
The CM are matrices with an even number of vertical stripes &lt;small&gt;(gray lines in the image)&lt;/small&gt; XORed with the unity matrix &lt;small&gt;(black dots in the image)&lt;/small&gt;.&lt;br&gt;
The order by the ''evil numbers'' ({{oeis|A001969}}) of the matrices, used in the image, corresponds to the lexicographical order of the compression vectors.

==Bit permutations==
{| align=&quot;right&quot;
|colspan=&quot;2&quot;|
{| align=right style=&quot;text-align=right; border:1px solid lightgray&quot; cellspacing=&quot;10&quot;
|[[Walsh permutation]]; [[Walsh permutation; bit permutation|bit permutation]] {{Rdrup}}
|}
|-
|style=&quot;vertical-align:top;&quot;|[[File:Inversion set 16; Walsh (2, 8, 1, 4).svg|thumb|Inversion set of ''wp''( 4, 1, 8, 2)&lt;br&gt;All columns are symmetrical.]]
|style=&quot;vertical-align:top;&quot;|[[File:Inversion set 16; Walsh (14, 13, 11, 7).svg|thumb|Inversion set of ''wp''(14,13,11, 7)]]
|-
|colspan=&quot;2&quot;|
{| align=right
|-
|style=&quot;vertical-align:top;&quot;|[[File:Symmetric group 4; cycle graph; details.svg|thumb|center|500px|[[w:Cycle graph (algebra)|Cycle graph]] of the bit permutations]]
|style=&quot;vertical-align:top;&quot;|[[File:Symmetric group 4; permutation list (0-based).svg|thumb|center|Permutations of 4 elements&lt;br&gt;{{Rdr}} [[Symmetric group S4|Symmetric group S&lt;sub&gt;4&lt;/sub&gt;]]]]
|}
|}


{| class=&quot;wikitable sortable&quot; style=&quot;text-align:right;&quot;
|-
!
!#
!class=&quot;unsortable&quot;|IS
!CV
|-
|style=&quot;background:white;{{text color default}};&quot;|&lt;big&gt;'''0'''&lt;/big&gt;
|0
|[[File:16 choose 2 array.svg|36px]]
|{{font|face=Courier|'''( 1, 2, 4, 8)'''}}
|-
|style=&quot;background:#33eb2a;{{text color default}};&quot;|1
|87181920722
|[[File:Inversion set 16; Walsh (2, 1, 4, 8).svg|36px]]
|{{font|face=Courier|'''( 2, 1, 4, 8)'''}}
|-
|style=&quot;background:#33eb2a;{{text color default}};&quot;|2
|13412045088
|[[File:Inversion set 16; wp( 1, 4, 2, 8).svg|36px]]
|{{font|face=Courier|'''( 1, 4, 2, 8)'''}}
|-
|style=&quot;background:white;{{text color default}};&quot;|3
|101069338632
|[[File:Inversion set 16; Walsh (2, 4, 1, 8).svg|36px]]
|{{font|face=Courier|'''( 4, 1, 2, 8)'''}}
|-
|style=&quot;background:white;{{text color default}};&quot;|4
|262496507810
|[[File:Inversion set 16; Walsh (4, 1, 2, 8).svg|36px]]
|{{font|face=Courier|'''( 2, 4, 1, 8)'''}}
|-
|style=&quot;background:#33eb2a;{{text color default}};&quot;|5
|269202530354
|[[File:Inversion set 16; Walsh (4, 2, 1, 8).svg|36px]]
|{{font|face=Courier|'''( 4, 2, 1, 8)'''}}
|-
|style=&quot;background:#33eb2a;{{text color default}};&quot;|6
|175795200
|[[File:Inversion set 16; Walsh (1, 2, 8, 4).svg|36px]]
|{{font|face=Courier|'''( 1, 2, 8, 4)'''}}
|-
|style=&quot;background:white;{{text color default}};&quot;|&lt;big&gt;'''7'''&lt;/big&gt;
|87357715922
|[[File:Inversion set 16; Walsh (2, 1, 8, 4).svg|36px]]
|{{font|face=Courier|'''( 2, 1, 8, 4)'''}}
|-
|style=&quot;background:white;{{text color default}};&quot;|8
|13588646400
|[[File:Inversion set 16; Walsh (1, 4, 8, 2).svg|36px]]
|{{font|face=Courier|'''( 1, 8, 2, 4)'''}}
|-
|style=&quot;background:#ffbb00;{{text color default}};&quot;|9
|101249608320
|[[File:Inversion set 16; Walsh (2, 4, 8, 1).svg|36px]]
|{{font|face=Courier|'''( 8, 1, 2, 4)'''}}
|-
|style=&quot;background:#ffbb00;{{text color default}};&quot;|10
|262679639762
|[[File:Inversion set 16; Walsh (4, 1, 8, 2).svg|36px]]
|{{font|face=Courier|'''( 2, 8, 1, 4)'''}}
|-
|style=&quot;background:white;{{text color default}};&quot;|11
|269386065362
|[[File:Inversion set 16; Walsh (4, 2, 8, 1).svg|36px]]
|{{font|face=Courier|'''( 8, 2, 1, 4)'''}}
|-
|style=&quot;background:white;{{text color default}};&quot;|12
|40237747488
|[[File:Inversion set 16; Walsh (1, 8, 2, 4).svg|36px]]
|{{font|face=Courier|'''( 1, 4, 8, 2)'''}}
|-
|style=&quot;background:#ffbb00;{{text color default}};&quot;|13
|127895041032
|[[File:Inversion set 16; Walsh (2, 8, 1, 4).svg|36px]]
|{{font|face=Courier|'''( 4, 1, 8, 2)'''}}
|-
|style=&quot;background:#33eb2a;{{text color default}};&quot;|14
|40325645088
|[[File:Inversion set 16; wp( 1, 8, 4, 2).svg|36px]]
|{{font|face=Courier|'''( 1, 8, 4, 2)'''}}
|-
|style=&quot;background:white;{{text color default}};&quot;|15
|127986607008
|[[File:Inversion set 16; Walsh (2, 8, 4, 1).svg|36px]]
|{{font|face=Courier|'''( 8, 1, 4, 2)'''}}
|-
|style=&quot;background:white;{{text color default}};&quot;|&lt;big&gt;'''16'''&lt;/big&gt;
|303216964872
|[[File:Inversion set 16; wp( 4, 8, 1, 2).svg|36px]]
|{{font|face=Courier|'''( 4, 8, 1, 2)'''}}
|-
|style=&quot;background:#ffbb00;{{text color default}};&quot;|17
|303260913672
|[[File:Inversion set 16; Walsh (4, 8, 2, 1).svg|36px]]
|{{font|face=Courier|'''( 8, 4, 1, 2)'''}}
|-
|style=&quot;background:#ffbb00;{{text color default}};&quot;|18
|613140355490
|[[File:Inversion set 16; Walsh (8, 1, 2, 4).svg|36px]]
|{{font|face=Courier|'''( 2, 4, 8, 1)'''}}
|-
|style=&quot;background:white;{{text color default}};&quot;|19
|619846378034
|[[File:Inversion set 16; Walsh (8, 2, 1, 4).svg|36px]]
|{{font|face=Courier|'''( 4, 2, 8, 1)'''}}
|-
|style=&quot;background:white;{{text color default}};&quot;|20
|613228253090
|[[File:Inversion set 16; Walsh (8, 1, 4, 2).svg|36px]]
|{{font|face=Courier|'''( 2, 8, 4, 1)'''}}
|-
|style=&quot;background:#33eb2a;{{text color default}};&quot;|21
|619934678690
|[[File:Inversion set 16; Walsh (8, 2, 4, 1).svg|36px]]
|{{font|face=Courier|'''( 8, 2, 4, 1)'''}}
|-
|style=&quot;background:#ffbb00;{{text color default}};&quot;|22
|633259229234
|[[File:Inversion set 16; wp( 4, 8, 2, 1).svg|36px]]
|{{font|face=Courier|'''( 4, 8, 2, 1)'''}}
|-
|style=&quot;background:white;{{text color default}};&quot;|&lt;big&gt;'''23'''&lt;/big&gt;
|633303178034
|[[File:Inversion set 16; Walsh (8, 4, 2, 1).svg|36px]]
|{{font|face=Courier|'''( 8, 4, 2, 1)'''}}
|}

{| class=&quot;collapsible collapsed&quot; style=&quot;width: 100%; border: 1px solid #cccccc; background:white;{{text color default}};&quot;
! bgcolor=&quot;#f0f0f0&quot;|{{font|color=#999999|F}}
|-
|
 Little-endian factorial numbers:
     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0
     0     0     1     0     0     0     1     0     0     0     1     0     0     0     1     0
     0     0     0     0     2     2     0     0     0     0     0     0     2     2     0     0
     0     0     0     0     3     2     1     0     0     0     0     0     3     2     1     0
     0     0     1     0     2     0     3     0     0     0     1     0     2     0     3     0
     0     0     1     0     3     1     3     0     0     0     1     0     3     1     3     0
     0     0     0     0     0     0     0     0     4     4     4     4     0     0     0     0
     0     0     1     0     0     0     1     0     4     4     5     4     0     0     1     0
     0     0     0     0     0     0     0     0     6     6     4     4     2     2     0     0
     0     0     0     0     0     0     0     0     7     6     5     4     3     2     1     0
     0     0     1     0     0     0     1     0     6     4     7     4     2     0     3     0
     0     0     1     0     0     0     1     0     7     5     7     4     3     1     3     0
     0     0     0     0     2     2     0     0     4     4     0     0     6     6     0     0
     0     0     0     0     3     2     1     0     4     4     0     0     7     6     1     0
     0     0     0     0     2     2     0     0     6     6     2     2     6     6     0     0
     0     0     0     0     2     2     0     0     7     6     3     2     7     6     1     0
     0     0     0     0     3     2     1     0     6     4     2     0     9     6     3     0
     0     0     0     0     3     2     1     0     7     5     3     1     9     6     3     0
     0     0     1     0     2     0     3     0     4     0     5     0     6     0     7     0
     0     0     1     0     3     1     3     0     4     0     5     0     7     1     7     0
     0     0     1     0     2     0     3     0     6     2     7     2     6     0     7     0
     0     0     1     0     2     0     3     0     7     3     7     2     7     1     7     0
     0     0     1     0     3     1     3     0     6     2     5     0     9     3     7     0
     0     0     1     0     3     1     3     0     7     3     6     1     9     3     7     0
 &amp;nbsp;
 Inversion numbers:
     0
     4
     8
    12
    12
    16
    16
    20
    24
    28
    28
    32
    24
    28
    32
    36
    36
    40
    28
    32
    36
    40
    40
    44
|}

==Nimber multiplication==
{| align=&quot;right&quot; style=&quot;text-align:right; border:1px solid lightgray&quot; cellspacing=&quot;10&quot;
| [[Walsh permutation]]; [[Walsh permutation; nimber multiplication|nimber multiplication]] {{rdrup}}
|}

Images: {{commons catlink|Inversion set of a 4-bit Walsh permutation; nimber multiplication}}

{{clear}}

{| class=&quot;wikitable sortable&quot; style=&quot;text-align:right&quot;
|-
! #
! CV
! Permutation
! F
! IN
|- style=&quot;font-family:Courier&quot;
|              0||( 1, 2, 4, 8)||(0, '''1''', 2, 3, &amp;nbsp; 4, 5, 6, 7, &amp;nbsp; 8, 9,10,11, &amp;nbsp;12,13,14,15) || (0, 0, 0, 0, &amp;nbsp; 0, 0, 0, 0, &amp;nbsp; 0, 0, 0, 0, &amp;nbsp; 0, 0, 0, '''0''') ||  0
|- style=&quot;font-family:Courier&quot;
| 13827898032492||( 2, 3, 8,12)||(0, '''2''', 3, 1, &amp;nbsp; 8,10,11, 9, &amp;nbsp;12,14,15,13, &amp;nbsp; 4, 6, 7, 5) || (0, 0, 0, 2, &amp;nbsp; 0, 0, 0, 2, &amp;nbsp; 0, 0, 0, 2, &amp;nbsp; 8, 8, 8,'''10''') || 40
|- style=&quot;font-family:Courier&quot;
|  7001306732168||( 3, 1,12, 4)||(0, '''3''', 1, 2, &amp;nbsp;12,15,13,14, &amp;nbsp; 4, 7, 5, 6, &amp;nbsp; 8,11, 9,10) || (0, 0, 1, 1, &amp;nbsp; 0, 0, 1, 1, &amp;nbsp; 4, 4, 5, 5, &amp;nbsp; 4, 4, 5, '''5''') || 40
|- style=&quot;font-family:Courier&quot;
| 19217812814448||( 4, 8, 6,11)||(0, '''4''', 8,12, &amp;nbsp; 6, 2,14,10, &amp;nbsp;11,15, 3, 7, &amp;nbsp;13, 9, 5, 1) || (0, 0, 0, 0, &amp;nbsp; 2, 4, 0, 2, &amp;nbsp; 2, 0, 8, 6, &amp;nbsp; 2, 6,10,'''14''') || 56
|- style=&quot;font-family:Courier&quot;
|  1630614229512||( 5,10, 2, 3)||(0, '''5''',10,15, &amp;nbsp; 2, 7, 8,13, &amp;nbsp; 3, 6, 9,12, &amp;nbsp; 1, 4,11,14) || (0, 0, 0, 0, &amp;nbsp; 3, 2, 2, 1, &amp;nbsp; 6, 5, 3, 2, &amp;nbsp;11, 9, 3, '''1''') || 48
|- style=&quot;font-family:Courier&quot;
| 15433122573720||( 6,11,14, 7)||(0, '''6''',11,13, &amp;nbsp;14, 8, 5, 3, &amp;nbsp; 7, 1,12,10, &amp;nbsp; 9,15, 2, 4) || (0, 0, 0, 0, &amp;nbsp; 0, 3, 5, 6, &amp;nbsp; 4, 8, 2, 4, &amp;nbsp; 5, 0,12,'''11''') || 60
|- style=&quot;font-family:Courier&quot;
|  5564988000144||( 7, 9,10,15)||(0, '''7''', 9,14, &amp;nbsp;10,13, 3, 4, &amp;nbsp;15, 8, 6, 1, &amp;nbsp; 5, 2,12,11) || (0, 0, 0, 0, &amp;nbsp; 1, 1, 5, 5, &amp;nbsp; 0, 5, 7,10, &amp;nbsp; 8,11, 3, '''4''') || 60
|- style=&quot;font-family:Courier&quot;
| 17445432818676||( 8,12,11,13)||(0, '''8''',12, 4, &amp;nbsp;11, 3, 7,15, &amp;nbsp;13, 5, 1, 9, &amp;nbsp; 6,14,10, 2) || (0, 0, 0, 2, &amp;nbsp; 1, 4, 3, 0, &amp;nbsp; 1, 6, 9, 4, &amp;nbsp; 7, 1, 5,'''13''') || 56
|- style=&quot;font-family:Courier&quot;
|  3551688986892||( 9,14,15, 5)||(0, '''9''',14, 7, &amp;nbsp;15, 6, 1, 8, &amp;nbsp; 5,12,11, 2, &amp;nbsp;10, 3, 4,13) || (0, 0, 0, 2, &amp;nbsp; 0, 4, 5, 3, &amp;nbsp; 6, 2, 3, 9, &amp;nbsp; 4,10,10, '''2''') || 60
|- style=&quot;font-family:Courier&quot;
| 10678227659604||(10,15, 3, 1)||(0,'''10''',15, 5, &amp;nbsp; 3, 9,12, 6, &amp;nbsp; 1,11,14, 4, &amp;nbsp; 2, 8,13, 7) || (0, 0, 0, 2, &amp;nbsp; 3, 2, 1, 4, &amp;nbsp; 7, 2, 1, 8, &amp;nbsp;10, 6, 2, '''8''') || 56
|- style=&quot;font-family:Courier&quot;
| 10169232103620||(11,13, 7, 9)||(0,'''11''',13, 6, &amp;nbsp; 7,12,10, 1, &amp;nbsp; 9, 2, 4,15, &amp;nbsp;14, 5, 3, 8) || (0, 0, 0, 2, &amp;nbsp; 2, 1, 3, 6, &amp;nbsp; 4, 7, 7, 0, &amp;nbsp; 1, 9,11, '''7''') || 60
|- style=&quot;font-family:Courier&quot;
| 15737148505928||(12, 4,13, 6)||(0,'''12''', 4, 8, &amp;nbsp;13, 1, 9, 5, &amp;nbsp; 6,10, 2,14, &amp;nbsp;11, 7,15, 3) || (0, 0, 1, 1, &amp;nbsp; 0, 4, 2, 4, &amp;nbsp; 4, 2, 8, 0, &amp;nbsp; 3, 7, 0,'''12''') || 48
|- style=&quot;font-family:Courier&quot;
|  5084423155256||(13, 6, 9,14)||(0,'''13''', 6,11, &amp;nbsp; 9, 4,15, 2, &amp;nbsp;14, 3, 8, 5, &amp;nbsp; 7,10, 1,12) || (0, 0, 1, 1, &amp;nbsp; 2, 4, 0, 6, &amp;nbsp; 1, 7, 5, 7, &amp;nbsp; 6, 4,13, '''3''') || 60
|- style=&quot;font-family:Courier&quot;
| 12454407149960||(14, 7, 5,10)||(0,'''14''', 7, 9, &amp;nbsp; 5,11, 2,12, &amp;nbsp;10, 4,13, 3, &amp;nbsp;15, 1, 8, 6) || (0, 0, 1, 1, &amp;nbsp; 3, 1, 5, 1, &amp;nbsp; 3, 7, 1, 9, &amp;nbsp; 0,12, 7, '''9''') || 60
|- style=&quot;font-family:Courier&quot;
|  8566683323240||(15, 5, 1, 2)||(0,'''15''', 5,10, &amp;nbsp; 1,14, 4,11, &amp;nbsp; 2,13, 7, 8, &amp;nbsp; 3,12, 6, 9) || (0, 0, 1, 1, &amp;nbsp; 3, 1, 4, 2, &amp;nbsp; 6, 2, 5, 5, &amp;nbsp; 9, 3, 8, '''6''') || 56
|}

[[Category:Walsh permutation]]</text>
      <sha1>g14fy67rcktq2dd3kcils64hfvbk4hm</sha1>
    </revision>
    <revision>
      <id>2819773</id>
      <parentid>2819772</parentid>
      <timestamp>2026-07-29T21:16:05Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
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      <text bytes="41086" sha1="c5t3ew74de2phqnbsqz0i640ne81bqd" xml:space="preserve">{| align=&quot;right&quot; style=&quot;text-align:right; border:1px solid lightgray&quot; cellspacing=&quot;10&quot;
| [[Walsh permutation]] {{rdrup}}
|-
| [[Inversion (discrete mathematics)]] {{rdrup}}
|}

[[File:Construction site ahead (17184445992).jpg|thumb|The vectors need to be corrected. See [[Walsh permutation#Notation warning]]. Done only for section [[#Nimber multiplication|Nimber multiplication]].]]

{|
|style=&quot;vertical-align:top;&quot;|
&lt;!---▶▶▶---&gt;{| style=&quot;border: 1px solid #cccccc&quot;
! bgcolor=&quot;#dddddd&quot; colspan=&quot;2&quot;|Abbreviations
|-
|
|-
|'''CV'''
|Compression vector
|-
|'''CM'''
|Compression matrix
|-
|'''Σ'''
|Number of ones in the compression matrix
|-
|'''IS'''
|Inversion set
|-
|'''#'''
|Index number of the finite permutation
|-
|'''IN'''
|Inversion number
|}&lt;!---◀◀◀---&gt;
|style=&quot;vertical-align:top;&quot;|__TOC__
|}

&lt;br style=&quot;clear: both;&quot;&gt;

==Stripes==
Among the ''n''-bit Walsh permutations are 2&lt;sup&gt;''n''&lt;/sup&gt; with striped inversion sets.&lt;br&gt;
As seen below they correspond to Walsh functions, i.e. the rows of the [[w:Walsh matrix|Walsh matrix]] of order 2&lt;sup&gt;''n''&lt;/sup&gt;.&lt;br&gt;
These permutations under [[w:Function composition|composition]] form the group Z&lt;sub&gt;2&lt;/sub&gt;&lt;sup&gt;''n''&lt;/sup&gt;, just like the row numbers under bitwise XOR.

=== 3-bit ===

An extract of the [[3-bit Walsh permutation; table|'''table''']] of [[3-bit Walsh permutation]]s:
{| class=&quot;wikitable sortable&quot; style=&quot;height:495px; text-align:center;&quot;
|-style=&quot;height:55px&quot;
!
!style=&quot;width:55px&quot;|CV
!style=&quot;width:55px&quot; class=&quot;unsortable&quot;|&amp;nbsp;&amp;nbsp;CM&amp;nbsp;&amp;nbsp;
!D
!Σ
!#
!Permutation&lt;br&gt;F
!IN
!class=&quot;unsortable&quot;|IS
!style=&quot;width:55px&quot; class=&quot;unsortable&quot;|PM
!class=&quot;unsortable&quot;|FPC
!Cy
|-   
|0 || {{font|color=#aaa|1 2 4}} || {{BIN1start}}{{BIN0}}{{BIN0;}}&lt;br&gt;{{BIN0}}{{BIN1}}{{BIN0;}}&lt;br&gt;{{BIN0}}{{BIN0}}{{BIN1end}} || + || 3 || align=&quot;right&quot;|  0 || {{font|color=#aaa|0 1 2 3 &amp;nbsp; 4 5 6 7}}&lt;br&gt;{{font|color=#adf|0  0  0  0 &amp;nbsp; 0  0  0  0}} || {{font|color=#adf|&amp;#8199;0}} ||{{Inversions8|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0}}  || {{Row8,0start}}{{Row8,1}}{{Row8,2}}{{Row8,3}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,4}}{{Row8,5}}{{Row8,6}}{{Row8,7end}} || [[File:Walsh permutation 124 Fano.svg|53px]] || align=&quot;left&quot;|{{sms|1 }}{{font|color=#c07|&amp;nbsp;&amp;#8199;0}}
|-   
|style=&quot;background:lightgray;{{text color default}};&quot;| 1 || 3 5 1  || {{BIN1start}}{{BIN1}}{{BIN1;}}&lt;br&gt;{{BIN1}}{{BIN0}}{{BIN0;}}&lt;br&gt;{{BIN0}}{{BIN1}}{{BIN0end}} || + || 5 || align=&quot;right&quot;|17576 || {{font|color=#aaa|0}}  7 1 6  &amp;nbsp;  2  {{font|color=#aaa|5}}  3 4 &lt;br&gt;{{font|color=#adf|0  0  1  1 &amp;nbsp; 2  2  3  3}} || {{font|color=#adf|12}} ||{{Inversions8|0|0|1|0|1|0|0|1|0|1|0|1|0|1|0|0|1|0|1|0|1|0|1|0|1|0|1|0}}  || {{Row8,0start}}{{Row8,7}}{{Row8,1}}{{Row8,6}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,2}}{{Row8,5}}{{Row8,3}}{{Row8,4end}} || || align=&quot;left&quot;|{{sms|3 }}{{font|color=#c07|&amp;nbsp;&amp;#8199;8}}&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;span style=&quot;font-size:60%&quot;&gt;(4+2)&lt;/span&gt;&gt;
|-   
|style=&quot;background:lightgray;{{text color default}};&quot;| 2 || 3 6 2  || {{BIN1start}}{{BIN0}}{{BIN0;}}&lt;br&gt;{{BIN1}}{{BIN1}}{{BIN1;}}&lt;br&gt;{{BIN0}}{{BIN1}}{{BIN0end}} || − || 5 || align=&quot;right&quot;|16854 || {{font|color=#aaa|0 1}}  7 6  &amp;nbsp;  2 3 5 4 &lt;br&gt;{{font|color=#adf|0  0  0  1 &amp;nbsp; 2  2  2  3}} || {{font|color=#adf|10}} ||{{Inversions8|0|0|0|0|0|1|0|0|1|1|0|0|1|1|0|0|0|1|1|0|0|0|0|1|1|0|0|1}}  || {{Row8,0start}}{{Row8,1}}{{Row8,7}}{{Row8,6}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,2}}{{Row8,3}}{{Row8,5}}{{Row8,4end}} || || align=&quot;left&quot;|{{sms|4 }}{{font|color=#c07|&amp;nbsp;&amp;#8199;9}}&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;span style=&quot;font-size:60%&quot;&gt;(3+3)&lt;/span&gt;&gt;
|-   
|3 || {{font|color=#aaa|1}}  7 3  || {{BIN1start}}{{BIN1}}{{BIN1;}}&lt;br&gt;{{BIN0}}{{BIN1}}{{BIN1;}}&lt;br&gt;{{BIN0}}{{BIN1}}{{BIN0end}} || − || 6 || align=&quot;right&quot;|22622 || {{font|color=#aaa|0}}  7 6 1  &amp;nbsp;  2  {{font|color=#aaa|5}}  4 3 &lt;br&gt;{{font|color=#adf|0  0  1  2 &amp;nbsp; 2  2  3  4}} || {{font|color=#adf|14}} ||{{Inversions8|0|0|1|0|1|1|0|1|1|0|0|1|1|0|0|0|1|1|0|0|1|0|1|1|0|0|1|1}}  || {{Row8,0start}}{{Row8,7}}{{Row8,6}}{{Row8,1}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,2}}{{Row8,5}}{{Row8,4}}{{Row8,3end}} || || align=&quot;left&quot;|{{sms|4 }}{{font|color=#c07|&amp;nbsp;&amp;#8199;9}}&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;span style=&quot;font-size:60%&quot;&gt;(3+3)&lt;/span&gt;&gt;
|-   
|style=&quot;background:lightgray;{{text color default}};&quot;| 4 || 5 6  {{font|color=#aaa|4}} || {{BIN1start}}{{BIN0}}{{BIN0;}}&lt;br&gt;{{BIN0}}{{BIN1}}{{BIN0;}}&lt;br&gt;{{BIN1}}{{BIN1}}{{BIN1end}} || + || 5 || align=&quot;right&quot;|16680 || {{font|color=#aaa|0 1 2 3}} &amp;nbsp;  7 6 5 4 &lt;br&gt;{{font|color=#adf|0  0  0  0 &amp;nbsp; 0  1  2  3}} || {{font|color=#adf|&amp;#8199;6}} ||{{Inversions8|0|0|0|0|0|0|0|0|0|0|0|0|0|0|1|0|0|0|0|1|1|0|0|0|0|1|1|1}}  || {{Row8,0start}}{{Row8,1}}{{Row8,2}}{{Row8,3}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,7}}{{Row8,6}}{{Row8,5}}{{Row8,4end}} || || align=&quot;left&quot;|{{sms|2 }}{{font|color=#c07|&amp;nbsp;&amp;#8199;3}}&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;span style=&quot;font-size:60%&quot;&gt;(2+2)&lt;/span&gt;&gt;
|-   
|5 || 7 1 5  || {{BIN1start}}{{BIN1}}{{BIN1;}}&lt;br&gt;{{BIN1}}{{BIN0}}{{BIN0;}}&lt;br&gt;{{BIN1}}{{BIN0}}{{BIN1end}} || − || 6 || align=&quot;right&quot;|22736 || {{font|color=#aaa|0}}  7 1 6  &amp;nbsp;  5 2 4 3 &lt;br&gt;{{font|color=#adf|0  0  1  1 &amp;nbsp; 2  3  3  4}} || {{font|color=#adf|14}} ||{{Inversions8|0|0|1|0|1|0|0|1|0|1|0|1|0|1|1|0|1|0|1|1|0|0|1|0|1|1|0|1}}  || {{Row8,0start}}{{Row8,7}}{{Row8,1}}{{Row8,6}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,5}}{{Row8,2}}{{Row8,4}}{{Row8,3end}} || || align=&quot;left&quot;|{{sms|6 }}{{font|color=#c07|&amp;nbsp;14}}&lt;sub&gt;b&lt;/sub&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;span style=&quot;font-size:60%&quot;&gt;(7)&lt;/span&gt;&gt;
|-   
|6 || 7  {{font|color=#aaa|2}}  6  || {{BIN1start}}{{BIN0}}{{BIN0;}}&lt;br&gt;{{BIN1}}{{BIN1}}{{BIN1;}}&lt;br&gt;{{BIN1}}{{BIN0}}{{BIN1end}} || + || 6 || align=&quot;right&quot;|23454 || {{font|color=#aaa|0 1}}  7 6  &amp;nbsp;  5 4 2 3 &lt;br&gt;{{font|color=#adf|0  0  0  1 &amp;nbsp; 2  3  4  4}} || {{font|color=#adf|14}} ||{{Inversions8|0|0|0|0|0|1|0|0|1|1|0|0|1|1|1|0|0|1|1|1|1|0|0|1|1|1|1|0}}  || {{Row8,0start}}{{Row8,1}}{{Row8,7}}{{Row8,6}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,5}}{{Row8,4}}{{Row8,2}}{{Row8,3end}} || [[File:Walsh permutation 165 Fano red.svg|53px]] || align=&quot;left&quot;|{{sms|3 }}{{font|color=#c07|&amp;nbsp;&amp;#8199;8}}&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;span style=&quot;font-size:60%&quot;&gt;(4+2)&lt;/span&gt;&gt;
|-   
|style=&quot;background:lightgray;{{text color default}};&quot;| 7 || 5 3 7  || {{BIN1start}}{{BIN1}}{{BIN1;}}&lt;br&gt;{{BIN0}}{{BIN1}}{{BIN1;}}&lt;br&gt;{{BIN1}}{{BIN0}}{{BIN1end}} || + || 7 || align=&quot;right&quot;|17702 || {{font|color=#aaa|0}}  7 6 1  &amp;nbsp;  5 2 3 4 &lt;br&gt;{{font|color=#adf|0  0  1  2 &amp;nbsp; 2  3  3  3}} || {{font|color=#adf|14}} ||{{Inversions8|0|0|1|0|1|1|0|1|1|0|0|1|1|0|1|0|1|1|0|1|0|0|1|1|0|1|0|0}}  || {{Row8,0start}}{{Row8,7}}{{Row8,6}}{{Row8,1}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,5}}{{Row8,2}}{{Row8,3}}{{Row8,4end}} || [[File:Walsh permutation 357 Fano red.svg|53px]] || align=&quot;left&quot;|{{sms|5 }}{{font|color=#c07|&amp;nbsp;14}}&lt;sub&gt;a&lt;/sub&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;span style=&quot;font-size:60%&quot;&gt;(7)&lt;/span&gt;&gt;
|}
{|
|Sorting by '''#''' gives the permutation (0,4,2,1,7,3,5,6) in the left column, which is ''reflection'' [[w:Function composition|∘]] [[w:Circular shift|''right shift'']] ∘ ''wp''(6,5,7) (with the compression matrix || {{Bin0start}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1end}} || ).
|}
Sorting by '''Permutation''' gives the (0,4,2,6, 1,5,3,7) in the left column, which is the bit-reversal permutation ''wp''(4,2,1).

=== 4-bit ===

[[File:Variadic logical XOR.svg|thumb|right|Binary Walsh matrix of order 16]]

{| class=&quot;wikitable sortable&quot; style=&quot;text-align:right;&quot;
|-
!
!#
!class=&quot;unsortable&quot;|IS
!CV
!class=&quot;unsortable&quot;|CM
!Σ
!Permutation&lt;br&gt;F
!IN
|-
|0
|0
|[[File:16 choose 2 array.svg|36px]]
|{{font|face=Courier|{{nowrap|'''( 1, 2, 4, 8)'''}}}}
|&lt;div style=&quot;white-space: nowrap&quot;&gt;{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin0}}{{Bin1end}}&lt;/div&gt;
|4
|{{font|face=Courier|{{nowrap|'''( 0, 1, 2, 3,&amp;nbsp; 4, 5, 6, 7,&amp;nbsp; 8, 9,10,11,&amp;nbsp;12,13,14,15)'''}}&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 0,&amp;nbsp; 0, 0, 0, 0,&amp;nbsp; 0, 0, 0, 0,&amp;nbsp; 0, 0, 0, 0)'''}}}}
|{{font|color=#adf|0}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|1
|9804424107176
|[[File:Inversion set 16; Walsh (15, 1, 2, 4).svg|36px]]
|{{font|face=Courier|'''( 3, 5, 9, 1)'''}}
|{{Bin1start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0end}}
|7
|{{font|face=Courier|'''( 0,15, 1,14,&amp;nbsp; 2,13, 3,12,&amp;nbsp; 4,11, 5,10,&amp;nbsp; 6, 9, 7, 8)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 1,&amp;nbsp; 2, 2, 3, 3,&amp;nbsp; 4, 4, 5, 5,&amp;nbsp; 6, 6, 7, 7)'''}}}}
|{{font|color=#adf|56}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|2
|9717242186454
|[[File:Inversion set 16; Walsh (1, 15, 2, 4).svg|36px]]
|{{font|face=Courier|'''( 3, 6,10, 2)'''}}
|{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0end}}
|7
|{{font|face=Courier|'''( 0, 1,15,14,&amp;nbsp; 2, 3,13,12,&amp;nbsp; 4, 5,11,10,&amp;nbsp; 6, 7, 9, 8)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 1,&amp;nbsp; 2, 2, 2, 3,&amp;nbsp; 4, 4, 4, 5,&amp;nbsp; 6, 6, 6, 7)'''}}}}
|{{font|color=#adf|52}}
|-
|3
|11112138397022
|[[File:Inversion set 16; wp( 1, 7,11, 3).svg|36px]]
|{{font|face=Courier|'''( 1, 7,11, 3)'''}}
|{{Bin1start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0end}}
|9
|{{font|face=Courier|'''( 0,15,14, 1,&amp;nbsp; 2,13,12, 3,&amp;nbsp; 4,11,10, 5,&amp;nbsp; 6, 9, 8, 7)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 2,&amp;nbsp; 2, 2, 3, 4,&amp;nbsp; 4, 4, 5, 6,&amp;nbsp; 6, 6, 7, 8)'''}}}}
|{{font|color=#adf|60}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|4
|9710017245480
|[[File:Inversion set 16; wp( 5, 6,12, 4).svg|36px]]
|{{font|face=Courier|'''( 5, 6,12, 4)'''}}
|{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0end}}
|7
|{{font|face=Courier|'''( 0, 1, 2, 3,&amp;nbsp;15,14,13,12,&amp;nbsp; 4, 5, 6, 7,&amp;nbsp;11,10, 9, 8)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 0,&amp;nbsp; 0, 1, 2, 3,&amp;nbsp; 4, 4, 4, 4,&amp;nbsp; 4, 5, 6, 7)'''}}}}
|{{font|color=#adf|44}}
|-
|5
|11118325501136
|[[File:Inversion set 16; Walsh (15, 1, 13, 4).svg|36px]]
|{{font|face=Courier|'''( 7, 1,13, 5)'''}}
|{{Bin1start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0end}}
|9
|{{font|face=Courier|'''( 0,15, 1,14,&amp;nbsp;13, 2,12, 3,&amp;nbsp; 4,11, 5,10,&amp;nbsp; 9, 6, 8, 7)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 1,&amp;nbsp; 2, 3, 3, 4,&amp;nbsp; 4, 4, 5, 5,&amp;nbsp; 6, 7, 7, 8)'''}}}}
|{{font|color=#adf|60}}
|-
|6
|11205500164254
|[[File:Inversion set 16; Walsh (1, 15, 13, 4).svg|36px]]
|{{font|face=Courier|'''( 7, 2,14, 6)'''}}
|{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0end}}
|9
|{{font|face=Courier|'''( 0, 1,15,14,&amp;nbsp;13,12, 2, 3,&amp;nbsp; 4, 5,11,10,&amp;nbsp; 9, 8, 6, 7)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 1,&amp;nbsp; 2, 3, 4, 4,&amp;nbsp; 4, 4, 4, 5,&amp;nbsp; 6, 7, 8, 8)'''}}}}
|{{font|color=#adf|60}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|7
|9810691044902
|[[File:Inversion set 16; Walsh (15, 14, 13, 4).svg|36px]]
|{{font|face=Courier|'''( 5, 3,15, 7)'''}}
|{{Bin1start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0end}}
|11
|{{font|face=Courier|'''( 0,15,14, 1,&amp;nbsp;13, 2, 3,12,&amp;nbsp; 4,11,10, 5,&amp;nbsp; 9, 6, 7, 8)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 2,&amp;nbsp; 2, 3, 3, 3,&amp;nbsp; 4, 4, 5, 6,&amp;nbsp; 6, 7, 7, 7)'''}}}}
|{{font|color=#adf|60}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|8
|9709968804480
|[[File:Inversion set 16; Walsh (1, 2, 4, 15).svg|36px]]
|{{font|face=Courier|'''( 9,10,12, 8)'''}}
|{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin1end}}
|7
|{{font|face=Courier|'''( 0, 1, 2, 3,&amp;nbsp; 4, 5, 6, 7,&amp;nbsp;15,14,13,12,&amp;nbsp;11,10, 9, 8)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 0,&amp;nbsp; 0, 0, 0, 0,&amp;nbsp;  0, 1, 2, 3,&amp;nbsp; 4, 5, 6, 7)'''}}}}
|{{font|color=#adf|28}}
|-
|9
|11118365775656
|[[File:Inversion set 16; Walsh (15, 1, 2, 11).svg|36px]]
|{{font|face=Courier|'''(11,13, 1, 9)'''}}
|{{Bin1start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin0}}{{Bin1end}}
|9
|{{font|face=Courier|'''( 0,15, 1,14,&amp;nbsp; 2,13, 3,12,&amp;nbsp;11, 4,10, 5,&amp;nbsp; 9, 6, 8, 7)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 1,&amp;nbsp; 2, 2, 3, 3,&amp;nbsp; 4, 5, 5, 6,&amp;nbsp; 6, 7, 7, 8)'''}}}}
|{{font|color=#adf|60}}
|-
|10
|11205547694934
|[[File:Inversion set 16; Walsh (1, 15, 2, 11).svg|36px]]
|{{font|face=Courier|'''(11,14, 2,10)'''}}
|{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin0}}{{Bin1end}}
|9
|{{font|face=Courier|'''( 0, 1,15,14,&amp;nbsp; 2, 3,13,12,&amp;nbsp;11,10, 4, 5,&amp;nbsp; 9, 8, 6, 7)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 1,&amp;nbsp; 2, 2, 2, 3,&amp;nbsp; 4, 5, 6, 6,&amp;nbsp; 6, 7, 8, 8)'''}}}}
|{{font|color=#adf|60}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|11
|9810651495902
|[[File:Inversion set 16; Walsh (15, 14, 2, 11).svg|36px]]
|{{font|face=Courier|'''( 9,15, 3,11)'''}}
|{{Bin1start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin0}}{{Bin1end}}
|11
|{{font|face=Courier|'''( 0,15,14, 1,&amp;nbsp; 2,13,12, 3,&amp;nbsp;11, 4, 5,10,&amp;nbsp; 9, 6, 7, 8)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 2,&amp;nbsp; 2, 2, 3, 4,&amp;nbsp; 4, 5, 5, 5,&amp;nbsp; 6, 7, 7, 7)'''}}}}
|{{font|color=#adf|60}}
|-
|12
|11212772635560
|[[File:Inversion set 16; Walsh (1, 2, 15, 11).svg|36px]]
|{{font|face=Courier|'''(13,14, 4,12)'''}}
|{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin0}}{{Bin1end}}
|9
|{{font|face=Courier|'''( 0, 1, 2, 3,&amp;nbsp;15,14,13,12,&amp;nbsp;11,10, 9, 8,&amp;nbsp; 4, 5, 6, 7)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 0,&amp;nbsp;  0, 1, 2, 3,&amp;nbsp; 4, 5, 6, 7,&amp;nbsp; 8, 8, 8, 8)'''}}}}
|{{font|color=#adf|60}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|13
|9804464392016
|[[File:Inversion set 16; Walsh (15, 1, 13, 11).svg|36px]]
|{{font|face=Courier|'''(15, 9, 5,13)'''}}
|{{Bin1start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin0}}{{Bin1end}}
|11
|{{font|face=Courier|'''( 0,15, 1,14,&amp;nbsp;13, 2,12, 3,&amp;nbsp;11, 4,10, 5,&amp;nbsp; 6, 9, 7, 8)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 1,&amp;nbsp; 2, 3, 3, 4,&amp;nbsp; 4, 5, 5, 6,&amp;nbsp; 6, 6, 7, 7)'''}}}}
|{{font|color=#adf|60}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|14
|9717289730334
|[[File:Inversion set 16; Walsh (1, 15, 13, 11).svg|36px]]
|{{font|face=Courier|'''(15,10, 6,14)'''}}
|{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin0}}{{Bin1end}}
|11
|{{font|face=Courier|'''( 0, 1,15,14,&amp;nbsp;13,12, 2, 3,&amp;nbsp;11,10, 4, 5,&amp;nbsp; 6, 7, 9, 8)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 1,&amp;nbsp; 2, 3, 4, 4,&amp;nbsp; 4, 5, 6, 6,&amp;nbsp; 6, 6, 6, 7)'''}}}}
|{{font|color=#adf|60}}
|-
|15
|11112098838182
|[[File:Inversion set 16; Walsh (15, 14, 13, 11).svg|36px]]
|{{font|face=Courier|'''(13,11, 7,15)'''}}
|{{Bin1start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin0}}{{Bin1end}}
|13
|{{font|face=Courier|'''( 0,15,14, 1,&amp;nbsp;13, 2, 3,12,&amp;nbsp;11, 4, 5,10,&amp;nbsp; 6, 9, 8, 7)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 2,&amp;nbsp; 2, 3, 3, 3,&amp;nbsp; 4, 5, 5, 5,&amp;nbsp; 6, 6, 7, 8)'''}}}}
|{{font|color=#adf|60}}
|}

{|
|Sorting by '''#''' gives the (0, 8,4,2,14, 1,13,11,7, 15,3,5,9, 6,10,12) in the left column, which is ''reflection'' [[w:Function composition|∘]] [[w:Circular shift|''right shift'']] ∘ ''wp''(12,10, 9, 7) (with the compression matrix || {{Bin0start}}{{Bin0}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin0}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin0end}} || ).
|}
Sorting by '''Permutation''' gives the (0,8,4,12, 2,10,6,14, 1,9,5,13, 3,11,7,15) in the left column, which is the [[w:Bit-reversal permutation|bit-reversal permutation]] ''wp''( 8, 4, 2, 1).

{| class=&quot;collapsible collapsed&quot; style=&quot;width: 100%; border: 1px solid #cccccc; background:white;{{text color default}};&quot;
! bgcolor=&quot;#f0f0f0&quot;|{{font|color=#999999|Matrices}}
|-
|
 Compression vectors:
     1     2     4     8
     3     5     9     1
     3     6    10     2
     1     7    11     3
     5     6    12     4
     7     1    13     5
     7     2    14     6
     5     3    15     7
     9    10    12     8
    11    13     1     9
    11    14     2    10
     9    15     3    11
    13    14     4    12
    15     9     5    13
    15    10     6    14
    13    11     7    15
 &amp;nbsp;
 Permutations:
     0     1     2     3     4     5     6     7     8     9    10    11    12    13    14    15
     0    15     1    14     2    13     3    12     4    11     5    10     6     9     7     8
     0     1    15    14     2     3    13    12     4     5    11    10     6     7     9     8
     0    15    14     1     2    13    12     3     4    11    10     5     6     9     8     7
     0     1     2     3    15    14    13    12     4     5     6     7    11    10     9     8
     0    15     1    14    13     2    12     3     4    11     5    10     9     6     8     7
     0     1    15    14    13    12     2     3     4     5    11    10     9     8     6     7
     0    15    14     1    13     2     3    12     4    11    10     5     9     6     7     8
     0     1     2     3     4     5     6     7    15    14    13    12    11    10     9     8
     0    15     1    14     2    13     3    12    11     4    10     5     9     6     8     7
     0     1    15    14     2     3    13    12    11    10     4     5     9     8     6     7
     0    15    14     1     2    13    12     3    11     4     5    10     9     6     7     8
     0     1     2     3    15    14    13    12    11    10     9     8     4     5     6     7
     0    15     1    14    13     2    12     3    11     4    10     5     6     9     7     8
     0     1    15    14    13    12     2     3    11    10     4     5     6     7     9     8
     0    15    14     1    13     2     3    12    11     4     5    10     6     9     8     7
 &amp;nbsp;
 Little-endian factorial numbers:
     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0
     0     0     1     1     2     2     3     3     4     4     5     5     6     6     7     7
     0     0     0     1     2     2     2     3     4     4     4     5     6     6     6     7
     0     0     1     2     2     2     3     4     4     4     5     6     6     6     7     8
     0     0     0     0     0     1     2     3     4     4     4     4     4     5     6     7
     0     0     1     1     2     3     3     4     4     4     5     5     6     7     7     8
     0     0     0     1     2     3     4     4     4     4     4     5     6     7     8     8
     0     0     1     2     2     3     3     3     4     4     5     6     6     7     7     7
     0     0     0     0     0     0     0     0     0     1     2     3     4     5     6     7
     0     0     1     1     2     2     3     3     4     5     5     6     6     7     7     8
     0     0     0     1     2     2     2     3     4     5     6     6     6     7     8     8
     0     0     1     2     2     2     3     4     4     5     5     5     6     7     7     7
     0     0     0     0     0     1     2     3     4     5     6     7     8     8     8     8
     0     0     1     1     2     3     3     4     4     5     5     6     6     6     7     7
     0     0     0     1     2     3     4     4     4     5     6     6     6     6     6     7
     0     0     1     2     2     3     3     3     4     5     5     5     6     6     7     8
|}

=== Compression matrices ===

[[File:Compression matrices of Walsh permutations with striped inversion sets.svg|left|800px]]
&lt;br style=&quot;clear: both;&quot;&gt;
The CM of the ''n''-th permutation is a matrix ''B'' with vertical stripes for the binary digits of ''n'' &lt;small&gt;(gray lines in the image)&lt;/small&gt;, XORed with a modified unity matrix ''U'' &lt;small&gt;(black dots in the image)&lt;/small&gt;.&lt;br&gt;
''U'' is modified by including an column row where ''n'' has its lowest binary digit ({{oeis|A001511}}). It pushes the other rows of the unity matrix to the right, and the last column &quot;out of the matrix&quot;.


==Bended stripes==

Among the ''n''-bit Walsh permutations are 2&lt;sup&gt;''n-1''&lt;/sup&gt; whose inversion sets show bended stripes.&lt;br&gt;
They correspond to Walsh functions, i.e. the rows of the [[w:Walsh matrix|Walsh matrix]] of order 2&lt;sup&gt;''n-1''&lt;/sup&gt;.&lt;br&gt;
These permutations under composition form the group Z&lt;sub&gt;2&lt;/sub&gt;&lt;sup&gt;''n''-1&lt;/sup&gt;, just like the row numbers under bitwise XOR.

=== 3-bit ===

{| class=&quot;wikitable sortable&quot; style=&quot;height:275px; text-align:center;&quot;
|-style=&quot;height:55px&quot;
!
!style=&quot;width:55px&quot;|CV
!style=&quot;width:55px&quot; class=&quot;unsortable&quot;|&amp;nbsp;&amp;nbsp;CM&amp;nbsp;&amp;nbsp;
!D
!#
!Permutation&lt;br&gt;F
!IN
!class=&quot;unsortable&quot;|IS
!style=&quot;width:55px&quot; class=&quot;unsortable&quot;|PM
!class=&quot;unsortable&quot;|FPC
!Cy
|-   
|0 || {{font|color=#aaa|1 2 4}} || {{BIN1start}}{{BIN0}}{{BIN0;}}&lt;br&gt;{{BIN0}}{{BIN1}}{{BIN0;}}&lt;br&gt;{{BIN0}}{{BIN0}}{{BIN1end}} || + || align=&quot;right&quot;|  0 || {{font|color=#aaa|0 1 2 3 &amp;nbsp; 4 5 6 7}}&lt;br&gt;{{font|color=#adf|0  0  0  0 &amp;nbsp; 0  0  0  0}} || {{font|color=#adf|&amp;#8199;0}} ||{{Inversions8|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0}}  || {{Row8,0start}}{{Row8,1}}{{Row8,2}}{{Row8,3}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,4}}{{Row8,5}}{{Row8,6}}{{Row8,7end}} || [[File:Walsh permutation 124 Fano.svg|53px]] || align=&quot;left&quot;|{{sms|1 }}{{font|color=#c07|&amp;nbsp;&amp;#8199;0}}
|-   
|style=&quot;background:lightgray;{{text color default}};&quot;|1 ||  4 7 1  || {{BIN0start}}{{BIN1}}{{BIN1;}}&lt;br&gt;{{BIN0}}{{BIN1}}{{BIN0;}}&lt;br&gt;{{BIN1}}{{BIN1}}{{BIN0end}} || − || align=&quot;right&quot;| 3776 || {{font|color=#aaa|0}}  6  {{font|color=#aaa|2}}  4  &amp;nbsp;  3  {{font|color=#aaa|5}}  1  {{font|color=#aaa|7}}&lt;br&gt;{{font|color=#adf|0  0  1  1 &amp;nbsp; 2  1  5  0}} || {{font|color=#adf|10}} ||{{Inversions8|0|0|1|0|1|0|0|1|0|1|0|1|0|0|0|0|1|1|1|1|1|0|0|0|0|0|0|0}}  || {{Row8,0start}}{{Row8,6}}{{Row8,2}}{{Row8,4}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,3}}{{Row8,5}}{{Row8,1}}{{Row8,7end}} || || align=&quot;left&quot;|{{sms|2 }}{{font|color=#c07|&amp;nbsp;&amp;#8199;3}}&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;span style=&quot;font-size:60%&quot;&gt;(2+2)&lt;/span&gt;&gt;
|-   
|style=&quot;background:lightgray;{{text color default}};&quot;|2 ||  7 4 2  || {{BIN1start}}{{BIN0}}{{BIN0;}}&lt;br&gt;{{BIN1}}{{BIN0}}{{BIN1;}}&lt;br&gt;{{BIN1}}{{BIN1}}{{BIN0end}} || − || align=&quot;right&quot;|  414 || {{font|color=#aaa|0 1}}  5 4  &amp;nbsp;  3 2  {{font|color=#aaa|6 7}}&lt;br&gt;{{font|color=#adf|0  0  0  1 &amp;nbsp; 2  3  0  0}} || {{font|color=#adf|&amp;#8199;6}} ||{{Inversions8|0|0|0|0|0|1|0|0|1|1|0|0|1|1|1|0|0|0|0|0|0|0|0|0|0|0|0|0}}  || {{Row8,0start}}{{Row8,1}}{{Row8,5}}{{Row8,4}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,3}}{{Row8,2}}{{Row8,6}}{{Row8,7end}} || || align=&quot;left&quot;|{{sms|2 }}{{font|color=#c07|&amp;nbsp;&amp;#8199;3}}&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;span style=&quot;font-size:60%&quot;&gt;(2+2)&lt;/span&gt;&gt;
|-   
|3 ||  2 1 7  || {{BIN0start}}{{BIN1}}{{BIN1;}}&lt;br&gt;{{BIN1}}{{BIN0}}{{BIN1;}}&lt;br&gt;{{BIN0}}{{BIN0}}{{BIN1end}} || − || align=&quot;right&quot;| 4142 || {{font|color=#aaa|0}}  6 5  {{font|color=#aaa|3 &amp;nbsp; 4}}  2 1  {{font|color=#aaa|7}}&lt;br&gt;{{font|color=#adf|0  0  1  2 &amp;nbsp; 2  4  5  0}} || {{font|color=#adf|14}} ||{{Inversions8|0|0|1|0|1|1|0|1|1|0|0|1|1|1|1|0|1|1|1|1|1|0|0|0|0|0|0|0}}  || {{Row8,0start}}{{Row8,6}}{{Row8,5}}{{Row8,3}}[[File:1x1.png|alt=]]&lt;br&gt;{{Row8,4}}{{Row8,2}}{{Row8,1}}{{Row8,7end}} || || align=&quot;left&quot;|{{sms|2 }}{{font|color=#c07|&amp;nbsp;&amp;#8199;3}}&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;span style=&quot;font-size:60%&quot;&gt;(2+2)&lt;/span&gt;&gt;
|}

Sorting by '''#''' or '''Permutation''' gives (0,2,1,3) in the left column, which is the bit-reversal permutation ''wp''(2,1).&lt;br&gt;
Sorting by '''CV''' gives (0,3,1,2) in the left column, which is ''wp''(3,1).


=== 4-bit ===

{| class=&quot;wikitable sortable&quot; style=&quot;text-align:right;&quot;
|-
!
!#
!class=&quot;unsortable&quot;|IS
!CV
!class=&quot;unsortable&quot;|CM
!Permutation&lt;br&gt;F
!IN
|-
|0
|0
|[[File:16 choose 2 array.svg|36px]]
|{{font|face=Courier|{{nowrap|'''( 1, 2, 4, 8)'''}}}}
|&lt;div style=&quot;white-space: nowrap&quot;&gt;{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin0}}{{Bin1end}}&lt;/div&gt;
|{{font|face=Courier|{{nowrap|'''( 0, 1, 2, 3,&amp;nbsp; 4, 5, 6, 7,&amp;nbsp; 8, 9,10,11,&amp;nbsp;12,13,14,15)'''}}&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 0,&amp;nbsp; 0, 0, 0, 0,&amp;nbsp; 0, 0, 0, 0,&amp;nbsp; 0, 0, 0, 0)'''}}}}
|{{font|color=#adf|0}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|1
|1144441325096
|[[File:Inversion set 16; Walsh (14, 2, 4, 7).svg|36px]]
|{{font|face=Courier|'''( 8,11,13, 1)'''}}
|{{Bin0start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin0end}}
|{{font|face=Courier|'''( 0,14, 2,12,&amp;nbsp; 4,10, 6, 8,&amp;nbsp; 7, 9, 5,11,&amp;nbsp; 3,13, 1,15)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 1,&amp;nbsp; 2, 2, 3, 3,&amp;nbsp; 4, 3, 7, 2,&amp;nbsp;10, 1,13, 0)'''}}}}
|{{font|color=#adf|52}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|2
|73376328534
|[[File:Inversion set 16; Walsh (1, 13, 4, 7).svg|36px]]
|{{font|face=Courier|'''(11, 8,14, 2)'''}}
|{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin0end}}
|{{font|face=Courier|'''( 0, 1,13,12,&amp;nbsp; 4, 5, 9, 8,&amp;nbsp; 7, 6,10,11,&amp;nbsp; 3, 2,14,15)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 1,&amp;nbsp; 2, 2, 2, 3,&amp;nbsp; 4, 5, 2, 2,&amp;nbsp;10,11, 0, 0)'''}}}}
|{{font|color=#adf|44}}
|-
|3
|1209107634782
|[[File:Inversion set 16; Walsh (14, 13, 4, 8).svg|36px]]
|{{font|face=Courier|'''( 2, 1, 7,11)'''}}
|{{Bin0start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin1}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin0}}{{Bin1end}}
|{{font|face=Courier|'''( 0,14,13, 3,&amp;nbsp; 4,10, 9, 7,&amp;nbsp; 8, 6, 5,11,&amp;nbsp;12, 2, 1,15)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 2,&amp;nbsp; 2, 2, 3, 4,&amp;nbsp; 4, 6, 7, 2,&amp;nbsp; 2,12,13, 0)'''}}}}
|{{font|color=#adf|60}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|4
|303182760
|[[File:Inversion set 16; Walsh (1, 2, 11, 7).svg|36px]]
|{{font|face=Courier|'''(13,14, 8, 4)'''}}
|{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin0}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin0end}}
|{{font|face=Courier|'''( 0, 1, 2, 3,&amp;nbsp;11,10, 9, 8,&amp;nbsp; 7, 6, 5, 4,&amp;nbsp;12,13,14,15)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 0,&amp;nbsp; 0, 1, 2, 3,&amp;nbsp; 4, 5, 6, 7,&amp;nbsp; 0, 0, 0, 0)'''}}}}
|{{font|color=#adf|28}}
|-
|5
|1144707321296
|[[File:Inversion set 16; Walsh (14, 2, 11, 8).svg|36px]]
|{{font|face=Courier|'''( 4, 7, 1,13)'''}}
|{{Bin0start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin1}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin0}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin0}}{{Bin1end}}
|{{font|face=Courier|'''( 0,14, 2,12,&amp;nbsp;11, 5, 9, 7,&amp;nbsp; 8, 6,10, 4,&amp;nbsp; 3,13, 1,15)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 1,&amp;nbsp; 2, 3, 3, 4,&amp;nbsp; 4, 6, 3, 9,&amp;nbsp;10, 1,13, 0)'''}}}}
|{{font|color=#adf|60}}
|-
|6
|73677162654
|[[File:Inversion set 16; Walsh (1, 13, 11, 8).svg|36px]]
|{{font|face=Courier|'''( 7, 4, 2,14)'''}}
|{{Bin1start}}{{Bin0}}{{Bin0}}{{Bin0;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin0}}{{Bin1;}}&lt;br&gt;{{Bin0}}{{Bin0}}{{Bin0}}{{Bin1end}}
|{{font|face=Courier|'''( 0, 1,13,12,&amp;nbsp;11,10, 6, 7,&amp;nbsp; 8, 9, 5, 4,&amp;nbsp; 3, 2,14,15)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 0, 1,&amp;nbsp; 2, 3, 4, 4,&amp;nbsp; 4, 4, 8, 9,&amp;nbsp;10,11, 0, 0)'''}}}}
|{{font|color=#adf|60}}
|-
|style=&quot;background:lightgray;{{text color default}};&quot;|7
|1209371443622
|[[File:Inversion set 16; Walsh (14, 13, 11, 7).svg|36px]]
|{{font|face=Courier|'''(14,13,11, 7)'''}}
|{{Bin0start}}{{Bin1}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin0}}{{Bin1}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin0}}{{Bin1;}}&lt;br&gt;{{Bin1}}{{Bin1}}{{Bin1}}{{Bin0end}}
|{{font|face=Courier|'''( 0,14,13, 3,&amp;nbsp;11, 5, 6, 8,&amp;nbsp; 7, 9,10, 4,&amp;nbsp;12, 2, 1,15)'''&lt;br&gt;{{font|color=#adf|'''( 0, 0, 1, 2,&amp;nbsp; 2, 3, 3, 3,&amp;nbsp; 4, 3, 3, 9,&amp;nbsp; 2,12,13, 0)'''}}}}
|{{font|color=#adf|60}}
|}

Sorting by '''#''' or '''Permutation''' gives (0,4,2,6, 1,5,3,7) in the left column, which is the bit-reversal permutation ''wp''(4,2,1).&lt;br&gt;
Sorting by '''CV''' gives (0,3,5,6, 1,2,4,7) in the left column, which is ''wp''(7,1,2).

===Compression matrices===

[[File:Compression matrices of Walsh permutations with bendedly striped inversion sets.svg|800px]]
&lt;br style=&quot;clear: both;&quot;&gt;
The CM are matrices with an even number of vertical stripes &lt;small&gt;(gray lines in the image)&lt;/small&gt; XORed with the unity matrix &lt;small&gt;(black dots in the image)&lt;/small&gt;.&lt;br&gt;
The order by the ''evil numbers'' ({{oeis|A001969}}) of the matrices, used in the image, corresponds to the lexicographical order of the compression vectors.

==Bit permutations==
{| align=&quot;right&quot;
|colspan=&quot;2&quot;|
{| align=right style=&quot;text-align=right; border:1px solid lightgray&quot; cellspacing=&quot;10&quot;
|[[Walsh permutation]]; [[Walsh permutation; bit permutation|bit permutation]] {{Rdrup}}
|}
|-
|style=&quot;vertical-align:top;&quot;|[[File:Inversion set 16; Walsh (2, 8, 1, 4).svg|thumb|Inversion set of ''wp''( 4, 1, 8, 2)&lt;br&gt;All columns are symmetrical.]]
|style=&quot;vertical-align:top;&quot;|[[File:Inversion set 16; Walsh (14, 13, 11, 7).svg|thumb|Inversion set of ''wp''(14,13,11, 7)]]
|-
|colspan=&quot;2&quot;|
{| align=right
|-
|style=&quot;vertical-align:top;&quot;|[[File:Symmetric group 4; cycle graph; details.svg|thumb|center|500px|[[w:Cycle graph (algebra)|Cycle graph]] of the bit permutations]]
|style=&quot;vertical-align:top;&quot;|[[File:Symmetric group 4; permutation list (0-based).svg|thumb|center|Permutations of 4 elements&lt;br&gt;{{Rdr}} [[Symmetric group S4|Symmetric group S&lt;sub&gt;4&lt;/sub&gt;]]]]
|}
|}


{| class=&quot;wikitable sortable&quot; style=&quot;text-align:right;&quot;
|-
!
!#
!class=&quot;unsortable&quot;|IS
!CV
|-
|style=&quot;background:white;{{text color default}};&quot;|&lt;big&gt;'''0'''&lt;/big&gt;
|0
|[[File:16 choose 2 array.svg|36px]]
|{{font|face=Courier|'''( 1, 2, 4, 8)'''}}
|-
|style=&quot;background:#33eb2a;{{text color default}};&quot;|1
|87181920722
|[[File:Inversion set 16; Walsh (2, 1, 4, 8).svg|36px]]
|{{font|face=Courier|'''( 2, 1, 4, 8)'''}}
|-
|style=&quot;background:#33eb2a;{{text color default}};&quot;|2
|13412045088
|[[File:Inversion set 16; wp( 1, 4, 2, 8).svg|36px]]
|{{font|face=Courier|'''( 1, 4, 2, 8)'''}}
|-
|style=&quot;background:white;{{text color default}};&quot;|3
|101069338632
|[[File:Inversion set 16; Walsh (2, 4, 1, 8).svg|36px]]
|{{font|face=Courier|'''( 4, 1, 2, 8)'''}}
|-
|style=&quot;background:white;{{text color default}};&quot;|4
|262496507810
|[[File:Inversion set 16; Walsh (4, 1, 2, 8).svg|36px]]
|{{font|face=Courier|'''( 2, 4, 1, 8)'''}}
|-
|style=&quot;background:#33eb2a;{{text color default}};&quot;|5
|269202530354
|[[File:Inversion set 16; Walsh (4, 2, 1, 8).svg|36px]]
|{{font|face=Courier|'''( 4, 2, 1, 8)'''}}
|-
|style=&quot;background:#33eb2a;{{text color default}};&quot;|6
|175795200
|[[File:Inversion set 16; Walsh (1, 2, 8, 4).svg|36px]]
|{{font|face=Courier|'''( 1, 2, 8, 4)'''}}
|-
|style=&quot;background:white;{{text color default}};&quot;|&lt;big&gt;'''7'''&lt;/big&gt;
|87357715922
|[[File:Inversion set 16; Walsh (2, 1, 8, 4).svg|36px]]
|{{font|face=Courier|'''( 2, 1, 8, 4)'''}}
|-
|style=&quot;background:white;{{text color default}};&quot;|8
|13588646400
|[[File:Inversion set 16; Walsh (1, 4, 8, 2).svg|36px]]
|{{font|face=Courier|'''( 1, 8, 2, 4)'''}}
|-
|style=&quot;background:#ffbb00;{{text color default}};&quot;|9
|101249608320
|[[File:Inversion set 16; Walsh (2, 4, 8, 1).svg|36px]]
|{{font|face=Courier|'''( 8, 1, 2, 4)'''}}
|-
|style=&quot;background:#ffbb00;{{text color default}};&quot;|10
|262679639762
|[[File:Inversion set 16; Walsh (4, 1, 8, 2).svg|36px]]
|{{font|face=Courier|'''( 2, 8, 1, 4)'''}}
|-
|style=&quot;background:white;{{text color default}};&quot;|11
|269386065362
|[[File:Inversion set 16; Walsh (4, 2, 8, 1).svg|36px]]
|{{font|face=Courier|'''( 8, 2, 1, 4)'''}}
|-
|style=&quot;background:white;{{text color default}};&quot;|12
|40237747488
|[[File:Inversion set 16; Walsh (1, 8, 2, 4).svg|36px]]
|{{font|face=Courier|'''( 1, 4, 8, 2)'''}}
|-
|style=&quot;background:#ffbb00;{{text color default}};&quot;|13
|127895041032
|[[File:Inversion set 16; Walsh (2, 8, 1, 4).svg|36px]]
|{{font|face=Courier|'''( 4, 1, 8, 2)'''}}
|-
|style=&quot;background:#33eb2a;{{text color default}};&quot;|14
|40325645088
|[[File:Inversion set 16; wp( 1, 8, 4, 2).svg|36px]]
|{{font|face=Courier|'''( 1, 8, 4, 2)'''}}
|-
|style=&quot;background:white;{{text color default}};&quot;|15
|127986607008
|[[File:Inversion set 16; Walsh (2, 8, 4, 1).svg|36px]]
|{{font|face=Courier|'''( 8, 1, 4, 2)'''}}
|-
|style=&quot;background:white;{{text color default}};&quot;|&lt;big&gt;'''16'''&lt;/big&gt;
|303216964872
|[[File:Inversion set 16; wp( 4, 8, 1, 2).svg|36px]]
|{{font|face=Courier|'''( 4, 8, 1, 2)'''}}
|-
|style=&quot;background:#ffbb00;{{text color default}};&quot;|17
|303260913672
|[[File:Inversion set 16; Walsh (4, 8, 2, 1).svg|36px]]
|{{font|face=Courier|'''( 8, 4, 1, 2)'''}}
|-
|style=&quot;background:#ffbb00;{{text color default}};&quot;|18
|613140355490
|[[File:Inversion set 16; Walsh (8, 1, 2, 4).svg|36px]]
|{{font|face=Courier|'''( 2, 4, 8, 1)'''}}
|-
|style=&quot;background:white;{{text color default}};&quot;|19
|619846378034
|[[File:Inversion set 16; Walsh (8, 2, 1, 4).svg|36px]]
|{{font|face=Courier|'''( 4, 2, 8, 1)'''}}
|-
|style=&quot;background:white;{{text color default}};&quot;|20
|613228253090
|[[File:Inversion set 16; Walsh (8, 1, 4, 2).svg|36px]]
|{{font|face=Courier|'''( 2, 8, 4, 1)'''}}
|-
|style=&quot;background:#33eb2a;{{text color default}};&quot;|21
|619934678690
|[[File:Inversion set 16; Walsh (8, 2, 4, 1).svg|36px]]
|{{font|face=Courier|'''( 8, 2, 4, 1)'''}}
|-
|style=&quot;background:#ffbb00;{{text color default}};&quot;|22
|633259229234
|[[File:Inversion set 16; wp( 4, 8, 2, 1).svg|36px]]
|{{font|face=Courier|'''( 4, 8, 2, 1)'''}}
|-
|style=&quot;background:white;{{text color default}};&quot;|&lt;big&gt;'''23'''&lt;/big&gt;
|633303178034
|[[File:Inversion set 16; Walsh (8, 4, 2, 1).svg|36px]]
|{{font|face=Courier|'''( 8, 4, 2, 1)'''}}
|}

{| class=&quot;collapsible collapsed&quot; style=&quot;width: 100%; border: 1px solid #cccccc; background:white;{{text color default}};&quot;
! bgcolor=&quot;#f0f0f0&quot;|{{font|color=#999999|F}}
|-
|
 Little-endian factorial numbers:
     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0
     0     0     1     0     0     0     1     0     0     0     1     0     0     0     1     0
     0     0     0     0     2     2     0     0     0     0     0     0     2     2     0     0
     0     0     0     0     3     2     1     0     0     0     0     0     3     2     1     0
     0     0     1     0     2     0     3     0     0     0     1     0     2     0     3     0
     0     0     1     0     3     1     3     0     0     0     1     0     3     1     3     0
     0     0     0     0     0     0     0     0     4     4     4     4     0     0     0     0
     0     0     1     0     0     0     1     0     4     4     5     4     0     0     1     0
     0     0     0     0     0     0     0     0     6     6     4     4     2     2     0     0
     0     0     0     0     0     0     0     0     7     6     5     4     3     2     1     0
     0     0     1     0     0     0     1     0     6     4     7     4     2     0     3     0
     0     0     1     0     0     0     1     0     7     5     7     4     3     1     3     0
     0     0     0     0     2     2     0     0     4     4     0     0     6     6     0     0
     0     0     0     0     3     2     1     0     4     4     0     0     7     6     1     0
     0     0     0     0     2     2     0     0     6     6     2     2     6     6     0     0
     0     0     0     0     2     2     0     0     7     6     3     2     7     6     1     0
     0     0     0     0     3     2     1     0     6     4     2     0     9     6     3     0
     0     0     0     0     3     2     1     0     7     5     3     1     9     6     3     0
     0     0     1     0     2     0     3     0     4     0     5     0     6     0     7     0
     0     0     1     0     3     1     3     0     4     0     5     0     7     1     7     0
     0     0     1     0     2     0     3     0     6     2     7     2     6     0     7     0
     0     0     1     0     2     0     3     0     7     3     7     2     7     1     7     0
     0     0     1     0     3     1     3     0     6     2     5     0     9     3     7     0
     0     0     1     0     3     1     3     0     7     3     6     1     9     3     7     0
 &amp;nbsp;
 Inversion numbers:
     0
     4
     8
    12
    12
    16
    16
    20
    24
    28
    28
    32
    24
    28
    32
    36
    36
    40
    28
    32
    36
    40
    40
    44
|}

==Nimber multiplication==
{| align=&quot;right&quot; style=&quot;text-align:right; border:1px solid lightgray&quot; cellspacing=&quot;10&quot;
| [[Walsh permutation]]; [[Walsh permutation; nimber multiplication|nimber multiplication]] {{rdrup}}
|}

Images: {{commons catlink|Inversion set of a 4-bit Walsh permutation; nimber multiplication}}

{{clear}}

{| class=&quot;wikitable sortable&quot; style=&quot;text-align:right&quot;
|-
! #
! CV
! Permutation
! F
! IN
|- style=&quot;font-family:Courier&quot;
|              0||( 1, 2, 4, 8)||(0, '''1''', 2, 3, &amp;nbsp; 4, 5, 6, 7, &amp;nbsp; 8, 9,10,11, &amp;nbsp;12,13,14,15) || (0, 0, 0, 0, &amp;nbsp; 0, 0, 0, 0, &amp;nbsp; 0, 0, 0, 0, &amp;nbsp; 0, 0, 0, '''0''') ||  0
|- style=&quot;font-family:Courier&quot;
| 13827898032492||( 2, 3, 8,12)||(0, '''2''', 3, 1, &amp;nbsp; 8,10,11, 9, &amp;nbsp;12,14,15,13, &amp;nbsp; 4, 6, 7, 5) || (0, 0, 0, 2, &amp;nbsp; 0, 0, 0, 2, &amp;nbsp; 0, 0, 0, 2, &amp;nbsp; 8, 8, 8,'''10''') || 40
|- style=&quot;font-family:Courier&quot;
|  7001306732168||( 3, 1,12, 4)||(0, '''3''', 1, 2, &amp;nbsp;12,15,13,14, &amp;nbsp; 4, 7, 5, 6, &amp;nbsp; 8,11, 9,10) || (0, 0, 1, 1, &amp;nbsp; 0, 0, 1, 1, &amp;nbsp; 4, 4, 5, 5, &amp;nbsp; 4, 4, 5, '''5''') || 40
|- style=&quot;font-family:Courier&quot;
| 19217812814448||( 4, 8, 6,11)||(0, '''4''', 8,12, &amp;nbsp; 6, 2,14,10, &amp;nbsp;11,15, 3, 7, &amp;nbsp;13, 9, 5, 1) || (0, 0, 0, 0, &amp;nbsp; 2, 4, 0, 2, &amp;nbsp; 2, 0, 8, 6, &amp;nbsp; 2, 6,10,'''14''') || 56
|- style=&quot;font-family:Courier&quot;
|  1630614229512||( 5,10, 2, 3)||(0, '''5''',10,15, &amp;nbsp; 2, 7, 8,13, &amp;nbsp; 3, 6, 9,12, &amp;nbsp; 1, 4,11,14) || (0, 0, 0, 0, &amp;nbsp; 3, 2, 2, 1, &amp;nbsp; 6, 5, 3, 2, &amp;nbsp;11, 9, 3, '''1''') || 48
|- style=&quot;font-family:Courier&quot;
| 15433122573720||( 6,11,14, 7)||(0, '''6''',11,13, &amp;nbsp;14, 8, 5, 3, &amp;nbsp; 7, 1,12,10, &amp;nbsp; 9,15, 2, 4) || (0, 0, 0, 0, &amp;nbsp; 0, 3, 5, 6, &amp;nbsp; 4, 8, 2, 4, &amp;nbsp; 5, 0,12,'''11''') || 60
|- style=&quot;font-family:Courier&quot;
|  5564988000144||( 7, 9,10,15)||(0, '''7''', 9,14, &amp;nbsp;10,13, 3, 4, &amp;nbsp;15, 8, 6, 1, &amp;nbsp; 5, 2,12,11) || (0, 0, 0, 0, &amp;nbsp; 1, 1, 5, 5, &amp;nbsp; 0, 5, 7,10, &amp;nbsp; 8,11, 3, '''4''') || 60
|- style=&quot;font-family:Courier&quot;
| 17445432818676||( 8,12,11,13)||(0, '''8''',12, 4, &amp;nbsp;11, 3, 7,15, &amp;nbsp;13, 5, 1, 9, &amp;nbsp; 6,14,10, 2) || (0, 0, 0, 2, &amp;nbsp; 1, 4, 3, 0, &amp;nbsp; 1, 6, 9, 4, &amp;nbsp; 7, 1, 5,'''13''') || 56
|- style=&quot;font-family:Courier&quot;
|  3551688986892||( 9,14,15, 5)||(0, '''9''',14, 7, &amp;nbsp;15, 6, 1, 8, &amp;nbsp; 5,12,11, 2, &amp;nbsp;10, 3, 4,13) || (0, 0, 0, 2, &amp;nbsp; 0, 4, 5, 3, &amp;nbsp; 6, 2, 3, 9, &amp;nbsp; 4,10,10, '''2''') || 60
|- style=&quot;font-family:Courier&quot;
| 10678227659604||(10,15, 3, 1)||(0,'''10''',15, 5, &amp;nbsp; 3, 9,12, 6, &amp;nbsp; 1,11,14, 4, &amp;nbsp; 2, 8,13, 7) || (0, 0, 0, 2, &amp;nbsp; 3, 2, 1, 4, &amp;nbsp; 7, 2, 1, 8, &amp;nbsp;10, 6, 2, '''8''') || 56
|- style=&quot;font-family:Courier&quot;
| 10169232103620||(11,13, 7, 9)||(0,'''11''',13, 6, &amp;nbsp; 7,12,10, 1, &amp;nbsp; 9, 2, 4,15, &amp;nbsp;14, 5, 3, 8) || (0, 0, 0, 2, &amp;nbsp; 2, 1, 3, 6, &amp;nbsp; 4, 7, 7, 0, &amp;nbsp; 1, 9,11, '''7''') || 60
|- style=&quot;font-family:Courier&quot;
| 15737148505928||(12, 4,13, 6)||(0,'''12''', 4, 8, &amp;nbsp;13, 1, 9, 5, &amp;nbsp; 6,10, 2,14, &amp;nbsp;11, 7,15, 3) || (0, 0, 1, 1, &amp;nbsp; 0, 4, 2, 4, &amp;nbsp; 4, 2, 8, 0, &amp;nbsp; 3, 7, 0,'''12''') || 48
|- style=&quot;font-family:Courier&quot;
|  5084423155256||(13, 6, 9,14)||(0,'''13''', 6,11, &amp;nbsp; 9, 4,15, 2, &amp;nbsp;14, 3, 8, 5, &amp;nbsp; 7,10, 1,12) || (0, 0, 1, 1, &amp;nbsp; 2, 4, 0, 6, &amp;nbsp; 1, 7, 5, 7, &amp;nbsp; 6, 4,13, '''3''') || 60
|- style=&quot;font-family:Courier&quot;
| 12454407149960||(14, 7, 5,10)||(0,'''14''', 7, 9, &amp;nbsp; 5,11, 2,12, &amp;nbsp;10, 4,13, 3, &amp;nbsp;15, 1, 8, 6) || (0, 0, 1, 1, &amp;nbsp; 3, 1, 5, 1, &amp;nbsp; 3, 7, 1, 9, &amp;nbsp; 0,12, 7, '''9''') || 60
|- style=&quot;font-family:Courier&quot;
|  8566683323240||(15, 5, 1, 2)||(0,'''15''', 5,10, &amp;nbsp; 1,14, 4,11, &amp;nbsp; 2,13, 7, 8, &amp;nbsp; 3,12, 6, 9) || (0, 0, 1, 1, &amp;nbsp; 3, 1, 4, 2, &amp;nbsp; 6, 2, 5, 5, &amp;nbsp; 9, 3, 8, '''6''') || 56
|}

[[Category:Walsh permutation]]</text>
      <sha1>c5t3ew74de2phqnbsqz0i640ne81bqd</sha1>
    </revision>
  </page>
  <page>
    <title>Understanding Arithmetic Circuits</title>
    <ns>0</ns>
    <id>139384</id>
    <revision>
      <id>2819754</id>
      <parentid>2819651</parentid>
      <timestamp>2026-07-29T14:04:20Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Adder */</comment>
      <origin>2819754</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3176" sha1="46gkusbebbe5m8gcpc5hrzd26r5lsvc" xml:space="preserve">
== Adder ==

* Binary Adder Architecture Exploration ( [[Media:Adder.20131113.pdf|pdf]]  )
 
{| class=&quot;wikitable&quot;
|-
!  Adder type      !! Overview !! Analysis !! VHDL Level Design  !! CMOS Level Design  
|-
| '''1. Ripple Carry Adder'''
|| [[Media:VLSI.Arith.1A.RCA.20250522.pdf|A]]||
|| [[Media:Adder.rca.20140313.pdf|pdf]]
|| [[Media:VLSI.Arith.1D.RCA.CMOS.20211108.pdf|pdf]]
|-
| '''2. Carry Lookahead Adder'''
|| [[Media:VLSI.Arith.2A.CLA.20260722.pdf|A]], [[Media:VLSI.Arith.2B.CLA.20260729.pdf|B]], [[Media:VLSI.Arith.2C.CLA.20260729.pdf|C]], [[Media:VLSI.Arith.2D.CLA.20260720.pdf|D]] ||
|| [[Media:Adder.cla.20140313.pdf|pdf]]||
|-
| '''3. Carry Save Adder''' 
|| [[Media:VLSI.Arith.1.A.CSave.20151209.pdf|A]]||
|| ||
|-
|| '''4. Carry Select Adder''' 
|| [[Media:VLSI.Arith.1.A.CSelA.20191002.pdf|A]]||
|| || 
|-
|| '''5. Carry Skip Adder'''
|| [[Media:VLSI.Arith.5A.CSkip.20250405.pdf|A]]||
|| 
|| [[Media:VLSI.Arith.5D.CSkip.CMOS.20211108.pdf|pdf]]
|- 
|| '''6. Carry Chain Adder'''
|| [[Media:VLSI.Arith.6A.CCA.20211109.pdf|A]]||
|| [[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|A]]||
|| [[Media:Adder.ksa.20140409.pdf|pdf]]||
|-
|| '''8. Prefix Adder'''
|| [[Media:VLSI.Arith.1.A.PFA.20140314.pdf|A]]||
|| || 
|-
|| '''9.1 Variable Block Adder'''
|| [[Media:VLSI.Arith.1A.VBA.20221110.pdf|A]], [[Media:VLSI.Arith.1B.VBA.20230911.pdf|B]], [[Media:VLSI.Arith.1C.VBA.20240622.pdf|C]], [[Media:VLSI.Arith.1C.VBA.20250218.pdf|D]]||
|| || 
|-
|| '''9.2 Multi-Level Variable Block Adder'''
|| [[Media:VLSI.Arith.1.A.VBA-Multi.20221031.pdf|A]]||
|| || 
|}
&lt;/br&gt;
 
=== 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
&lt;/br&gt;

== Barrel Shifter ==
* Barrel Shifter Architecture Exploration ([[Media:Bshift.20131105.pdf|bshfit.vhdl]], [[Media:Bshift.makefile.20131109.pdf|bshfit.makefile]])
&lt;/br&gt;

'''Mux Based Barrel Shifter'''
* Analysis  ([[Media:Arith.BShfiter.20151207.pdf|pdf]]) 
* Implementation 
&lt;/br&gt;

== Multiplier == 
=== Array Multipliers ===
* Analysis  ([[Media:VLSI.Arith.1.A.Mult.20151209.pdf|pdf]])
&lt;/br&gt;

=== 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]])
&lt;/br&gt;

=== Booth Multipliers ===
* [[Media:RNS4.BoothEncode.20161005.pdf|Booth Encoding Note]]
* Booth Multiplier Note ([[Media:BoothMult.20160929.pdf|H1.pdf]])
&lt;/br&gt;

== Divider == 
* Binary Divider ([[Media:VLSI.Arith.1.A.Divider.20131217.pdf|pdf]])&lt;/br&gt;

&lt;/br&gt;
&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

[[Category:Digital Circuit Design]]
[[Category:FPGA]]</text>
      <sha1>46gkusbebbe5m8gcpc5hrzd26r5lsvc</sha1>
    </revision>
  </page>
  <page>
    <title>Complex analysis in plain view</title>
    <ns>0</ns>
    <id>171005</id>
    <revision>
      <id>2819760</id>
      <parentid>2819656</parentid>
      <timestamp>2026-07-29T14:16:07Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Geometric Series Examples */</comment>
      <origin>2819760</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5296" sha1="8i49788vmi2zima442zyzsebb54fwg7" xml:space="preserve">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 &lt;math&gt;x+iy&lt;/math&gt;, where &lt;math&gt;i=\sqrt{-1}&lt;/math&gt;, 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 &lt;math&gt;L&lt;/math&gt;-functions, modular forms, elliptic functions, etc. &lt;blockquote&gt;The shortest path between two truths in the real domain passes through the complex domain.  — [[wikipedia:Jacques_Hadamard|Jacques Hadamard]]&lt;/blockquote&gt;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]]) 

* Permutations ([[Media:Laurent.6.Permutation.6A.20230711.pdf|A.pdf]], [[Media:Laurent.5.Permutation.6B.20251225.pdf|B.pdf]], [[Media:Laurent.5.Permutation.6C.20260729.pdf|C.pdf]], [[Media:Laurent.5.Permutation.6C.20240528.pdf|D.pdf]]) 

* Applications ([[Media:Laurent.5.Application.6B.20220723.pdf|A.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]]</text>
      <sha1>8i49788vmi2zima442zyzsebb54fwg7</sha1>
    </revision>
  </page>
  <page>
    <title>Haskell programming in plain view</title>
    <ns>0</ns>
    <id>203942</id>
    <revision>
      <id>2819798</id>
      <parentid>2819645</parentid>
      <timestamp>2026-07-30T05:00:09Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Lambda Calculus */</comment>
      <origin>2819798</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6273" sha1="ego7dbkqwlnjmqei4rac7e5r5r8e8tx" xml:space="preserve">
==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]])
&lt;/br&gt;

==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]])

&lt;/br&gt;

==Using GHCi==
* Getting started ([[Media:GHCi.Start.1.A.20170605.pdf |pdf]])
&lt;/br&gt;

==Using Libraries==
* Library ([[Media:Library.1.A.20170605.pdf |pdf]])
&lt;/br&gt;
&lt;/br&gt;


==Types==
* Constructors ([[Media:Background.1.A.Constructor.20180904.pdf |pdf]])
* TypeClasses ([[Media:Background.1.B.TypeClass.20180904.pdf |pdf]])
* 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]])

==Functions==
* Functions ([[Media:Background.1.C.Function.20180712.pdf |pdf]])
* Operators ([[Media:Background.1.E.Operator.20180707.pdf |pdf]])
* Continuation Passing Style ([[Media:MP3.1D.Mut.Continuation.20220110.pdf |pdf]])

==Expressions==
* Expressions I ([[Media:Background.1.D.Expression.20180707.pdf |pdf]])
* Expressions II ([[Media:MP3.1E.Mut.Expression.20220628.pdf |pdf]])
* Non-terminating Expressions ([[Media:MP3.1F.Mut.Non-terminating.20220616.pdf |pdf]])
&lt;/br&gt;
&lt;/br&gt;

==Lambda Calculus==
* Lambda Calculus - informal description ([[Media:LCal.1A.informal.20220831.pdf |pdf]])
* Lambda Calculus - Formal definition ([[Media:LCal.2A.formal.20221015.pdf |pdf]])
* Expression Reduction ([[Media:LCal.3A.reduction.20220920.pdf |pdf]])
* Normal Forms ([[Media:LCal.4A.Normal.20220903.pdf |pdf]])
* Encoding Datatypes 
:- Church Numerals ([[Media:LCal.5A.Numeral.20230627.pdf |pdf]])
:- Church Booleans ([[Media:LCal.6A.Boolean.20230815.pdf |pdf]])
:- Functions ([[Media:LCal.7A.Function.20231230.pdf |pdf]])
:- Combinators ([[Media:LCal.8A.Combinator.20241202.pdf |pdf]])
:- Recursions ([[Media:LCal.9A.Recursion.20260729.pdf |A]], [[Media:LCal.9B.Recursion.20260330.pdf |B]])
&lt;/br&gt;
&lt;/br&gt;

==Function Oriented Typeclasses== 
=== 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 
: - 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 &amp; 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 &amp; 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]])
&lt;/br&gt;

==Polymorphism==
* Polymorphism Overview ([[Media:Poly.1.A.20180220.pdf |pdf]])
&lt;/br&gt;

==Concurrent Haskell ==
&lt;/br&gt;


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:Haskell|programming in plain view]]</text>
      <sha1>ego7dbkqwlnjmqei4rac7e5r5r8e8tx</sha1>
    </revision>
    <revision>
      <id>2819800</id>
      <parentid>2819798</parentid>
      <timestamp>2026-07-30T05:04:27Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Lambda Calculus */</comment>
      <origin>2819800</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6273" sha1="7da03omx231hpooncd0f5e4ikeq07b4" xml:space="preserve">
==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]])
&lt;/br&gt;

==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]])

&lt;/br&gt;

==Using GHCi==
* Getting started ([[Media:GHCi.Start.1.A.20170605.pdf |pdf]])
&lt;/br&gt;

==Using Libraries==
* Library ([[Media:Library.1.A.20170605.pdf |pdf]])
&lt;/br&gt;
&lt;/br&gt;


==Types==
* Constructors ([[Media:Background.1.A.Constructor.20180904.pdf |pdf]])
* TypeClasses ([[Media:Background.1.B.TypeClass.20180904.pdf |pdf]])
* 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]])

==Functions==
* Functions ([[Media:Background.1.C.Function.20180712.pdf |pdf]])
* Operators ([[Media:Background.1.E.Operator.20180707.pdf |pdf]])
* Continuation Passing Style ([[Media:MP3.1D.Mut.Continuation.20220110.pdf |pdf]])

==Expressions==
* Expressions I ([[Media:Background.1.D.Expression.20180707.pdf |pdf]])
* Expressions II ([[Media:MP3.1E.Mut.Expression.20220628.pdf |pdf]])
* Non-terminating Expressions ([[Media:MP3.1F.Mut.Non-terminating.20220616.pdf |pdf]])
&lt;/br&gt;
&lt;/br&gt;

==Lambda Calculus==
* Lambda Calculus - informal description ([[Media:LCal.1A.informal.20220831.pdf |pdf]])
* Lambda Calculus - Formal definition ([[Media:LCal.2A.formal.20221015.pdf |pdf]])
* Expression Reduction ([[Media:LCal.3A.reduction.20220920.pdf |pdf]])
* Normal Forms ([[Media:LCal.4A.Normal.20220903.pdf |pdf]])
* Encoding Datatypes 
:- Church Numerals ([[Media:LCal.5A.Numeral.20230627.pdf |pdf]])
:- Church Booleans ([[Media:LCal.6A.Boolean.20230815.pdf |pdf]])
:- Functions ([[Media:LCal.7A.Function.20231230.pdf |pdf]])
:- Combinators ([[Media:LCal.8A.Combinator.20241202.pdf |pdf]])
:- Recursions ([[Media:LCal.9A.Recursion.20260730.pdf |A]], [[Media:LCal.9B.Recursion.20260330.pdf |B]])
&lt;/br&gt;
&lt;/br&gt;

==Function Oriented Typeclasses== 
=== 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 
: - 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 &amp; 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 &amp; 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]])
&lt;/br&gt;

==Polymorphism==
* Polymorphism Overview ([[Media:Poly.1.A.20180220.pdf |pdf]])
&lt;/br&gt;

==Concurrent Haskell ==
&lt;/br&gt;


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:Haskell|programming in plain view]]</text>
      <sha1>7da03omx231hpooncd0f5e4ikeq07b4</sha1>
    </revision>
  </page>
  <page>
    <title>Python programming in plain view</title>
    <ns>0</ns>
    <id>212733</id>
    <revision>
      <id>2819762</id>
      <parentid>2819569</parentid>
      <timestamp>2026-07-29T17:40:43Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Using Libraries */</comment>
      <origin>2819762</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2696" sha1="gtvbbiiydweuqjdmw389arfia40ld46" xml:space="preserve">==''' Part I '''==
&lt;!----------------------------------------------------------------------&gt;
===  Introduction ===

* Overview 
* Memory 
* Number 

&lt;!----------------------------------------------------------------------&gt;


=== Python for C programmers ===

* Hello, World! ([[Media:CProg.Hello.1A.20230406.pdf |pdf]])
* Statement Level  ([[Media:CProg.Statement.1A.20230509.pdf |pdf]])
* Output with print
* Formatted output
* File IO

&lt;!----------------------------------------------------------------------&gt;


=== Using Libraries ===

* Scripts ([[Media:Python.Work2.Script.1A.20231129.pdf |pdf]])
* Modules ([[Media:Python.Work2.Module.1A.20231216.pdf |pdf]])
* Packages ([[Media:Python.Work2.Package.1A.20241207.pdf |pdf]])
* Libraries ([[Media:Python.Work2.Library.1A.20260728.pdf |A]], [[Media:Python.Work2.Library.1B.20260720.pdf |B]])
* Namespaces ([[Media:Python.Work2.Scope.1A.20231021.pdf |pdf]])


&lt;!----------------------------------------------------------------------&gt;

===   Handling Repetition ===
 
* Control ([[Media:Python.Repeat1.Control.1.A.20230314.pdf |pdf]])

* Loop ([[Media:Repeat2.Loop.1A.20230401.pdf |pdf]])

&lt;!----------------------------------------------------------------------&gt;

=== Handling a Big Work ===

* Functions ([[Media:Python.Work1.Function.1A.20230529.pdf |pdf]])
* Lambda ([[Media:Python.Work2.Lambda.1A.20230705.pdf |pdf]])
* Type Annotations ([[Media:Python.Work2.AtypeAnnot.1A.20230817.pdf |pdf]])

  
&lt;!----------------------------------------------------------------------&gt;

=== Handling Series of Data ===

* Arrays ([[Media:Python.Series1.Array.1A.pdf |pdf]])
* Tuples ([[Media:Python.Series2.Tuple.1A.pdf |pdf]])
* Lists ([[Media:Python.Series3.List.1A.pdf |pdf]])
* Tuples ([[Media:Python.Series4.Tuple.1A.pdf |pdf]])
* Sets ([[Media:Python.Series5.Set.1A.pdf |pdf]])
* Dictionary ([[Media:Python.Series6.Dictionary.1A.pdf |pdf]])

&lt;!----------------------------------------------------------------------&gt;

=== Handling Various Kinds of Data ===
  
* Types 
* Operators ([[Media:Python.Data3.Operators.1.A.pdf |pdf]])
* Files ([[Media:Python.Data4.File.1.A.pdf |pdf]])


&lt;!----------------------------------------------------------------------&gt;

=== Class and Objects ===
  
* Classes &amp; Objects ([[Media:Python.Work2.Class.1A.20230906.pdf |pdf]])
* Inheritance

 
&lt;!----------------------------------------------------------------------&gt;
&lt;/br&gt;

== Python in Numerical Analysis  ==
 
&lt;/br&gt;
&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

==External links==
* [http://www.southampton.ac.uk/~fangohr/training/python/pdfs/Python-for-Computational-Science-and-Engineering.pdf Python and Computational Science and Engineering]</text>
      <sha1>gtvbbiiydweuqjdmw389arfia40ld46</sha1>
    </revision>
    <revision>
      <id>2819764</id>
      <parentid>2819762</parentid>
      <timestamp>2026-07-29T17:41:44Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Using Libraries */</comment>
      <origin>2819764</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2696" sha1="46t5cavtxh6f4fyblwjf3g6jv1q3yo4" xml:space="preserve">==''' Part I '''==
&lt;!----------------------------------------------------------------------&gt;
===  Introduction ===

* Overview 
* Memory 
* Number 

&lt;!----------------------------------------------------------------------&gt;


=== Python for C programmers ===

* Hello, World! ([[Media:CProg.Hello.1A.20230406.pdf |pdf]])
* Statement Level  ([[Media:CProg.Statement.1A.20230509.pdf |pdf]])
* Output with print
* Formatted output
* File IO

&lt;!----------------------------------------------------------------------&gt;


=== Using Libraries ===

* Scripts ([[Media:Python.Work2.Script.1A.20231129.pdf |pdf]])
* Modules ([[Media:Python.Work2.Module.1A.20231216.pdf |pdf]])
* Packages ([[Media:Python.Work2.Package.1A.20241207.pdf |pdf]])
* Libraries ([[Media:Python.Work2.Library.1A.20260729.pdf |A]], [[Media:Python.Work2.Library.1B.20260720.pdf |B]])
* Namespaces ([[Media:Python.Work2.Scope.1A.20231021.pdf |pdf]])


&lt;!----------------------------------------------------------------------&gt;

===   Handling Repetition ===
 
* Control ([[Media:Python.Repeat1.Control.1.A.20230314.pdf |pdf]])

* Loop ([[Media:Repeat2.Loop.1A.20230401.pdf |pdf]])

&lt;!----------------------------------------------------------------------&gt;

=== Handling a Big Work ===

* Functions ([[Media:Python.Work1.Function.1A.20230529.pdf |pdf]])
* Lambda ([[Media:Python.Work2.Lambda.1A.20230705.pdf |pdf]])
* Type Annotations ([[Media:Python.Work2.AtypeAnnot.1A.20230817.pdf |pdf]])

  
&lt;!----------------------------------------------------------------------&gt;

=== Handling Series of Data ===

* Arrays ([[Media:Python.Series1.Array.1A.pdf |pdf]])
* Tuples ([[Media:Python.Series2.Tuple.1A.pdf |pdf]])
* Lists ([[Media:Python.Series3.List.1A.pdf |pdf]])
* Tuples ([[Media:Python.Series4.Tuple.1A.pdf |pdf]])
* Sets ([[Media:Python.Series5.Set.1A.pdf |pdf]])
* Dictionary ([[Media:Python.Series6.Dictionary.1A.pdf |pdf]])

&lt;!----------------------------------------------------------------------&gt;

=== Handling Various Kinds of Data ===
  
* Types 
* Operators ([[Media:Python.Data3.Operators.1.A.pdf |pdf]])
* Files ([[Media:Python.Data4.File.1.A.pdf |pdf]])


&lt;!----------------------------------------------------------------------&gt;

=== Class and Objects ===
  
* Classes &amp; Objects ([[Media:Python.Work2.Class.1A.20230906.pdf |pdf]])
* Inheritance

 
&lt;!----------------------------------------------------------------------&gt;
&lt;/br&gt;

== Python in Numerical Analysis  ==
 
&lt;/br&gt;
&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

==External links==
* [http://www.southampton.ac.uk/~fangohr/training/python/pdfs/Python-for-Computational-Science-and-Engineering.pdf Python and Computational Science and Engineering]</text>
      <sha1>46t5cavtxh6f4fyblwjf3g6jv1q3yo4</sha1>
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    <title>User:Platos Cave (physics)/Simulation Hypothesis/Electron (mathematical)</title>
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      <id>2819753</id>
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      <timestamp>2026-07-29T13:47:09Z</timestamp>
      <contributor>
        <username>Platos Cave (physics)</username>
        <id>2562653</id>
      </contributor>
      <origin>2819753</origin>
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      <text bytes="20785" sha1="braleq3olq5pxwfv02dzuporlzq9ml0" xml:space="preserve">{{Original research}}


'''The mathematical electron model'''

In the mathematical electron model &lt;ref&gt;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 }}&lt;/ref&gt;, the electron is a geometrical object denoted by the formula (ψ), itself the geometry of 3 numbers; the (inverse) [[w:fine-structure constant | fine structure constant alpha]] = 137.035 999 139 (CODATA 2014), a mathematical constant Omega, and pi (Omega itself is a construct of pi and Euler's number [[w:E_(mathematical_constant) |e]] = 2.718281828459). 

:&lt;math&gt;\psi = 4\pi^2(2^6 3 \pi^2 \alpha^{-1} \Omega^5)^3 = .23895453... \times 10^{23},\;unit = 1&lt;/math&gt; (unit-less)

:&lt;math&gt;\Omega = \sqrt{ \left(\pi^e e^{(1-e)}\right)}  = 2.0071349543... &lt;/math&gt;


Although dimensionless, this formula ψ encodes the information required to make the physical electron parameters (wavelength, frequency, mass, charge). It does this by embedding within its own geometry objects for [[w:Planck_units |Planck units]] analogues; M for mass &lt;math&gt;m_P&lt;/math&gt;, L for length &lt;math&gt;l_p&lt;/math&gt;, T for time &lt;math&gt;t_p&lt;/math&gt; and A for charge &lt;math&gt;A&lt;/math&gt;, and ψ also dictates the frequency of these geometrical (Planck) objects. For example; 

[[w:Compton wavelength | electron wavelength]]
:&lt;math&gt;\lambda_e = 2\pi l_p \psi&lt;/math&gt;  

[[w:electron mass | electron mass]] 
:&lt;math&gt;m_e = \frac{m_P}{\psi}&lt;/math&gt; 


=== Geometrical objects MLTA ===
{{main|User:Platos_Cave_(physics)/Simulation_Hypothesis/Planck_units_(geometrical)}}

Key:
:&lt;math&gt;\frac{ P^{15} T^2}{M^{12}} = \pi^2 \Omega^{15},\;\;\text{units} = \left(\frac{{kg m}}{{s}}\right)^{15/2} \times \frac{{s}^2}{{kg}^{12}} = \frac{{m}^{15/2}}{{kg}^{9/2} \,{s}^{11/2}} = 1&lt;/math&gt; (natural Planck units)
:&lt;math&gt;v = 11 843 707.905 ...,\; units = \frac{m}{s}&lt;/math&gt;
:&lt;math&gt;r = 0.712 562 514 304 ...,\; units = (\frac{kg.m}{s})^{1/4}&lt;/math&gt;
{| class=&quot;wikitable&quot;
|+ Table 1. Geometrical objects (reference table)
! Attribute
! Formula
! Object
! Scalars
! Numerical
! &lt;math&gt;\theta&lt;/math&gt;
! [[v:User:Platos_Cave_(physics)/Simulation_Hypothesis/Planck_units_(geometrical)#Dimensionless_f(x) |Units*]]
|-
| M (mass) 
| ''fundamental'' 
| &lt;math&gt;(1)&lt;/math&gt;
| &lt;math&gt;\dfrac{r^4}{v}&lt;/math&gt;
| 0.217672825e-7
| &lt;math&gt;u^{15}&lt;/math&gt;
| &lt;math&gt;kg&lt;/math&gt;
|-
| T (time) 
| ''fundamental''  
| &lt;math&gt;(\pi)&lt;/math&gt;
| &lt;math&gt;\dfrac{r^9}{v^6}\;j^2&lt;/math&gt;
| 0.539051838e-43
| &lt;math&gt;u^{-30}&lt;/math&gt;
| &lt;math&gt;s&lt;/math&gt;
|-
| P (sqrt of momentum)  
| ''fundamental'' 
| &lt;math&gt;(\Omega)&lt;/math&gt;
| &lt;math&gt;r^2&lt;/math&gt;
| 1.019113428
| &lt;math&gt;u^{16}&lt;/math&gt;
| &lt;math&gt;\sqrt{\frac{kg\;m}{s}}&lt;/math&gt;
|-
| V (velocity)  
| &lt;math&gt;\dfrac{2\pi P^2}{M}&lt;/math&gt; 
| &lt;math&gt;(2\pi\Omega^2)&lt;/math&gt;
| &lt;math&gt;v&lt;/math&gt;
| 299792458
| &lt;math&gt;u^{17}&lt;/math&gt;
| &lt;math&gt;\frac{m}{s}&lt;/math&gt;
|-
| L (length)  
| &lt;math&gt;VT &lt;/math&gt; 
| &lt;math&gt;(2\pi^2\Omega^2)&lt;/math&gt;
| &lt;math&gt;\dfrac{r^9}{v^5}\;j^2&lt;/math&gt;
| 0.161603675e-34
| &lt;math&gt;u^{-13}&lt;/math&gt;
| &lt;math&gt;m&lt;/math&gt;
|-
| A (ampere) 
| &lt;math&gt;\dfrac{2^4 V^3 \alpha}{P^3}&lt;/math&gt; 
| &lt;math&gt;(2^7\pi^3 \alpha \Omega^3)&lt;/math&gt;
| &lt;math&gt;\dfrac{v^3}{r^6}&lt;/math&gt;
| 0.297221257e25
| &lt;math&gt;u^{3}&lt;/math&gt;
| &lt;math&gt;\frac{m^{3/2}}{kg^{3/2} s^{3/2}}&lt;/math&gt;
|-
| K (temperature) 
| &lt;math&gt;\dfrac{AV}{2\pi}&lt;/math&gt; 
| &lt;math&gt;(2^7 \pi^3 \alpha \Omega^5)&lt;/math&gt;
| &lt;math&gt;\dfrac{v^4}{r^6}&lt;/math&gt;
| 0.141814520e33
| &lt;math&gt;u^{20}&lt;/math&gt;
| &lt;math&gt;\left(\frac{A \; m}{s}\right)&lt;/math&gt;
|}


=== Scalars ===
As alpha and Omega can be assigned numerical values, so too the MLTA objects can be expressed numerically. We can then convert these objects to their Planck unit equivalents by including a dimensioned scalar. For example; 

:If we use scalar ''k''&lt;sub&gt;SI&lt;/sub&gt; to convert ''M'' (M=1) to the SI Planck mass (M*''k''&lt;sub&gt;SI&lt;/sub&gt; = &lt;math&gt;m_P&lt;/math&gt;), then ''k''&lt;sub&gt;SI&lt;/sub&gt; = 0.2176728e-7kg ([[w:SI_units |SI units]])

:SI ''c'' = object V * scalar ''v''&lt;sub&gt;SI&lt;/sub&gt; = 299792458m/s ([[w:SI_units |SI units]])

:Imperial ''c'' = object V * scalar ''v''&lt;sub&gt;imp&lt;/sub&gt; = 186282miles/s ([[w:Imperial_units |imperial units]])


==== Scalar relationships ====
Scalars that translate to the SI unit system must therefore carry not only the numerical conversion values but also the unit (for scalars we will refer to the SI unit system by default), i.e.: scalar ''v'' = 11843707.905 m/s. This also means that the scalars must also follow the unit number relationship θ. 

{| class=&quot;wikitable&quot;
|+Table 2. Geometrical units
! Attribute
! Geometrical object
! Scalar 
! Unit ''u''&lt;sup&gt;θ&lt;/sup&gt;
|-
| mass 
| &lt;math&gt;M = (1)&lt;/math&gt;  
| ''k''
| &lt;math&gt;u^{15}&lt;/math&gt;
|-
| time 
| &lt;math&gt;T = (\pi)&lt;/math&gt;  
| ''t''
| &lt;math&gt;u^{-30}&lt;/math&gt;
|-
| [[v:User:Platos Cave (physics)/Simulation_Hypothesis/Sqrt_Planck_momentum | sqrt(momentum)]]
| &lt;math&gt;P = (\Omega)&lt;/math&gt;
| ''r''&lt;sup&gt;2&lt;/sup&gt;
| &lt;math&gt;u^{16}&lt;/math&gt;
|-
| velocity  
| &lt;math&gt;V = (2\pi\Omega^2)&lt;/math&gt;  
| ''v''
| &lt;math&gt;u^{17}&lt;/math&gt;
|-
| length  
| &lt;math&gt;L = (2\pi^2\Omega^2)&lt;/math&gt; 
| ''l''
| &lt;math&gt;u^{-13}&lt;/math&gt;
|-
| ampere 
| &lt;math&gt;A = (\frac{2^7 \pi^3 \Omega^3}{a})&lt;/math&gt; 
| ''q''
| &lt;math&gt;u^3&lt;/math&gt;
|}


Here are examples where units = 1, as such ''only 2 scalars are required'', for example, if we know the numerical value for ''q'' and for ''l'' then we know the numerical value for ''t'' ('''t = q&lt;sup&gt;3&lt;/sup&gt;l&lt;sup&gt;3&lt;/sup&gt;'''), and from ''l'' and ''t'' we know the value for ''k''. 

:&lt;math&gt;\frac{u^{3*3} u^{-13*3}}{u^{-30}}\;(\frac{q^3 l^3}{t}) = \frac{u^{-13*15}}{u^{15*9} u^{-30*11}} \;(\frac{l^{15}}{k^9 t^{11}}) = \;...\; =1&lt;/math&gt;


In other words, once any 2 scalars have been assigned values, the other scalars are then defined by default, consequently the CODATA 2014 values are used here as only 2 constants (c, [[w:permeability of vacuum|μ&lt;sub&gt;0&lt;/sub&gt;]]) are assigned exact values, following the [[w:2019 redefinition of SI base units|2019 redefinition of SI base units]] a total of 4 constants now have independently exact values assigned which is problematic in terms of this model. 


Scalars ''r'' (θ = 8) and ''v'' (θ = 17) are chosen here for demonstration as they can be derived directly from the 2 constants with exact values; ''c'' and ''μ&lt;sub&gt;0&lt;/sub&gt;''.
:&lt;math&gt;c = 299792458&lt;/math&gt; m/s
:&lt;math&gt;\mu_0 = 4\pi / 10^7&lt;/math&gt;
:&lt;math&gt;v = \frac{c}{2 \pi \Omega^2}= 11 843 707.905 ...,\; units = \frac{m}{s}&lt;/math&gt;
:&lt;math&gt;r^7 = \frac{2^{11} \pi^5 \Omega^4 \mu_0}{a};\; r = 0.712 562 514 304 ...,\; units = (\frac{kg.m}{s})^{1/4}&lt;/math&gt;

{| class=&quot;wikitable&quot;
|+Table 3. Geometrical objects
! attribute
! geometrical object
! unit number θ
! scalar r(8), v(17)
|-
| mass 
| &lt;math&gt;M = (1)&lt;/math&gt;
| 15 = 8*4-17
| &lt;math&gt;k = \frac{r^4}{v}&lt;/math&gt;
|-
| time 
| &lt;math&gt;T = (\pi)&lt;/math&gt;
|  -30 = 8*9-17*6 
| &lt;math&gt;t = \frac{r^9}{v^6}&lt;/math&gt;
|-
| sqrt(momentum)  
| &lt;math&gt;P = (\Omega)&lt;/math&gt;
| 16
| &lt;math&gt;r^2&lt;/math&gt;
|-
| velocity  
| &lt;math&gt;V = (2\pi\Omega^2)&lt;/math&gt;
| 17
| &lt;math&gt;v&lt;/math&gt;
|-
| length  
| &lt;math&gt;L = (2\pi^2\Omega^2)&lt;/math&gt; 
| -13 = 8*9-17*5
| &lt;math&gt;l = \frac{r^9}{v^5}&lt;/math&gt;
|-
| ampere 
| &lt;math&gt;A = (\frac{2^7 \pi^3 \Omega^3}{a})&lt;/math&gt; 
| 3 = 17*3-8*6
| &lt;math&gt;\frac{v^3}{r^6}&lt;/math&gt;
|}


=== Mathematical electron ===

The mathematical electron formula ψ incorporates the dimensioned Planck units but itself is dimension-less (units = scalars = 1). Here ψ is defined in terms of ''σ&lt;sub&gt;e&lt;/sub&gt;'', where AL is an ampere-meter (ampere-length = ''e*c'' are the units for a [[w:magnetic monopole | magnetic monopole]]). 

:&lt;math&gt;T = \pi,\; unit = u^{-30},\;scalars = \frac{r^9}{v^6}&lt;/math&gt;

:&lt;math&gt;\sigma_{e} = \frac{3 \alpha_{inv}^2 A L}{2\pi^2} = {2^7 3 \pi^3 \alpha_{inv} \Omega^5},\; unit = u^{(3 \;-13 \;= \;-10)},\; scalars = \frac{r^3}{v^2}&lt;/math&gt;

:&lt;math&gt;\psi = \frac{\sigma_{e}^3}{2 T} = \frac{(2^7 3 \pi^3 \alpha_{inv} \Omega^5)^3}{2\pi},\; unit = \frac{(u^{-10})^3}{u^{-30}} = 1, scalars = (\frac{r^3}{v^2})^3 \frac{v^6}{r^9} = 1&lt;/math&gt;

:&lt;math&gt;\psi = 4\pi^2(2^6 3 \pi^2 \alpha_{inv} \Omega^5)^3 = .23895453...x10^{23},\;unit = 1&lt;/math&gt; (unit-less)

Both units and scalars cancel. 


===== Electron parameters =====
We can solve the electron parameters; electron mass, wavelength, frequency, charge ... as the frequency of the Planck units themselves, and this frequency is ψ.
:&lt;math&gt;v = 11 843 707.905 ...,\; units = \frac{m}{s}&lt;/math&gt;
:&lt;math&gt;r = 0.712 562 514 304 ...,\; units = (\frac{kg.m}{s})^{1/4}&lt;/math&gt;

[[w:Compton wavelength | electron wavelength]] λ&lt;sub&gt;e&lt;/sub&gt; = 2.4263102367e-12m (CODATA 2014)
:&lt;math&gt;\lambda_e^* = 2\pi L \psi&lt;/math&gt; = 2.4263102386e-12m  (L ⇔ [[w:Planck length | Planck length]])


[[w:electron mass | electron mass]] m&lt;sub&gt;e&lt;/sub&gt; = 9.10938356e-31kg (CODATA 2014)
:&lt;math&gt;m_e^* = \frac{M}{\psi}&lt;/math&gt; = 9.1093823211e-31kg (M ⇔ [[w:Planck mass | Planck mass]]) 


[[w:elementary charge | elementary charge]] e = 1.6021766208e-19C (CODATA 2014)   
:&lt;math&gt;e^* = A\;T&lt;/math&gt; = 1.6021765130e-19 (T ⇔ [[w:Planck time | Planck time]])


[[w:Rydberg constant | Rydberg constant]] R = 10973731.568508/m  (CODATA 2014)
:&lt;math&gt;R^* = (\frac{m_e}{4 \pi L \alpha_{inv}^2 M}) = \frac{1}{2^{23} 3^3 \pi^{11} \alpha_{inv}^5 \Omega^{17}}\frac{v^5}{r^9}\;u^{13}&lt;/math&gt; = 10973731.568508


From the above formulas, we see that wavelength is ψ units of Planck length, frequency is ψ units of Planck time ... however the electron mass is only 1 unit of Planck mass.


===== Electron Mass =====
Particle mass is a unit of Planck mass that occurs only once per ψ units of Planck time, the other parameters are continuums of the Planck units.
:units &lt;math&gt;\psi = \frac{(AL)^3}{T}&lt;/math&gt; = 1

This may be interpreted as; for ψ units of Planck time the electron has wavelength L, charge A ... and then the AL combine with time T (A&lt;sup&gt;3&lt;/sup&gt;L&lt;sup&gt;3&lt;/sup&gt;/T) and the units (and scalars) cancel. The electron is now mass (for 1 unit of Planck time). In this consideration, the electron is an event that oscillates over time between an electric wave state (duration ψ units of Planck time) to a unit of Planck mass point state (1 unit of Planck time). The electron is a quantum scale event, it does not exist at the discrete Planck scale (and so therefore neither does the quantum scale).

As electron mass is the frequency of the geometrical Planck mass M = 1, which is a point (and so with point co-ordinates), then we have a model for a [[w:black hole electron |black-hole electron]], the electron function ψ centered around this unit of Planck mass. When the wave-state (A*L)&lt;sup&gt;3&lt;/sup&gt;/T units collapse, this black-hole center (point) is exposed for 1 unit of (Planck) time. The electron is 'now' (a unit of Planck) mass.

Mass in this consideration is not a constant property of the particle, rather the measured particle mass ''m'' would refer to the average mass, the average occurrence of the discrete Planck mass point-state over time. The formula ''E = hf'' is a measure of the frequency ''f'' of occurrence of [[w:Planck constant |Planck's constant ''h'']] and applies to the electric wave-state. As for each wave-state there is a corresponding mass point-state, then for a particle ''hf == mc2''. Notably however the ''c'' term is a fixed constant unlike the ''f'' term, and so the ''m'' term is the frequency term, it is referring to an average mass (mass which is measured over time) rather than a constant mass (mass as a constant property of the particle at unit Planck time). Thus as noted, when we refer to mass as a constant property, we are referring to average mass at the quantum scale, and the electron as a quantum-state particle.

If the [[v:User:Platos_Cave_(physics)/Simulation_Hypothesis/Planck_unit_scaffolding |scaffolding of the universe]] includes units of Planck mass '''M''', then it is not necessary for a particle itself to have mass, what we define as electron mass could be the absence of electron &lt;ref&gt;Macleod, Malcolm J.; {{Cite journal |title= 1. Planck unit scaffolding correlates with the Cosmic Microwave Background |journal=SSRN |date=Feb 2011 |  doi=10.2139/ssrn.3333513}}&lt;/ref&gt;.  


===== Quarks and Spin=====
The charge on the electron derives from the embedded ampere A and length L, the electron formula ψ itself is dimensionless. These AL magnetic monopoles would seem to be analogous to quarks (there are 3 monopoles per electron), but due to the symmetry and so stability of the geometrical ψ there is no clear fracture point by which an electron could decay, and so this would be difficult to test. We can however conjecture on what a quark solution might look like, the advantage with this approach being that we do not need to introduce new 'entities' for our quarks, the Planck units embedded within the electron suffice &lt;ref&gt;Macleod, M.J. {{Cite journal |title= 7. Geometric Origin of Quarks and Spin, the Mathematical Electron extended |journal=RG | doi=10.13140/RG.2.2.21695.16808/1 }}&lt;/ref&gt;.

====== Quarks======
Electron formula
:&lt;math&gt;\psi = 2^{20} \pi^8 3^3 \alpha_{inv}^3 \Omega^{15},\; unit = 1, scalars = 1&lt;/math&gt;

Time
:&lt;math&gt;T = \pi \frac{r^9}{v^6},\; u^{-30}&lt;/math&gt;


AL magnetic monopole
:&lt;math&gt;\sigma_{e} = \frac{3 \alpha_{inv}^2 A L}{2\pi^2} = {2^7 3 \pi^3 \alpha_{inv} \Omega^5},\; u^{-10}, \;scalars = \frac{r^3}{v^2}&lt;/math&gt;

:&lt;math&gt;\psi = \frac{\sigma_{e}^3}{2 T} = \frac{(2^7 3 \pi^3 \alpha_{inv} \Omega^5)^3}{2\pi} = 2^{20} 3^3 \pi^8 \alpha_{inv}^3 \Omega^{15},\; unit = \frac{(u^{-10})^3}{u^{-30}} = 1, scalars = (\frac{r^3}{v^2})^3 \frac{v^6}{r^9} = 1&lt;/math&gt;

If &lt;math&gt;\sigma_{e}&lt;/math&gt; could equate to a quark with an [[w:electric charge|electric charge]] of {{sfrac|-1|3}}[[w:elementary charge|''e'']], then it would be an analogue of the '''D''' quark. 3 of these D quarks would constitute the electron as DDD = (AL)*(AL)*(AL). 


We would assume that the charge on the [[w:positron |positron]] (anti-matter electron) is just the inverse of the above, however there is 1 problem, the AL (A; θ=3, L; θ=-13) units = -10, and if we look at the [[v:User:Platos Cave (physics)/Simulation_Hypothesis/Planck_units_(geometrical)#Table_of_Constants |table of constants]], there is no 'units = +10' combination that can include A. We cannot make an inverse electron. However we can make a [[w:Planck temperature|Planck temperature T&lt;sub&gt;p&lt;/sub&gt;]] AV ''monopole'' (ampere-velocity).

:&lt;math&gt;T_p = \frac{2^7 \pi^3 \Omega^5}{\alpha_{inv}},\; u^{20}, \;scalars = \frac{r^9}{v^6}&lt;/math&gt;

:&lt;math&gt;\sigma_{t} = \frac{3 \alpha_{inv}^2 T_p}{2\pi} = \frac{3 \alpha_{inv}^2 A V}{2\pi^2} = ({2^6 3 \pi^2 \alpha_{inv} \Omega^5}),\; u^{20},\;scalars = \frac{v^4}{r^6}&lt;/math&gt;

:&lt;math&gt;\psi = (2T) \sigma_{t}^2 \sigma_{e} = 2^{20} 3^3 \pi^8 \alpha_{inv}^3 \Omega^{15},\; unit = (u^{-30}) (u^{20})^2 (u^{-10}) = 1, scalars = (\frac{r^9}{v^6}) (\frac{v^4}{r^6})^2   \frac{r^3}{v^2} = 1&lt;/math&gt;


The units for &lt;math&gt;\sigma_{t}&lt;/math&gt; = +20, and so if units = -10 equates to {{sfrac|-1|3}}e, then we may conjecture that units = +20  equates to {{sfrac|2|3}}e, which would be the analogue of the '''U''' quark. Our plus charge now becomes DUU, and so although the positron has the same wavelength, frequency, mass and charge magnitude as the electron (both solve to ψ), internally its charge structure resembles that of the proton, the positron is not simply an inverse of the electron. This could have implications for the missing anti-matter, and for why the charge magnitude of the proton is ''exactly'' the charge magnitude of the electron.

:&lt;math&gt;D = \sigma_{e},\; unit = u^{-10},\; charge = \frac{-1e}{3}, \;scalars = \frac{r^3}{v^2}&lt;/math&gt;

:&lt;math&gt;U = \sigma_{t},\; unit = u^{20},\; charge = \frac{2e}{3}, \;scalars = \frac{v^4}{r^6}&lt;/math&gt;


Numerically:

Adding a proton and electron gives (proton) UUD &amp; DDD (electron) = 2(UDD) = 20 -10 -10 = 0 (zero charge), scalars = 0.

Converting between U and D via U &amp; DDD (electron) = 20 -10 -10 -10 = -10 (D), scalars = &lt;math&gt;\frac{r^3}{v^2}&lt;/math&gt;


====== Spin ======
[[v:User:Platos_Cave_(physics)/Simulation_Hypothesis/Relativity |Relativity at the Planck scale]] can be described by a translation between 2 co-ordinate systems; an expanding (in Planck steps at the speed of light) 4-axis hyper-sphere projecting onto a 3-D space &lt;ref&gt;Macleod, M.J. {{Cite journal |title= 2. Relativity as the mathematics of perspective in a hyper-sphere universe |journal=SSRN | doi=10.2139/ssrn.3334282 }}&lt;/ref&gt;. In this scenario, particles (with mass) are pulled along by the expansion of the hyper-sphere, this then requires particles to have an axis; generically labeled N-S, with the N denoting the direction of particle travel within the hyper-sphere. Changing the direction of travel involves changing the orientation of the particle N-S axis. We can link that external N--S axis to the internal monopole (DDD) geometry, and from this show how the three internal phases produce the spin-½ transformation law under spatial rotations about the N-S direction &lt;ref&gt;Macleod, M.J. {{Cite journal |title= 7. Geometric Origin of Quarks and Spin, the Mathematical Electron extended |journal=RG | doi=10.13140/RG.2.2.21695.16808/1 }}&lt;/ref&gt;.


==Article series==
* [[https://simulationuniverse.org/ simulationuniverse.org]]: Home page

===The 3 levels===
A model overview from the sub Planck scale to the container universe
* [[https://doi.org/10.13140/RG.2.2.36449.26723 Planck unit scaffolding to Cosmic Microwave Background correlation]]: Part 1. The Cosmological scale
* [[https://doi.org/10.13140/RG.2.2.12830.09283/1  The Minimal Complexity Algorithm of the Planck Scaffolding]]: Part 2. The sub-Planck scale
* [[https://doi.org/10.13140/RG.2.2.32962.75200/1 The Heavens and the Free Will postulate]]: Part 3. The Heavens (the container universe)

===Wiki series===
Introductions to principle concepts (for more complete discussion see General articles)
* [[User:Platos_Cave_(physics)/Simulation_Hypothesis/Planck_units_(geometrical)]]: Planck units MLTPA as geometrical objects
* [[User:Platos_Cave_(physics)/Simulation_Hypothesis/Gravity_via_Atomic_orbitals]]: Gravity as a function of atomic orbitals 
* [[User:Platos_Cave_(physics)/Simulation_Hypothesis/Relativity]]: Relativity as a translation between 2 co-ordinate systems
* [[User:Platos_Cave_(physics)/Simulation_Hypothesis/Planck_unit_scaffolding]]: CMB and a Planck unit universe scaffolding
* [[User:Platos_Cave_(physics)/Simulation_Hypothesis/Sqrt_Planck_momentum]]: Link between charge and mass
* [[User:Platos_Cave_(physics)/Simulation_Hypothesis/God_(programmer)]]: Introduction to a Planck scale Programmer God Simulation Hypothesis model

===General articles===
The following are HTML transcripts of journal articles
* [[https://simulationuniverse.org/1-Planck-unit-CMB.html 1-Planck-unit-CMB.html]]: Constructs the universe frame from Planck units 
* [[https://simulationuniverse.org/2-Relativity-hypersphere.html 2-Relativity-hypersphere.html]]: Relativity as the mathematics of perspective 
* [[https://simulationuniverse.org/3-Gravitational-orbitals.html 3-Gravitational-orbitals.html]]: Gravity as sum of n-body rotating orbital particle-particle pairs
* [[https://simulationuniverse.org/4-Atomic-orbitals.html 4-Atomic-orbitals.html]]: Atomic orbitals as single rotating orbital particle-particle pairs
* [[https://simulationuniverse.org/5-w_axis.html 5-w_axis.html]]: Imaginary number axis (radiation domain)
* [[https://simulationuniverse.org/6-Physical-constant-anomalies.html 6-Physical-constant-anomalies.html]]: Statistical analysis of physical constant anomalies
* [[https://simulationuniverse.org/7-Monopole-quarks.html 7-Monopole-quarks.html]]: Quarks as monopoles
* [[https://simulationuniverse.org/8-Holographic-universe.html 8-Holographic-universe.html]]: Hypersphere surface as 2-D analogue


== External links == 
* [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 (Book)
* [https://link.springer.com/article/10.1134/S0202289308020011/ Dirac-Kerr-Newman black-hole electron] -Alexander Burinskii (article)
* [https://www.imdb.com/title/tt0133093/ The Matrix, (1999)]
* [https://plato.stanford.edu/entries/pythagoras/ Pythagoras &quot;all is number&quot;] - Stanford University
* [[w:Simulation Hypothesis | Simulation Hypothesis]]
* [[w:Mathematical universe hypothesis | Mathematical universe hypothesis]]
* [[w:Philosophy of mathematics | Philosophy of mathematics]]
* [[w:Philosophy of physics | Philosophy of physics]]
* [[w:Platonism | Platonism]]


== References == 
{{Reflist}}


[[Category: Physics]]
[[Category: Philosophy of science]]
__INDEX__</text>
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      <text bytes="1149" sha1="qclry2dkzs2nmur8un7mprvyvf8melo" xml:space="preserve">{{Course search|style=image}}

During human digestion, proteins are broken down in the stomach to smaller polypeptide chains via hydrochloric acid and protease actions. This is crucial for the absorption of the essential amino acids that cannot be biosynthesized by the body.

=== In The Mouth ===

==== Chemical Digestion ====
Saliva contains carbohydrases. Carbohydrates is partially digested by carbohydrates into partially digested carbohydrates.

==== Physical Digestion ====
Teeth and jaw muscles cut and crush food into smaller pieces.

=== In the Oesopohagus ===

==== Chemical Digestion ====
Gullet does not release any digestive juice. Carbohydrates from mouth continues to be digested by carbohydrases that are mixed with the food.

==== Physical Digestion ====
Peristalsis&lt;ref&gt;'''Peristalsis''' is a series of wave-like muscle contractions that move food through the digestive system.&lt;/ref&gt; physically breaks down food into smaller pieces.

在人类消化过程中，蛋白质通过盐酸和蛋白酶作用在胃中分解成较小的多肽链。 这对于吸收不能被身体生物合成的必需氨基酸至关重要。

{{CourseCat}}</text>
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      <text bytes="7786" sha1="tm6oifwrs2sakkks9r115fs1fwa67mo" xml:space="preserve">{{Original research}}


In the [[v:User:Platos_Cave_(physics)/Simulation_Hypothesis/Electron_(mathematical) |mathematical electron]] &lt;ref&gt;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 }}&lt;/ref&gt; model, the [[w:Planck units |Planck units]] are assigned geometrical objects, themselves the geometry of alpha (the [[w:fine-structure constant | fine structure constant '''α''']]) and 2 mathematical constants; Omega and pi. These geometrical objects (mass M = 1, time T = π, P = Ω ...) embed their unit function (the geometry of the length object L embeds the function length and so on ...). By converting the physical constants (''h'', ''c'', ''e'', ''m''&lt;sub&gt;e&lt;/sub&gt; ...) also to geometrical forms, and as these forms are also the geometry of alpha and Omega, then we can reverse engineer them and find a solution for alpha. Using CODATA 2014 the best fit alpha = 1/137.0359931388... 


Key:
:&lt;math&gt;\Omega = \sqrt{ \left(\pi^e e^{(1-e)}\right)}  = 2.007\;134\;9543... &lt;/math&gt;
:&lt;math&gt;\frac{ P^{15} T^2}{M^{12}} = \pi^2 \Omega^{15},\;\;\text{units} = \left(\frac{{kg m}}{{s}}\right)^{15/2} \times \frac{{s}^2}{{kg}^{12}} = \frac{{m}^{15/2}}{{kg}^{9/2} \,{s}^{11/2}} = 1&lt;/math&gt; (natural Planck units)
:&lt;math&gt;v = 11 843 707.905 ...,\; units = \frac{m}{s}&lt;/math&gt;
:&lt;math&gt;r = 0.712 562 514 304 ...,\; units = (\frac{kg.m}{s})^{1/4}&lt;/math&gt;
{| class=&quot;wikitable&quot;
|+ Table 1. Geometrical objects (reference table)
! Attribute
! Formula
! Object
! Scalars
! Numerical
! &lt;math&gt;\theta&lt;/math&gt;
! [[v:User:Platos_Cave_(physics)/Simulation_Hypothesis/Planck_units_(geometrical)#Dimensionless_f(x) |Units*]]
|-
| M (mass) 
| ''fundamental'' 
| &lt;math&gt;(1)&lt;/math&gt;
| &lt;math&gt;\dfrac{r^4}{v}&lt;/math&gt;
| 0.217672825e-7
| &lt;math&gt;u^{15}&lt;/math&gt;
| &lt;math&gt;kg&lt;/math&gt;
|-
| T (time) 
| ''fundamental''  
| &lt;math&gt;(\pi)&lt;/math&gt;
| &lt;math&gt;\dfrac{r^9}{v^6}\;j^2&lt;/math&gt;
| 0.539051838e-43
| &lt;math&gt;u^{-30}&lt;/math&gt;
| &lt;math&gt;s&lt;/math&gt;
|-
| P (sqrt of momentum)  
| ''fundamental'' 
| &lt;math&gt;(\Omega)&lt;/math&gt;
| &lt;math&gt;r^2&lt;/math&gt;
| 1.019113428
| &lt;math&gt;u^{16}&lt;/math&gt;
| &lt;math&gt;\sqrt{\frac{kg\;m}{s}}&lt;/math&gt;
|-
| V (velocity)  
| &lt;math&gt;\dfrac{2\pi P^2}{M}&lt;/math&gt; 
| &lt;math&gt;(2\pi\Omega^2)&lt;/math&gt;
| &lt;math&gt;v&lt;/math&gt;
| 299792458
| &lt;math&gt;u^{17}&lt;/math&gt;
| &lt;math&gt;\frac{m}{s}&lt;/math&gt;
|-
| L (length)  
| &lt;math&gt;VT &lt;/math&gt; 
| &lt;math&gt;(2\pi^2\Omega^2)&lt;/math&gt;
| &lt;math&gt;\dfrac{r^9}{v^5}\;j^2&lt;/math&gt;
| 0.161603675e-34
| &lt;math&gt;u^{-13}&lt;/math&gt;
| &lt;math&gt;m&lt;/math&gt;
|-
| A (ampere) 
| &lt;math&gt;\dfrac{2^4 V^3 \alpha}{P^3}&lt;/math&gt; 
| &lt;math&gt;(2^7\pi^3 \alpha \Omega^3)&lt;/math&gt;
| &lt;math&gt;\dfrac{v^3}{r^6}&lt;/math&gt;
| 0.297221257e25
| &lt;math&gt;u^{3}&lt;/math&gt;
| &lt;math&gt;\frac{m^{3/2}}{kg^{3/2} s^{3/2}}&lt;/math&gt;
|-
| K (temperature) 
| &lt;math&gt;\dfrac{AV}{2\pi}&lt;/math&gt; 
| &lt;math&gt;(2^7 \pi^3 \alpha \Omega^5)&lt;/math&gt;
| &lt;math&gt;\dfrac{v^4}{r^6}&lt;/math&gt;
| 0.141814520e33
| &lt;math&gt;u^{20}&lt;/math&gt;
| &lt;math&gt;\left(\frac{A \; m}{s}\right)&lt;/math&gt;
|}


=== Diophantine System with Constraints ===

We seek integer solutions for the unit-number exponents &lt;math&gt;(\theta_P, \theta_T, \theta_M)&lt;/math&gt; satisfying the constraints derived from the lattice invariants:

&lt;math&gt;
\begin{cases}
15\theta_P + 2\theta_T - 12\theta_M = 0 \\
2\theta_M + \theta_T = 0
\end{cases}
&lt;/math&gt;

with the additional conditions:
* &lt;math&gt;\theta_P, \theta_T, \theta_M \in \mathbb{Z} \setminus \{0\}&lt;/math&gt; (non-zero integers),
* &lt;math&gt;-30 \le \theta_P, \theta_T, \theta_M \le 30&lt;/math&gt;,
* &lt;math&gt;\theta_P, \theta_T, \theta_M&lt;/math&gt; are all distinct.

=== Algebraic solution ===

From the second equation we obtain
&lt;math&gt;\theta_T = -2\theta_M.&lt;/math&gt;

Substitute this into the first equation:
&lt;math&gt;15\theta_P + 2(-2\theta_M) - 12\theta_M = 0 \;\Longrightarrow\; 15\theta_P - 4\theta_M - 12\theta_M = 0 \;\Longrightarrow\; 15\theta_P = 16\theta_M.&lt;/math&gt;

Since &lt;math&gt;\gcd(15,16)=1&lt;/math&gt;, we must have &lt;math&gt;\theta_P = 16k&lt;/math&gt; and &lt;math&gt;\theta_M = 15k&lt;/math&gt; for some integer &lt;math&gt;k&lt;/math&gt;.
Thus,
&lt;math&gt;\theta_P = 16k,\qquad \theta_M = 15k,\qquad \theta_T = -2(15k) = -30k.&lt;/math&gt;

=== Applying bounds and distinctness ===

The bounds &lt;math&gt;-30 \le \theta_P, \theta_T, \theta_M \le 30&lt;/math&gt; require
&lt;math&gt;|16k| \le 30,\quad |15k| \le 30,\quad |{-30k}| \le 30,&lt;/math&gt;
so &lt;math&gt;|k| \le 1&lt;/math&gt;.  Since all variables are non-zero, &lt;math&gt;k \neq 0&lt;/math&gt;. Hence &lt;math&gt;k = \pm 1&lt;/math&gt;.

For &lt;math&gt;k=1&lt;/math&gt;:
&lt;math&gt;\theta_P = 16,\quad \theta_T = -30,\quad \theta_M = 15.&lt;/math&gt;

For &lt;math&gt;k=-1&lt;/math&gt;:
&lt;math&gt;\theta_P = -16,\quad \theta_T = 30,\quad \theta_M = -15.&lt;/math&gt;

In both triples &lt;math&gt;\theta_P \neq \theta_T&lt;/math&gt;, &lt;math&gt;\theta_T \neq \theta_M&lt;/math&gt;, &lt;math&gt;\theta_P \neq \theta_M&lt;/math&gt;, so the distinctness condition is satisfied.

=== Conclusion ===

The only integer solutions fulfilling all constraints are

&lt;math&gt;{(\theta_P, \theta_T, \theta_M) = (16,\,-30,\,15) \quad\text{and}\quad (-16,\,30,\,-15)}.&lt;/math&gt;

The positive solution &lt;math&gt;(16, -30, 15)&lt;/math&gt; corresponds exactly to the unit numbers presented in Table 1.


==Article series==
* [[https://simulationuniverse.org/ simulationuniverse.org]]: Home page

===The 3 levels===
A model overview from the sub Planck scale to the container universe
* [[https://doi.org/10.13140/RG.2.2.36449.26723 Planck unit scaffolding to Cosmic Microwave Background correlation]]: Part 1. The Cosmological scale
* [[https://doi.org/10.13140/RG.2.2.12830.09283/1  The Minimal Complexity Algorithm of the Planck Scaffolding]]: Part 2. The sub-Planck scale
* [[https://doi.org/10.13140/RG.2.2.32962.75200/1 The Heavens and the Free Will postulate]]: Part 3. The Heavens (the container universe)

===Wiki series===
Introductions to principle concepts (for more complete discussion see General articles)
* [[User:Platos_Cave_(physics)/Simulation_Hypothesis/Planck_units_(geometrical)]]: Planck units MLTPA as geometrical objects
* [[User:Platos_Cave_(physics)/Simulation_Hypothesis/Gravity_via_Atomic_orbitals]]: Gravity as a function of atomic orbitals 
* [[User:Platos_Cave_(physics)/Simulation_Hypothesis/Relativity]]: Relativity as a translation between 2 co-ordinate systems
* [[User:Platos_Cave_(physics)/Simulation_Hypothesis/Planck_unit_scaffolding]]: CMB and a Planck unit universe scaffolding
* [[User:Platos_Cave_(physics)/Simulation_Hypothesis/Sqrt_Planck_momentum]]: Link between charge and mass
* [[User:Platos_Cave_(physics)/Simulation_Hypothesis/God_(programmer)]]: Introduction to a Planck scale Programmer God Simulation Hypothesis model

===General articles===
The following are HTML transcripts of journal articles
* [[https://simulationuniverse.org/1-Planck-unit-CMB.html 1-Planck-unit-CMB.html]]: Constructs the universe frame from Planck units 
* [[https://simulationuniverse.org/2-Relativity-hypersphere.html 2-Relativity-hypersphere.html]]: Relativity as the mathematics of perspective 
* [[https://simulationuniverse.org/3-Gravitational-orbitals.html 3-Gravitational-orbitals.html]]: Gravity as sum of n-body rotating orbital particle-particle pairs
* [[https://simulationuniverse.org/4-Atomic-orbitals.html 4-Atomic-orbitals.html]]: Atomic orbitals as single rotating orbital particle-particle pairs
* [[https://simulationuniverse.org/5-w_axis.html 5-w_axis.html]]: Imaginary number axis (radiation domain)
* [[https://simulationuniverse.org/6-Physical-constant-anomalies.html 6-Physical-constant-anomalies.html]]: Statistical analysis of physical constant anomalies
* [[https://simulationuniverse.org/7-Monopole-quarks.html 7-Monopole-quarks.html]]: Quarks as monopoles
* [[https://simulationuniverse.org/8-Holographic-universe.html 8-Holographic-universe.html]]: Hypersphere surface as 2-D analogue


==References==
{{Reflist}}


[[Category:Physics| ]]
[[Category:Philosophy of science| ]]</text>
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{{Motivation and emotion/Tutorials/In development}}
&lt;!-- {{Motivation and emotion/Tutorials/Complete2}} --&gt;
&lt;!-- {{Motivation and emotion/Tutorials/In development}} --&gt;
&lt;!-- {{Note|For more info, see [[Motivation and emotion/Tutorials/Introduction]]}} --&gt;
==Overview==
This tutorial:
# explains how [[Motivation and emotion/Tutorials|tutorials]] will work and what will be covered
# shows how to select a topic for the [[Motivation and emotion/Assessment/Major project|major project]]
# involves students in sharing about their selected topic or interests
# demonstrates how to add a template to help scaffold the topic development

==Welcome==
# Welcome and tutor introduction
# Tutorial content will typically involve:
## Discussion of key concepts from the textbook and lecture
## Group activities that expand and explore key concepts
## Project work which develops skills for the topic development, book chapter, and exam
# Overview of [[Motivation and emotion/Tutorials|tutorial topics]]

;Virtual room tour:
# Chat
# Microphone (headset recommended) / webcam
# Raise hand
# Status and feedback
# Whiteboard tools
# Breakout rooms
# Polls

==Icebreaker==
These get-to-know-you activities provide an opportunity to meet and greet and start to explore some examples of motivations and emotions.

;Virtual
# Speedfriending: 3 minutes to meet, greet, and share about what you're looking forward to (or not) in [[motivation and emotion]]

;In-person
# [[w:sociometry|sociometric]] exercise or whiteboard discussion:
## What job/career do you hope to be doing in 5 to 10 years' time? (examples of motivations)
## How have you been feeling so far today? (Mood/Emotion) (examples of emotions)
&lt;!-- Ask participants to arrange themselves in a line or groups which represent our similarities and differences in motivation and emotion. For each question:
* Ask participants to introduce themselves to the people next to them / in their group.
* Link the sociometric data and responses to concepts in motivation and emotion.
--&gt;

==Topic check-in and whiparound==

===Check-in===
Poll - What stage are you up to?
# Huh? What do I do?
# I created a Wikiversity user account
# I signed up to a book topic
# I have a topic and started editing
# I'm all over it!

===Whiparound===
#[https://goalbookapp.com/toolkit/v/strategy/whiparound Whiparound] discussion based on each person's book chapter interests and progress. Ask each student to share:
## their name
## topic they've chosen (or what areas they are interested in)
# According to each person's stage of development:
## Very briefly discuss or expand on each topic, suggest related resources or ideas etc.
## If students are unsure about their interests, probe about what they are curious about - or at least find out, are they more interested in motivation or emotion etc.?
# Key points:
## Topics are typically allocated to either motivation or emotion (with a few listed under motivation and emotion). Be aware of which broad category the selected topic fits into.
## Some topics are broader (requiring a zoomed out overview without getting too deep in the weeds) and others are narrower (requiring more comprehensive, indepth coverage).
## Broader topics can be overwhelming if there is a lot of material; an option in this case is to [[Motivation and emotion/Assessment/Selection#New topics|negotiate]] a narrowing of the topic title and/or sub-title.
## Narrower topics for which there is a lack of sufficient literature may benefit from negotiating a broadening of the topic title and/or sub-title.

==Project tasks==

===Create a Wikiversity account===
# [[Create an account|Create an account]]:
## Register a user name and password. Include an email for password recovery.
## Edits to Wikiversity are publicly available, so choose a name that suits your privacy requirements (e.g., can be a real name, pseudonym, or student number).
## Start at http://en.wikiversity.org (English Wikiversity) - Create account (top-right)
## Or go direct to: [[Special:Userlogin/signup|create a Wikiversity user account]]
# Log in (top-right)

===Topic selection===
# Topic selection is [[wikt:first in, best dressed|first in, best dressed]]
# Sign up to a pre-approved topic in the {{Motivation and emotion/Book}} by:
## Clicking &quot;Edit&quot; or &quot;Edit source&quot;
## Replace &quot;User name&quot; alongside your topic of choice with your Wikiversity user name
## Publish the page
## Alternatively, [[Motivation and emotion/Assessment/Selection#New topics|negotiate a topic]]

===Quick start template===
# It is recommended (but not required) that you add the [[Template:Motivation and emotion/Book chapter structure|quick start template]] to the topic page.
# To do this, click &quot;Create source&quot; or &quot;Edit source&quot; on the target page, paste '''[[Template:Motivation and emotion/Book chapter structure|&lt;nowiki&gt;{{subst:ME/BCS}}&lt;/nowiki&gt;]]''', then Publish.
## If you get a spam filter warning it is probably because you have a new account. You can ask the tutor to add the template for you. Or publish some smaller edits first (e.g., on your user page). This helps the system learn to trust you, then try inserting the template again.

==Next steps==
# Sign up for (or [[Motivation and emotion/Assessment/Selection#New topics|negotiate]]) a book chapter topic by the end of Week 2.
# Draft the [[Motivation and emotion/Assessment/Topic|topic development]] which is a detailed plan for a full chapter, consisting of main headings with bullet-points about key points, on Wikiversity by the end of Week 3.

==Recording==
&lt;!-- * [ Tutorial 1] (2026) --&gt;
* [https://au-lti.bbcollab.com/recording/71cf60b37faf4d40b17f9107a417e0ee Tutorial 1] (2025)&lt;!--
* [https://au-lti.bbcollab.com/recording/81cb0a731ef44fe1ab82b529fd3a836a Tutorial 1] (2024)
* [https://au-lti.bbcollab.com/recording/ace491a49cdd4d8f8441c7b357fa60dc Tutorial 1] (2023)
* [https://au-lti.bbcollab.com/recording/91c93320558847a8a6717e2a0ad4b6dd Tutorial 1 recording] (2022)
* [https://au-lti.bbcollab.com/recording/332ed2ad09824fb88c799b92cd64eaaa Tutorial 1 recording] (2021) --&gt;

==See also==
; Topic selection
* [[Motivation and emotion/Assessment/Selection|Topic selection guidelines]]

;Topic development
* [[Motivation and emotion/Assessment/Topic|Topic development guidelines]]
* [[Motivation and emotion/Book/2026|Book chapter table of contents]] (2026)

;Lecture
* [[Motivation and emotion/Lectures/Introduction|Introduction]]

;Tutorial
* [[{{#titleparts:{{PAGENAME}}|2}}/Wiki editing|Wiki editing]] (Next tutorial)
&lt;!--
;Admin
* [[/Instructor notes/]]
* [[Introduction to psychology/Psy102/Tutorials/Motivation|Motivation tutorial]] (Psychology 102)
--&gt;

==External links==
* [https://cogniti.canberra.edu.au/agents/6a3caf85b2fed9a95789241a/chat?k=WfY1yx6AB7yF_9XqiyYENzUBNCGfV1e2FCBodCDD2xU Chat with the motivation and emotion book chapter topic generator] (Cogniti)

&lt;noinclude&gt;{{Motivation and emotion/Tutorials/Navigation}}&lt;/noinclude&gt;
[[Category:Motivation and emotion/Tutorials/Topic selection]]</text>
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'''Natural Planck units as geometrical objects (the mathematical electron model)'''

The physical constants form the scaffolding around which the theories of physics are erected, and they define the fabric of our universe, but science has no idea why they take the special numerical values that they do, for these constants follow no discernible pattern. The desire to explain the constants has been one of the driving forces behind efforts to develop a complete unified description of nature, or &quot;theory of everything&quot;. Physicists have hoped that such a theory would show that each of the constants of nature ''could have only one logically possible value''. It would reveal an underlying order to the seeming arbitrariness of nature &lt;ref&gt;J. Barrow, J. Webb {{Cite journal |title= Inconsistent constants |journal=Scientific American |volume=292 |pages=56 |date=2005}}&lt;/ref&gt;.


In the [[v:User:Platos_Cave_(physics)/Simulation_Hypothesis/Electron_(mathematical) |mathematical electron]] &lt;ref&gt;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 }}&lt;/ref&gt; model, the [[w:Planck units |Planck units]] are assigned geometrical objects, themselves the geometry of alpha (the [[w:fine-structure constant | fine structure constant '''α''']]) and 2 mathematical constants; Omega and pi. These geometrical objects (mass M = 1, time T = π, P = Ω ...) embed their unit function (the geometry of the length object L embeds the function length and so on ...). By converting the physical constants (''h'', ''c'', ''e'', ''m''&lt;sub&gt;e&lt;/sub&gt; ...) also to geometrical forms, and as these forms are also the geometry of alpha and Omega, then we can reverse engineer them and find a solution for alpha; using CODATA 2014 the best fit alpha = 1/137.0359931388... 

As the geometries of dimensionless constants, these objects are also dimensionless and so are independent of any system of units, and of any numerical system, and so could qualify as &quot;natural units&quot; ;
{{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 &quot;natural units&quot;... -Max Planck &lt;ref&gt;Planck (1899), p. 479.&lt;/ref&gt;&lt;ref name=&quot;TOM&quot;&gt;*Tomilin, K. A., 1999, &quot;[http://www.ihst.ru/personal/tomilin/papers/tomil.pdf Natural Systems of Units: To the Centenary Anniversary of the Planck System]&quot;, 287–296.&lt;/ref&gt;}}  


=== Planck unit limitations ===
The SI Planck units are measured; [[w:Planck mass |Planck mass]] in ''kg'', [[w:Planck length |Planck length]] in ''m'', [[w:Planck time |Planck time]] in ''s'' ... . These units have numerical values, this precludes finding a relation between these SI units, for numerical values are simply dimensionless frequencies, 299792458 could refer to the speed of light 299792458m/s or equally to the number of apples in a container (299792458 apples). Numbers such as 299792458 carry no unit-specific information, and so we are required to add a descriptive (such as m/s). Without foundations, any coincidences can be dismissed as numerology and the base units (kg, m, s, A...) become independent units by default.


=== Geometrical objects ===
This inherent restriction can be resolved by assigning to each unit a geometrical object for which the geometry embeds the attribute (for example, the geometry of the time object T embeds the function time and so a descriptive unit ''s'' = seconds is not required). We may then combine these objects Lego-style to form more complex objects; from electrons to galaxies, while still retaining the underlying attributes (of mass M, wavelength L, frequency T ...). An apple has mass because its 'geometry' includes the geometrical object for mass M.

We define a physical constant as a constant embedded within the universe &quot;source code&quot; (it is a given). A mathematical constant however can be derived within the boundaries of the universe itself; π for example can be constructed from integers in series. 
&lt;math display=&quot;block&quot;&gt;\sum_{n=1}^\infty \frac{1}{n^2} = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \cdots. &lt;/math&gt;

The mathematical constant Omega is built around a base-15 geometry, this is a requirement to construct the full set of Planck units. Omega itself is the geometry of π and Euler's number [[w:E_(mathematical_constant) |e]] = 2.718281828459... (which can also be constructed by integers in series);
:&lt;math&gt;\Omega = \sqrt{ \left(\pi^e e^{(1-e)}\right)}  = 2.007\;134\;9543... &lt;/math&gt;


=== Base 15 geometry ===
Key:
:&lt;math&gt;\frac{ P^{15} T^2}{M^{12}} = \pi^2 \Omega^{15},\;\;\text{units} = \left(\frac{{kg m}}{{s}}\right)^{15/2} \times \frac{{s}^2}{{kg}^{12}} = \frac{{m}^{15/2}}{{kg}^{9/2} \,{s}^{11/2}} = 1&lt;/math&gt; (natural Planck units)
:&lt;math&gt;v = 11 843 707.905 ...,\; units = \frac{m}{s}&lt;/math&gt;
:&lt;math&gt;r = 0.712 562 514 304 ...,\; units = (\frac{kg.m}{s})^{1/4}&lt;/math&gt;
{| class=&quot;wikitable&quot;
|+ Table 1. Geometrical objects (reference table)
! Attribute
! Formula
! Object
! Scalars
! Numerical
! &lt;math&gt;\theta&lt;/math&gt;
! [[v:User:Platos_Cave_(physics)/Simulation_Hypothesis/Planck_units_(geometrical)#Dimensionless_f(x) |Units*]]
|-
| M (mass) 
| ''fundamental'' 
| &lt;math&gt;(1)&lt;/math&gt;
| &lt;math&gt;\dfrac{r^4}{v}&lt;/math&gt;
| 0.217672825e-7
| &lt;math&gt;u^{15}&lt;/math&gt;
| &lt;math&gt;kg&lt;/math&gt;
|-
| T (time) 
| ''fundamental''  
| &lt;math&gt;(\pi)&lt;/math&gt;
| &lt;math&gt;\dfrac{r^9}{v^6}\;j^2&lt;/math&gt;
| 0.539051838e-43
| &lt;math&gt;u^{-30}&lt;/math&gt;
| &lt;math&gt;s&lt;/math&gt;
|-
| P (sqrt of momentum)  
| ''fundamental'' 
| &lt;math&gt;(\Omega)&lt;/math&gt;
| &lt;math&gt;r^2&lt;/math&gt;
| 1.019113428
| &lt;math&gt;u^{16}&lt;/math&gt;
| &lt;math&gt;\sqrt{\frac{kg\;m}{s}}&lt;/math&gt;
|-
| V (velocity)  
| &lt;math&gt;\dfrac{2\pi P^2}{M}&lt;/math&gt; 
| &lt;math&gt;(2\pi\Omega^2)&lt;/math&gt;
| &lt;math&gt;v&lt;/math&gt;
| 299792458
| &lt;math&gt;u^{17}&lt;/math&gt;
| &lt;math&gt;\frac{m}{s}&lt;/math&gt;
|-
| L (length)  
| &lt;math&gt;VT &lt;/math&gt; 
| &lt;math&gt;(2\pi^2\Omega^2)&lt;/math&gt;
| &lt;math&gt;\dfrac{r^9}{v^5}\;j^2&lt;/math&gt;
| 0.161603675e-34
| &lt;math&gt;u^{-13}&lt;/math&gt;
| &lt;math&gt;m&lt;/math&gt;
|-
| A (ampere) 
| &lt;math&gt;\dfrac{2^4 V^3 \alpha}{P^3}&lt;/math&gt; 
| &lt;math&gt;(2^7\pi^3 \alpha \Omega^3)&lt;/math&gt;
| &lt;math&gt;\dfrac{v^3}{r^6}&lt;/math&gt;
| 0.297221257e25
| &lt;math&gt;u^{3}&lt;/math&gt;
| &lt;math&gt;\frac{m^{3/2}}{kg^{3/2} s^{3/2}}&lt;/math&gt;
|-
| K (temperature) 
| &lt;math&gt;\dfrac{AV}{2\pi}&lt;/math&gt; 
| &lt;math&gt;(2^7 \pi^3 \alpha \Omega^5)&lt;/math&gt;
| &lt;math&gt;\dfrac{v^4}{r^6}&lt;/math&gt;
| 0.141814520e33
| &lt;math&gt;u^{20}&lt;/math&gt;
| &lt;math&gt;\left(\frac{A \; m}{s}\right)&lt;/math&gt;
|}


==== Diophantine System with Constraints ====

We seek integer solutions for the unit-number exponents &lt;math&gt;(\theta_P, \theta_T, \theta_M)&lt;/math&gt; satisfying the constraints derived from the lattice invariants:

&lt;math&gt;
\begin{cases}
15\theta_P + 2\theta_T - 12\theta_M = 0 \\
2\theta_M + \theta_T = 0
\end{cases}
&lt;/math&gt;

with the additional conditions:
* &lt;math&gt;\theta_P, \theta_T, \theta_M \in \mathbb{Z} \setminus \{0\}&lt;/math&gt; (non-zero integers),
* &lt;math&gt;-30 \le \theta_P, \theta_T, \theta_M \le 30&lt;/math&gt;,
* &lt;math&gt;\theta_P, \theta_T, \theta_M&lt;/math&gt; are all distinct.

==== Algebraic solution ====

From the second equation we obtain
&lt;math&gt;\theta_T = -2\theta_M.&lt;/math&gt;

Substitute this into the first equation:
&lt;math&gt;15\theta_P + 2(-2\theta_M) - 12\theta_M = 0 \;\Longrightarrow\; 15\theta_P - 4\theta_M - 12\theta_M = 0 \;\Longrightarrow\; 15\theta_P = 16\theta_M.&lt;/math&gt;

Since &lt;math&gt;\gcd(15,16)=1&lt;/math&gt;, we must have &lt;math&gt;\theta_P = 16k&lt;/math&gt; and &lt;math&gt;\theta_M = 15k&lt;/math&gt; for some integer &lt;math&gt;k&lt;/math&gt;.
Thus,
&lt;math&gt;\theta_P = 16k,\qquad \theta_M = 15k,\qquad \theta_T = -2(15k) = -30k.&lt;/math&gt;

==== Applying bounds and distinctness ====

The bounds &lt;math&gt;-30 \le \theta_P, \theta_T, \theta_M \le 30&lt;/math&gt; require
&lt;math&gt;|16k| \le 30,\quad |15k| \le 30,\quad |{-30k}| \le 30,&lt;/math&gt;
so &lt;math&gt;|k| \le 1&lt;/math&gt;.  Since all variables are non-zero, &lt;math&gt;k \neq 0&lt;/math&gt;. Hence &lt;math&gt;k = \pm 1&lt;/math&gt;.

For &lt;math&gt;k=1&lt;/math&gt;:
&lt;math&gt;\theta_P = 16,\quad \theta_T = -30,\quad \theta_M = 15.&lt;/math&gt;

For &lt;math&gt;k=-1&lt;/math&gt;:
&lt;math&gt;\theta_P = -16,\quad \theta_T = 30,\quad \theta_M = -15.&lt;/math&gt;

In both triples &lt;math&gt;\theta_P \neq \theta_T&lt;/math&gt;, &lt;math&gt;\theta_T \neq \theta_M&lt;/math&gt;, &lt;math&gt;\theta_P \neq \theta_M&lt;/math&gt;, so the distinctness condition is satisfied.

==== Conclusion ====

The only integer solutions fulfilling all constraints are

&lt;math&gt;{(\theta_P, \theta_T, \theta_M) = (16,\,-30,\,15) \quad\text{and}\quad (-16,\,30,\,-15)}.&lt;/math&gt;

The positive solution &lt;math&gt;(16, -30, 15)&lt;/math&gt; corresponds exactly to the unit numbers presented in Table 1.


=== Scalars ===
As alpha and Omega can be assigned numerical values, so too the MLTA objects can be expressed numerically. We can then convert these objects to their Planck unit equivalents by including a dimensioned scalar. For example; 

:If we use scalar ''k''&lt;sub&gt;SI&lt;/sub&gt; to convert ''M'' (M=1) to the SI Planck mass (M*''k''&lt;sub&gt;SI&lt;/sub&gt; = &lt;math&gt;m_P&lt;/math&gt;), then ''k''&lt;sub&gt;SI&lt;/sub&gt; = 0.2176728e-7kg ([[w:SI_units |SI units]])

:SI ''c'' = object V * scalar ''v''&lt;sub&gt;SI&lt;/sub&gt; = 299792458m/s ([[w:SI_units |SI units]])

:Imperial ''c'' = object V * scalar ''v''&lt;sub&gt;imp&lt;/sub&gt; = 186282miles/s ([[w:Imperial_units |imperial units]])


==== Scalar relationships ====
Scalars that translate to the SI unit system must therefore carry not only the numerical conversion values but also the unit (for scalars we will refer to the SI unit system by default), i.e.: scalar ''v'' = 11843707.905 m/s. This also means that the scalars must also follow the unit number relationship θ. 

{| class=&quot;wikitable&quot;
|+Table 2. Geometrical units
! Attribute
! Geometrical object
! Scalar 
! Unit ''u''&lt;sup&gt;θ&lt;/sup&gt;
|-
| mass 
| &lt;math&gt;M = (1)&lt;/math&gt;  
| ''k''
| &lt;math&gt;u^{15}&lt;/math&gt;
|-
| time 
| &lt;math&gt;T = (\pi)&lt;/math&gt;  
| ''t''
| &lt;math&gt;u^{-30}&lt;/math&gt;
|-
| [[v:User:Platos Cave (physics)/Simulation_Hypothesis/Sqrt_Planck_momentum | sqrt(momentum)]]
| &lt;math&gt;P = (\Omega)&lt;/math&gt;
| ''r''&lt;sup&gt;2&lt;/sup&gt;
| &lt;math&gt;u^{16}&lt;/math&gt;
|-
| velocity  
| &lt;math&gt;V = (2\pi\Omega^2)&lt;/math&gt;  
| ''v''
| &lt;math&gt;u^{17}&lt;/math&gt;
|-
| length  
| &lt;math&gt;L = (2\pi^2\Omega^2)&lt;/math&gt; 
| ''l''
| &lt;math&gt;u^{-13}&lt;/math&gt;
|-
| ampere 
| &lt;math&gt;A = (\frac{2^7 \pi^3 \Omega^3}{a})&lt;/math&gt; 
| ''q''
| &lt;math&gt;u^3&lt;/math&gt;
|}


Here are examples where units = 1, as such ''only 2 scalars are required'', for example, if we know the numerical value for ''q'' and for ''l'' then we know the numerical value for ''t'' ('''t = q&lt;sup&gt;3&lt;/sup&gt;l&lt;sup&gt;3&lt;/sup&gt;'''), and from ''l'' and ''t'' we know the value for ''k''. 

:&lt;math&gt;\frac{u^{3*3} u^{-13*3}}{u^{-30}}\;(\frac{q^3 l^3}{t}) = \frac{u^{-13*15}}{u^{15*9} u^{-30*11}} \;(\frac{l^{15}}{k^9 t^{11}}) = \;...\; =1&lt;/math&gt;


In other words, once any 2 scalars have been assigned values, the other scalars are then defined by default, consequently the CODATA 2014 values are used here as only 2 constants (c, [[w:permeability of vacuum|μ&lt;sub&gt;0&lt;/sub&gt;]]) are assigned exact values, following the [[w:2019 redefinition of SI base units|2019 redefinition of SI base units]] a total of 4 constants now have independently exact values assigned which is problematic in terms of this model. 


Scalars ''r'' (θ = 8) and ''v'' (θ = 17) are chosen here for demonstration as they can be derived directly from the 2 constants with exact values; ''c'' and ''μ&lt;sub&gt;0&lt;/sub&gt;''.
:&lt;math&gt;c = 299792458&lt;/math&gt; m/s
:&lt;math&gt;\mu_0 = 4\pi / 10^7&lt;/math&gt;
:&lt;math&gt;v = \frac{c}{2 \pi \Omega^2}= 11 843 707.905 ...,\; units = \frac{m}{s}&lt;/math&gt;
:&lt;math&gt;r^7 = \frac{2^{11} \pi^5 \Omega^4 \mu_0}{a};\; r = 0.712 562 514 304 ...,\; units = (\frac{kg.m}{s})^{1/4}&lt;/math&gt;

{| class=&quot;wikitable&quot;
|+Table 3. Geometrical objects
! attribute
! geometrical object
! unit number θ
! scalar r(8), v(17)
|-
| mass 
| &lt;math&gt;M = (1)&lt;/math&gt;
| 15 = 8*4-17
| &lt;math&gt;k = \frac{r^4}{v}&lt;/math&gt;
|-
| time 
| &lt;math&gt;T = (\pi)&lt;/math&gt;
|  -30 = 8*9-17*6 
| &lt;math&gt;t = \frac{r^9}{v^6}&lt;/math&gt;
|-
| sqrt(momentum)  
| &lt;math&gt;P = (\Omega)&lt;/math&gt;
| 16
| &lt;math&gt;r^2&lt;/math&gt;
|-
| velocity  
| &lt;math&gt;V = (2\pi\Omega^2)&lt;/math&gt;
| 17
| &lt;math&gt;v&lt;/math&gt;
|-
| length  
| &lt;math&gt;L = (2\pi^2\Omega^2)&lt;/math&gt; 
| -13 = 8*9-17*5
| &lt;math&gt;l = \frac{r^9}{v^5}&lt;/math&gt;
|-
| ampere 
| &lt;math&gt;A = (\frac{2^7 \pi^3 \Omega^3}{a})&lt;/math&gt; 
| 3 = 17*3-8*6
| &lt;math&gt;\frac{v^3}{r^6}&lt;/math&gt;
|}


{| class=&quot;wikitable&quot;
|+ Table 4. Comparison; SI and θ
! constant
! θ (SI unit)
! MLTVA
! scalar r(8), v(17)
|-
| ''c'' 
| &lt;math&gt;\frac{m}{s}&lt;/math&gt; (-13+30 = {{font color|red|white|17}})
| ''c*'' = &lt;math&gt;V*v&lt;/math&gt; 
| {{font color|red|white|17}}
|-
| ''h'' 
| &lt;math&gt;\frac{kg \;m^2}{s}&lt;/math&gt; (15-26+30={{font color|red|white|19}})
| ''h*'' = &lt;math&gt;2 \pi M V L * \frac{r^{13}}{v^5}&lt;/math&gt; 
| 8*13-17*5={{font color|red|white|19}}
|-
| ''G''
| &lt;math&gt;\frac{m^3}{kg \;s^2}&lt;/math&gt; (-39-15+60={{font color|red|white|6}})
| ''G*'' = &lt;math&gt;\frac{V^2 L}{M} * \frac{r^5}{v^2}&lt;/math&gt; 
| 8*5-17*2={{font color|red|white|6}}
|-
| ''e'' 
| &lt;math&gt;C = A s&lt;/math&gt; (3-30={{font color|red|white|-27}})
| ''e*'' = &lt;math&gt;A T * \frac{r^3}{v^3}&lt;/math&gt; 
| 8*3-17*3={{font color|red|white|-27}}
|-
| ''k&lt;sub&gt;B&lt;/sub&gt;'' 
| &lt;math&gt;\frac{kg \;m^2}{s^2 \;K}&lt;/math&gt; (15-26+60-20={{font color|red|white|29}})
| ''k&lt;sub&gt;B&lt;/sub&gt;*'' = &lt;math&gt;\frac{2 \pi V M}{A} * \frac{r^{10}}{v^3}&lt;/math&gt; 
| 8*10-17*3={{font color|red|white|29}}
|-
| ''μ&lt;sub&gt;0&lt;/sub&gt;'' 
| &lt;math&gt;\frac{kg \;m}{s^2 \;A^2}&lt;/math&gt; (15-13+60-6={{font color|red|white|56}})
| ''μ&lt;sub&gt;0&lt;/sub&gt;*'' = &lt;math&gt;\frac{4 \pi V^2 M}{a L A^2} * r^7&lt;/math&gt; 
| 8*7={{font color|red|white|56}}
|}


====CODATA 2014====
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&lt;sub&gt;B&lt;/sub&gt;''), consequently here we are using CODATA 2014 values. This is because only 2 dimensioned physical constants can be assigned exact values, once 2 constants (2 scalars) have been assigned values, then all other constants are defined by default. In CODATA 2014 2 constants have exact values; &lt;math&gt;c&lt;/math&gt; and the [[w:Vacuum permeability | vacuum permeability]] &lt;math&gt;\mu_0&lt;/math&gt;. 


=== Dimensionless f(x) ===
From our unit number relationship we can build a generic dimensionless formula f&lt;sub&gt;x&lt;/sub&gt;;

&lt;math&gt;f_x = \frac{kg^9 s^{11}}{m^{15}} = \frac{(\frac{r^4}{v})^9 (\frac{r^9}{v^6})^{11}}{(\frac{r^9}{v^5})^{15}} = 1&lt;/math&gt;

This f&lt;sub&gt;x&lt;/sub&gt;, although embedded within it are the dimensioned structures for mass, time and length (in the above ratio), according to the scalars it would be a dimensionless mathematical structure, units = 1. Thus we may create as much mass, time and length as we wish, the only proviso being that they are created in f&lt;sub&gt;x&lt;/sub&gt; style ratios, so that regardless of how massive, old and large our universe becomes, it is still in sum total dimensionless. 

Defining the dimensioned quantities ''r'', ''v'' in SI unit terms.

:&lt;math&gt;r = (\frac{kg\;m}{s})^{1/4}&lt;/math&gt;

:&lt;math&gt;v = \frac{m}{s}&lt;/math&gt;


Mass
:&lt;math&gt;\frac{r^4}{v} = (\frac{kg\;m}{s})\;(\frac{s}{m}) = kg&lt;/math&gt;


Length
:&lt;math&gt;(r^9)^4 = \frac{kg^9\;m^9}{s^9} &lt;/math&gt;

:&lt;math&gt;(\frac{1}{v^5})^4 = \frac{s^{20}}{m^{20}}&lt;/math&gt;

:&lt;math&gt;(\frac{r^9}{v^5})^4 = \frac{kg^9 s^{11}}{m^{11}} = m^4 \frac{kg^9 s^{11}}{m^{15}} = m^4 f_x = m^4&lt;/math&gt;


Time
:&lt;math&gt;(r^9)^4 = \frac{kg^9\;m^9}{s^9} &lt;/math&gt;

:&lt;math&gt;(\frac{1}{v^6})^4 = \frac{s^{24}}{m^{24}}&lt;/math&gt;

:&lt;math&gt;(\frac{r^9}{v^6})^4 = \frac{kg^9 s^{15}}{m^{15}} = s^4 \frac{kg^9 s^{11}}{m^{15}} = s^4 f_x = s^4&lt;/math&gt;


We may note that at the macro-level (of planets and stars) these f&lt;sub&gt;x&lt;/sub&gt; ratio are not found, and so this level is the domain of the observed physical universe, however at the quantum level, f&lt;sub&gt;x&lt;/sub&gt; ratio do appear, ψ&lt;sub&gt;electron&lt;/sub&gt; is an example, the mathematical and physical domains then blurring. This would also explain why physics can measure precisely the parameters of the electron (wavelength, mass ...), but has never found the electron itself.


=== Fine structure constant ===
Classically the fine structure constant can be expressed by this formula.

:&lt;math&gt;\frac{2 h}{\mu_0 e^2 c} = \color{red}\alpha^{-1} \color{black}&lt;/math&gt;


If we insert the geometrical analogues in this formula then alpha emerges, units and scalars cancel, serving to validate the unit number relationship and the geometries.

:&lt;math&gt;\frac{2 (h^*)}{(\mu_0^*) (e^*)^2 (c^*)} = 2({2^3 \pi^4 \Omega^4})/(\frac{a}{2^{11} \pi^5 \Omega^4})(\frac{2^{7} \pi^4 \Omega^3}{a})^2(2 \pi \Omega^2) = 
 \color{red}a \color{black}&lt;/math&gt;

:&lt;math&gt;units \;\frac{u^{19}}{u^{56} (u^{-27})^2 u^{17}} = 1&lt;/math&gt;

:&lt;math&gt;scalars \;(\frac{r^{13}}{v^5})(\frac{1}{r^7})(\frac{v^6}{r^6})(\frac{1}{v}) = 1&lt;/math&gt;

Thus proving that &lt;math&gt; \color{red}\alpha\color{black} = \color{red}\alpha \color{black}&lt;/math&gt;


=== Electron formula ===
{{main|User:Platos Cave (physics)/Simulation_Hypothesis/Electron (mathematical)}}
The ''electron object'' (''formula ψ'') is a mathematical particle (units and scalars cancel). 

:&lt;math&gt;\psi = 4\pi^2(2^6 3 \pi^2 a \Omega^5)^3 = .23895453...x10^{23}&lt;/math&gt; units = 1


In this example, embedded within the electron are the objects for charge, length and time ALT. AL as an ampere-meter (ampere-length) are the units for a [[w:magnetic monopole | magnetic monopole]].

:&lt;math&gt;T = \pi \frac{r^9}{v^6},\; u^{-30}&lt;/math&gt;

:&lt;math&gt;\sigma_{e} = \frac{3 a^2 A L}{2\pi^2} = {2^7 3 \pi^3 a \Omega^5}\frac{r^3}{v^2},\; u^{-10}&lt;/math&gt;

:&lt;math&gt;\psi = \frac{\sigma_{e}^3}{2 T} = \frac{(2^7 3 \pi^3 a \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&lt;/math&gt;


Associated with the electron are dimensioned parameters, these parameters however are a function of the MLTA units, the formula ψ dictating the frequency of these units.

[[w:electron mass | electron mass]]     &lt;math&gt;m_e^* = \frac{M}{\psi}&lt;/math&gt; (M as [[w:Planck mass | Planck mass]] = &lt;math&gt;\frac{r^4}{v})&lt;/math&gt; 

[[w:Compton wavelength | electron wavelength]] &lt;math&gt;\lambda_e^* = 2\pi L \psi&lt;/math&gt; (L as [[w:Planck length | Planck length]] = &lt;math&gt;2\pi\Omega^2\frac{r^9}{v^5})&lt;/math&gt;

[[w:elementary charge | elementary charge]]    &lt;math&gt;e^* = A\;T&lt;/math&gt; (T as [[w:Planck time | Planck time]]) = &lt;math&gt;\frac{2^7 \pi^4 \Omega^3}{\alpha}\frac{r^3}{v^3}&lt;/math&gt; 

[[w:Rydberg constant | Rydberg constant]] &lt;math&gt;R^* = (\frac{m_e}{4 \pi L a^2 M}) = \frac{1}{2^{23} 3^3 \pi^{11} a^5 \Omega^{17}}\frac{v^5}{r^9}\;u^{13}&lt;/math&gt; 


=== Omega ===
There is a natural number solution for Ω that is a square root implying that Ω can have a plus or a minus solution, and this agrees with the requirements of this theory (in the mass domain Ω occurs as Ω&lt;sup&gt;2&lt;/sup&gt; = plus only, in the charge domain Ω occurs as Ω&lt;sup&gt;3&lt;/sup&gt; = can be plus or minus; see [[v:User:Platos_Cave_(physics)/Simulation_Hypothesis/Sqrt_Planck_momentum | sqrt(momentum)]]). 

:&lt;math&gt;\Omega = \sqrt{ \left(\pi^e e^{(1-e)}\right)}  = 2.007\;134\;9543... &lt;/math&gt;

As we saw in table 1., Omega occurs within a base-15 geometry &lt;ref&gt;https://simulationuniverse.org/article-1b_tau_function.html Omega and the sub Planck scale&lt;/ref&gt;. 


=== Dimensionless combinations ===
According to the unit number relationship, we can also combine the physical constants in combinations where the unit numbers cancel, in this model these combinations are dimensionless, however they still retain SI units. If the model is correct (if the combinations are dimensionless) then the scalars will also have cancelled and numerically the solutions using CODATA and Geometrical objects will approach equality (barring uncertainties). These combinations can be used to test the veracity of the MLTA geometries as natural Planck units.


Example:

:&lt;math&gt;\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}} = &lt;/math&gt; {{font color|green|yellow| '''0.228 473 759... 10&lt;sup&gt;-58&lt;/sup&gt;'''}}

:&lt;math&gt;\frac{h^3}{e^{13} c^{24}} =&lt;/math&gt; {{font color|green|yellow| '''0.228 473 639... 10&lt;sup&gt;-58&lt;/sup&gt;'''}}, units = &lt;math&gt;\frac{kg^3 s^8}{m^{18} A^{13}}&lt;/math&gt;, units = 1 (15*3-30*8+13*18-3*13 = 0)

 The geometry &lt;math&gt;\color{red}(\Omega^{15})^n\color{black}&lt;/math&gt; (integer n ≥ 0) is common to all ratios where units and scalars cancel (i.e.: only combinations with &lt;math&gt;\Omega^0, \Omega^{15}, \Omega^{30}, \Omega^{45}&lt;/math&gt;... will be dimensionless). Note also, the most precise correlations are those using the most precisely known constants.


{| class=&quot;wikitable&quot;
|+Table 5. Dimensionless combinations
! CODATA 2014 mean
! (α, Ω) mean
! units = 1
! scalars = 1
|-
| &lt;math&gt;\frac{k_B e c}{h} =&lt;/math&gt; {{font color|green|yellow|'''1.000 8254'''}}
| &lt;math&gt;\frac{(k_B^*) (e^*) (c^*)}{(h^*)}&lt;/math&gt; = {{font color|green|yellow|'''1.0'''}}
| &lt;math&gt;\frac{ (u^{29}) (u^{-27}) (u^{17}) }{ (u^{19}) } = 1&lt;/math&gt;
| &lt;math&gt;(\frac{r^{10}}{v^3}) (\frac{r^3}{v^3}) (v) / (\frac{r^{13}}{v^5}) = 1&lt;/math&gt;
|-
| &lt;math&gt;\frac{h^3}{e^{13} c^{24}} =&lt;/math&gt; {{font color|green|yellow| '''0.228 473 639... 10&lt;sup&gt;-58&lt;/sup&gt;'''}}
| &lt;math&gt;\frac{(h^*)^3}{(e^*)^{13} (c^*)^{24}} = \frac{\alpha^{13}}{2^{106} \pi^{64} (\color{red}\Omega^{15})^5\color{black}} =&lt;/math&gt; {{font color|green|yellow| '''0.228 473 759... 10&lt;sup&gt;-58&lt;/sup&gt;'''}}
| &lt;math&gt;\frac{(u^{19})^{3}}{(u^{-27})^{13} (u^{17})^{24}} = 1&lt;/math&gt;
| &lt;math&gt;(\frac{r^{13}}{v^5})^3 / (\frac{r^3}{v^3})^{13} (v^{24}) = 1&lt;/math&gt;
|-
| &lt;math&gt;\frac{c^9 e^4}{m_e^3} =&lt;/math&gt; {{font color|green|yellow| '''0.170 514 342... 10&lt;sup&gt;92&lt;/sup&gt;'''}}
| &lt;math&gt;\frac{(c^*)^9 (e^*)^4}{(m_e^*)^3} = 2^{97} \pi^{49} 3^9 \alpha^5 (\color{red}\Omega^{15})^5\color{black}=&lt;/math&gt; {{font color|green|yellow| '''0.170 514 368... 10&lt;sup&gt;92&lt;/sup&gt;'''}}
| &lt;math&gt;\frac{ (u^{29}) (u^{-27}) (u^{17}) }{ (u^{19}) } = 1&lt;/math&gt;
| &lt;math&gt;(v^9) (\frac{r^3}{v^3})^4 / (\frac{r^4}{v})^3 = 1&lt;/math&gt; 
|-
| &lt;math&gt;\frac{k_B}{e^2 m_e c^4} =&lt;/math&gt; {{font color|green|yellow| '''73 095 507 858.'''}}
| &lt;math&gt;\frac{(k_B^*)}{(e^*)^2 (m_e^*) (c^*)^4} = \frac{3^3 \alpha^6}{2^3 \pi^5} =&lt;/math&gt; {{font color|green|yellow| '''73 035 235 897.'''}} 
| &lt;math&gt;\frac{(u^{29})}{(u^{-27})^2 (u^{15}) (u^{17})^4} = 1&lt;/math&gt;
| &lt;math&gt;(\frac{r^{10}}{v^3}) / (\frac{r^3}{v^3})^2 (\frac{r^4}{v}) (v)^4 = 1&lt;/math&gt;
|}


=== Derivation via CODATA ===
In this section, we show how to numerically solve the least precise dimensioned physical constants (''G'', ''h'', ''e'', ''m''&lt;sub&gt;e&lt;/sub&gt;, ''k''&lt;sub&gt;B&lt;/sub&gt; ...) in terms of the 3 most precise dimensioned physical constants); [[w:Speed of light | speed of light]] '''c''' (exact value), [[w:Vacuum permeability | vacuum permeability]] '''μ&lt;sub&gt;0&lt;/sub&gt;''' (exact value), [[w:Rydberg constant | Rydberg constant]] '''R''' (12-13 digits). 


We first look for combinations in which the unit numbers are equal, and then add dimensionless numbers as required. For example;
:&lt;math&gt;{(h^*)}^3 = (2^3 \pi^4 \Omega^4 \frac{r^{13} u^{19}}{v^5})^3 = \frac{3^{19} \pi^{12} \Omega^{12} r^{39}u^{57}}{v^{15}},\; \theta = 57&lt;/math&gt;

:&lt;math&gt;\frac{2\pi^{10} {(\mu_0^*)}^3} {3^6 {(c^*)}^5 \alpha^{13} {(R^*)}^2} = \frac{3^{19} \pi^{12} \Omega^{12} r^{39} u^{57}}{v^{15}},\; \theta = 57&lt;/math&gt;


We then replace the geometrical with the SI (''c'', ''μ&lt;sub&gt;0&lt;/sub&gt;'', ''R'') 

&lt;math&gt;{(h^*)}^3 = \frac{2\pi^{10} {\mu_0}^3} {3^6 {c}^5 a^{13} {R}^2}&lt;/math&gt;


{| class=&quot;wikitable&quot;
|+Table 7. R, c, μ&lt;sub&gt;0&lt;/sub&gt;, a ... 
! constant
! formula*
! θ
! Units
|-
| [[w:Planck constant | Planck constant]]
| &lt;math&gt;{(h^*)}^3 = \frac{2\pi^{10} {\mu_0}^3} {3^6 {c}^5 a^{13} {R}^2}&lt;/math&gt;
| &lt;math&gt;\frac{kg^3}{A^6 s}&lt;/math&gt;, 15*3-3*6+30 = {{font color|red|white|57}}
| &lt;math&gt;\frac{kg \;m^2}{s}&lt;/math&gt;, θ = 15-13*2+30 = {{font color|red|white|19}}
|-
| [[w:Gravitational constant | Gravitational constant]]
| &lt;math&gt;{(G^*)}^5 = \frac{\pi^3 {\mu_0}}{2^{20} 3^6 a^{11} {R}^2}&lt;/math&gt;
| &lt;math&gt;\frac{kg\; m^3}{A^2 s^2}&lt;/math&gt;, 15-13*3-3*2+30*2 = {{font color|red|white|30}}
| &lt;math&gt;\frac{m^3}{kg \;s^2}&lt;/math&gt;, θ = -13*3-15+30*2 = {{font color|red|white|6}}
|-
| [[w:Elementary charge | Elementary charge]]
| &lt;math&gt;{(e^*)}^3 = \frac{4 \pi^5}{3^3 {c}^4 a^8 {R}}&lt;/math&gt;
| &lt;math&gt;\frac{s^3}{m^3}&lt;/math&gt;, -30*4+13*3 = {{font color|red|white|-81}}
| &lt;math&gt;A s&lt;/math&gt;, θ = 3-30 = {{font color|red|white|-27}}
|-
| [[w:Boltzmann constant | Boltzmann constant]]
| &lt;math&gt;{(k_B^*)}^3 = \frac{\pi^5 {\mu_0}^3}{3^3 2 {c}^4 a^5 {R}}&lt;/math&gt;
| &lt;math&gt;\frac{kg^3}{s^2 A^6}&lt;/math&gt;, 15*3+30*2-3*6 = {{font color|red|white|87}}
| &lt;math&gt;\frac{kg \;m^2}{s^2 \;K}&lt;/math&gt;, θ = 15-26+60-20 = {{font color|red|white|29}}
|-
| [[w:Electron mass | Electron mass]]
| &lt;math&gt;{(m_e^*)}^3 = \frac{16 \pi^{10} {R} {\mu_0}^3}{3^6 {c}^8 a^7}&lt;/math&gt;
| &lt;math&gt;\frac{kg^3 s^2}{m^6 A^6}&lt;/math&gt;, 15*3-30*2+13*6-3*6 = {{font color|red|white|45}}
| &lt;math&gt;kg&lt;/math&gt;, θ = {{font color|red|white|15}}
|-
| [[w:Planck length | Planck length]]
| &lt;math&gt;({l_p^*})^{15} = \frac{\pi^{22} {\mu_0}^9}{2^{35} 3^{24} a^{49} c^{35} R^8}&lt;/math&gt;
| &lt;math&gt;\frac{kg^9 s^{17}}{m^{18}A^{18}}&lt;/math&gt;, 15*9-30*17+13*18-3*18 = {{font color|red|white|-195}}
| &lt;math&gt;m&lt;/math&gt;, θ = {{font color|red|white|-13}}
|-
| [[w:Planck mass | Planck mass]]
| &lt;math&gt;({m_P^*})^{15} = \frac{2^{25} \pi^{13} {\mu_0}^6}{3^6 c^5 a^{16} R^2}&lt;/math&gt;
| u = &lt;math&gt;\frac{kg^6 m^3}{s^7 A^{12}}&lt;/math&gt;, 15*6-13*3+30*7-3*12 = {{font color|red|white|225}}
| &lt;math&gt;kg&lt;/math&gt;, θ = {{font color|red|white|15}}
|}


=== Table of Constants ===
note: &lt;math&gt;\color{red}(u^{15})^n\color{black}&lt;/math&gt; constants have no Omega term.

{| class=&quot;wikitable&quot;
|+Table 9. Dimensioned constants; geometrical vs CODATA 2014
! Constant
! In Planck units
! Geometrical object
! SI calculated (r, v, Ω, α&lt;sup&gt;*&lt;/sup&gt;)
! SI CODATA 2014 &lt;ref&gt;[http://www.codata.org/] | CODATA, The Committee on Data for Science and Technology | (2014)&lt;/ref&gt;
|-
| [[w:Speed of light | Speed of light]]
| V
|  &lt;math&gt;c^* = (2\pi\Omega^2)v,\;u^{17} &lt;/math&gt; 
| ''c&lt;sup&gt;*&lt;/sup&gt;'' = 299 792 458, unit = u&lt;sup&gt;17&lt;/sup&gt; 
| ''c'' =  299 792 458 (exact)
|-
| [[w:Fine structure constant | Fine structure constant]]
|  
| 
| ''α&lt;sup&gt;*&lt;/sup&gt;'' = 137.035 999 139 (mean)
| ''α'' = 137.035 999 139(31)
|-
| [[w:Rydberg constant | Rydberg constant]]
| &lt;math&gt;R^* = (\frac{m_e}{4 \pi L \alpha^2 M})&lt;/math&gt;  
| &lt;math&gt;R^* = \frac{1}{2^{23} 3^3 \pi^{11} \alpha^5 \Omega^{17}}\frac{v^5}{r^9},\;u^{13} &lt;/math&gt; 
| ''R&lt;sup&gt;*&lt;/sup&gt;'' = 10 973 731.568 508, unit = u&lt;sup&gt;13&lt;/sup&gt;  
| ''R'' = 10 973 731.568 508(65)
|-
| [[w:Vacuum permeability | Vacuum permeability]]
| &lt;math&gt;\mu_0^* = \frac{4 \pi V^2 M}{\alpha L A^2}&lt;/math&gt; 
| &lt;math&gt;\mu_0^* = \frac{\alpha}{2^{11} \pi^5 \Omega^4} r^7,\; u^{56}&lt;/math&gt;
| ''μ&lt;sub&gt;0&lt;/sub&gt;&lt;sup&gt;*&lt;/sup&gt;'' = 4π/10^7, unit = u&lt;sup&gt;56&lt;/sup&gt; 
| ''μ&lt;sub&gt;0&lt;/sub&gt;'' = 4π/10^7 (exact)
|-
| [[w:Vacuum permittivity | Vacuum permittivity]]
| &lt;math&gt;\epsilon_0^* = \frac{1}{\mu_0^* (c^*)^2}&lt;/math&gt; 
| &lt;math&gt;\epsilon_0^* = \frac{2^9 \pi^3}{\alpha}\frac{1}{r^7 v^2},\; \color{red}1/(u^{15})^6\color{black} = u^{-90}&lt;/math&gt;
|  
|
|-
| [[w:Planck constant | Planck constant]]
| &lt;math&gt;h^* = 2 \pi M V L&lt;/math&gt;
| &lt;math&gt;h^* = 2^3 \pi^4 \Omega^4 \frac{r^{13}}{v^5},\; u^{19}&lt;/math&gt;
| ''h&lt;sup&gt;*&lt;/sup&gt;'' = 6.626 069 134 e-34, unit = u&lt;sup&gt;19&lt;/sup&gt; 
| ''h'' = 6.626 070 040(81) e-34
|-
| [[w:Gravitational constant | Gravitational constant]]
| &lt;math&gt;G^* = \frac{V^2 L}{M}&lt;/math&gt;
| &lt;math&gt;G^* = 2^3 \pi^4 \Omega^6 \frac{r^5}{v^2},\; u^{6}&lt;/math&gt; 
| ''G&lt;sup&gt;*&lt;/sup&gt;'' =  6.672 497 192 29 e11, unit = u&lt;sup&gt;6&lt;/sup&gt;
| ''G'' = 6.674 08(31) e-11
|-
| [[w:Elementary charge | Elementary charge]]
| &lt;math&gt;e^* = A T&lt;/math&gt;
| &lt;math&gt;e^* = \frac{2^7 \pi^4 \Omega^3}{\alpha}\frac{r^3}{v^3},\; u^{-27}&lt;/math&gt;
| ''e&lt;sup&gt;*&lt;/sup&gt;'' = 1.602 176 511 30 e-19, unit = u&lt;sup&gt;-27&lt;/sup&gt;
| ''e'' = 1.602 176 620 8(98) e-19
|-
| [[w:Boltzmann constant | Boltzmann constant]]
| &lt;math&gt;k_B^* = \frac{2 \pi V M}{A}&lt;/math&gt; 
| &lt;math&gt;k_B^* = \frac{\alpha}{2^5 \pi \Omega} \frac{r^{10}}{v^3},\; u^{29}&lt;/math&gt; 
| ''k&lt;sub&gt;B&lt;/sub&gt;&lt;sup&gt;*&lt;/sup&gt;'' = 1.379 510 147 52 e-23, unit = u&lt;sup&gt;29&lt;/sup&gt;
| ''k&lt;sub&gt;B&lt;/sub&gt;'' = 1.380 648 52(79) e-23
|-
| [[w:Electron mass | Electron mass]]
| 
| &lt;math&gt;m_e^* = \frac{M}{\psi},\; u^{15}&lt;/math&gt; 
| ''m&lt;sub&gt;e&lt;/sub&gt;&lt;sup&gt;*&lt;/sup&gt;'' = 9.109 382 312 56 e-31, unit = u&lt;sup&gt;15&lt;/sup&gt;
| ''m&lt;sub&gt;e&lt;/sub&gt;'' = 9.109 383 56(11) e-31
|-
| [[w:Classical electron radius | Classical electron radius]]
|  
| &lt;math&gt;\lambda_e^* = 2\pi L \psi,\; u^{-13}&lt;/math&gt;  
| ''λ&lt;sub&gt;e&lt;/sub&gt;&lt;sup&gt;*&lt;/sup&gt;'' = 2.426 310 2366 e-12, unit = u&lt;sup&gt;-13&lt;/sup&gt;
| ''λ&lt;sub&gt;e&lt;/sub&gt;'' = 2.426 310 236 7(11) e-12
|-
| [[w:Planck temperature | Planck temperature]]
| &lt;math&gt;T_p^* = \frac{A V}{\pi}&lt;/math&gt;
| &lt;math&gt;T_p^* = \frac{2^7 \pi^3 \Omega^5}{\alpha} \frac{v^4}{r^6} ,\; u^{20} &lt;/math&gt;
| ''T&lt;sub&gt;p&lt;/sub&gt;&lt;sup&gt;*&lt;/sup&gt;'' =  1.418 145 219 e32, unit = u&lt;sup&gt;20&lt;/sup&gt;
| ''T&lt;sub&gt;p&lt;/sub&gt;'' = 1.416 784(16) e32
|-
| [[w:Planck mass | Planck mass]]
| M
| &lt;math&gt;m_P^* = (1)\frac{r^4}{v} ,\; \color{red}\color{red}(u^{15})^1\color{black}&lt;/math&gt;
| ''m&lt;sub&gt;P&lt;/sub&gt;&lt;sup&gt;*&lt;/sup&gt;'' =  .217 672 817 580 e-7, unit = u&lt;sup&gt;15&lt;/sup&gt; 
| ''m&lt;sub&gt;P&lt;/sub&gt;'' = .217 647 0(51) e-7
|-
| [[w:Planck length | Planck length]]
| L
| &lt;math&gt;l_p^* = (2\pi^2\Omega^2)\frac{r^9}{v^5},\;u^{-13} &lt;/math&gt; 
| ''l&lt;sub&gt;p&lt;/sub&gt;&lt;sup&gt;*&lt;/sup&gt;'' = .161 603 660 096 e-34, unit = u&lt;sup&gt;-13&lt;/sup&gt;
| ''l&lt;sub&gt;p&lt;/sub&gt;'' = .161 622 9(38) e-34
|-
| [[w:Planck time | Planck time]]
| T
| &lt;math&gt;t_p^* = (\pi)\frac{r^9}{v^6} ,\; \color{red}\color{red}1/(u^{15})^2\color{black} &lt;/math&gt;
| ''t&lt;sub&gt;p&lt;/sub&gt;&lt;sup&gt;*&lt;/sup&gt;'' = 5.390 517 866 e-44, unit = u&lt;sup&gt;-30&lt;/sup&gt;
| ''t&lt;sub&gt;p&lt;/sub&gt;'' = 5.391 247(60) e-44
|-
| [[w:Ampere | Ampere]]
| &lt;math&gt;A = \frac{16 V^3}{\alpha P^3}&lt;/math&gt;
| &lt;math&gt;A^* = \frac{2^7\pi^3\Omega^3}{\alpha}\frac{v^3}{r^6} ,\; u^3 &lt;/math&gt;
| A&lt;sup&gt;*&lt;/sup&gt; = 0.297 221 e25, unit = u&lt;sup&gt;3&lt;/sup&gt;
| ''e/t&lt;sub&gt;p&lt;/sub&gt;'' = 0.297 181 e25 
|-
| [[w:Quantum Hall effect | Von Klitzing constant ]]
| &lt;math&gt;R_K^* = (\frac{h}{e^2})^*&lt;/math&gt;  
|  &lt;math&gt;R_K^* = \frac{\alpha^2}{2^{11} \pi^4 \Omega^2} r^7 v ,\; u^{73}&lt;/math&gt; 
| ''R&lt;sub&gt;K&lt;/sub&gt;&lt;sup&gt;*&lt;/sup&gt;'' = 25812.807 455 59, unit = u&lt;sup&gt;73&lt;/sup&gt;
| ''R&lt;sub&gt;K&lt;/sub&gt;'' = 25812.807 455 5(59)
|-
| [[w:Gyromagnetic ratio | Gyromagnetic ratio]]
| 
| &lt;math&gt;\gamma_e/2\pi = \frac{g l_p^* m_P^*}{2 k_B^* m_e^*},\; unit = u^{-42}&lt;/math&gt;  
| ''γ&lt;sub&gt;e&lt;/sub&gt;/2π&lt;sup&gt;*&lt;/sup&gt;'' = 28024.953 55, unit = u&lt;sup&gt;-42&lt;/sup&gt;
| ''γ&lt;sub&gt;e&lt;/sub&gt;/2π'' = 28024.951 64(17) 
|}
 

=== Scalars (general)===

:&lt;math&gt;M = m_P = (1)k;\; k = m_P = .217\;672\;817\;58... \;10^{-7},\; u^{15}\; (kg)&lt;/math&gt;

:&lt;math&gt;T = t_p = {\pi}t;\; t = \frac{t_p}{\pi} = .171\;585\;512\;84... 10^{-43},\; u^{-30}\; (s)&lt;/math&gt;

:&lt;math&gt;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)&lt;/math&gt;

:&lt;math&gt;V = c = {2\pi\Omega^2}v;\; v = \frac{c}{2\pi\Omega^2} = 11\;843\;707.905... ,\; u^{17}\; (m/s)&lt;/math&gt;

:&lt;math&gt;A = e/t_p = (\frac{2^7 \pi^3 \Omega^3}{a})q = .126\;918\;588\;59... 10^{23},\; u^{3}\; (A)&lt;/math&gt;


===== MT to LPVA =====

In this example LPVA are derived from MT. The formulas for MT;

:&lt;math&gt;M = (1)k,\; unit = u^{15}&lt;/math&gt;

:&lt;math&gt;T = (\pi) t,\; unit = u^{-30}&lt;/math&gt;

Replacing scalars ''pvlq'' with ''kt''

:&lt;math&gt;P = (\Omega)\;\frac{k^{12/15}}{t^{2/15}},\; unit = u^{12/15*15-2/15*(-30)=16}&lt;/math&gt;

:&lt;math&gt;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}  &lt;/math&gt;

:&lt;math&gt;L = T V = (2 \pi^2 \Omega^2) \; k^{9/15} t^{11/15},\; unit = u^{9/15*15+11/15*(-30)=-13}&lt;/math&gt;

:&lt;math&gt;A = \frac{2^4 V^3}{a P^3} = \left(\frac{2^7 \pi^3 \Omega^3}{a}\right)\; \frac{1}{k^{3/5} t^{2/5}},\; unit = 

u^{9/15*(-15)+6/15*30=3}    &lt;/math&gt;


===== PV to MTLA =====

In this example MLTA are derived from PV. The formulas for PV; 

:&lt;math&gt;P = (\Omega)p,\; unit = u^{16}&lt;/math&gt;

:&lt;math&gt;V = (2\pi\Omega^2)v,\; unit = u^{17}&lt;/math&gt;

Replacing scalars ''kltq'' with ''pv''

:&lt;math&gt;M = \frac{2\pi P^2}{V} = (1)\frac{p^2}{v},\; unit = u^{16*2-17=15} &lt;/math&gt;

:&lt;math&gt;T = (\pi) \frac{p^{9/2}}{v^6},\; unit = u^{16*9/2-17*6=-30} &lt;/math&gt;

:&lt;math&gt;L = T V = (2\pi^2\Omega^2)\frac{p^{9/2}}{v^5},\; unit = u^{16*9/2-17*5=-13}&lt;/math&gt;

:&lt;math&gt;A = \frac{2^4 V^3}{a P^3} = (\frac{2^7 \pi^3 \Omega^3}{a})\frac{v^3}{p^3},\; unit = u^{17*3-16*3=3}&lt;/math&gt;


==Article series==
* [[https://simulationuniverse.org/ simulationuniverse.org]]: Home page

===The 3 levels===
A model overview from the sub Planck scale to the container universe
* [[https://doi.org/10.13140/RG.2.2.36449.26723 Planck unit scaffolding to Cosmic Microwave Background correlation]]: Part 1. The Cosmological scale
* [[https://doi.org/10.13140/RG.2.2.12830.09283/1  The Minimal Complexity Algorithm of the Planck Scaffolding]]: Part 2. The sub-Planck scale
* [[https://doi.org/10.13140/RG.2.2.32962.75200/1 The Heavens and the Free Will postulate]]: Part 3. The Heavens (the container universe)

===Wiki series===
Introductions to principle concepts (for more complete discussion see General articles)
* [[User:Platos_Cave_(physics)/Simulation_Hypothesis/Gravity_via_Atomic_orbitals]]: Gravity as a function of atomic orbitals 
* [[User:Platos_Cave_(physics)/Simulation_Hypothesis/Relativity]]: Relativity as a translation between 2 co-ordinate systems
* [[User:Platos_Cave_(physics)/Simulation_Hypothesis/Planck_unit_scaffolding]]: CMB and a Planck unit universe scaffolding
* [[User:Platos_Cave_(physics)/Simulation_Hypothesis/Sqrt_Planck_momentum]]: Link between charge and mass
* [[User:Platos_Cave_(physics)/Simulation_Hypothesis/God_(programmer)]]: Introduction to a Planck scale Programmer God Simulation Hypothesis model

===General articles===
The following are HTML transcripts of journal articles
* [[https://simulationuniverse.org/1-Planck-unit-CMB.html 1-Planck-unit-CMB.html]]: Constructs the universe frame from Planck units 
* [[https://simulationuniverse.org/2-Relativity-hypersphere.html 2-Relativity-hypersphere.html]]: Relativity as the mathematics of perspective 
* [[https://simulationuniverse.org/3-Gravitational-orbitals.html 3-Gravitational-orbitals.html]]: Gravity as sum of n-body rotating orbital particle-particle pairs
* [[https://simulationuniverse.org/4-Atomic-orbitals.html 4-Atomic-orbitals.html]]: Atomic orbitals as single rotating orbital particle-particle pairs
* [[https://simulationuniverse.org/5-w_axis.html 5-w_axis.html]]: Imaginary number axis (radiation domain)
* [[https://simulationuniverse.org/6-Physical-constant-anomalies.html 6-Physical-constant-anomalies.html]]: Statistical analysis of physical constant anomalies
* [[https://simulationuniverse.org/7-Monopole-quarks.html 7-Monopole-quarks.html]]: Quarks as monopoles
* [[https://simulationuniverse.org/8-Holographic-universe.html 8-Holographic-universe.html]]: Hypersphere surface as 2-D analogue


== External links == 
* [https://theprogrammergod.com/ Overview of the Programmer God (mathematical electron model)]


== References == 
{{Reflist}}


[[Category: Physics]]
[[Category: Philosophy of science]]
__INDEX__</text>
      <sha1>d13ddtf6tspjixnify6hyoha3p63anf</sha1>
    </revision>
  </page>
  <page>
    <title>C language in plain view</title>
    <ns>0</ns>
    <id>285380</id>
    <revision>
      <id>2819757</id>
      <parentid>2819654</parentid>
      <timestamp>2026-07-29T14:10:43Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Applications */</comment>
      <origin>2819757</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9878" sha1="7yma0qas8y95vkgq8qlk3jnrcavqanf" xml:space="preserve">=== 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 &amp; Variables ([[Media:C03.Func2.Variable.1.A.20161222.pdf |A.pdf]], [[Media:C03.Func2.Variable.1.B.20161222.pdf |B.pdf]])
* Functions &amp; Pointers ([[Media:C03.Func3.Pointer.1.A.20161122.pdf |A.pdf]], [[Media:C03.Func3.Pointer.1.B.20161122.pdf |B.pdf]])
* Functions &amp; 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 ====
* Pointers ([[Media:C04.S1.Pointer.1A.20240524.pdf |A.pdf]], [[Media:C04.Series2.Pointer.1.B.20161115.pdf |B.pdf]])
* Arrays ([[Media:C04.S2.Array.1A.20240514.pdf |A.pdf]], [[Media:C04.Series1.Array.1.B.20161115.pdf |B.pdf]])
* Array Pointers ([[Media:C04.S3.ArrayPointer.1A.20240208.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]])
* Multi-dimensional Arrays ([[Media:C04.Series4.MultiDim.1.A.20221130.pdf |A.pdf]], [[Media:C04.Series4.MultiDim.1.B.1111.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]])

==== 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]])

==== Applications ==== 
* Address-of and de-reference operators ([[Media:C04.SA0.PtrOperator.1A.20260729.pdf |A.pdf]])
* Applications of Pointers ([[Media:C04.SA1.AppPointer.1A.20241121.pdf |A.pdf]])
* Applications of Arrays ([[Media:C04.SA2.AppArray.1A.20240715.pdf |A.pdf]])
* Applications of Array Pointers ([[Media:C04.SA3.AppArrayPointer.1A.20240210.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]])

=== 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


&lt;!----------------------------------------------------------------------&gt;
&lt;/br&gt;
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]])
 
&lt;br&gt;

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 language]]

&lt;/br&gt;</text>
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  </page>
  <page>
    <title>Motivation and emotion/Assessment/Selection</title>
    <ns>0</ns>
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    <revision>
      <id>2819825</id>
      <parentid>2819737</parentid>
      <timestamp>2026-07-30T08:07:01Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <minor />
      <origin>2819825</origin>
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      <format>text/x-wiki</format>
      <text bytes="6146" sha1="l21ldq6qmxotj0ukic9hs9clgki3wk7" xml:space="preserve">{{title|Topic selection — Guidelines}}
&lt;!--&lt;div style=&quot;text-align: center;&quot;&gt;''Select major project topic'' --&gt;
&lt;!-- ---------------------------------- ---&gt;
&lt;!-- Count down --&gt;
&lt;!-- ---------------------------------- ---&gt;
&lt;!-- {{countdown
 |year       = 2024
 |month      = 08
 |day        = 04
 |hour       = 23
 |minute     = 0
 |second     = 0
 |event = this assessment is due
}} --&gt;
&lt;!-- {{Motivation and emotion/Assessment/In development}} --&gt;
&lt;!-- Show this during semester {{:Motivation and emotion/Assessment/Chapter/Contents}}&lt;/div&gt; --&gt;
{{TOCright}}
==Overview==
&lt;!--
* Weight: 0%
* Due: {{/Due}}
* Ungraded early assessment exercise
* Tasks --&gt;
# [[Special:CreateAccount|Create a Wikiversity account]]
# Sign up to (or negotiate) an approved topic for the major project — see {{Motivation and emotion/Book}}
# Ask clarifying questions
&lt;!--
==Marking and feedback==
* No marks
* Feedback will be provided to:
** Approve topic selection
** Respond to student questions
** Provide suggestions
* Feedback will be available via {{Motivation and emotion/Canvas}} before the [[Motivation and emotion/Assessment/Topic|topic development]] due date
* Follow up if you don't understand the feedback

==Extensions and late submissions==
* No extensions or late submissions
* If you don't submit, go ahead and sign up to an approved topic, and move on to the [[Motivation and emotion/Assessment/Topic|topic development exercise]]

==Learning outcomes==
How the unit's [[Motivation and emotion/About/Learning outcomes|learning outcomes]] are addressed by this assessment exercise:
{| border=1 cellpadding=5 cellspacing=&quot;0&quot; background:transparent style=&quot;width:90%; margin: auto;&quot;
|-
| style=&quot;width:40%;&quot; | '''Learning outcome'''
| style=&quot;width:60%;&quot; | '''Assessment task''' 
|-
| Critically apply knowledge of motivation or emotion to an indepth understanding of a specific topic in this field.
| Select an appropriate, unique, specific motivation or emotion topic for the major project 
|}

==Graduate attributes==
How the unit's [[Motivation and emotion/About/Graduate attributes|graduate attributes]] are addressed by this assessment exercise:
{| border=1 cellpadding=5 cellspacing=&quot;0&quot; background:transparent style=&quot;width:90%; margin: auto;&quot;
|-
| style=&quot;width:40%;&quot; | '''Graduate attribute'''
| style=&quot;width:60%;&quot; | '''Assessment task''' 
|-
| style=&quot;vertical-align:top;&quot; | Be professional — communicate effectively
| style=&quot;vertical-align:top;&quot; | Communicate by signing up to a topic, submitting for approval, and asking clarifying questions
|-
| style=&quot;vertical-align:top;&quot; | Be professional — display initiative and drive, and use organisation skills to plan and manage workload
| style=&quot;vertical-align:top;&quot; | Get organised by selecting a topic
|-
| style=&quot;vertical-align:top;&quot; | Be a lifelong learner — evaluate and adopt new technology
| style=&quot;vertical-align:top;&quot; | Create a Wikiversity account and make at least 1 edit by signing up to a topic
|}
--&gt;

==Instructions==
Create a Wikiversity account and sign up to a pre-approved major project topic in the {{Motivation and emotion/Book}} or negotiate a new topic. This topic will be used for the [[Motivation and emotion/Assessment/Topic|topic development]] and [[Motivation and emotion/Assessment/Chapter|book chapter]]. Ask any clarifying questions.

===[[Special:CreateAccount|Create a Wikiversity account]]===
* User names start with a capital letter
* Can use real name, pseudonym, student number etc.

===Sign up to a topic===
'''Pre-approved topics''' are available via the {{Motivation and emotion/Book}}
* How to sign up:
** Click &quot;Edit&quot; or &quot;Edit source&quot;
** Replace '''[[User:User Name|User Name]]''' alongside the topic of interest with your Wikiversity user name
** Click &quot;Publish&quot;
&lt;!-- * Conduct an initial literature search. If the topic is too broad or narrow, it may be difficult to satisfy the [[#Marking criteria|marking criteria]] --&gt;
* Pre-approved topics can be modified by emailing a revised title and sub-title to the [[Motivation and emotion/About/Staff|unit convener]]
* You can change topics by moving your username to a topic without an author
&lt;!-- * More topics will be added, but students are also encouraged to propose topics --&gt;

===New topics===
'''New or modified topics''' must be approved by the [[Motivation and emotion/About/Staff|unit convener]].
* Ways to get some ideas:
** [https://cogniti.canberra.edu.au/agents/6a3caf85b2fed9a95789241a/chat?k=WfY1yx6AB7yF_9XqiyYENzUBNCGfV1e2FCBodCDD2xU Chat with the motivation and emotion book chapter topic generator] (Cogniti)
** Look through chapters written in [[Motivation and emotion/Book|previous years]]
** Check out [[Motivation and emotion/Book/Ideas for topics|similar projects]] on the internet
* New topics must:
** '''Be unique''': Topic must not already be sufficiently covered by a [[Motivation and emotion/Book|previous motivation and motivation book chapter]]. Search before making a proposal. How does the proposed topic build on, or differ from, previous work?
** '''Align with the project theme''': Topic must be related to related to [[motivation]] or [[emotion]] and fit the overarching book theme which is to help people to understand and improve their motivational and emotional lives using psychological science
** '''Have appropriate scope''': If the topic is too narrow, or if there is a lack of psychological theory and research, it will be difficult to satisfy the [[#Marking criteria|marking criteria]]
* To propose a new topic, email the [[Motivation and emotion/About/Staff|unit convener]] with these details:
** Title
** Sub-title (in the form of a question) — see [[Motivation and emotion/Book|examples]]
** Wikiversity user name
** Details of any related previous book chapter topics (check via this [[Motivation and emotion/Book|search box]])

===Ask clarifying questions===
* What questions could you ask to help you to successfully tackle the major project?

==See also==
* [[Motivation and emotion/Tutorials/Topic selection|Tutorial 01: Topic selection]]
{{Motivation and emotion/Assessment/Navigation}}
[[Category:Motivation and emotion/Assessment/Selection| ]]</text>
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    <revision>
      <id>2819826</id>
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&lt;!--&lt;div style=&quot;text-align: center;&quot;&gt;''Select major project topic'' --&gt;
&lt;!-- ---------------------------------- ---&gt;
&lt;!-- Count down --&gt;
&lt;!-- ---------------------------------- ---&gt;
&lt;!-- {{countdown
 |year       = 2024
 |month      = 08
 |day        = 04
 |hour       = 23
 |minute     = 0
 |second     = 0
 |event = this assessment is due
}} --&gt;
&lt;!-- {{Motivation and emotion/Assessment/In development}} --&gt;
&lt;!-- Show this during semester {{:Motivation and emotion/Assessment/Chapter/Contents}}&lt;/div&gt; --&gt;
{{TOCright}}
==Overview==
&lt;!--
* Weight: 0%
* Due: {{/Due}}
* Ungraded early assessment exercise
* Tasks --&gt;
# [[Special:CreateAccount|Create a Wikiversity account]]
# Sign up to (or negotiate) an approved topic for the [[Motivation and emotion/Assessment/Major project|major project]] — see {{Motivation and emotion/Book}}
# Ask clarifying questions
&lt;!--
==Marking and feedback==
* No marks
* Feedback will be provided to:
** Approve topic selection
** Respond to student questions
** Provide suggestions
* Feedback will be available via {{Motivation and emotion/Canvas}} before the [[Motivation and emotion/Assessment/Topic|topic development]] due date
* Follow up if you don't understand the feedback

==Extensions and late submissions==
* No extensions or late submissions
* If you don't submit, go ahead and sign up to an approved topic, and move on to the [[Motivation and emotion/Assessment/Topic|topic development exercise]]

==Learning outcomes==
How the unit's [[Motivation and emotion/About/Learning outcomes|learning outcomes]] are addressed by this assessment exercise:
{| border=1 cellpadding=5 cellspacing=&quot;0&quot; background:transparent style=&quot;width:90%; margin: auto;&quot;
|-
| style=&quot;width:40%;&quot; | '''Learning outcome'''
| style=&quot;width:60%;&quot; | '''Assessment task''' 
|-
| Critically apply knowledge of motivation or emotion to an indepth understanding of a specific topic in this field.
| Select an appropriate, unique, specific motivation or emotion topic for the major project 
|}

==Graduate attributes==
How the unit's [[Motivation and emotion/About/Graduate attributes|graduate attributes]] are addressed by this assessment exercise:
{| border=1 cellpadding=5 cellspacing=&quot;0&quot; background:transparent style=&quot;width:90%; margin: auto;&quot;
|-
| style=&quot;width:40%;&quot; | '''Graduate attribute'''
| style=&quot;width:60%;&quot; | '''Assessment task''' 
|-
| style=&quot;vertical-align:top;&quot; | Be professional — communicate effectively
| style=&quot;vertical-align:top;&quot; | Communicate by signing up to a topic, submitting for approval, and asking clarifying questions
|-
| style=&quot;vertical-align:top;&quot; | Be professional — display initiative and drive, and use organisation skills to plan and manage workload
| style=&quot;vertical-align:top;&quot; | Get organised by selecting a topic
|-
| style=&quot;vertical-align:top;&quot; | Be a lifelong learner — evaluate and adopt new technology
| style=&quot;vertical-align:top;&quot; | Create a Wikiversity account and make at least 1 edit by signing up to a topic
|}
--&gt;

==Instructions==
Create a Wikiversity account and sign up to a pre-approved major project topic in the {{Motivation and emotion/Book}} or negotiate a new topic. This topic will be used for the [[Motivation and emotion/Assessment/Topic|topic development]] and [[Motivation and emotion/Assessment/Chapter|book chapter]]. Ask any clarifying questions.

===[[Special:CreateAccount|Create a Wikiversity account]]===
* User names start with a capital letter
* Can use real name, pseudonym, student number etc.

===Sign up to a topic===
'''Pre-approved topics''' are available via the {{Motivation and emotion/Book}}
* How to sign up:
** Click &quot;Edit&quot; or &quot;Edit source&quot;
** Replace '''[[User:User Name|User Name]]''' alongside the topic of interest with your Wikiversity user name
** Click &quot;Publish&quot;
&lt;!-- * Conduct an initial literature search. If the topic is too broad or narrow, it may be difficult to satisfy the [[#Marking criteria|marking criteria]] --&gt;
* Pre-approved topics can be modified by emailing a revised title and sub-title to the [[Motivation and emotion/About/Staff|unit convener]]
* You can change topics by moving your username to a topic without an author
&lt;!-- * More topics will be added, but students are also encouraged to propose topics --&gt;

===New topics===
'''New or modified topics''' must be approved by the [[Motivation and emotion/About/Staff|unit convener]].
* Ways to get some ideas:
** [https://cogniti.canberra.edu.au/agents/6a3caf85b2fed9a95789241a/chat?k=WfY1yx6AB7yF_9XqiyYENzUBNCGfV1e2FCBodCDD2xU Chat with the motivation and emotion book chapter topic generator] (Cogniti)
** Look through chapters written in [[Motivation and emotion/Book|previous years]]
** Check out [[Motivation and emotion/Book/Ideas for topics|similar projects]] on the internet
* New topics must:
** '''Be unique''': Topic must not already be sufficiently covered by a [[Motivation and emotion/Book|previous motivation and motivation book chapter]]. Search before making a proposal. How does the proposed topic build on, or differ from, previous work?
** '''Align with the project theme''': Topic must be related to related to [[motivation]] or [[emotion]] and fit the overarching book theme which is to help people to understand and improve their motivational and emotional lives using psychological science
** '''Have appropriate scope''': If the topic is too narrow, or if there is a lack of psychological theory and research, it will be difficult to satisfy the [[#Marking criteria|marking criteria]]
* To propose a new topic, email the [[Motivation and emotion/About/Staff|unit convener]] with these details:
** Title
** Sub-title (in the form of a question) — see [[Motivation and emotion/Book|examples]]
** Wikiversity user name
** Details of any related previous book chapter topics (check via this [[Motivation and emotion/Book|search box]])

===Ask clarifying questions===
* What questions could you ask to help you to successfully tackle the major project?

==See also==
* [[Motivation and emotion/Tutorials/Topic selection|Tutorial 01: Topic selection]]
{{Motivation and emotion/Assessment/Navigation}}
[[Category:Motivation and emotion/Assessment/Selection| ]]</text>
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    </revision>
    <revision>
      <id>2819827</id>
      <parentid>2819826</parentid>
      <timestamp>2026-07-30T08:10:34Z</timestamp>
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        <username>Jtneill</username>
        <id>10242</id>
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      <text bytes="5939" sha1="i7bqeh7gr2ledjh4o8nqfot79hofmk4" xml:space="preserve">{{title|Topic selection — Guidelines}}
&lt;!--&lt;div style=&quot;text-align: center;&quot;&gt;''Select major project topic'' --&gt;
&lt;!-- ---------------------------------- ---&gt;
&lt;!-- Count down --&gt;
&lt;!-- ---------------------------------- ---&gt;
&lt;!-- {{countdown
 |year       = 2024
 |month      = 08
 |day        = 04
 |hour       = 23
 |minute     = 0
 |second     = 0
 |event = this assessment is due
}} --&gt;
&lt;!-- {{Motivation and emotion/Assessment/In development}} --&gt;
&lt;!-- Show this during semester {{:Motivation and emotion/Assessment/Chapter/Contents}}&lt;/div&gt; --&gt;
{{TOCright}}
==Overview==
&lt;!--
* Weight: 0%
* Due: {{/Due}}
* Ungraded early assessment exercise
* Tasks --&gt;
# [[Special:CreateAccount|Create a Wikiversity account]]
# Sign up to (or negotiate) an approved topic for the [[Motivation and emotion/Assessment/Major project|major project]] — see {{Motivation and emotion/Book}}
# Ask clarifying questions
&lt;!--
==Marking and feedback==
* No marks
* Feedback will be provided to:
** Approve topic selection
** Respond to student questions
** Provide suggestions
* Feedback will be available via {{Motivation and emotion/Canvas}} before the [[Motivation and emotion/Assessment/Topic|topic development]] due date
* Follow up if you don't understand the feedback

==Extensions and late submissions==
* No extensions or late submissions
* If you don't submit, go ahead and sign up to an approved topic, and move on to the [[Motivation and emotion/Assessment/Topic|topic development exercise]]

==Learning outcomes==
How the unit's [[Motivation and emotion/About/Learning outcomes|learning outcomes]] are addressed by this assessment exercise:
{| border=1 cellpadding=5 cellspacing=&quot;0&quot; background:transparent style=&quot;width:90%; margin: auto;&quot;
|-
| style=&quot;width:40%;&quot; | '''Learning outcome'''
| style=&quot;width:60%;&quot; | '''Assessment task''' 
|-
| Critically apply knowledge of motivation or emotion to an indepth understanding of a specific topic in this field.
| Select an appropriate, unique, specific motivation or emotion topic for the major project 
|}

==Graduate attributes==
How the unit's [[Motivation and emotion/About/Graduate attributes|graduate attributes]] are addressed by this assessment exercise:
{| border=1 cellpadding=5 cellspacing=&quot;0&quot; background:transparent style=&quot;width:90%; margin: auto;&quot;
|-
| style=&quot;width:40%;&quot; | '''Graduate attribute'''
| style=&quot;width:60%;&quot; | '''Assessment task''' 
|-
| style=&quot;vertical-align:top;&quot; | Be professional — communicate effectively
| style=&quot;vertical-align:top;&quot; | Communicate by signing up to a topic, submitting for approval, and asking clarifying questions
|-
| style=&quot;vertical-align:top;&quot; | Be professional — display initiative and drive, and use organisation skills to plan and manage workload
| style=&quot;vertical-align:top;&quot; | Get organised by selecting a topic
|-
| style=&quot;vertical-align:top;&quot; | Be a lifelong learner — evaluate and adopt new technology
| style=&quot;vertical-align:top;&quot; | Create a Wikiversity account and make at least 1 edit by signing up to a topic
|}
--&gt;

==[[Special:CreateAccount|Create a Wikiversity account]]==
* User names start with a capital letter
* Can use real name, pseudonym, student number etc.

==Sign up to a topic==
'''Pre-approved topics''' are available via the {{Motivation and emotion/Book}}
* How to sign up:
** Click &quot;Edit&quot; or &quot;Edit source&quot;
** Replace '''[[User:User Name|User Name]]''' alongside the topic of interest with your Wikiversity user name
** Click &quot;Publish&quot;
&lt;!-- * Conduct an initial literature search. If the topic is too broad or narrow, it may be difficult to satisfy the [[#Marking criteria|marking criteria]] --&gt;
* Pre-approved topics can be modified by emailing a revised title and sub-title to the [[Motivation and emotion/About/Staff|unit convener]]
* You can change topics by moving your username to a topic without an author
&lt;!-- * More topics will be added, but students are also encouraged to propose topics --&gt;

==Negotiate a topic==
'''New or modified topics''' must be approved by the [[Motivation and emotion/About/Staff|unit convener]].
* Ways to get some ideas:
** [https://cogniti.canberra.edu.au/agents/6a3caf85b2fed9a95789241a/chat?k=WfY1yx6AB7yF_9XqiyYENzUBNCGfV1e2FCBodCDD2xU Chat with the motivation and emotion book chapter topic generator] (Cogniti)
** Look through chapters written in [[Motivation and emotion/Book|previous years]]
** Check out [[Motivation and emotion/Book/Ideas for topics|similar projects]] on the internet
* New topics must:
** '''Be unique''': Topic must not already be sufficiently covered by a [[Motivation and emotion/Book|previous motivation and motivation book chapter]]. Search before making a proposal. How does the proposed topic build on, or differ from, previous work?
** '''Align with the project theme''': Topic must be related to related to [[motivation]] or [[emotion]] and fit the overarching book theme which is to help people to understand and improve their motivational and emotional lives using psychological science
** '''Have appropriate scope''': If the topic is too narrow, or if there is a lack of psychological theory and research, it will be difficult to satisfy the [[#Marking criteria|marking criteria]]
* To propose a new topic, email the [[Motivation and emotion/About/Staff|unit convener]] with these details:
** Title
** Sub-title (in the form of a question) — see [[Motivation and emotion/Book|examples]]
** Wikiversity user name
** Details of any related previous book chapter topics (check via this [[Motivation and emotion/Book|search box]])

==Ask clarifying questions==
* What questions could you ask to help you to successfully tackle the major project?
* Where and how could you ask these questions to optimise quality and quantity of feedback?

==See also==
* [[Motivation and emotion/Tutorials/Topic selection|Tutorial 01: Topic selection]]
{{Motivation and emotion/Assessment/Navigation}}
[[Category:Motivation and emotion/Assessment/Selection| ]]</text>
      <sha1>i7bqeh7gr2ledjh4o8nqfot79hofmk4</sha1>
    </revision>
    <revision>
      <id>2819828</id>
      <parentid>2819827</parentid>
      <timestamp>2026-07-30T08:12:12Z</timestamp>
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        <id>10242</id>
      </contributor>
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      <text bytes="6126" sha1="ndq0lczz59mbmaa0gpiiuaz32tq129g" xml:space="preserve">{{title|Topic selection — Guidelines}}
&lt;!--&lt;div style=&quot;text-align: center;&quot;&gt;''Select major project topic'' --&gt;
&lt;!-- ---------------------------------- ---&gt;
&lt;!-- Count down --&gt;
&lt;!-- ---------------------------------- ---&gt;
&lt;!-- {{countdown
 |year       = 2024
 |month      = 08
 |day        = 04
 |hour       = 23
 |minute     = 0
 |second     = 0
 |event = this assessment is due
}} --&gt;
&lt;!-- {{Motivation and emotion/Assessment/In development}} --&gt;
&lt;!-- Show this during semester {{:Motivation and emotion/Assessment/Chapter/Contents}}&lt;/div&gt; --&gt;
{{TOCright}}
==Overview==
&lt;!--
* Weight: 0%
* Due: {{/Due}}
* Ungraded early assessment exercise
* Tasks --&gt;
# [[Special:CreateAccount|Create a Wikiversity account]]
# Sign up to (or negotiate) an approved topic for the [[Motivation and emotion/Assessment/Major project|major project]] — see {{Motivation and emotion/Book}} — or negotiate a new topic
# Ask clarifying questions

This topic will be used for the [[Motivation and emotion/Assessment/Topic|topic development]] and [[Motivation and emotion/Assessment/Chapter|book chapter]].

==[[Special:CreateAccount|Create a Wikiversity account]]==
* User names start with a capital letter
* Can use real name, pseudonym, student number etc.

==Sign up to a topic==
'''Pre-approved topics''' are available via the {{Motivation and emotion/Book}}
* How to sign up:
** Click &quot;Edit&quot; or &quot;Edit source&quot;
** Replace '''[[User:User Name|User Name]]''' alongside the topic of interest with your Wikiversity user name
** Click &quot;Publish&quot;
&lt;!-- * Conduct an initial literature search. If the topic is too broad or narrow, it may be difficult to satisfy the [[#Marking criteria|marking criteria]] --&gt;
* Pre-approved topics can be modified by emailing a revised title and sub-title to the [[Motivation and emotion/About/Staff|unit convener]]
* You can change topics by moving your username to a topic without an author
&lt;!-- * More topics will be added, but students are also encouraged to propose topics --&gt;

==Negotiate a topic==
'''New or modified topics''' must be approved by the [[Motivation and emotion/About/Staff|unit convener]].
* Ways to get some ideas:
** [https://cogniti.canberra.edu.au/agents/6a3caf85b2fed9a95789241a/chat?k=WfY1yx6AB7yF_9XqiyYENzUBNCGfV1e2FCBodCDD2xU Chat with the motivation and emotion book chapter topic generator] (Cogniti)
** Look through chapters written in [[Motivation and emotion/Book|previous years]]
** Check out [[Motivation and emotion/Book/Ideas for topics|similar projects]] on the internet
* New topics must:
** '''Be unique''': Topic must not already be sufficiently covered by a [[Motivation and emotion/Book|previous motivation and motivation book chapter]]. Search before making a proposal. How does the proposed topic build on, or differ from, previous work?
** '''Align with the project theme''': Topic must be related to related to [[motivation]] or [[emotion]] and fit the overarching book theme which is to help people to understand and improve their motivational and emotional lives using psychological science
** '''Have appropriate scope''': If the topic is too narrow, or if there is a lack of psychological theory and research, it will be difficult to satisfy the [[#Marking criteria|marking criteria]]
* To propose a new topic, email the [[Motivation and emotion/About/Staff|unit convener]] with these details:
** Title
** Sub-title (in the form of a question) — see [[Motivation and emotion/Book|examples]]
** Wikiversity user name
** Details of any related previous book chapter topics (check via this [[Motivation and emotion/Book|search box]])

==Ask clarifying questions==
* What questions could you ask to help you to successfully tackle the major project?
* Where and how could you ask these questions to optimise quality and quantity of feedback?
&lt;!--
==Marking and feedback==
* No marks
* Feedback will be provided to:
** Approve topic selection
** Respond to student questions
** Provide suggestions
* Feedback will be available via {{Motivation and emotion/Canvas}} before the [[Motivation and emotion/Assessment/Topic|topic development]] due date
* Follow up if you don't understand the feedback

==Extensions and late submissions==
* No extensions or late submissions
* If you don't submit, go ahead and sign up to an approved topic, and move on to the [[Motivation and emotion/Assessment/Topic|topic development exercise]]

==Learning outcomes==
How the unit's [[Motivation and emotion/About/Learning outcomes|learning outcomes]] are addressed by this assessment exercise:
{| border=1 cellpadding=5 cellspacing=&quot;0&quot; background:transparent style=&quot;width:90%; margin: auto;&quot;
|-
| style=&quot;width:40%;&quot; | '''Learning outcome'''
| style=&quot;width:60%;&quot; | '''Assessment task''' 
|-
| Critically apply knowledge of motivation or emotion to an indepth understanding of a specific topic in this field.
| Select an appropriate, unique, specific motivation or emotion topic for the major project 
|}

==Graduate attributes==
How the unit's [[Motivation and emotion/About/Graduate attributes|graduate attributes]] are addressed by this assessment exercise:
{| border=1 cellpadding=5 cellspacing=&quot;0&quot; background:transparent style=&quot;width:90%; margin: auto;&quot;
|-
| style=&quot;width:40%;&quot; | '''Graduate attribute'''
| style=&quot;width:60%;&quot; | '''Assessment task''' 
|-
| style=&quot;vertical-align:top;&quot; | Be professional — communicate effectively
| style=&quot;vertical-align:top;&quot; | Communicate by signing up to a topic, submitting for approval, and asking clarifying questions
|-
| style=&quot;vertical-align:top;&quot; | Be professional — display initiative and drive, and use organisation skills to plan and manage workload
| style=&quot;vertical-align:top;&quot; | Get organised by selecting a topic
|-
| style=&quot;vertical-align:top;&quot; | Be a lifelong learner — evaluate and adopt new technology
| style=&quot;vertical-align:top;&quot; | Create a Wikiversity account and make at least 1 edit by signing up to a topic
|}
--&gt;
==See also==
* [[Motivation and emotion/Tutorials/Topic selection|Tutorial 01: Topic selection]]
{{Motivation and emotion/Assessment/Navigation}}
[[Category:Motivation and emotion/Assessment/Selection| ]]</text>
      <sha1>ndq0lczz59mbmaa0gpiiuaz32tq129g</sha1>
    </revision>
    <revision>
      <id>2819829</id>
      <parentid>2819828</parentid>
      <timestamp>2026-07-30T08:13:31Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
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&lt;!--&lt;div style=&quot;text-align: center;&quot;&gt;''Select major project topic'' --&gt;
&lt;!-- ---------------------------------- ---&gt;
&lt;!-- Count down --&gt;
&lt;!-- ---------------------------------- ---&gt;
&lt;!-- {{countdown
 |year       = 2024
 |month      = 08
 |day        = 04
 |hour       = 23
 |minute     = 0
 |second     = 0
 |event = this assessment is due
}} --&gt;
&lt;!-- {{Motivation and emotion/Assessment/In development}} --&gt;
&lt;!-- Show this during semester {{:Motivation and emotion/Assessment/Chapter/Contents}}&lt;/div&gt; --&gt;
{{TOCright}}
==Overview==
&lt;!--
* Weight: 0%
* Due: {{/Due}}
* Ungraded early assessment exercise
* Tasks --&gt;
# [[Special:CreateAccount|Create a Wikiversity account]]
# Sign up to (or negotiate) an approved topic for the [[Motivation and emotion/Assessment/Major project|major project]] — see {{Motivation and emotion/Book}} — or negotiate a new topic
# Ask clarifying questions

⤿ The selected topic will then be used for the [[Motivation and emotion/Assessment/Topic|topic development]] and [[Motivation and emotion/Assessment/Chapter|book chapter]].

==[[Special:CreateAccount|Create a Wikiversity account]]==
* User names start with a capital letter
* Can use real name, pseudonym, student number etc.

==Sign up to a topic==
'''Pre-approved topics''' are available via the {{Motivation and emotion/Book}}
* How to sign up:
** Click &quot;Edit&quot; or &quot;Edit source&quot;
** Replace '''[[User:User Name|User Name]]''' alongside the topic of interest with your Wikiversity user name
** Click &quot;Publish&quot;
&lt;!-- * Conduct an initial literature search. If the topic is too broad or narrow, it may be difficult to satisfy the [[#Marking criteria|marking criteria]] --&gt;
* Pre-approved topics can be modified by emailing a revised title and sub-title to the [[Motivation and emotion/About/Staff|unit convener]]
* You can change topics by moving your username to a topic without an author
&lt;!-- * More topics will be added, but students are also encouraged to propose topics --&gt;

==Negotiate a topic==
'''New or modified topics''' must be approved by the [[Motivation and emotion/About/Staff|unit convener]].
* Ways to get some ideas:
** [https://cogniti.canberra.edu.au/agents/6a3caf85b2fed9a95789241a/chat?k=WfY1yx6AB7yF_9XqiyYENzUBNCGfV1e2FCBodCDD2xU Chat with the motivation and emotion book chapter topic generator] (Cogniti)
** Look through chapters written in [[Motivation and emotion/Book|previous years]]
** Check out [[Motivation and emotion/Book/Ideas for topics|similar projects]] on the internet
* New topics must:
** '''Be unique''': Topic must not already be sufficiently covered by a [[Motivation and emotion/Book|previous motivation and motivation book chapter]]. Search before making a proposal. How does the proposed topic build on, or differ from, previous work?
** '''Align with the project theme''': Topic must be related to related to [[motivation]] or [[emotion]] and fit the overarching book theme which is to help people to understand and improve their motivational and emotional lives using psychological science
** '''Have appropriate scope''': If the topic is too narrow, or if there is a lack of psychological theory and research, it will be difficult to satisfy the [[#Marking criteria|marking criteria]]
* To propose a new topic, email the [[Motivation and emotion/About/Staff|unit convener]] with these details:
** Title
** Sub-title (in the form of a question) — see [[Motivation and emotion/Book|examples]]
** Wikiversity user name
** Details of any related previous book chapter topics (check via this [[Motivation and emotion/Book|search box]])

==Ask clarifying questions==
* What questions could you ask to help you to successfully tackle the major project?
* Where and how could you ask these questions to optimise quality and quantity of feedback?
&lt;!--
==Marking and feedback==
* No marks
* Feedback will be provided to:
** Approve topic selection
** Respond to student questions
** Provide suggestions
* Feedback will be available via {{Motivation and emotion/Canvas}} before the [[Motivation and emotion/Assessment/Topic|topic development]] due date
* Follow up if you don't understand the feedback

==Extensions and late submissions==
* No extensions or late submissions
* If you don't submit, go ahead and sign up to an approved topic, and move on to the [[Motivation and emotion/Assessment/Topic|topic development exercise]]

==Learning outcomes==
How the unit's [[Motivation and emotion/About/Learning outcomes|learning outcomes]] are addressed by this assessment exercise:
{| border=1 cellpadding=5 cellspacing=&quot;0&quot; background:transparent style=&quot;width:90%; margin: auto;&quot;
|-
| style=&quot;width:40%;&quot; | '''Learning outcome'''
| style=&quot;width:60%;&quot; | '''Assessment task''' 
|-
| Critically apply knowledge of motivation or emotion to an indepth understanding of a specific topic in this field.
| Select an appropriate, unique, specific motivation or emotion topic for the major project 
|}

==Graduate attributes==
How the unit's [[Motivation and emotion/About/Graduate attributes|graduate attributes]] are addressed by this assessment exercise:
{| border=1 cellpadding=5 cellspacing=&quot;0&quot; background:transparent style=&quot;width:90%; margin: auto;&quot;
|-
| style=&quot;width:40%;&quot; | '''Graduate attribute'''
| style=&quot;width:60%;&quot; | '''Assessment task''' 
|-
| style=&quot;vertical-align:top;&quot; | Be professional — communicate effectively
| style=&quot;vertical-align:top;&quot; | Communicate by signing up to a topic, submitting for approval, and asking clarifying questions
|-
| style=&quot;vertical-align:top;&quot; | Be professional — display initiative and drive, and use organisation skills to plan and manage workload
| style=&quot;vertical-align:top;&quot; | Get organised by selecting a topic
|-
| style=&quot;vertical-align:top;&quot; | Be a lifelong learner — evaluate and adopt new technology
| style=&quot;vertical-align:top;&quot; | Create a Wikiversity account and make at least 1 edit by signing up to a topic
|}
--&gt;
==See also==
* [[Motivation and emotion/Tutorials/Topic selection|Tutorial 01: Topic selection]]
{{Motivation and emotion/Assessment/Navigation}}
[[Category:Motivation and emotion/Assessment/Selection| ]]</text>
      <sha1>eo44r056wuodi6p7ui3zvcysmyn77yu</sha1>
    </revision>
    <revision>
      <id>2819830</id>
      <parentid>2819829</parentid>
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&lt;!--&lt;div style=&quot;text-align: center;&quot;&gt;''Select major project topic'' --&gt;
&lt;!-- ---------------------------------- ---&gt;
&lt;!-- Count down --&gt;
&lt;!-- ---------------------------------- ---&gt;
&lt;!-- {{countdown
 |year       = 2024
 |month      = 08
 |day        = 04
 |hour       = 23
 |minute     = 0
 |second     = 0
 |event = this assessment is due
}} --&gt;
&lt;!-- {{Motivation and emotion/Assessment/In development}} --&gt;
&lt;!-- Show this during semester {{:Motivation and emotion/Assessment/Chapter/Contents}}&lt;/div&gt; --&gt;
{{TOCright}}
==Overview==
&lt;!--
* Weight: 0%
* Due: {{/Due}}
* Ungraded early assessment exercise
* Tasks --&gt;
# [[Special:CreateAccount|Create a Wikiversity account]]
# Sign up to (or negotiate) an approved topic for the [[Motivation and emotion/Assessment/Major project|major project]] — see {{Motivation and emotion/Book}} — or negotiate a new topic
# Ask clarifying questions
# ⤿ The selected topic will then be used for the [[Motivation and emotion/Assessment/Topic|topic development]] and [[Motivation and emotion/Assessment/Chapter|book chapter]].

==[[Special:CreateAccount|Create a Wikiversity account]]==
* User names start with a capital letter
* Can use real name, pseudonym, student number etc.

==Sign up to a topic==
'''Pre-approved topics''' are available via the {{Motivation and emotion/Book}}
* How to sign up:
** Click &quot;Edit&quot; or &quot;Edit source&quot;
** Replace '''[[User:User Name|User Name]]''' alongside the topic of interest with your Wikiversity user name
** Click &quot;Publish&quot;
&lt;!-- * Conduct an initial literature search. If the topic is too broad or narrow, it may be difficult to satisfy the [[#Marking criteria|marking criteria]] --&gt;
* Pre-approved topics can be modified by emailing a revised title and sub-title to the [[Motivation and emotion/About/Staff|unit convener]]
* You can change topics by moving your username to a topic without an author
&lt;!-- * More topics will be added, but students are also encouraged to propose topics --&gt;

==Negotiate a topic==
'''New or modified topics''' must be approved by the [[Motivation and emotion/About/Staff|unit convener]].
* Ways to get some ideas:
** [https://cogniti.canberra.edu.au/agents/6a3caf85b2fed9a95789241a/chat?k=WfY1yx6AB7yF_9XqiyYENzUBNCGfV1e2FCBodCDD2xU Chat with the motivation and emotion book chapter topic generator] (Cogniti)
** Look through chapters written in [[Motivation and emotion/Book|previous years]]
** Check out [[Motivation and emotion/Book/Ideas for topics|similar projects]] on the internet
* New topics must:
** '''Be unique''': Topic must not already be sufficiently covered by a [[Motivation and emotion/Book|previous motivation and motivation book chapter]]. Search before making a proposal. How does the proposed topic build on, or differ from, previous work?
** '''Align with the project theme''': Topic must be related to related to [[motivation]] or [[emotion]] and fit the overarching book theme which is to help people to understand and improve their motivational and emotional lives using psychological science
** '''Have appropriate scope''': If the topic is too narrow, or if there is a lack of psychological theory and research, it will be difficult to satisfy the [[#Marking criteria|marking criteria]]
* To propose a new topic, email the [[Motivation and emotion/About/Staff|unit convener]] with these details:
** Title
** Sub-title (in the form of a question) — see [[Motivation and emotion/Book|examples]]
** Wikiversity user name
** Details of any related previous book chapter topics (check via this [[Motivation and emotion/Book|search box]])

==Ask clarifying questions==
* What questions could you ask to help you to successfully tackle the major project?
* Where and how could you ask these questions to optimise quality and quantity of feedback?
&lt;!--
==Marking and feedback==
* No marks
* Feedback will be provided to:
** Approve topic selection
** Respond to student questions
** Provide suggestions
* Feedback will be available via {{Motivation and emotion/Canvas}} before the [[Motivation and emotion/Assessment/Topic|topic development]] due date
* Follow up if you don't understand the feedback

==Extensions and late submissions==
* No extensions or late submissions
* If you don't submit, go ahead and sign up to an approved topic, and move on to the [[Motivation and emotion/Assessment/Topic|topic development exercise]]

==Learning outcomes==
How the unit's [[Motivation and emotion/About/Learning outcomes|learning outcomes]] are addressed by this assessment exercise:
{| border=1 cellpadding=5 cellspacing=&quot;0&quot; background:transparent style=&quot;width:90%; margin: auto;&quot;
|-
| style=&quot;width:40%;&quot; | '''Learning outcome'''
| style=&quot;width:60%;&quot; | '''Assessment task''' 
|-
| Critically apply knowledge of motivation or emotion to an indepth understanding of a specific topic in this field.
| Select an appropriate, unique, specific motivation or emotion topic for the major project 
|}

==Graduate attributes==
How the unit's [[Motivation and emotion/About/Graduate attributes|graduate attributes]] are addressed by this assessment exercise:
{| border=1 cellpadding=5 cellspacing=&quot;0&quot; background:transparent style=&quot;width:90%; margin: auto;&quot;
|-
| style=&quot;width:40%;&quot; | '''Graduate attribute'''
| style=&quot;width:60%;&quot; | '''Assessment task''' 
|-
| style=&quot;vertical-align:top;&quot; | Be professional — communicate effectively
| style=&quot;vertical-align:top;&quot; | Communicate by signing up to a topic, submitting for approval, and asking clarifying questions
|-
| style=&quot;vertical-align:top;&quot; | Be professional — display initiative and drive, and use organisation skills to plan and manage workload
| style=&quot;vertical-align:top;&quot; | Get organised by selecting a topic
|-
| style=&quot;vertical-align:top;&quot; | Be a lifelong learner — evaluate and adopt new technology
| style=&quot;vertical-align:top;&quot; | Create a Wikiversity account and make at least 1 edit by signing up to a topic
|}
--&gt;
==See also==
* [[Motivation and emotion/Tutorials/Topic selection|Tutorial 01: Topic selection]]
{{Motivation and emotion/Assessment/Navigation}}
[[Category:Motivation and emotion/Assessment/Selection| ]]</text>
      <sha1>3x0sqb8172s2bnkpy4o0j93wp6ext8q</sha1>
    </revision>
    <revision>
      <id>2819831</id>
      <parentid>2819830</parentid>
      <timestamp>2026-07-30T08:14:47Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <minor />
      <comment>/* Create a Wikiversity account */</comment>
      <origin>2819831</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6159" sha1="rizc1j0wjroyf8ntthxf2yhd9r7gnoh" xml:space="preserve">{{title|Topic selection — Guidelines}}
&lt;!--&lt;div style=&quot;text-align: center;&quot;&gt;''Select major project topic'' --&gt;
&lt;!-- ---------------------------------- ---&gt;
&lt;!-- Count down --&gt;
&lt;!-- ---------------------------------- ---&gt;
&lt;!-- {{countdown
 |year       = 2024
 |month      = 08
 |day        = 04
 |hour       = 23
 |minute     = 0
 |second     = 0
 |event = this assessment is due
}} --&gt;
&lt;!-- {{Motivation and emotion/Assessment/In development}} --&gt;
&lt;!-- Show this during semester {{:Motivation and emotion/Assessment/Chapter/Contents}}&lt;/div&gt; --&gt;
{{TOCright}}
==Overview==
&lt;!--
* Weight: 0%
* Due: {{/Due}}
* Ungraded early assessment exercise
* Tasks --&gt;
# [[Special:CreateAccount|Create a Wikiversity account]]
# Sign up to (or negotiate) an approved topic for the [[Motivation and emotion/Assessment/Major project|major project]] — see {{Motivation and emotion/Book}} — or negotiate a new topic
# Ask clarifying questions
# ⤿ The selected topic will then be used for the [[Motivation and emotion/Assessment/Topic|topic development]] and [[Motivation and emotion/Assessment/Chapter|book chapter]].

==[[Special:CreateAccount|Create a Wikiversity account]]==
* All user names start with a capital letter
* Can use your real name, pseudonym, student number, and so on

==Sign up to a topic==
'''Pre-approved topics''' are available via the {{Motivation and emotion/Book}}
* How to sign up:
** Click &quot;Edit&quot; or &quot;Edit source&quot;
** Replace '''[[User:User Name|User Name]]''' alongside the topic of interest with your Wikiversity user name
** Click &quot;Publish&quot;
&lt;!-- * Conduct an initial literature search. If the topic is too broad or narrow, it may be difficult to satisfy the [[#Marking criteria|marking criteria]] --&gt;
* Pre-approved topics can be modified by emailing a revised title and sub-title to the [[Motivation and emotion/About/Staff|unit convener]]
* You can change topics by moving your username to a topic without an author
&lt;!-- * More topics will be added, but students are also encouraged to propose topics --&gt;

==Negotiate a topic==
'''New or modified topics''' must be approved by the [[Motivation and emotion/About/Staff|unit convener]].
* Ways to get some ideas:
** [https://cogniti.canberra.edu.au/agents/6a3caf85b2fed9a95789241a/chat?k=WfY1yx6AB7yF_9XqiyYENzUBNCGfV1e2FCBodCDD2xU Chat with the motivation and emotion book chapter topic generator] (Cogniti)
** Look through chapters written in [[Motivation and emotion/Book|previous years]]
** Check out [[Motivation and emotion/Book/Ideas for topics|similar projects]] on the internet
* New topics must:
** '''Be unique''': Topic must not already be sufficiently covered by a [[Motivation and emotion/Book|previous motivation and motivation book chapter]]. Search before making a proposal. How does the proposed topic build on, or differ from, previous work?
** '''Align with the project theme''': Topic must be related to related to [[motivation]] or [[emotion]] and fit the overarching book theme which is to help people to understand and improve their motivational and emotional lives using psychological science
** '''Have appropriate scope''': If the topic is too narrow, or if there is a lack of psychological theory and research, it will be difficult to satisfy the [[#Marking criteria|marking criteria]]
* To propose a new topic, email the [[Motivation and emotion/About/Staff|unit convener]] with these details:
** Title
** Sub-title (in the form of a question) — see [[Motivation and emotion/Book|examples]]
** Wikiversity user name
** Details of any related previous book chapter topics (check via this [[Motivation and emotion/Book|search box]])

==Ask clarifying questions==
* What questions could you ask to help you to successfully tackle the major project?
* Where and how could you ask these questions to optimise quality and quantity of feedback?
&lt;!--
==Marking and feedback==
* No marks
* Feedback will be provided to:
** Approve topic selection
** Respond to student questions
** Provide suggestions
* Feedback will be available via {{Motivation and emotion/Canvas}} before the [[Motivation and emotion/Assessment/Topic|topic development]] due date
* Follow up if you don't understand the feedback

==Extensions and late submissions==
* No extensions or late submissions
* If you don't submit, go ahead and sign up to an approved topic, and move on to the [[Motivation and emotion/Assessment/Topic|topic development exercise]]

==Learning outcomes==
How the unit's [[Motivation and emotion/About/Learning outcomes|learning outcomes]] are addressed by this assessment exercise:
{| border=1 cellpadding=5 cellspacing=&quot;0&quot; background:transparent style=&quot;width:90%; margin: auto;&quot;
|-
| style=&quot;width:40%;&quot; | '''Learning outcome'''
| style=&quot;width:60%;&quot; | '''Assessment task''' 
|-
| Critically apply knowledge of motivation or emotion to an indepth understanding of a specific topic in this field.
| Select an appropriate, unique, specific motivation or emotion topic for the major project 
|}

==Graduate attributes==
How the unit's [[Motivation and emotion/About/Graduate attributes|graduate attributes]] are addressed by this assessment exercise:
{| border=1 cellpadding=5 cellspacing=&quot;0&quot; background:transparent style=&quot;width:90%; margin: auto;&quot;
|-
| style=&quot;width:40%;&quot; | '''Graduate attribute'''
| style=&quot;width:60%;&quot; | '''Assessment task''' 
|-
| style=&quot;vertical-align:top;&quot; | Be professional — communicate effectively
| style=&quot;vertical-align:top;&quot; | Communicate by signing up to a topic, submitting for approval, and asking clarifying questions
|-
| style=&quot;vertical-align:top;&quot; | Be professional — display initiative and drive, and use organisation skills to plan and manage workload
| style=&quot;vertical-align:top;&quot; | Get organised by selecting a topic
|-
| style=&quot;vertical-align:top;&quot; | Be a lifelong learner — evaluate and adopt new technology
| style=&quot;vertical-align:top;&quot; | Create a Wikiversity account and make at least 1 edit by signing up to a topic
|}
--&gt;
==See also==
* [[Motivation and emotion/Tutorials/Topic selection|Tutorial 01: Topic selection]]
{{Motivation and emotion/Assessment/Navigation}}
[[Category:Motivation and emotion/Assessment/Selection| ]]</text>
      <sha1>rizc1j0wjroyf8ntthxf2yhd9r7gnoh</sha1>
    </revision>
    <revision>
      <id>2819832</id>
      <parentid>2819831</parentid>
      <timestamp>2026-07-30T08:16:46Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <minor />
      <comment>/* Sign up to a topic */</comment>
      <origin>2819832</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6224" sha1="e1z97pbuveifbv2fa0mn20ppogz55pt" xml:space="preserve">{{title|Topic selection — Guidelines}}
&lt;!--&lt;div style=&quot;text-align: center;&quot;&gt;''Select major project topic'' --&gt;
&lt;!-- ---------------------------------- ---&gt;
&lt;!-- Count down --&gt;
&lt;!-- ---------------------------------- ---&gt;
&lt;!-- {{countdown
 |year       = 2024
 |month      = 08
 |day        = 04
 |hour       = 23
 |minute     = 0
 |second     = 0
 |event = this assessment is due
}} --&gt;
&lt;!-- {{Motivation and emotion/Assessment/In development}} --&gt;
&lt;!-- Show this during semester {{:Motivation and emotion/Assessment/Chapter/Contents}}&lt;/div&gt; --&gt;
{{TOCright}}
==Overview==
&lt;!--
* Weight: 0%
* Due: {{/Due}}
* Ungraded early assessment exercise
* Tasks --&gt;
# [[Special:CreateAccount|Create a Wikiversity account]]
# Sign up to (or negotiate) an approved topic for the [[Motivation and emotion/Assessment/Major project|major project]] — see {{Motivation and emotion/Book}} — or negotiate a new topic
# Ask clarifying questions
# ⤿ The selected topic will then be used for the [[Motivation and emotion/Assessment/Topic|topic development]] and [[Motivation and emotion/Assessment/Chapter|book chapter]].

==[[Special:CreateAccount|Create a Wikiversity account]]==
* All user names start with a capital letter
* Can use your real name, pseudonym, student number, and so on

==Sign up to a topic==
'''Pre-approved topics''' are listed in the {{Motivation and emotion/Book}}
* '''How to sign up''':
** Click &quot;Edit&quot; or &quot;Edit source&quot;
** Replace '''[[User:User Name|User Name]]''' alongside the topic of interest with your Wikiversity user name
** Click &quot;Publish&quot;
&lt;!-- * Conduct an initial literature search. If the topic is too broad or narrow, it may be difficult to satisfy the [[#Marking criteria|marking criteria]] --&gt;
* To modify a pre-approved topic, email a revised title and sub-title to the [[Motivation and emotion/About/Staff|unit convener]]
* You can change topics you are assigned to by editng the table of contents and moving your username to any other topic which does not have an author
&lt;!-- * More topics will be added, but students are also encouraged to propose topics --&gt;

==Negotiate a topic==
'''New or modified topics''' must be approved by the [[Motivation and emotion/About/Staff|unit convener]].
* Ways to get some ideas:
** [https://cogniti.canberra.edu.au/agents/6a3caf85b2fed9a95789241a/chat?k=WfY1yx6AB7yF_9XqiyYENzUBNCGfV1e2FCBodCDD2xU Chat with the motivation and emotion book chapter topic generator] (Cogniti)
** Look through chapters written in [[Motivation and emotion/Book|previous years]]
** Check out [[Motivation and emotion/Book/Ideas for topics|similar projects]] on the internet
* New topics must:
** '''Be unique''': Topic must not already be sufficiently covered by a [[Motivation and emotion/Book|previous motivation and motivation book chapter]]. Search before making a proposal. How does the proposed topic build on, or differ from, previous work?
** '''Align with the project theme''': Topic must be related to related to [[motivation]] or [[emotion]] and fit the overarching book theme which is to help people to understand and improve their motivational and emotional lives using psychological science
** '''Have appropriate scope''': If the topic is too narrow, or if there is a lack of psychological theory and research, it will be difficult to satisfy the [[#Marking criteria|marking criteria]]
* To propose a new topic, email the [[Motivation and emotion/About/Staff|unit convener]] with these details:
** Title
** Sub-title (in the form of a question) — see [[Motivation and emotion/Book|examples]]
** Wikiversity user name
** Details of any related previous book chapter topics (check via this [[Motivation and emotion/Book|search box]])

==Ask clarifying questions==
* What questions could you ask to help you to successfully tackle the major project?
* Where and how could you ask these questions to optimise quality and quantity of feedback?
&lt;!--
==Marking and feedback==
* No marks
* Feedback will be provided to:
** Approve topic selection
** Respond to student questions
** Provide suggestions
* Feedback will be available via {{Motivation and emotion/Canvas}} before the [[Motivation and emotion/Assessment/Topic|topic development]] due date
* Follow up if you don't understand the feedback

==Extensions and late submissions==
* No extensions or late submissions
* If you don't submit, go ahead and sign up to an approved topic, and move on to the [[Motivation and emotion/Assessment/Topic|topic development exercise]]

==Learning outcomes==
How the unit's [[Motivation and emotion/About/Learning outcomes|learning outcomes]] are addressed by this assessment exercise:
{| border=1 cellpadding=5 cellspacing=&quot;0&quot; background:transparent style=&quot;width:90%; margin: auto;&quot;
|-
| style=&quot;width:40%;&quot; | '''Learning outcome'''
| style=&quot;width:60%;&quot; | '''Assessment task''' 
|-
| Critically apply knowledge of motivation or emotion to an indepth understanding of a specific topic in this field.
| Select an appropriate, unique, specific motivation or emotion topic for the major project 
|}

==Graduate attributes==
How the unit's [[Motivation and emotion/About/Graduate attributes|graduate attributes]] are addressed by this assessment exercise:
{| border=1 cellpadding=5 cellspacing=&quot;0&quot; background:transparent style=&quot;width:90%; margin: auto;&quot;
|-
| style=&quot;width:40%;&quot; | '''Graduate attribute'''
| style=&quot;width:60%;&quot; | '''Assessment task''' 
|-
| style=&quot;vertical-align:top;&quot; | Be professional — communicate effectively
| style=&quot;vertical-align:top;&quot; | Communicate by signing up to a topic, submitting for approval, and asking clarifying questions
|-
| style=&quot;vertical-align:top;&quot; | Be professional — display initiative and drive, and use organisation skills to plan and manage workload
| style=&quot;vertical-align:top;&quot; | Get organised by selecting a topic
|-
| style=&quot;vertical-align:top;&quot; | Be a lifelong learner — evaluate and adopt new technology
| style=&quot;vertical-align:top;&quot; | Create a Wikiversity account and make at least 1 edit by signing up to a topic
|}
--&gt;
==See also==
* [[Motivation and emotion/Tutorials/Topic selection|Tutorial 01: Topic selection]]
{{Motivation and emotion/Assessment/Navigation}}
[[Category:Motivation and emotion/Assessment/Selection| ]]</text>
      <sha1>e1z97pbuveifbv2fa0mn20ppogz55pt</sha1>
    </revision>
    <revision>
      <id>2819833</id>
      <parentid>2819832</parentid>
      <timestamp>2026-07-30T08:17:41Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <minor />
      <comment>/* Sign up to a topic */</comment>
      <origin>2819833</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6247" sha1="avvsjfnt6q4x7hcecysm56tckygghbf" xml:space="preserve">{{title|Topic selection — Guidelines}}
&lt;!--&lt;div style=&quot;text-align: center;&quot;&gt;''Select major project topic'' --&gt;
&lt;!-- ---------------------------------- ---&gt;
&lt;!-- Count down --&gt;
&lt;!-- ---------------------------------- ---&gt;
&lt;!-- {{countdown
 |year       = 2024
 |month      = 08
 |day        = 04
 |hour       = 23
 |minute     = 0
 |second     = 0
 |event = this assessment is due
}} --&gt;
&lt;!-- {{Motivation and emotion/Assessment/In development}} --&gt;
&lt;!-- Show this during semester {{:Motivation and emotion/Assessment/Chapter/Contents}}&lt;/div&gt; --&gt;
{{TOCright}}
==Overview==
&lt;!--
* Weight: 0%
* Due: {{/Due}}
* Ungraded early assessment exercise
* Tasks --&gt;
# [[Special:CreateAccount|Create a Wikiversity account]]
# Sign up to (or negotiate) an approved topic for the [[Motivation and emotion/Assessment/Major project|major project]] — see {{Motivation and emotion/Book}} — or negotiate a new topic
# Ask clarifying questions
# ⤿ The selected topic will then be used for the [[Motivation and emotion/Assessment/Topic|topic development]] and [[Motivation and emotion/Assessment/Chapter|book chapter]].

==[[Special:CreateAccount|Create a Wikiversity account]]==
* All user names start with a capital letter
* Can use your real name, pseudonym, student number, and so on

==Sign up to a topic==
'''Pre-approved topics''' are listed in the {{Motivation and emotion/Book}}
* '''How to sign up''':
** Click &quot;Edit&quot; or &quot;Edit source&quot;
** Replace '''[[User:User Name|User Name]]''' alongside the topic of interest with your Wikiversity user name
** Click &quot;Publish&quot;
&lt;!-- * Conduct an initial literature search. If the topic is too broad or narrow, it may be difficult to satisfy the [[#Marking criteria|marking criteria]] --&gt;
* To modify a pre-approved topic, email a revised title and sub-title to the [[Motivation and emotion/About/Staff|unit convener]]
* You can change the topic you are assigned to by editng the table of contents page and moving your username from the current topic to a topic which does not have an author
&lt;!-- * More topics will be added, but students are also encouraged to propose topics --&gt;

==Negotiate a topic==
'''New or modified topics''' must be approved by the [[Motivation and emotion/About/Staff|unit convener]].
* Ways to get some ideas:
** [https://cogniti.canberra.edu.au/agents/6a3caf85b2fed9a95789241a/chat?k=WfY1yx6AB7yF_9XqiyYENzUBNCGfV1e2FCBodCDD2xU Chat with the motivation and emotion book chapter topic generator] (Cogniti)
** Look through chapters written in [[Motivation and emotion/Book|previous years]]
** Check out [[Motivation and emotion/Book/Ideas for topics|similar projects]] on the internet
* New topics must:
** '''Be unique''': Topic must not already be sufficiently covered by a [[Motivation and emotion/Book|previous motivation and motivation book chapter]]. Search before making a proposal. How does the proposed topic build on, or differ from, previous work?
** '''Align with the project theme''': Topic must be related to related to [[motivation]] or [[emotion]] and fit the overarching book theme which is to help people to understand and improve their motivational and emotional lives using psychological science
** '''Have appropriate scope''': If the topic is too narrow, or if there is a lack of psychological theory and research, it will be difficult to satisfy the [[#Marking criteria|marking criteria]]
* To propose a new topic, email the [[Motivation and emotion/About/Staff|unit convener]] with these details:
** Title
** Sub-title (in the form of a question) — see [[Motivation and emotion/Book|examples]]
** Wikiversity user name
** Details of any related previous book chapter topics (check via this [[Motivation and emotion/Book|search box]])

==Ask clarifying questions==
* What questions could you ask to help you to successfully tackle the major project?
* Where and how could you ask these questions to optimise quality and quantity of feedback?
&lt;!--
==Marking and feedback==
* No marks
* Feedback will be provided to:
** Approve topic selection
** Respond to student questions
** Provide suggestions
* Feedback will be available via {{Motivation and emotion/Canvas}} before the [[Motivation and emotion/Assessment/Topic|topic development]] due date
* Follow up if you don't understand the feedback

==Extensions and late submissions==
* No extensions or late submissions
* If you don't submit, go ahead and sign up to an approved topic, and move on to the [[Motivation and emotion/Assessment/Topic|topic development exercise]]

==Learning outcomes==
How the unit's [[Motivation and emotion/About/Learning outcomes|learning outcomes]] are addressed by this assessment exercise:
{| border=1 cellpadding=5 cellspacing=&quot;0&quot; background:transparent style=&quot;width:90%; margin: auto;&quot;
|-
| style=&quot;width:40%;&quot; | '''Learning outcome'''
| style=&quot;width:60%;&quot; | '''Assessment task''' 
|-
| Critically apply knowledge of motivation or emotion to an indepth understanding of a specific topic in this field.
| Select an appropriate, unique, specific motivation or emotion topic for the major project 
|}

==Graduate attributes==
How the unit's [[Motivation and emotion/About/Graduate attributes|graduate attributes]] are addressed by this assessment exercise:
{| border=1 cellpadding=5 cellspacing=&quot;0&quot; background:transparent style=&quot;width:90%; margin: auto;&quot;
|-
| style=&quot;width:40%;&quot; | '''Graduate attribute'''
| style=&quot;width:60%;&quot; | '''Assessment task''' 
|-
| style=&quot;vertical-align:top;&quot; | Be professional — communicate effectively
| style=&quot;vertical-align:top;&quot; | Communicate by signing up to a topic, submitting for approval, and asking clarifying questions
|-
| style=&quot;vertical-align:top;&quot; | Be professional — display initiative and drive, and use organisation skills to plan and manage workload
| style=&quot;vertical-align:top;&quot; | Get organised by selecting a topic
|-
| style=&quot;vertical-align:top;&quot; | Be a lifelong learner — evaluate and adopt new technology
| style=&quot;vertical-align:top;&quot; | Create a Wikiversity account and make at least 1 edit by signing up to a topic
|}
--&gt;
==See also==
* [[Motivation and emotion/Tutorials/Topic selection|Tutorial 01: Topic selection]]
{{Motivation and emotion/Assessment/Navigation}}
[[Category:Motivation and emotion/Assessment/Selection| ]]</text>
      <sha1>avvsjfnt6q4x7hcecysm56tckygghbf</sha1>
    </revision>
    <revision>
      <id>2819834</id>
      <parentid>2819833</parentid>
      <timestamp>2026-07-30T08:18:07Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <minor />
      <comment>/* Sign up to a topic */</comment>
      <origin>2819834</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6243" sha1="46jb73fb46ojml42wdu0ndivyflf4ha" xml:space="preserve">{{title|Topic selection — Guidelines}}
&lt;!--&lt;div style=&quot;text-align: center;&quot;&gt;''Select major project topic'' --&gt;
&lt;!-- ---------------------------------- ---&gt;
&lt;!-- Count down --&gt;
&lt;!-- ---------------------------------- ---&gt;
&lt;!-- {{countdown
 |year       = 2024
 |month      = 08
 |day        = 04
 |hour       = 23
 |minute     = 0
 |second     = 0
 |event = this assessment is due
}} --&gt;
&lt;!-- {{Motivation and emotion/Assessment/In development}} --&gt;
&lt;!-- Show this during semester {{:Motivation and emotion/Assessment/Chapter/Contents}}&lt;/div&gt; --&gt;
{{TOCright}}
==Overview==
&lt;!--
* Weight: 0%
* Due: {{/Due}}
* Ungraded early assessment exercise
* Tasks --&gt;
# [[Special:CreateAccount|Create a Wikiversity account]]
# Sign up to (or negotiate) an approved topic for the [[Motivation and emotion/Assessment/Major project|major project]] — see {{Motivation and emotion/Book}} — or negotiate a new topic
# Ask clarifying questions
# ⤿ The selected topic will then be used for the [[Motivation and emotion/Assessment/Topic|topic development]] and [[Motivation and emotion/Assessment/Chapter|book chapter]].

==[[Special:CreateAccount|Create a Wikiversity account]]==
* All user names start with a capital letter
* Can use your real name, pseudonym, student number, and so on

==Sign up to a topic==
'''Pre-approved topics''' are listed in the {{Motivation and emotion/Book}}
* '''How to sign up''':
** Click &quot;Edit&quot; or &quot;Edit source&quot;
** Replace &quot;[[User:User Name|User Name]]&quot; alongside the topic of interest with your Wikiversity user name
** Click &quot;Publish&quot;
&lt;!-- * Conduct an initial literature search. If the topic is too broad or narrow, it may be difficult to satisfy the [[#Marking criteria|marking criteria]] --&gt;
* To modify a pre-approved topic, email a revised title and sub-title to the [[Motivation and emotion/About/Staff|unit convener]]
* You can change the topic you are assigned to by editng the table of contents page and moving your username from the current topic to a topic which does not have an author
&lt;!-- * More topics will be added, but students are also encouraged to propose topics --&gt;

==Negotiate a topic==
'''New or modified topics''' must be approved by the [[Motivation and emotion/About/Staff|unit convener]].
* Ways to get some ideas:
** [https://cogniti.canberra.edu.au/agents/6a3caf85b2fed9a95789241a/chat?k=WfY1yx6AB7yF_9XqiyYENzUBNCGfV1e2FCBodCDD2xU Chat with the motivation and emotion book chapter topic generator] (Cogniti)
** Look through chapters written in [[Motivation and emotion/Book|previous years]]
** Check out [[Motivation and emotion/Book/Ideas for topics|similar projects]] on the internet
* New topics must:
** '''Be unique''': Topic must not already be sufficiently covered by a [[Motivation and emotion/Book|previous motivation and motivation book chapter]]. Search before making a proposal. How does the proposed topic build on, or differ from, previous work?
** '''Align with the project theme''': Topic must be related to related to [[motivation]] or [[emotion]] and fit the overarching book theme which is to help people to understand and improve their motivational and emotional lives using psychological science
** '''Have appropriate scope''': If the topic is too narrow, or if there is a lack of psychological theory and research, it will be difficult to satisfy the [[#Marking criteria|marking criteria]]
* To propose a new topic, email the [[Motivation and emotion/About/Staff|unit convener]] with these details:
** Title
** Sub-title (in the form of a question) — see [[Motivation and emotion/Book|examples]]
** Wikiversity user name
** Details of any related previous book chapter topics (check via this [[Motivation and emotion/Book|search box]])

==Ask clarifying questions==
* What questions could you ask to help you to successfully tackle the major project?
* Where and how could you ask these questions to optimise quality and quantity of feedback?
&lt;!--
==Marking and feedback==
* No marks
* Feedback will be provided to:
** Approve topic selection
** Respond to student questions
** Provide suggestions
* Feedback will be available via {{Motivation and emotion/Canvas}} before the [[Motivation and emotion/Assessment/Topic|topic development]] due date
* Follow up if you don't understand the feedback

==Extensions and late submissions==
* No extensions or late submissions
* If you don't submit, go ahead and sign up to an approved topic, and move on to the [[Motivation and emotion/Assessment/Topic|topic development exercise]]

==Learning outcomes==
How the unit's [[Motivation and emotion/About/Learning outcomes|learning outcomes]] are addressed by this assessment exercise:
{| border=1 cellpadding=5 cellspacing=&quot;0&quot; background:transparent style=&quot;width:90%; margin: auto;&quot;
|-
| style=&quot;width:40%;&quot; | '''Learning outcome'''
| style=&quot;width:60%;&quot; | '''Assessment task''' 
|-
| Critically apply knowledge of motivation or emotion to an indepth understanding of a specific topic in this field.
| Select an appropriate, unique, specific motivation or emotion topic for the major project 
|}

==Graduate attributes==
How the unit's [[Motivation and emotion/About/Graduate attributes|graduate attributes]] are addressed by this assessment exercise:
{| border=1 cellpadding=5 cellspacing=&quot;0&quot; background:transparent style=&quot;width:90%; margin: auto;&quot;
|-
| style=&quot;width:40%;&quot; | '''Graduate attribute'''
| style=&quot;width:60%;&quot; | '''Assessment task''' 
|-
| style=&quot;vertical-align:top;&quot; | Be professional — communicate effectively
| style=&quot;vertical-align:top;&quot; | Communicate by signing up to a topic, submitting for approval, and asking clarifying questions
|-
| style=&quot;vertical-align:top;&quot; | Be professional — display initiative and drive, and use organisation skills to plan and manage workload
| style=&quot;vertical-align:top;&quot; | Get organised by selecting a topic
|-
| style=&quot;vertical-align:top;&quot; | Be a lifelong learner — evaluate and adopt new technology
| style=&quot;vertical-align:top;&quot; | Create a Wikiversity account and make at least 1 edit by signing up to a topic
|}
--&gt;
==See also==
* [[Motivation and emotion/Tutorials/Topic selection|Tutorial 01: Topic selection]]
{{Motivation and emotion/Assessment/Navigation}}
[[Category:Motivation and emotion/Assessment/Selection| ]]</text>
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&lt;!--&lt;div style=&quot;text-align: center;&quot;&gt;''Select major project topic'' --&gt;
&lt;!-- ---------------------------------- ---&gt;
&lt;!-- Count down --&gt;
&lt;!-- ---------------------------------- ---&gt;
&lt;!-- {{countdown
 |year       = 2024
 |month      = 08
 |day        = 04
 |hour       = 23
 |minute     = 0
 |second     = 0
 |event = this assessment is due
}} --&gt;
&lt;!-- {{Motivation and emotion/Assessment/In development}} --&gt;
&lt;!-- Show this during semester {{:Motivation and emotion/Assessment/Chapter/Contents}}&lt;/div&gt; --&gt;
{{TOCright}}
==Overview==
&lt;!--
* Weight: 0%
* Due: {{/Due}}
* Ungraded early assessment exercise
* Tasks --&gt;
# [[Special:CreateAccount|Create a Wikiversity account]]
# Sign up to (or negotiate) an approved topic for the [[Motivation and emotion/Assessment/Major project|major project]] — see {{Motivation and emotion/Book}} — or negotiate a new topic
# Ask clarifying questions
# ⤿ The selected topic will then be used for the [[Motivation and emotion/Assessment/Topic|topic development]] and [[Motivation and emotion/Assessment/Chapter|book chapter]].

==[[Special:CreateAccount|Create a Wikiversity account]]==
* All user names start with a capital letter
* Can use your real name, pseudonym, student number, and so on

==Sign up to a topic==
'''Pre-approved topics''' are listed in the {{Motivation and emotion/Book}}
* '''How to sign up''':
** Click &quot;Edit&quot; or &quot;Edit source&quot;
** Replace &quot;[[User:User Name|User Name]]&quot; alongside the topic of interest with your Wikiversity user name
** Click &quot;Publish&quot;
&lt;!-- * Conduct an initial literature search. If the topic is too broad or narrow, it may be difficult to satisfy the [[#Marking criteria|marking criteria]] --&gt;
* To modify a pre-approved topic, email a revised title and sub-title to the [[Motivation and emotion/About/Staff|unit convener]]
* You can change the topic you are assigned to by editng the table of contents page and moving your username from the current topic to a topic which does not have an author
&lt;!-- * More topics will be added, but students are also encouraged to propose topics --&gt;

==Negotiate a topic==
'''New or modified topics''' must be approved by the [[Motivation and emotion/About/Staff|unit convener]].
* Ways to get some ideas:
** [https://cogniti.canberra.edu.au/agents/6a3caf85b2fed9a95789241a/chat?k=WfY1yx6AB7yF_9XqiyYENzUBNCGfV1e2FCBodCDD2xU Chat with the motivation and emotion book chapter topic generator] (Cogniti)
** Look through chapters written in [[Motivation and emotion/Book|previous years]]
** Check out [[Motivation and emotion/Book/Ideas for topics|similar projects]] on the internet
* New topics must:
** '''Be unique''': The topic must not already be sufficiently covered by a [[Motivation and emotion/Book|previous motivation and motivation book chapter]]. Search before making a proposal. How does the proposed topic build on, or differ from, previous work?
** '''Align with the project theme''': Topic must be related to related to [[motivation]] or [[emotion]] and fit the overarching book theme which is to ''help people to understand and improve their motivational and emotional lives using psychological science''.
** '''Have appropriate scope''': If the topic is too narrow, or if there is a lack of psychological theory and research, it will be difficult to satisfy the [[#Marking criteria|marking criteria]]. If the topic is too broad, it will be unwieldy and lack sufficiently focused applicability.
* To propose a new topic, email the [[Motivation and emotion/About/Staff|unit convener]] with these details:
** Title
** Sub-title (in the form of a question) — see [[Motivation and emotion/Book|examples]]
** Wikiversity user name
** Details of any related previous book chapter topics (check via this [[Motivation and emotion/Book|search box]])

==Ask clarifying questions==
* What questions could you ask to help you to successfully tackle the major project?
* Where and how could you ask these questions to optimise quality and quantity of feedback?
&lt;!--
==Marking and feedback==
* No marks
* Feedback will be provided to:
** Approve topic selection
** Respond to student questions
** Provide suggestions
* Feedback will be available via {{Motivation and emotion/Canvas}} before the [[Motivation and emotion/Assessment/Topic|topic development]] due date
* Follow up if you don't understand the feedback

==Extensions and late submissions==
* No extensions or late submissions
* If you don't submit, go ahead and sign up to an approved topic, and move on to the [[Motivation and emotion/Assessment/Topic|topic development exercise]]

==Learning outcomes==
How the unit's [[Motivation and emotion/About/Learning outcomes|learning outcomes]] are addressed by this assessment exercise:
{| border=1 cellpadding=5 cellspacing=&quot;0&quot; background:transparent style=&quot;width:90%; margin: auto;&quot;
|-
| style=&quot;width:40%;&quot; | '''Learning outcome'''
| style=&quot;width:60%;&quot; | '''Assessment task''' 
|-
| Critically apply knowledge of motivation or emotion to an indepth understanding of a specific topic in this field.
| Select an appropriate, unique, specific motivation or emotion topic for the major project 
|}

==Graduate attributes==
How the unit's [[Motivation and emotion/About/Graduate attributes|graduate attributes]] are addressed by this assessment exercise:
{| border=1 cellpadding=5 cellspacing=&quot;0&quot; background:transparent style=&quot;width:90%; margin: auto;&quot;
|-
| style=&quot;width:40%;&quot; | '''Graduate attribute'''
| style=&quot;width:60%;&quot; | '''Assessment task''' 
|-
| style=&quot;vertical-align:top;&quot; | Be professional — communicate effectively
| style=&quot;vertical-align:top;&quot; | Communicate by signing up to a topic, submitting for approval, and asking clarifying questions
|-
| style=&quot;vertical-align:top;&quot; | Be professional — display initiative and drive, and use organisation skills to plan and manage workload
| style=&quot;vertical-align:top;&quot; | Get organised by selecting a topic
|-
| style=&quot;vertical-align:top;&quot; | Be a lifelong learner — evaluate and adopt new technology
| style=&quot;vertical-align:top;&quot; | Create a Wikiversity account and make at least 1 edit by signing up to a topic
|}
--&gt;
==See also==
* [[Motivation and emotion/Tutorials/Topic selection|Tutorial 01: Topic selection]]
{{Motivation and emotion/Assessment/Navigation}}
[[Category:Motivation and emotion/Assessment/Selection| ]]</text>
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&lt;!--&lt;div style=&quot;text-align: center;&quot;&gt;''Select major project topic'' --&gt;
&lt;!-- ---------------------------------- ---&gt;
&lt;!-- Count down --&gt;
&lt;!-- ---------------------------------- ---&gt;
&lt;!-- {{countdown
 |year       = 2024
 |month      = 08
 |day        = 04
 |hour       = 23
 |minute     = 0
 |second     = 0
 |event = this assessment is due
}} --&gt;
&lt;!-- {{Motivation and emotion/Assessment/In development}} --&gt;
&lt;!-- Show this during semester {{:Motivation and emotion/Assessment/Chapter/Contents}}&lt;/div&gt; --&gt;
{{TOCright}}
==Overview==
&lt;!--
* Weight: 0%
* Due: {{/Due}}
* Ungraded early assessment exercise
* Tasks --&gt;
# [[Special:CreateAccount|Create a Wikiversity account]]
# Sign up to (or negotiate) an approved topic for the [[Motivation and emotion/Assessment/Major project|major project]] — see {{Motivation and emotion/Book}} — or negotiate a new topic
# Ask clarifying questions
# ⤿ The selected topic will then be used for the [[Motivation and emotion/Assessment/Topic|topic development]] and [[Motivation and emotion/Assessment/Chapter|book chapter]].

==[[Special:CreateAccount|Create a Wikiversity account]]==
* All user names start with a capital letter
* Can use your real name, pseudonym, student number, and so on

==Sign up to a topic==
'''Pre-approved topics''' are listed in the {{Motivation and emotion/Book}}
* '''How to sign up''':
** Click &quot;Edit&quot; or &quot;Edit source&quot;
** Replace &quot;[[User:User Name|User Name]]&quot; alongside the topic of interest with your Wikiversity user name
** Click &quot;Publish&quot;
&lt;!-- * Conduct an initial literature search. If the topic is too broad or narrow, it may be difficult to satisfy the [[#Marking criteria|marking criteria]] --&gt;
* To modify a pre-approved topic, email a revised title and sub-title to the [[Motivation and emotion/About/Staff|unit convener]]
* You can change the topic you are assigned to by editng the table of contents page and moving your username from the current topic to a topic which does not have an author
&lt;!-- * More topics will be added, but students are also encouraged to propose topics --&gt;

==Negotiate a topic==
'''New or modified topics''' must be approved by the [[Motivation and emotion/About/Staff|unit convener]].
* Ways to get some ideas:
** [https://cogniti.canberra.edu.au/agents/6a3caf85b2fed9a95789241a/chat?k=WfY1yx6AB7yF_9XqiyYENzUBNCGfV1e2FCBodCDD2xU Chat with the motivation and emotion book chapter topic generator] (Cogniti)
** Look through chapters written in [[Motivation and emotion/Book|previous years]]
** Check out [[Motivation and emotion/Book/Ideas for topics|similar projects]] on the internet
* New topics must:
** '''Be unique''': The topic must not already be sufficiently covered by a [[Motivation and emotion/Book|previous motivation and motivation book chapter]]. Search before making a proposal. How does the proposed topic build on, or differ from, previous work?
** '''Align with the project theme''': Topic must be related to related to [[motivation]] or [[emotion]] and fit the overarching book theme which is to ''help people to understand and improve their motivational and emotional lives using psychological science''.
** '''Have appropriate scope''': If the topic is too narrow, or if there is a lack of psychological theory and research, it will be difficult to satisfy the [[#Marking criteria|marking criteria]]. If the topic is too broad, it will be unwieldy and lack sufficiently focused applicability.
* To propose a new topic, email the [[Motivation and emotion/About/Staff|unit convener]] with these details:
** Title
** Sub-title (in the form of a question) — see [[Motivation and emotion/Book|examples]]
** Wikiversity user name
** Details of any related previous book chapter topics (check via this [[Motivation and emotion/Book|search box]])

==Ask clarifying questions==
* What questions could you ask to help you to successfully tackle the major project?
* Where and how could you ask these questions to optimise quality and quantity of feedback?
&lt;!--
==Marking and feedback==
* No marks
* Feedback will be provided to:
** Approve topic selection
** Respond to student questions
** Provide suggestions
* Feedback will be available via {{Motivation and emotion/Canvas}} before the [[Motivation and emotion/Assessment/Topic|topic development]] due date
* Follow up if you don't understand the feedback

==Extensions and late submissions==
* No extensions or late submissions
* If you don't submit, go ahead and sign up to an approved topic, and move on to the [[Motivation and emotion/Assessment/Topic|topic development exercise]]

==Learning outcomes==
How the unit's [[Motivation and emotion/About/Learning outcomes|learning outcomes]] are addressed by this assessment exercise:
{| border=1 cellpadding=5 cellspacing=&quot;0&quot; background:transparent style=&quot;width:90%; margin: auto;&quot;
|-
| style=&quot;width:40%;&quot; | '''Learning outcome'''
| style=&quot;width:60%;&quot; | '''Assessment task''' 
|-
| Critically apply knowledge of motivation or emotion to an indepth understanding of a specific topic in this field.
| Select an appropriate, unique, specific motivation or emotion topic for the major project 
|}

==Graduate attributes==
How the unit's [[Motivation and emotion/About/Graduate attributes|graduate attributes]] are addressed by this assessment exercise:
{| border=1 cellpadding=5 cellspacing=&quot;0&quot; background:transparent style=&quot;width:90%; margin: auto;&quot;
|-
| style=&quot;width:40%;&quot; | '''Graduate attribute'''
| style=&quot;width:60%;&quot; | '''Assessment task''' 
|-
| style=&quot;vertical-align:top;&quot; | Be professional — communicate effectively
| style=&quot;vertical-align:top;&quot; | Communicate by signing up to a topic, submitting for approval, and asking clarifying questions
|-
| style=&quot;vertical-align:top;&quot; | Be professional — display initiative and drive, and use organisation skills to plan and manage workload
| style=&quot;vertical-align:top;&quot; | Get organised by selecting a topic
|-
| style=&quot;vertical-align:top;&quot; | Be a lifelong learner — evaluate and adopt new technology
| style=&quot;vertical-align:top;&quot; | Create a Wikiversity account and make at least 1 edit by signing up to a topic
|}
--&gt;
==See also==
* [[Motivation and emotion/Tutorials/Topic selection|Tutorial 1: Topic selection]]
{{Motivation and emotion/Assessment/Navigation}}
[[Category:Motivation and emotion/Assessment/Selection| ]]</text>
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    <title>Motivation and emotion/Assessment/Using generative AI</title>
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        <id>10242</id>
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      <text bytes="7630" sha1="0hwachg1vx9huqs3aeemxtc2m46bu9r" xml:space="preserve">&lt;noinclude&gt;{{title|Using generative AI guidelines}}
__TOC__

&lt;noinclude&gt;==In a nutshell==&lt;/noinclude&gt;
* '''[https://www.canberra.edu.au/about-uc/l-and-t/genai-and-assessment-at-uc Permitted]''' - The ethical use of GenAI is allowed in completing the [[Motivation and emotion/Assessment/Topic|topic development]] and [[Motivation and emotion/Assessment/Chapter|book chapter]].
* '''[https://www.canberra.edu.au/about-uc/l-and-t/genai-and-assessment-at-uc Restricted]''' - The use of GenAI is NOT allowed in completing the [[Motivation and emotion/Assessment/Exam|exam]].

To use genAI appropriately:
* Acknowledge genAI use in [[Wikiversity:FAQ/Editing/Edit summary|
Wikiversity edit summaries]] with a publicly accessible link to the conversation or tool used and prompt details.
* [[w:Fact-checking|Fact-check]] genAI content.
* Only cite peer-reviewed sources which you have consulted.
* Human-rewrite genAI content to enhance quality.&lt;includeonly&gt;
* [[Motivation and emotion/Assessment/Using generative AI|More detail ...]]&lt;/includeonly&gt;&lt;noinclude&gt;

==Summary==
[[File:Deeply engrossed in puzzle.png|thumb|220x220px|'''Figure 1'''. &lt;!-- An image of an elderly woman deeply engrossed in her daily crossword puzzle. --&gt;This image was generated by [[Motivation and emotion|Motivation and Emotion]] student [[User:JorjaFive|JorjaFive]] using [[w:Midjourney|Midjourney]] and uploaded to [[commons:|Wikimedia Commons]] for use in the [[Motivation and emotion/Book/2023/Flourishing in the elderly|flourishing in the elderly]] chapter.]]

[[w:Generative AI|Generative artificial intelligence]] (genAI) use is '''permitted''' for the [[Motivation and emotion/Assessment/Major project|major project]] (topic development and book chapter) and '''restricted''' for the [[Motivation and emotion/Assessment/Exam|exam]].

GenAI tools can help brainstorm, explain concepts, develop a structure, synthesise ideas, and improve the quality of written expression. GenAI tools can aid, but should not replace, independent thinking and reading of primary sources.

Acknowledge genAI use in [[Wikiversity:FAQ/Editing/Edit summary|Wikiversity edit summaries]]. Academia is based on transparency. Follow the principle that &quot;''more acknowledgment is better than less''&quot;. However, acknowledgement is not required for low-level tasks such as improving spelling and grammar or using Studiosity Writing Feedback+.

You are responsible for content you submit. Be aware of limitations of genAI tools such as inaccuracies, biases, and incomplete content. Fact-check all claims and only cite peer-reviewed citations which you consulted.

The best results are obtained from genAI tools through carefully crafted prompting based on reading of primary sources, with human rewriting to enhance quality. 

If you are unsure, post to [[Motivation and emotion/About/Discussion|discussions]], so we can all learn together.

==Detailed guidelines==
===Use ethically, with caution===
Learning to use genAI tools (such as [[w:ChatGPT|ChatGPT]], [[w:Claude (language model)|Claude]],
[[w:Gemini (chatbot)|Gemini]], and [[w:Microsoft Copilot|Microsoft Copilot]]) responsibly and ethically is an emerging skill. GenAI tools can be used to enhance academic work, but should be used judiciously, as a supplementary tool, rather than as a replacement for independent thinking and academic inquiry.

GenAI tools may be used to assist in preparation of the major project ([[Motivation and emotion/Assessment/Topic|topic development]] and [[Motivation and emotion/Assessment/Chapter|book chapter]]). 

===How to use===
Recommended uses of genAI tools include:
* brainstorming
* explaining key concepts
* developing a structure
* synthesising complex ideas
* rephrasing to improve [[Motivation and emotion/Assessment/Chapter/Readability|readability]] and the quality of written expression
* correcting spelling and grammar
* image generation (e.g., see Figure 1)
* scenario development
* critical feedback and suggestions for improvement

===How to acknowledge===
[[File:Wikipedia Edit Summary dialog in VisualEditor.png|thumb|450px|'''Figure 2'''. If contributing genAI content, include the tool and prompt details in the edit summary, with a link to the conversation]]

[[File:Edit summary for genAI content.png|thumb|450px|'''Figure 3'''. Example page history which demonstrates best practice edit summaries for contributing and revising genAI content]]

Use of GenAI tools must be clearly acknowledged in [[Wikiversity:FAQ/Editing/Edit summary|Wikiversity edit summaries]] (e.g., see Figure 2), otherwise it is a violation of academic integrity.

Best practice is to include a publicly accessible link to the chatbot conversation (e.g., [https://help.openai.com/en/articles/7925741-chatgpt-shared-links-faq ChatGPT shared links FAQ]). If a link can't be shared, then provide details about the tool and the prompt in the [[Wikiversity:FAQ/Editing/Edit summary|edit summary]], (e.g., &quot;ChatGPT May 24 2025 Version. ''Prompt detail or summary''&quot;) (see Figure 3). The chatbot conversation should ''not'' be used as a citation and listed in the references because it is not a reliable, primary, peer-reviewed source.

These practices help to ensure that the use of genAI is clear. Transparency is key to good practice in academia. If in doubt, err on the side of providing acknowledgement. However, there is no need to acknowledge genAI use for low-level tasks such as fixing grammar and spelling or using Studiosity Writing Feedback+.

===Limitations===
Be aware of the limitations of genAI tools. Content they generate may be inaccurate, biased, incomplete, or otherwise problematic. Minimal effort reading and prompting will yield low quality results. Refine prompts based on critical reading and thinking to get better outcomes. You are entirely responsible for the accuracy and quality of any content you submit.

===Fact-check and cite===
Always fact check. Regardless of whether genAI has been used, all claims need to be supported by verified peer-reviewed citations which you have consulted. Guide and craft genAI responses based on your reading of peer-reviewed theory and research. Low-energy or unreflective reuse of genAI text without further investigation and human-writing based on reviewing primary, peer-reviewed academic literature is likely to lead to a poor quality result. GenAI tools work best for topics which you already understand. 

===Going forth===
Despite these warnings, you are encouraged to explore use of genAI tools to help develop higher quality work.

If you are unsure about how to use genAI effectively or how to acknowledge its use appropriately, ask in [[Motivation and emotion/About/Discussion|discussions]], so we can all learn together.

==Example==
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Affiliation_motivation_across_cultures&amp;action=history Affiliation and motivation] (Book chapter, 2025)

==Learn about genAI==
* [https://canberra.libguides.com/genai GenAI for students] (University of Canberra Library)
* [https://techcrunch.com/2024/06/01/what-is-ai-how-does-ai-work/ WTF is AI?] provides a useful introduction and non-technical overview about how genAI works, what it is capable of, limitations, and issues

==See also==
* [[b:Wikibooks:Artificial Intelligence|Wikibooks:Artificial Intelligence]] (policy)
* [[w:Wikipedia:Large language models|Wikipedia:Large language models]] (information page)
* [[Wikiversity:Artificial intelligence|Wikiversity:Artificial intelligence]] (policy)

[[Category:Motivation and emotion/Assessment]]
[[Category:Generative artificial intelligence]]
&lt;/noinclude&gt;</text>
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__TOC__

&lt;noinclude&gt;==In a nutshell==&lt;/noinclude&gt;
* '''[https://www.canberra.edu.au/about-uc/l-and-t/genai-and-assessment-at-uc Permitted]''' - The ethical use of GenAI is allowed in completing the [[Motivation and emotion/Assessment/Topic|topic development]] and [[Motivation and emotion/Assessment/Chapter|book chapter]].
* '''[https://www.canberra.edu.au/about-uc/l-and-t/genai-and-assessment-at-uc Restricted]''' - The use of GenAI is NOT allowed in completing the [[Motivation and emotion/Assessment/Exam|exam]].

To use genAI appropriately:
* Acknowledge genAI use in [[Wikiversity:FAQ/Editing/Edit summary|
Wikiversity edit summaries]] with a publicly accessible link to the conversation or tool used and prompt details.
* [[w:Fact-checking|Fact-check]] genAI content.
* Only cite peer-reviewed sources which you have consulted.
* Human-rewrite genAI content to enhance quality.&lt;includeonly&gt;
* [[Motivation and emotion/Assessment/Using generative AI|More detail ...]]&lt;/includeonly&gt;&lt;noinclude&gt;

==Summary==
[[File:Deeply engrossed in puzzle.png|thumb|220x220px|'''Figure 1'''. &lt;!-- An image of an elderly woman deeply engrossed in her daily crossword puzzle. --&gt;This image was generated by [[Motivation and emotion|Motivation and Emotion]] student [[User:JorjaFive|JorjaFive]] using [[w:Midjourney|Midjourney]] and uploaded to [[commons:|Wikimedia Commons]] for use in the [[Motivation and emotion/Book/2023/Flourishing in the elderly|flourishing in the elderly]] chapter.]]

[[w:Generative AI|Generative artificial intelligence]] (genAI) use is '''permitted''' for the [[Motivation and emotion/Assessment/Major project|major project]] (topic development and book chapter) and '''restricted''' for the [[Motivation and emotion/Assessment/Exam|exam]].

GenAI tools can help brainstorm, explain concepts, develop a structure, synthesise ideas, and improve the quality of written expression. GenAI tools can aid, but should not replace, independent thinking and reading of primary sources.

Acknowledge genAI use in [[Wikiversity:FAQ/Editing/Edit summary|Wikiversity edit summaries]]. Academia is based on transparency. Follow the principle that &quot;''more acknowledgment is better than less''&quot;. However, acknowledgement is not required for using the Cogniti [https://cogniti.canberra.edu.au/agents/6a3caf85b2fed9a95789241a/chat?k=WfY1yx6AB7yF_9XqiyYENzUBNCGfV1e2FCBodCDD2xU book chapter topic generator],  Studiosity Writing Feedback+, or for low-level tasks such as improving spelling and grammar.

You are responsible for content you submit. Be aware of limitations of genAI tools such as inaccuracies, biases, and incomplete content. Fact-check all claims and only cite peer-reviewed citations which you consulted.

The best results are obtained from genAI tools through carefully crafted prompting based on reading of primary sources, with human rewriting to enhance quality. 

If you are unsure, post to [[Motivation and emotion/About/Discussion|discussions]], so we can all learn together.

==Detailed guidelines==
===Use ethically, with caution===
Learning to use genAI tools (such as [[w:ChatGPT|ChatGPT]], [[w:Claude (language model)|Claude]],
[[w:Gemini (chatbot)|Gemini]], and [[w:Microsoft Copilot|Microsoft Copilot]]) responsibly and ethically is an emerging skill. GenAI tools can be used to enhance academic work, but should be used judiciously, as a supplementary tool, rather than as a replacement for independent thinking and academic inquiry.

GenAI tools may be used to assist in preparation of the major project ([[Motivation and emotion/Assessment/Topic|topic development]] and [[Motivation and emotion/Assessment/Chapter|book chapter]]). 

===How to use===
Recommended uses of genAI tools include:
* brainstorming
* explaining key concepts
* developing a structure
* synthesising complex ideas
* rephrasing to improve [[Motivation and emotion/Assessment/Chapter/Readability|readability]] and the quality of written expression
* correcting spelling and grammar
* image generation (e.g., see Figure 1)
* scenario development
* critical feedback and suggestions for improvement

===How to acknowledge===
[[File:Wikipedia Edit Summary dialog in VisualEditor.png|thumb|450px|'''Figure 2'''. If contributing genAI content, include the tool and prompt details in the edit summary, with a link to the conversation]]

[[File:Edit summary for genAI content.png|thumb|450px|'''Figure 3'''. Example page history which demonstrates best practice edit summaries for contributing and revising genAI content]]

Use of GenAI tools must be clearly acknowledged in [[Wikiversity:FAQ/Editing/Edit summary|Wikiversity edit summaries]] (e.g., see Figure 2), otherwise it is a violation of academic integrity.

Best practice is to include a publicly accessible link to the chatbot conversation (e.g., [https://help.openai.com/en/articles/7925741-chatgpt-shared-links-faq ChatGPT shared links FAQ]). If a link can't be shared, then provide details about the tool and the prompt in the [[Wikiversity:FAQ/Editing/Edit summary|edit summary]], (e.g., &quot;ChatGPT May 24 2025 Version. ''Prompt detail or summary''&quot;) (see Figure 3). The chatbot conversation should ''not'' be used as a citation and listed in the references because it is not a reliable, primary, peer-reviewed source.

These practices help to ensure that the use of genAI is clear. Transparency is key to good practice in academia. If in doubt, err on the side of providing acknowledgement. However, there is no need to acknowledge genAI use for low-level tasks such as fixing grammar and spelling or using Studiosity Writing Feedback+.

===Limitations===
Be aware of the limitations of genAI tools. Content they generate may be inaccurate, biased, incomplete, or otherwise problematic. Minimal effort reading and prompting will yield low quality results. Refine prompts based on critical reading and thinking to get better outcomes. You are entirely responsible for the accuracy and quality of any content you submit.

===Fact-check and cite===
Always fact check. Regardless of whether genAI has been used, all claims need to be supported by verified peer-reviewed citations which you have consulted. Guide and craft genAI responses based on your reading of peer-reviewed theory and research. Low-energy or unreflective reuse of genAI text without further investigation and human-writing based on reviewing primary, peer-reviewed academic literature is likely to lead to a poor quality result. GenAI tools work best for topics which you already understand. 

===Going forth===
Despite these warnings, you are encouraged to explore use of genAI tools to help develop higher quality work.

If you are unsure about how to use genAI effectively or how to acknowledge its use appropriately, ask in [[Motivation and emotion/About/Discussion|discussions]], so we can all learn together.

==Example==
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Affiliation_motivation_across_cultures&amp;action=history Affiliation and motivation] (Book chapter, 2025)

==Learn about genAI==
* [https://canberra.libguides.com/genai GenAI for students] (University of Canberra Library)
* [https://techcrunch.com/2024/06/01/what-is-ai-how-does-ai-work/ WTF is AI?] provides a useful introduction and non-technical overview about how genAI works, what it is capable of, limitations, and issues

==See also==
* [[b:Wikibooks:Artificial Intelligence|Wikibooks:Artificial Intelligence]] (policy)
* [[w:Wikipedia:Large language models|Wikipedia:Large language models]] (information page)
* [[Wikiversity:Artificial intelligence|Wikiversity:Artificial intelligence]] (policy)

[[Category:Motivation and emotion/Assessment]]
[[Category:Generative artificial intelligence]]
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&lt;noinclude&gt;==In a nutshell==&lt;/noinclude&gt;
* '''[https://www.canberra.edu.au/about-uc/l-and-t/genai-and-assessment-at-uc Permitted]''' - The ethical use of GenAI is allowed in completing the [[Motivation and emotion/Assessment/Topic|topic development]] and [[Motivation and emotion/Assessment/Chapter|book chapter]].
* '''[https://www.canberra.edu.au/about-uc/l-and-t/genai-and-assessment-at-uc Restricted]''' - The use of GenAI is NOT allowed in completing the [[Motivation and emotion/Assessment/Exam|exam]].

To use genAI appropriately:
* Acknowledge genAI use in [[Wikiversity:FAQ/Editing/Edit summary|
Wikiversity edit summaries]] with a publicly accessible link to the conversation or tool used and prompt details.
* [[w:Fact-checking|Fact-check]] genAI content.
* Only cite peer-reviewed sources which you have consulted.
* Human-rewrite genAI content to enhance quality.&lt;includeonly&gt;
* [[Motivation and emotion/Assessment/Using generative AI|More detail ...]]&lt;/includeonly&gt;&lt;noinclude&gt;

==Summary==
[[File:Deeply engrossed in puzzle.png|thumb|220x220px|'''Figure 1'''. &lt;!-- An image of an elderly woman deeply engrossed in her daily crossword puzzle. --&gt;This image was generated by [[Motivation and emotion|Motivation and Emotion]] student [[User:JorjaFive|JorjaFive]] using [[w:Midjourney|Midjourney]] and uploaded to [[commons:|Wikimedia Commons]] for use in the [[Motivation and emotion/Book/2023/Flourishing in the elderly|flourishing in the elderly]] chapter.]]

[[w:Generative AI|Generative artificial intelligence]] (genAI) use is '''permitted''' for the [[Motivation and emotion/Assessment/Major project|major project]] (topic development and book chapter) and '''restricted''' for the [[Motivation and emotion/Assessment/Exam|exam]].

GenAI tools can help brainstorm, explain concepts, develop a structure, synthesise ideas, and improve the quality of written expression. GenAI tools can aid, but should not replace, independent thinking and reading of primary sources.

Acknowledge genAI use in [[Wikiversity:FAQ/Editing/Edit summary|Wikiversity edit summaries]]. Academia is based on transparency. Follow the principle that &quot;''more acknowledgment is better than less''&quot;. However, acknowledgement is not required for using the Cogniti [https://cogniti.canberra.edu.au/agents/6a3caf85b2fed9a95789241a/chat?k=WfY1yx6AB7yF_9XqiyYENzUBNCGfV1e2FCBodCDD2xU book chapter topic generator],  Studiosity Writing Feedback+, or for low-level tasks such as improving spelling and grammar.

You are responsible for content you submit. Be aware of limitations of genAI tools such as inaccuracies, biases, and incomplete content. Fact-check all claims and only cite peer-reviewed citations which you consulted.

The best results are obtained from genAI tools through carefully crafted prompting based on reading of primary sources, with human rewriting to enhance quality. 

If you are unsure, post to [[Motivation and emotion/About/Discussion|discussions]], so we can all learn together.

==Detailed guidelines==
===Use ethically, with caution===
Learning to use genAI tools (such as [[w:ChatGPT|ChatGPT]], [[w:Claude (language model)|Claude]],
[[w:Gemini (chatbot)|Gemini]], and [[w:Microsoft Copilot|Microsoft Copilot]]) responsibly and ethically is an emerging skill. GenAI tools can be used to enhance academic work, but should be used judiciously, as a supplementary tool, rather than as a replacement for independent thinking and academic inquiry.

GenAI tools may be used to assist in preparation of the major project ([[Motivation and emotion/Assessment/Topic|topic development]] and [[Motivation and emotion/Assessment/Chapter|book chapter]]). These tools can include the Cogniti book chapter topic generator, Studiosity Writing Feedback+, and frontier models.

===How to use===
Recommended uses of genAI tools include:
* brainstorming
* explaining key concepts
* developing a structure
* synthesising complex ideas
* rephrasing to improve [[Motivation and emotion/Assessment/Chapter/Readability|readability]] and the quality of written expression
* correcting spelling and grammar
* image generation (e.g., see Figure 1)
* scenario development
* critical feedback and suggestions for improvement

===How to acknowledge===
[[File:Wikipedia Edit Summary dialog in VisualEditor.png|thumb|450px|'''Figure 2'''. If contributing genAI content, include the tool and prompt details in the edit summary, with a link to the conversation]]

[[File:Edit summary for genAI content.png|thumb|450px|'''Figure 3'''. Example page history which demonstrates best practice edit summaries for contributing and revising genAI content]]

Use of GenAI tools must be clearly acknowledged in [[Wikiversity:FAQ/Editing/Edit summary|Wikiversity edit summaries]] (e.g., see Figure 2), otherwise it is a violation of academic integrity.

Best practice is to include a publicly accessible link to the chatbot conversation (e.g., [https://help.openai.com/en/articles/7925741-chatgpt-shared-links-faq ChatGPT shared links FAQ]). If a link can't be shared, then provide details about the tool and the prompt in the [[Wikiversity:FAQ/Editing/Edit summary|edit summary]], (e.g., &quot;ChatGPT May 24 2025 Version. ''Prompt detail or summary''&quot;) (see Figure 3). The chatbot conversation should ''not'' be used as a citation and listed in the references because it is not a reliable, primary, peer-reviewed source.

These practices help to ensure that the use of genAI is clear. Transparency is key to good practice in academia. If in doubt, err on the side of providing acknowledgement. However, there is no need to acknowledge genAI use for low-level tasks such as fixing grammar and spelling or using Studiosity Writing Feedback+.

===Limitations===
Be aware of the limitations of genAI tools. Content they generate may be inaccurate, biased, incomplete, or otherwise problematic. Minimal effort reading and prompting will yield low quality results. Refine prompts based on critical reading and thinking to get better outcomes. You are entirely responsible for the accuracy and quality of any content you submit.

===Fact-check and cite===
Always fact check. Regardless of whether genAI has been used, all claims need to be supported by verified peer-reviewed citations which you have consulted. Guide and craft genAI responses based on your reading of peer-reviewed theory and research. Low-energy or unreflective reuse of genAI text without further investigation and human-writing based on reviewing primary, peer-reviewed academic literature is likely to lead to a poor quality result. GenAI tools work best for topics which you already understand. 

===Going forth===
Despite these warnings, you are encouraged to explore use of genAI tools to help develop higher quality work.

If you are unsure about how to use genAI effectively or how to acknowledge its use appropriately, ask in [[Motivation and emotion/About/Discussion|discussions]], so we can all learn together.

==Example==
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Affiliation_motivation_across_cultures&amp;action=history Affiliation and motivation] (Book chapter, 2025)

==Learn about genAI==
* [https://canberra.libguides.com/genai GenAI for students] (University of Canberra Library)
* [https://techcrunch.com/2024/06/01/what-is-ai-how-does-ai-work/ WTF is AI?] provides a useful introduction and non-technical overview about how genAI works, what it is capable of, limitations, and issues

==See also==
* [[b:Wikibooks:Artificial Intelligence|Wikibooks:Artificial Intelligence]] (policy)
* [[w:Wikipedia:Large language models|Wikipedia:Large language models]] (information page)
* [[Wikiversity:Artificial intelligence|Wikiversity:Artificial intelligence]] (policy)

[[Category:Motivation and emotion/Assessment]]
[[Category:Generative artificial intelligence]]
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__TOC__

&lt;noinclude&gt;==In a nutshell==&lt;/noinclude&gt;
* '''[https://www.canberra.edu.au/about-uc/l-and-t/genai-and-assessment-at-uc Permitted]''' - The ethical use of GenAI is allowed in completing the [[Motivation and emotion/Assessment/Topic|topic development]] and [[Motivation and emotion/Assessment/Chapter|book chapter]].
* '''[https://www.canberra.edu.au/about-uc/l-and-t/genai-and-assessment-at-uc Restricted]''' - The use of GenAI is NOT allowed in completing the [[Motivation and emotion/Assessment/Exam|exam]].

To use genAI appropriately:
* Acknowledge genAI use in [[Wikiversity:FAQ/Editing/Edit summary|
Wikiversity edit summaries]] with a publicly accessible link to the conversation or tool used and prompt details.
* [[w:Fact-checking|Fact-check]] genAI content.
* Only cite peer-reviewed sources which you have consulted.
* Human-rewrite genAI content to enhance quality.&lt;includeonly&gt;
* [[Motivation and emotion/Assessment/Using generative AI|More detail ...]]&lt;/includeonly&gt;&lt;noinclude&gt;

==Summary==
[[File:Deeply engrossed in puzzle.png|thumb|220x220px|'''Figure 1'''. &lt;!-- An image of an elderly woman deeply engrossed in her daily crossword puzzle. --&gt;This image was generated by [[Motivation and emotion|Motivation and Emotion]] student [[User:JorjaFive|JorjaFive]] using [[w:Midjourney|Midjourney]] and uploaded to [[commons:|Wikimedia Commons]] for use in the [[Motivation and emotion/Book/2023/Flourishing in the elderly|flourishing in the elderly]] chapter.]]

[[w:Generative AI|Generative artificial intelligence]] (genAI) use is '''permitted''' for the [[Motivation and emotion/Assessment/Major project|major project]] (topic development and book chapter) and '''restricted''' for the [[Motivation and emotion/Assessment/Exam|exam]].

GenAI tools can help brainstorm, explain concepts, develop a structure, synthesise ideas, and improve the quality of written expression. GenAI tools can aid, but should not replace, independent thinking and reading of primary sources.

Acknowledge genAI use in [[Wikiversity:FAQ/Editing/Edit summary|Wikiversity edit summaries]]. Academia is based on transparency. Follow the principle that &quot;''more acknowledgment is better than less''&quot;. However, acknowledgement is not required for using the Cogniti [https://cogniti.canberra.edu.au/agents/6a3caf85b2fed9a95789241a/chat?k=WfY1yx6AB7yF_9XqiyYENzUBNCGfV1e2FCBodCDD2xU book chapter topic generator],  Studiosity Writing Feedback+, or for low-level tasks such as improving spelling and grammar.

You are responsible for content you submit. Be aware of limitations of genAI tools such as inaccuracies, biases, and incomplete content. Fact-check all claims and only cite peer-reviewed citations which you consulted.

The best results are obtained from genAI tools through carefully crafted prompting based on reading of primary sources, with human rewriting to enhance quality. 

If you are unsure, post to [[Motivation and emotion/About/Discussion|discussions]], so we can all learn together.

==Detailed guidelines==
===Use ethically, with caution===
Learning to use genAI tools (such as [[w:ChatGPT|ChatGPT]], [[w:Claude (language model)|Claude]],
[[w:Gemini (chatbot)|Gemini]], and [[w:Microsoft Copilot|Microsoft Copilot]]) responsibly and ethically is an emerging skill. GenAI tools can be used to enhance academic work, but should be used judiciously, as a supplementary tool, rather than as a replacement for independent thinking and academic inquiry.

GenAI tools may be used to assist in preparation of the major project ([[Motivation and emotion/Assessment/Topic|topic development]] and [[Motivation and emotion/Assessment/Chapter|book chapter]]). These tools can include the Cogniti book chapter topic generator, Studiosity Writing Feedback+, and frontier models.

===How to use===
Recommended uses of genAI tools include:
* brainstorming
* explaining key concepts
* developing a structure
* synthesising complex ideas
* rephrasing to improve [[Motivation and emotion/Assessment/Chapter/Readability|readability]] and quality of written expression
* correcting spelling and grammar
* image generation (e.g., see Figure 1)
* scenario development
* critical feedback and suggestions for improvement

===How to acknowledge===
[[File:Wikipedia Edit Summary dialog in VisualEditor.png|thumb|450px|'''Figure 2'''. If contributing genAI content, include the tool and prompt details in the edit summary, with a link to the conversation]]

[[File:Edit summary for genAI content.png|thumb|450px|'''Figure 3'''. Example page history which demonstrates best practice edit summaries for contributing and revising genAI content]]

Use of GenAI tools must be clearly acknowledged in [[Wikiversity:FAQ/Editing/Edit summary|Wikiversity edit summaries]] (e.g., see Figure 2), otherwise it is a violation of academic integrity.

Best practice is to include a publicly accessible link to the chatbot conversation (e.g., [https://help.openai.com/en/articles/7925741-chatgpt-shared-links-faq ChatGPT shared links FAQ]). If a link can't be shared, then provide details about the tool and the prompt in the [[Wikiversity:FAQ/Editing/Edit summary|edit summary]], (e.g., &quot;ChatGPT May 24 2025 Version. ''Prompt detail or summary''&quot;) (see Figure 3). The chatbot conversation should ''not'' be used as a citation and listed in the references because it is not a reliable, primary, peer-reviewed source.

These practices help to ensure that the use of genAI is clear. Transparency is key to good practice in academia. If in doubt, err on the side of providing acknowledgement. However, there is no need to acknowledge genAI use for low-level tasks such as fixing grammar and spelling or using Studiosity Writing Feedback+.

===Limitations===
Be aware of the limitations of genAI tools. Content they generate may be inaccurate, biased, incomplete, or otherwise problematic. Minimal effort reading and prompting will yield low quality results. Refine prompts based on critical reading and thinking to get better outcomes. You are entirely responsible for the accuracy and quality of any content you submit.

===Fact-check and cite===
Always fact check. Regardless of whether genAI has been used, all claims need to be supported by verified peer-reviewed citations which you have consulted. Guide and craft genAI responses based on your reading of peer-reviewed theory and research. Low-energy or unreflective reuse of genAI text without further investigation and human-writing based on reviewing primary, peer-reviewed academic literature is likely to lead to a poor quality result. GenAI tools work best for topics which you already understand. 

===Going forth===
Despite these warnings, you are encouraged to explore use of genAI tools to help develop higher quality work.

If you are unsure about how to use genAI effectively or how to acknowledge its use appropriately, ask in [[Motivation and emotion/About/Discussion|discussions]], so we can all learn together.

==Example==
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Affiliation_motivation_across_cultures&amp;action=history Affiliation and motivation] (Book chapter, 2025)

==Learn about genAI==
* [https://canberra.libguides.com/genai GenAI for students] (University of Canberra Library)
* [https://techcrunch.com/2024/06/01/what-is-ai-how-does-ai-work/ WTF is AI?] provides a useful introduction and non-technical overview about how genAI works, what it is capable of, limitations, and issues

==See also==
* [[b:Wikibooks:Artificial Intelligence|Wikibooks:Artificial Intelligence]] (policy)
* [[w:Wikipedia:Large language models|Wikipedia:Large language models]] (information page)
* [[Wikiversity:Artificial intelligence|Wikiversity:Artificial intelligence]] (policy)

[[Category:Motivation and emotion/Assessment]]
[[Category:Generative artificial intelligence]]
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__TOC__

&lt;noinclude&gt;==In a nutshell==&lt;/noinclude&gt;
* '''[https://www.canberra.edu.au/about-uc/l-and-t/genai-and-assessment-at-uc Permitted]''' - The ethical use of GenAI is allowed in completing the [[Motivation and emotion/Assessment/Topic|topic development]] and [[Motivation and emotion/Assessment/Chapter|book chapter]].
* '''[https://www.canberra.edu.au/about-uc/l-and-t/genai-and-assessment-at-uc Restricted]''' - The use of GenAI is NOT allowed in completing the [[Motivation and emotion/Assessment/Exam|exam]].

To use genAI appropriately:
* Acknowledge genAI use in [[Wikiversity:FAQ/Editing/Edit summary|
Wikiversity edit summaries]] with a publicly accessible link to the conversation or tool used and prompt details.
* [[w:Fact-checking|Fact-check]] genAI content.
* Only cite peer-reviewed sources which you have consulted.
* Human-rewrite genAI content to enhance quality.&lt;includeonly&gt;
* [[Motivation and emotion/Assessment/Using generative AI|More detail ...]]&lt;/includeonly&gt;&lt;noinclude&gt;

==Summary==
[[File:Deeply engrossed in puzzle.png|thumb|220x220px|'''Figure 1'''. &lt;!-- An image of an elderly woman deeply engrossed in her daily crossword puzzle. --&gt;This image was generated by [[Motivation and emotion|Motivation and Emotion]] student [[User:JorjaFive|JorjaFive]] using [[w:Midjourney|Midjourney]] and uploaded to [[commons:|Wikimedia Commons]] for use in the [[Motivation and emotion/Book/2023/Flourishing in the elderly|flourishing in the elderly]] chapter.]]

[[w:Generative AI|Generative artificial intelligence]] (genAI) use is '''permitted''' for the [[Motivation and emotion/Assessment/Major project|major project]] (topic development and book chapter) and '''restricted''' for the [[Motivation and emotion/Assessment/Exam|exam]].

GenAI tools can help brainstorm, explain concepts, develop a structure, synthesise ideas, and improve the quality of written expression. GenAI tools can aid, but should not replace, independent thinking and reading of primary sources.

Acknowledge genAI use in [[Wikiversity:FAQ/Editing/Edit summary|Wikiversity edit summaries]]. Academia is based on transparency. Follow the principle that &quot;''more acknowledgment is better than less''&quot;. However, acknowledgement is not required for using the Cogniti [https://cogniti.canberra.edu.au/agents/6a3caf85b2fed9a95789241a/chat?k=WfY1yx6AB7yF_9XqiyYENzUBNCGfV1e2FCBodCDD2xU book chapter topic generator],  Studiosity Writing Feedback+, or for low-level tasks such as improving spelling and grammar.

You are responsible for content you submit. Be aware of limitations of genAI tools such as inaccuracies, biases, and incomplete content. Fact-check all claims and only cite peer-reviewed citations which you consulted.

The best results are obtained from genAI tools through carefully crafted prompting based on reading of primary sources, with human rewriting to enhance quality. 

If you are unsure, post to [[Motivation and emotion/About/Discussion|discussions]], so we can all learn together.

==Detailed guidelines==
===Use ethically, with caution===
Learning to use genAI tools (such as [[w:ChatGPT|ChatGPT]], [[w:Claude (language model)|Claude]],
[[w:Gemini (chatbot)|Gemini]], and [[w:Microsoft Copilot|Microsoft Copilot]]) responsibly and ethically is an emerging skill. GenAI tools can be used to enhance academic work, but should be used judiciously, as a supplementary tool, rather than as a replacement for independent thinking and academic inquiry.

GenAI tools may be used to assist in preparation of the major project ([[Motivation and emotion/Assessment/Topic|topic development]] and [[Motivation and emotion/Assessment/Chapter|book chapter]]). These tools can include the Cogniti book chapter topic generator, Studiosity Writing Feedback+, and frontier models.

===How to use===
Recommended uses of genAI tools include:
* brainstorming
* explaining key concepts
* developing a structure
* synthesising complex ideas
* rephrasing to improve [[Motivation and emotion/Assessment/Chapter/Readability|readability]] and quality of written expression
* correcting spelling and grammar
* image generation (e.g., see Figure 1)
* scenario development
* critical feedback and suggestions for improvement

===How to acknowledge===
[[File:Wikipedia Edit Summary dialog in VisualEditor.png|thumb|450px|'''Figure 2'''. If contributing genAI content, include the tool and prompt details in the edit summary, with a link to the conversation]]

[[File:Edit summary for genAI content.png|thumb|450px|'''Figure 3'''. Example page history which demonstrates best practice edit summaries for contributing and revising genAI content]]

Use of GenAI tools must be clearly acknowledged in [[Wikiversity:FAQ/Editing/Edit summary|Wikiversity edit summaries]] (e.g., see Figure 2), otherwise it is a violation of academic integrity.

Best practice is to include a publicly accessible link to the chatbot conversation (e.g., [https://help.openai.com/en/articles/7925741-chatgpt-shared-links-faq ChatGPT shared links FAQ]). If a link can't be shared, then provide details about the tool and the prompt in the [[Wikiversity:FAQ/Editing/Edit summary|edit summary]], (e.g., &quot;ChatGPT May 24 2025 Version. ''Prompt detail or summary''&quot;) (see Figure 3). The chatbot conversation should ''not'' be used as a citation and listed in the references because it is not a reliable, primary, peer-reviewed source.

These practices help to ensure that the use of genAI is clear. Transparency is key to good practice in academia. If in doubt, err on the side of providing acknowledgement. However, there is no need to acknowledge genAI use of the Cogniti book chapter topic generator, Studiosity Writing Feedback+, or for low-level tasks such as fixing grammar and spelling.

===Limitations===
Be aware of the limitations of genAI tools. Content they generate may be inaccurate, biased, incomplete, or otherwise problematic. Minimal effort reading and prompting will yield low quality results. Refine prompts based on critical reading and thinking to get better outcomes. You are entirely responsible for the accuracy and quality of any content you submit.

===Fact-check and cite===
Always fact check. Regardless of whether genAI has been used, all claims need to be supported by verified peer-reviewed citations which you have consulted. Guide and craft genAI responses based on your reading of peer-reviewed theory and research. Low-energy or unreflective reuse of genAI text without further investigation and human-writing based on reviewing primary, peer-reviewed academic literature is likely to lead to a poor quality result. GenAI tools work best for topics which you already understand. 

===Going forth===
Despite these warnings, you are encouraged to explore use of genAI tools to help develop higher quality work.

If you are unsure about how to use genAI effectively or how to acknowledge its use appropriately, ask in [[Motivation and emotion/About/Discussion|discussions]], so we can all learn together.

==Example==
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Affiliation_motivation_across_cultures&amp;action=history Affiliation and motivation] (Book chapter, 2025)

==Learn about genAI==
* [https://canberra.libguides.com/genai GenAI for students] (University of Canberra Library)
* [https://techcrunch.com/2024/06/01/what-is-ai-how-does-ai-work/ WTF is AI?] provides a useful introduction and non-technical overview about how genAI works, what it is capable of, limitations, and issues

==See also==
* [[b:Wikibooks:Artificial Intelligence|Wikibooks:Artificial Intelligence]] (policy)
* [[w:Wikipedia:Large language models|Wikipedia:Large language models]] (information page)
* [[Wikiversity:Artificial intelligence|Wikiversity:Artificial intelligence]] (policy)

[[Category:Motivation and emotion/Assessment]]
[[Category:Generative artificial intelligence]]
&lt;/noinclude&gt;</text>
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      <text bytes="7946" sha1="lse9ziyxln7f1y8w8jj2gj102bw8g6b" xml:space="preserve">&lt;noinclude&gt;{{title|Using generative AI guidelines}}
__TOC__

&lt;noinclude&gt;==In a nutshell==&lt;/noinclude&gt;
* '''[https://www.canberra.edu.au/about-uc/l-and-t/genai-and-assessment-at-uc Permitted]''' - The ethical use of GenAI is allowed in completing the [[Motivation and emotion/Assessment/Topic|topic development]] and [[Motivation and emotion/Assessment/Chapter|book chapter]].
* '''[https://www.canberra.edu.au/about-uc/l-and-t/genai-and-assessment-at-uc Restricted]''' - The use of GenAI is NOT allowed in completing the [[Motivation and emotion/Assessment/Exam|exam]].

To use genAI appropriately:
* Acknowledge genAI use in [[Wikiversity:FAQ/Editing/Edit summary|
Wikiversity edit summaries]] with a publicly accessible link to the conversation or tool used and prompt details.
* [[w:Fact-checking|Fact-check]] genAI content.
* Only cite peer-reviewed sources which you have consulted.
* Human-rewrite genAI content to enhance quality.&lt;includeonly&gt;
* [[Motivation and emotion/Assessment/Using generative AI|More detail ...]]&lt;/includeonly&gt;&lt;noinclude&gt;

==Summary==
[[File:Deeply engrossed in puzzle.png|thumb|220x220px|'''Figure 1'''. &lt;!-- An image of an elderly woman deeply engrossed in her daily crossword puzzle. --&gt;This image was generated by [[Motivation and emotion|Motivation and Emotion]] student [[User:JorjaFive|JorjaFive]] using [[w:Midjourney|Midjourney]] and uploaded to [[commons:|Wikimedia Commons]] for use in the [[Motivation and emotion/Book/2023/Flourishing in the elderly|flourishing in the elderly]] chapter.]]

[[w:Generative AI|Generative artificial intelligence]] (genAI) use is '''permitted''' for the [[Motivation and emotion/Assessment/Major project|major project]] (topic development and book chapter) and '''restricted''' for the [[Motivation and emotion/Assessment/Exam|exam]].

GenAI tools can help brainstorm, explain concepts, develop a structure, synthesise ideas, and improve the quality of written expression. GenAI tools can aid, but should not replace, independent thinking and reading of primary sources.

Acknowledge genAI use in [[Wikiversity:FAQ/Editing/Edit summary|Wikiversity edit summaries]]. Academia is based on transparency. Follow the principle that &quot;''more acknowledgment is better than less''&quot;. However, acknowledgement is not required for using the Cogniti [https://cogniti.canberra.edu.au/agents/6a3caf85b2fed9a95789241a/chat?k=WfY1yx6AB7yF_9XqiyYENzUBNCGfV1e2FCBodCDD2xU book chapter topic generator],  Studiosity Writing Feedback+, or for low-level tasks such as improving spelling and grammar.

You are responsible for content you submit. Be aware of limitations of genAI tools such as inaccuracies, biases, and incomplete content. Fact-check all claims and only cite peer-reviewed citations which you consulted.

The best results are obtained from genAI tools through carefully crafted prompting based on reading of primary sources, with human rewriting to enhance quality. 

If you are unsure, post to [[Motivation and emotion/About/Discussion|discussions]], so we can all learn together.

==Detailed guidelines==
===Use ethically, with caution===
Learning to use genAI tools (such as [[w:ChatGPT|ChatGPT]], [[w:Claude (language model)|Claude]],
[[w:Gemini (chatbot)|Gemini]], and [[w:Microsoft Copilot|Microsoft Copilot]]) responsibly and ethically is an emerging skill. GenAI tools can be used to enhance academic work, but should be used judiciously, as a supplementary tool, rather than as a replacement for independent thinking and academic inquiry.

GenAI tools may be used to assist in preparation of the major project ([[Motivation and emotion/Assessment/Topic|topic development]] and [[Motivation and emotion/Assessment/Chapter|book chapter]]). These tools can include the Cogniti book chapter topic generator, Studiosity Writing Feedback+, and frontier models.

===How to use===
Recommended uses of genAI tools include:
* brainstorming
* explaining key concepts
* developing a structure
* synthesising complex ideas
* rephrasing to improve [[Motivation and emotion/Assessment/Chapter/Readability|readability]] and quality of written expression
* correcting spelling and grammar
* image generation (e.g., see Figure 1)
* scenario development
* critical feedback and suggestions for improvement

===How to acknowledge===
[[File:Wikipedia Edit Summary dialog in VisualEditor.png|thumb|450px|'''Figure 2'''. If contributing genAI content, include the tool and prompt details in the edit summary, with a link to the conversation]]

[[File:Edit summary for genAI content.png|thumb|450px|'''Figure 3'''. Example page history which demonstrates best practice edit summaries for contributing and revising genAI content]]

Use of GenAI tools must be clearly acknowledged in [[Wikiversity:FAQ/Editing/Edit summary|Wikiversity edit summaries]] (e.g., see Figure 2), otherwise it is a violation of academic integrity.

Best practice is to include a publicly accessible link to the chatbot conversation (e.g., [https://help.openai.com/en/articles/7925741-chatgpt-shared-links-faq ChatGPT shared links FAQ]). If a link can't be shared, then provide details about the tool and the prompt in the [[Wikiversity:FAQ/Editing/Edit summary|edit summary]], (e.g., &quot;ChatGPT May 24 2025 Version. ''Prompt detail or summary''&quot;) (see Figure 3). The chatbot conversation should ''not'' be used as a citation and listed in the references because it is not a reliable, primary, peer-reviewed source.

These practices help to ensure that the use of genAI is clear. Transparency is key to good practice in academia. If in doubt, err on the side of providing acknowledgement. However, there is no need to acknowledge genAI use of the Cogniti book chapter topic generator, Studiosity Writing Feedback+, or for low-level tasks such as fixing grammar and spelling.

===Limitations===
Be aware of the limitations of genAI tools. Content they generate may be inaccurate, biased, incomplete, or otherwise problematic. Minimal effort reading and prompting will yield low quality results. Refine prompts based on critical reading and thinking to get better outcomes. You are entirely responsible for the accuracy and quality of any content you submit.

===Fact-check and cite===
Always fact-check. Regardless of whether genAI has been used, all claims need to be supported by verified peer-reviewed citations which you have consulted. Guide and craft genAI responses based on your reading of peer-reviewed theory and research. Low-energy or unreflective reuse of genAI text without further investigation and human-writing based on reviewing primary, peer-reviewed academic literature is likely to lead to a poor quality result. GenAI tools work best for topics which you already understand.

===Going forth===
Despite these warnings, you are encouraged to explore use of genAI tools to help develop higher quality work.

If you are unsure about how to use genAI effectively or how to acknowledge its use appropriately, ask in [[Motivation and emotion/About/Discussion|discussions]], so we can all learn together.

==Example==
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Affiliation_motivation_across_cultures&amp;action=history Affiliation and motivation] (Book chapter, 2025)

==Learn about genAI==
* [https://canberra.libguides.com/genai GenAI for students] (University of Canberra Library)
* [https://techcrunch.com/2024/06/01/what-is-ai-how-does-ai-work/ WTF is AI?] provides a useful introduction and non-technical overview about how genAI works, what it is capable of, limitations, and issues

==See also==
* [[b:Wikibooks:Artificial Intelligence|Wikibooks:Artificial Intelligence]] (policy)
* [[w:Wikipedia:Large language models|Wikipedia:Large language models]] (information page)
* [[Wikiversity:Artificial intelligence|Wikiversity:Artificial intelligence]] (policy)

[[Category:Motivation and emotion/Assessment]]
[[Category:Generative artificial intelligence]]
&lt;/noinclude&gt;</text>
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    <title>User:Dc.samizdat/Golden chords of the 120-cell</title>
    <ns>2</ns>
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      <text bytes="87782" sha1="7puqgxwbhi0qb1n55ew8rnkn23j2eyi" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell) 4-simplex, is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). Consequently the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== The 16-cell 4-orthoplex ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. Their procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it also moves a distance of &lt;math&gt;-1&lt;/math&gt;. Fontaine and Hurley also demonstrated the significance of &lt;math&gt;1/r_i&lt;/math&gt; in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length &lt;math&gt;1/r_i&lt;/math&gt; rather than &lt;math&gt;1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Each chord is a distinct 4-vector with a length and a direction. Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation, where vertices circle over the chords of an &lt;math&gt;r_i&lt;/math&gt; polygon. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns. The angle between two &lt;math&gt;r_i&lt;/math&gt; chords is &lt;math&gt;180^\circ - i \times 45^\circ&lt;/math&gt;.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon {8/1} which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of the 16-cell form an ''edge polygon'' {8/2}=2{4}. The two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt; (it is &lt;math&gt;6\pi&lt;/math&gt; in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes twelve 45° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic revolution each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

We shall refer to this isoclinic rotation as the ''great square rotation characteristic of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_3&lt;/math&gt; {8/3} star polygon, which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== The 8-cell tesseract ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedrons]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in the great square rotation characteristic of the 16-cell, with the same effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords. They form a circular double helix which intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common and their corresponding pairs of vertices 180° apart. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding pairs of vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}. The unit-radius planar {12}-gon has chords of length:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

The &lt;math&gt;r_1&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt; chords form 8 Petrie dodecagons which zig-zag back and forth, in the left and right rotational directions, between two Clifford parallel great hexagons formed by &lt;math&gt;r_2&lt;/math&gt; chords. The 8 Petrie dodecagons can be divided four ways into 2 disjoint Petrie dodecagons {24/2}=2{12}.

The &lt;math&gt;r_2&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt; chords form 16 great hexagons, which can be divided four ways into 4 Clifford parallel great hexagons {24/4}=4{6}.

The &lt;math&gt;r_3&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form 18 great squares, which can be divided three ways into 6 Clifford parallel great squares {24/6}=6{4}, including one pair of completely orthogonal great squares from each of the three 16-cells.

The &lt;math&gt;r_4&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords form 32 great triangles, which can be divided four ways into 8 disjoint great triangles {24/8}=8{3} inscribed in 4 Clifford parallel great hexagons.

The &lt;math&gt;r_5&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords form 8 circular helix Clifford polygons, visible as a green {12/5} dodecagram in the orthogonal projection. An isoclinic rotation of the 24-cell in 4 invariant &lt;math&gt;r_2&lt;/math&gt; hexagon planes moves the vertices along 2 Clifford parallel circular isoclines {24/2}=2{12/5} over &lt;math&gt;r_5&lt;/math&gt; chords.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges of Clifford parallel great squares, also isocline chords in great square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} shows three octagram isoclines of &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt; &lt;/small&gt;chords in the 24-cell]]

We can rotate the 24-cell isoclinically in 6 Clifford parallel invariant great square planes containing 16-cell edges, in the great square rotation characteristic of the 16-cell, with the same effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords in the 24-cell]]We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing 24-cell edges, over &lt;math&gt;r_{5}&lt;/math&gt; isocline chords. This is the ''great hexagon rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_5&lt;/math&gt; {12/5} star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A complete 24-cell great circle edge plane revolution requires 720° like a complete 16-cell great circle edge plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each.  An isoclinic rotation by 60° in an invariant central plane containing a 24-cell edge takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as 4 invariant great hexagon planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 12 &lt;math&gt;r_5&lt;/math&gt; &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. The rotational curve over each &lt;math&gt;r_5&lt;/math&gt; 120° chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |6 distinct 180° chord pairs make 6 distinct isoclinic rotations
|-
! colspan=&quot;3&quot; |Edge chords
!Invariant planes
! colspan=&quot;3&quot; |Isocline chords
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_1&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;{24/1}={24}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;{24/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{11}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|165°
|15°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_2&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,1).svg|100px]]&lt;br&gt;{24/2}=2{12}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;{24/10}=2{12/5}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|150°
|30°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_3&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;{24/3}=3{8}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;{24/9}=3{8/3}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{9}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|135°
|45°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;{24/12}=12{2}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;{24/8}=8{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{8}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;{24/5}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;{24/7}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{7}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|105°
|75°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_6&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{6}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

By examining the chords &lt;math&gt;r_i&lt;/math&gt; of the 24-cell's Petrie {12}-gon we have found two distinct isoclinic rotations, the great square rotation characteristic of the 16-cell and the great hexagon rotation characteristic of the 24-cell. If we examine the chords &lt;math&gt;t_i&lt;/math&gt; of the 24-cell's {24}-gon we find these, and also four other distinct isoclinic rotations.  

Each row of the table is a distinct isoclinic rotation of the 24-cell characterized by a pair of chords whose arc-lengths sum to 180°. The short edge chord and long isocline chord each have their characteristic {24/n}-gon. The edge chords form the rotation's edge polygons, over which vertices circle as the whole polygon moves orthogonally. The isocline chords form the rotation's Clifford polygons, which are stationary in 4-space as vertices circle over them. The rotational angle between successive edge chords and the rotational angle between successive isocline chords also sum to 180°. The invariant planes of the rotation intersect 0, 2, 4, or 6 vertices.  

Each isoclinic rotation takes two chiral forms. There is a ''right rotation'' and a ''left rotation'' for each row of the table. In the left rotation the roles of the edge chord polygon and the isocline chord polygon, as described above, are reversed: the edge chord polygon is stationary in 4-space as vertices circle over it, and the isocline chord polygon is an invariant configuration of moving vertices.  

{{Clear}}

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt; the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has chords of length:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are:

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Edge chords
! Section
! colspan=&quot;3&quot; |Isocline chords
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{15}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|180°
|0°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_1&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{14}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: palegreen;&quot; |
|0.618~
|1.902~
|- style=&quot;background: palegreen;&quot; |
|168°
|12°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_2&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{13}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: gainsboro;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
|156°
|24°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |[[File:V1 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{12}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
|144°
|36°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{11}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|132°
|48°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |[[File:V2 dodecahedron.png|100px]]&lt;br&gt;Dodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{6}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |[[File:V3 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{9}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: yellow;&quot; |
|108°
|72°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{7}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |[[File:V4 icosidodecahedron.png|100px]]&lt;br&gt;Icosidodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{8}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|96°
|84°
|}

The list of 600-cell chords &lt;math&gt;r_{i}&lt;/math&gt; can be rearranged into a table of 8 rows with a pair of 180° complements in each row. The short edge chord and long isocline chord each have their characteristic {30/n}-gon. Each row identifies a distinct isoclinic rotation of the 600-cell.

Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;\phi^{-1}&lt;/math&gt; is a icosahedron vertex figure, and the largest section at radial distance &lt;math&gt;\sqrt{2}&lt;/math&gt; is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;\sqrt{2}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance &lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section).

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell,  with the same effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once.

The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This ''great square rotation characteristic of the 600-cell'' takes place over &lt;math&gt;r_7&lt;/math&gt; edge chords and &lt;math&gt;r_8&lt;/math&gt; isocline chords. The {30/7} edge polygon is a skew helix of circumference &lt;math&gt;14\pi&lt;/math&gt; with each &lt;math&gt;r_7&lt;/math&gt; edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference &lt;math&gt;16\pi&lt;/math&gt; with each &lt;math&gt;r_8&lt;/math&gt; isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}
[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over &lt;math&gt;\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;\sqrt{3}&lt;/math&gt; isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. 

The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over &lt;math&gt;r_{4}=\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords This ''great hexagon rotation characteristic of the 600-cell'' has period 30 and visits every vertex of a 600-cell Petrie polygon. Its {30/11} Clifford polygon is a skew helix where each &lt;math&gt;r_{11}&lt;/math&gt; isocline chord is the &lt;math&gt;\sqrt{3}&lt;/math&gt; diagonal of a great hexagon of a distinct 24-cell. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° &lt;math&gt;r_{11}&lt;/math&gt; isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference &lt;math&gt;22\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° &lt;math&gt;r_{3}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{12}&lt;/math&gt; isocline chords. This ''great pentagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each &lt;math&gt;r_{12}&lt;/math&gt; chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference &lt;math&gt;4\pi&lt;/math&gt; over five &lt;math&gt;r_{12}&lt;/math&gt; chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. 

The rotation of the 600-cell by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions.

The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° &lt;math&gt;r_{2}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. The rotational curve over each 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference &lt;math&gt;26\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}

== Finally the 120-cell ==

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |30 chords (15 180° pairs) make 15 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Edge chords
! Section
! colspan=&quot;3&quot; |Isocline chords
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|180°
|0°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}
|164.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|{{radic|3.927~}}
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1.982~
|- style=&quot;background: palegreen;&quot; |
|168°
|12°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|154.8~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|{{radic|3.809~}}
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.952~
|- style=&quot;background: gainsboro;&quot; |
|156°
|24°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
|144°
|36°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|138.6~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|{{radic|3.5}}
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|1.871~
|- style=&quot;background: gainsboro;&quot; |
|138°
|42°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|135.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|{{radic|3.427~}}
|- style=&quot;background: palegreen;&quot; |
|0.757~
|1.851~
|- style=&quot;background: palegreen;&quot; |
|132°
|48°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|130.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|{{radic|3.309~}}
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|1.819~
|- style=&quot;background: gainsboro;&quot; |
|128°
|52°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|124°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|{{radic|3.118~}}
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|1.766~
|- style=&quot;background: gainsboro;&quot; |
|124°
|56°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|113.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|{{radic|2.809~}}
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|1.676~
|- style=&quot;background: gainsboro;&quot; |
|116°
|64°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|110.2~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|{{radic|2.691~}}
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|1.640~
|- style=&quot;background: gainsboro;&quot; |
|112°
|68°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: yellow;&quot; |
|108°
|72°
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|104.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|{{radic|2.5}}
|- style=&quot;background: palegreen;&quot; |
|1.224~
|1.581~
|- style=&quot;background: palegreen;&quot; |
|96°
|84°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|98.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|{{radic|2.309~}}
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|1.520~
|- style=&quot;background: gainsboro;&quot; |
|°
|°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|95.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|{{radic|2.191~}}
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|1.480~
|- style=&quot;background: gainsboro;&quot; |
|°
|°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. 

The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of thirty 120-cell chords]] &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. He showed that in spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section at radial distance &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. 

Only 8 of the 30 chords in the table occur in the 600-cell. The 120-cell's additional chords arise originally from the regular 5-cell 4-simplex, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is the entirety of what's new in the 120-cell.

The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure.

... 

{{Clear}}

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of reflections in ''n'' dimensions. Application of the Fontaine and Hurley procedure to the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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== Proposal to rehost Wikinews here ==

As many of you know, and mentioned here at the Colloquium, our sister project Wikinews recently closed, with all 31 active editions made read-only. [[User:BigKrow]] has asked about the prospect of writing news stories here and I suggested that since we already have [[School:Journalism]] and some resources related to the [[:Category:Journalism|broader topic of journalism]]. I would like to propose that we have continued and indefinite space for {{w|citizen journalism}} by essentially repurposing Wikinews into a sub-project here. The only special infrastructure that Wikinews required was [[:mw:Extension:DynamicPageList]], which was deactivated and caused issues due to a lack of maintenance.

I will add this proposal to the site banner, but I recognize that that may be a conflict of interest, so if anyone requests that I remove it, I will. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:30, 14 May 2026 (UTC)
:I would like to see this conversation go for at least 30 days to establish a consensus. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:35, 14 May 2026 (UTC)
::A few days shy of 30, it seems obvious that this is not going to pass. So I '''withdraw''' as presumptively '''failed'''. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:14, 9 June 2026 (UTC)

===Votes===
*{{support}} as proposer (with BK's inspiration). I think that an ongoing experiment in citizen journalism is a fit and appropriate use of this site. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:35, 14 May 2026 (UTC)
*{{support}}, hope to seeing ideas about this, and thank you @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 11:08, 14 May 2026 (UTC)
*{{support}} Other than perhaps inflating the total number of pages reported, I see the idea of &quot;practicing journalism&quot; a worthy and relevant activity within the domain of Wikiversity. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 21:41, 14 May 2026 (UTC)
*{{support}} Conditional on development of (a) community guidelines that ensure alignment with Wikiversity's purpose, and (b) clear, nested page-naming structures for projects. More detail below. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:48, 15 May 2026 (UTC)
*{{contra}} This proposal doesn't seem interested in expanding educational materials in journalism, but rather in providing space and protection for Wikinews contributors. But this is contrary to the goals of Wikiversity, and I'm not sure it's a good idea, even with regard to WMF. If WMF decides to close a project and another community lets it run on its domain, that's a bit of an undermining of WMF's and the community's decisions. Given that Wikiversity has had several conflicts with other communities and WMF in its history, I'm against it.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:59, 15 May 2026 (UTC)
*{{contra}} This seems like a proposal to continue the mission of WikiNews, but not a proposal specifically to improve Wikiversity. I concur with Juandev's comments. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 20:29, 30 May 2026 (UTC)
* {{oppose}} per above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:05, 1 June 2026 (UTC)
*{{oppose}} Wikiversity isn’t Wikinews and it also isn’t a dumping ground for anything not covered by other projects. It was already suggested, rather bafflingly, that Wikinews parasitize Wikipedia as a host. If it were allowed to freeload off of Wikiversity it would simply promote a view I and likely many others have— that Wikiversity (as it currently exists) has no standards and mostly just exists to host subpar content that wouldn’t be tolerated on any other Wikimedia site. Wikinews needs a new, non-Wikimedia host, and Wikiversity needs to get its act together by enforcing a minimum scope and standard for what it allows. --[[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:16, 4 June 2026 (UTC)
* {{oppose}} per above. Wikiversity&lt;math&gt;\not=&lt;/math&gt; Wikinews - not a good idea to mix the scope of projects. --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 12:03, 8 June 2026 (UTC)
* {{abstain}} I will abstain since I'm not an active Wikiversity contributor. But I just feel like Wikinews had a very clear and specific goal of providing news, and Wikiversity is just a different project with different goals. For me, it would be odd to rehost Wikinews here. But please do not count my vote, this is only a comment. --[[User:Antimundo|Antimundo]] ([[User talk:Antimundo|discuss]] • [[Special:Contributions/Antimundo|contribs]]) 13:19, 6 June 2026 (UTC)
* {{oppose}} Although I think it's a pity that Wikinews is closed. --[[User:Dick Bos|Dick Bos]] ([[User talk:Dick Bos|discuss]] • [[Special:Contributions/Dick Bos|contribs]]) 19:06, 8 June 2026 (UTC)
*{{support}} In 2018 I initiated [[:Category:Videoconferences on media and democracy]] as a platform for disseminating public affairs events. In 2021 I officially initiated a podcast series on &quot;Media &amp; Democracy&quot; syndicated for the [[w:List of Pacifica Radio stations and affiliates|Pacifica radio network]]. In 2024 I converted it from irregular to fortnightly. I think this is all educational and supports the Wikiversity education mission, and I think that &quot;rehost Wikinews here&quot; would be appropriate. (I had some experience with Wikinews a few years ago. I felt it was too tightly controlled: Article submissions went stale, because I could not get official permission to publish and I could not get the information needed to understand what I was supposed to do to obtain the official permission. I would be opposed to rehosting Wikinews here if the policy similarly made it unreasonably difficult for volunteer contributor to get the information needed to meet the journalistic standards imposed by the overworked editors.) {{unsigned|DavidMCEddy}}

===Comments and questions===
:Definitely worthy of discussion, so I have no problem with the proposal in the sitenotice.
:Initial questions:
:* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages?
:* What are &quot;active editions&quot;?
:* How can Wikiversity navigate the concerns that lead to the closure of Wikinews?
:* Are any changes to the scope of Wikinews proposed?
:* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension?
:** e.g., I'm not sure that a page like [[User:BigKrow/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource.
:-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 05:52, 14 May 2026 (UTC)
:* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages?
::*No, not at this time.
:* What are &quot;active editions&quot;?
::*There were 30 other active editions of Wikinews in addition to English (e.g. [[:n:es:]]) at the time of universal closure (2026-05-04).
:* How can Wikiversity navigate the concerns that lead to the closure of Wikinews?
::*One of the biggest issues was the problems with DPL, which is now irrelevant. Another was the lack of activity, which can be ameliorated by having it be part of an existing project instead of its own domain (e.g. some editions of Wikipedia host their own Wikinews already and those projects were not impacted by the closure).
:* Are any changes to the scope of Wikinews proposed?
::*Not at this juncture. I would also propose as far as implemention goes that we would request a new namespace and that the material be more-or-less sequestered into its own ongoing project, like Wikijournal is or like the Cookbook and Wikijunior are at our sister [[:b:]].
:* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension?
:** e.g., I'm not sure that a page like [[Story/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource.
::*The process of citizen journalists practicing their craft in real-time and collaborating with others to do so is itself an education activity. We would essentially be hosting a real-time experiment in citizen journalism, online communities, and collaborative learning in addition to the prospect of spreading educational information from someone actually reading the news. I would propose that we could also make a more deliberate attempt to engage with learning &lt;em&gt;about&lt;/em&gt; what does and doesn't work with collaborative news writing by experimentation (e.g. audio news, syndicating to other sites, incorporating freely-licensed news from other sources, writing hyper-local news, writing briefs versus longer-term reportage) and also seeing if the problems noted in the Task Force report that recommended closure can be overcome. Note that we have already done some local investigation about and learning about wiki-based journalism on Wikinews here at [[Journalism studies and Wikinews]]. We could continue that learning and refine the process, including incorporating journalism students from universities. As for tensions, Wikinews is the only sister project that must be done with a quick turn-around: if you take a long time to [[:s:|transcribe a book]], that's just how long it takes, but if you take a long time to write news, it ceases to be news entirely. Wikiversity has been a very slow-growing project that has definitely had some successes but has generally come together over a long period with most learning resources being individual passion projects (or sometimes, frankly, crankery) which would not work with collaborative news that requires more than just a single editor writing whatever he feels like.
::Please let me know any other questions/concerns and any other editors feel free to give your own perspective. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 06:13, 14 May 2026 (UTC)
:::Thanks, Justin — it is food for thought.
:::In attempting to understand how we've arrived here, I've summarised some of the background on this page: [[Wikinews]].
:::Perhaps it could be helpful to flesh out more of the vision / ideas / possibilities / challenges on that page? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:49, 14 May 2026 (UTC)

:::*Having given it some thought, in principle, I support hosting [[citizen journalism]] on Wikiversity where it is clearly connected to a learning project and/or constitutes original research, both of which align strongly with [[Wikiversity:Mission|Wikiversity’s educational mission]].
:::*My chief concern is the potential for news content that is not clearly linked to the purpose of Wikiversity. To avoid this, some community-agreed guidelines would be prudent. These need not be overly restrictive; they should support boldness and experimentation while helping ensure alignment with Wikiversity's purpose.
:::*Given the reported low and declining activity on Wikinews, it seems unlikely that English Wikiversity would be overwhelmed by an influx of news-related editing. My impression is that English Wikinews was the most active edition, but even so, many contributors are likely to disperse to other projects or cease editing altogether. A modest migration of interested editors to Wikiversity seems manageable.
:::*At this stage, I do not think a dedicated namespace is necessary. Subpages under [[Wikinews]] or nested pages under relevant learning or research projects, or user-space draft pages should be suitable. I agree that [[Wikijournal]] offers a useful model, as do several existing course structures on Wikiversity.
:::*I support [[User:Koavf]]’s suggestions about framing Wikinews activity explicitly around learning. This would create a distinctive space for experimenting with collaborative news production in ways that are pedagogically meaningful. I agree that the [[journalism studies and Wikinews]] project developed by David and Leigh Blackall through the University of Wollongong is an excellent example of the intersection between Wikiversity and Wikinews. The [[Wikinews]] page could evolve into a hub for such projects.
:::*I've tidied the [[:Category:Wikinews|Wikinews category]] and merged some content into the [[Wikinews]] page. As part of a reinvigoration effort, please review these and related resources such as [[:Category:Journalism]] and [[School:Journalism]].
:::*A further argument in favour of this initiative is that Wikipedia explicitly excludes both news reporting and original research. So, there is value in maintaining spaces within the Wikimedia ecosystem where these forms of knowledge production can be openly developed and curated. Such work can, in turn, generate valuable evidence and source material that may later inform Wikipedia articles.
:::*The closure of WMF-hosted Wikinews does not imply that open wiki-based news curation lacks value. Indeed, the closure documentation appears supportive of experimentation with alternative news models across Wikimedia projects, including through Wikipedia and Wikidata. In that context, Wikiversity seems a natural home for a Wikinews experiment, provided it is clearly grounded in learning and/or research.
:::-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:39, 15 May 2026 (UTC)
My understanding towards Wikinews' failure is that everything takes too long to be approved for the publish status, which means that any breaking news would have already become days-old stale news. Wikinews has a brand recognition (for right or wrong reasons) than Wikiversity and I wonder how effective Wikiversity can attract the &quot;Wikinews refugees&quot; to edit here. And just a quick note on the governance. Since each Wikiversity language operates independently, each language has to vote &amp; adopt this proposal independently. [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 13:47, 15 May 2026 (UTC)

:Your assessment about Wikinews is partially correct. I referenced it earlier, but to be explicit, there is a [[:m:Proposal for Closing Wikinews|report by a task force on sister projects]] that outlines their concerns. There are a few, one of which was the nature of the staleness of news. Thanks also for clarifying that this proposal is only relevant to en.wv and is not binding or even proposed for other editions of Wikiversity. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:54, 15 May 2026 (UTC)
*Note: I am not a regular here, and just visit Wikiversity for the WikiJournal project. Challenges of Wikinews included that it required timely reporting and fact-checking processes which differed greatly from the well-established ones in Wikipedia. Here in Wikiversity, there is the WikiJournal project, and that can take some some forms of journalism, just not breaking news reporting. I am in favor of salvaging parts of Wikinews if helpful. Could it, would it be feasible to adapt Wikijournal to accept some forms of news journalism, but just not the timed news reporting? For example, WikiJournal already is doing conference proceedings, and could likely do related event reports even months after the event ended. It could probably accept long-form investigative reporting, which is a sort of news that is not breaking news. I am not sure what the possibilities are, but I would prefer to build up systems that already work rather than import systems which had problems elsewhere. Thanks. [[User:Bluerasberry|&lt;span style=&quot;background:#cedff2;color:#11e&quot;&gt;''' Blue Rasberry '''&lt;/span&gt;]][[User talk:Bluerasberry|&lt;span style=&quot;cursor:help&quot;&gt;&lt;span style=&quot;background:#cedff2;color:#11e&quot;&gt;(talk)&lt;/span&gt;&lt;/span&gt;]] 19:17, 22 May 2026 (UTC)
*:I agree that there are certain kinds of journalism that are perfectly valid and not time-bound like breaking news reporting, so that won't suffer from the issues noted before. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 21:15, 22 May 2026 (UTC)
*::@[[User:Bluerasberry|Bluerasberry]] WikiJournal is not interested in taking on news journalism. WikiJournal is publishing conference proceedings at the request of some Wikimedian educators, and conference proceedings is what a &quot;regular&quot; journal publishes. News journalism is quite different from this, and if WikiJournal starts to deviate towards publishing news journalism, it will create barrier towards future initiatives like being indexed in Medline or Web of Science, and may risk being delisted from Scopus. [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 22:43, 5 June 2026 (UTC)
*:::Thats a good point. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:09, 9 June 2026 (UTC)</text>
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== Proposal to rehost Wikinews here ==

As many of you know, and mentioned here at the Colloquium, our sister project Wikinews recently closed, with all 31 active editions made read-only. [[User:BigKrow]] has asked about the prospect of writing news stories here and I suggested that since we already have [[School:Journalism]] and some resources related to the [[:Category:Journalism|broader topic of journalism]]. I would like to propose that we have continued and indefinite space for {{w|citizen journalism}} by essentially repurposing Wikinews into a sub-project here. The only special infrastructure that Wikinews required was [[:mw:Extension:DynamicPageList]], which was deactivated and caused issues due to a lack of maintenance.

I will add this proposal to the site banner, but I recognize that that may be a conflict of interest, so if anyone requests that I remove it, I will. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:30, 14 May 2026 (UTC)
:I would like to see this conversation go for at least 30 days to establish a consensus. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:35, 14 May 2026 (UTC)
::A few days shy of 30, it seems obvious that this is not going to pass. So I '''withdraw''' as presumptively '''failed'''. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:14, 9 June 2026 (UTC)

===Votes===
*{{support}} as proposer (with BK's inspiration). I think that an ongoing experiment in citizen journalism is a fit and appropriate use of this site. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:35, 14 May 2026 (UTC)
*{{support}}, hope to seeing ideas about this, and thank you @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 11:08, 14 May 2026 (UTC)
*{{support}} Other than perhaps inflating the total number of pages reported, I see the idea of &quot;practicing journalism&quot; a worthy and relevant activity within the domain of Wikiversity. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 21:41, 14 May 2026 (UTC)
*{{support}} Conditional on development of (a) community guidelines that ensure alignment with Wikiversity's purpose, and (b) clear, nested page-naming structures for projects. More detail below. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:48, 15 May 2026 (UTC)
*{{contra}} This proposal doesn't seem interested in expanding educational materials in journalism, but rather in providing space and protection for Wikinews contributors. But this is contrary to the goals of Wikiversity, and I'm not sure it's a good idea, even with regard to WMF. If WMF decides to close a project and another community lets it run on its domain, that's a bit of an undermining of WMF's and the community's decisions. Given that Wikiversity has had several conflicts with other communities and WMF in its history, I'm against it.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:59, 15 May 2026 (UTC)
*{{contra}} This seems like a proposal to continue the mission of WikiNews, but not a proposal specifically to improve Wikiversity. I concur with Juandev's comments. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 20:29, 30 May 2026 (UTC)
* {{oppose}} per above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:05, 1 June 2026 (UTC)
*{{oppose}} Wikiversity isn’t Wikinews and it also isn’t a dumping ground for anything not covered by other projects. It was already suggested, rather bafflingly, that Wikinews parasitize Wikipedia as a host. If it were allowed to freeload off of Wikiversity it would simply promote a view I and likely many others have— that Wikiversity (as it currently exists) has no standards and mostly just exists to host subpar content that wouldn’t be tolerated on any other Wikimedia site. Wikinews needs a new, non-Wikimedia host, and Wikiversity needs to get its act together by enforcing a minimum scope and standard for what it allows. --[[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:16, 4 June 2026 (UTC)
* {{oppose}} per above. Wikiversity&lt;math&gt;\not=&lt;/math&gt; Wikinews - not a good idea to mix the scope of projects. --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 12:03, 8 June 2026 (UTC)
* {{abstain}} I will abstain since I'm not an active Wikiversity contributor. But I just feel like Wikinews had a very clear and specific goal of providing news, and Wikiversity is just a different project with different goals. For me, it would be odd to rehost Wikinews here. But please do not count my vote, this is only a comment. --[[User:Antimundo|Antimundo]] ([[User talk:Antimundo|discuss]] • [[Special:Contributions/Antimundo|contribs]]) 13:19, 6 June 2026 (UTC)
* {{oppose}} Although I think it's a pity that Wikinews is closed. --[[User:Dick Bos|Dick Bos]] ([[User talk:Dick Bos|discuss]] • [[Special:Contributions/Dick Bos|contribs]]) 19:06, 8 June 2026 (UTC)
*{{support}} In 2018 I initiated [[:Category:Videoconferences on media and democracy]] as a platform for disseminating public affairs events. In 2021 I officially initiated a podcast series on &quot;Media &amp; Democracy&quot; syndicated for the [[w:List of Pacifica Radio stations and affiliates|Pacifica radio network]]. In 2024 I converted it from irregular to fortnightly. I think this is all educational and supports the Wikiversity education mission, and I think that &quot;rehost Wikinews here&quot; would be appropriate. (I had some experience with Wikinews a few years ago. I felt it was too tightly controlled: Article submissions went stale, because I could not get official permission to publish and I could not get the information needed to understand what I was supposed to do to obtain the official permission. I would be opposed to rehosting Wikinews here if the policy similarly made it unreasonably difficult for volunteer contributor to get the information needed to meet the journalistic standards imposed by the overworked editors.) {{unsigned|DavidMCEddy}}

===Comments and questions===
:Definitely worthy of discussion, so I have no problem with the proposal in the sitenotice.
:Initial questions:
:* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages?
:* What are &quot;active editions&quot;?
:* How can Wikiversity navigate the concerns that lead to the closure of Wikinews?
:* Are any changes to the scope of Wikinews proposed?
:* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension?
:** e.g., I'm not sure that a page like [[User:BigKrow/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource.
:-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 05:52, 14 May 2026 (UTC)
:* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages?
::*No, not at this time.
:* What are &quot;active editions&quot;?
::*There were 30 other active editions of Wikinews in addition to English (e.g. [[:n:es:]]) at the time of universal closure (2026-05-04).
:* How can Wikiversity navigate the concerns that lead to the closure of Wikinews?
::*One of the biggest issues was the problems with DPL, which is now irrelevant. Another was the lack of activity, which can be ameliorated by having it be part of an existing project instead of its own domain (e.g. some editions of Wikipedia host their own Wikinews already and those projects were not impacted by the closure).
:* Are any changes to the scope of Wikinews proposed?
::*Not at this juncture. I would also propose as far as implemention goes that we would request a new namespace and that the material be more-or-less sequestered into its own ongoing project, like Wikijournal is or like the Cookbook and Wikijunior are at our sister [[:b:]].
:* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension?
:** e.g., I'm not sure that a page like [[Story/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource.
::*The process of citizen journalists practicing their craft in real-time and collaborating with others to do so is itself an education activity. We would essentially be hosting a real-time experiment in citizen journalism, online communities, and collaborative learning in addition to the prospect of spreading educational information from someone actually reading the news. I would propose that we could also make a more deliberate attempt to engage with learning &lt;em&gt;about&lt;/em&gt; what does and doesn't work with collaborative news writing by experimentation (e.g. audio news, syndicating to other sites, incorporating freely-licensed news from other sources, writing hyper-local news, writing briefs versus longer-term reportage) and also seeing if the problems noted in the Task Force report that recommended closure can be overcome. Note that we have already done some local investigation about and learning about wiki-based journalism on Wikinews here at [[Journalism studies and Wikinews]]. We could continue that learning and refine the process, including incorporating journalism students from universities. As for tensions, Wikinews is the only sister project that must be done with a quick turn-around: if you take a long time to [[:s:|transcribe a book]], that's just how long it takes, but if you take a long time to write news, it ceases to be news entirely. Wikiversity has been a very slow-growing project that has definitely had some successes but has generally come together over a long period with most learning resources being individual passion projects (or sometimes, frankly, crankery) which would not work with collaborative news that requires more than just a single editor writing whatever he feels like.
::Please let me know any other questions/concerns and any other editors feel free to give your own perspective. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 06:13, 14 May 2026 (UTC)
:::Thanks, Justin — it is food for thought.
:::In attempting to understand how we've arrived here, I've summarised some of the background on this page: [[Wikinews]].
:::Perhaps it could be helpful to flesh out more of the vision / ideas / possibilities / challenges on that page? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:49, 14 May 2026 (UTC)

:::*Having given it some thought, in principle, I support hosting [[citizen journalism]] on Wikiversity where it is clearly connected to a learning project and/or constitutes original research, both of which align strongly with [[Wikiversity:Mission|Wikiversity’s educational mission]].
:::*My chief concern is the potential for news content that is not clearly linked to the purpose of Wikiversity. To avoid this, some community-agreed guidelines would be prudent. These need not be overly restrictive; they should support boldness and experimentation while helping ensure alignment with Wikiversity's purpose.
:::*Given the reported low and declining activity on Wikinews, it seems unlikely that English Wikiversity would be overwhelmed by an influx of news-related editing. My impression is that English Wikinews was the most active edition, but even so, many contributors are likely to disperse to other projects or cease editing altogether. A modest migration of interested editors to Wikiversity seems manageable.
:::*At this stage, I do not think a dedicated namespace is necessary. Subpages under [[Wikinews]] or nested pages under relevant learning or research projects, or user-space draft pages should be suitable. I agree that [[Wikijournal]] offers a useful model, as do several existing course structures on Wikiversity.
:::*I support [[User:Koavf]]’s suggestions about framing Wikinews activity explicitly around learning. This would create a distinctive space for experimenting with collaborative news production in ways that are pedagogically meaningful. I agree that the [[journalism studies and Wikinews]] project developed by David and Leigh Blackall through the University of Wollongong is an excellent example of the intersection between Wikiversity and Wikinews. The [[Wikinews]] page could evolve into a hub for such projects.
:::*I've tidied the [[:Category:Wikinews|Wikinews category]] and merged some content into the [[Wikinews]] page. As part of a reinvigoration effort, please review these and related resources such as [[:Category:Journalism]] and [[School:Journalism]].
:::*A further argument in favour of this initiative is that Wikipedia explicitly excludes both news reporting and original research. So, there is value in maintaining spaces within the Wikimedia ecosystem where these forms of knowledge production can be openly developed and curated. Such work can, in turn, generate valuable evidence and source material that may later inform Wikipedia articles.
:::*The closure of WMF-hosted Wikinews does not imply that open wiki-based news curation lacks value. Indeed, the closure documentation appears supportive of experimentation with alternative news models across Wikimedia projects, including through Wikipedia and Wikidata. In that context, Wikiversity seems a natural home for a Wikinews experiment, provided it is clearly grounded in learning and/or research.
:::-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:39, 15 May 2026 (UTC)
My understanding towards Wikinews' failure is that everything takes too long to be approved for the publish status, which means that any breaking news would have already become days-old stale news. Wikinews has a brand recognition (for right or wrong reasons) than Wikiversity and I wonder how effective Wikiversity can attract the &quot;Wikinews refugees&quot; to edit here. And just a quick note on the governance. Since each Wikiversity language operates independently, each language has to vote &amp; adopt this proposal independently. [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 13:47, 15 May 2026 (UTC)

:Your assessment about Wikinews is partially correct. I referenced it earlier, but to be explicit, there is a [[:m:Proposal for Closing Wikinews|report by a task force on sister projects]] that outlines their concerns. There are a few, one of which was the nature of the staleness of news. Thanks also for clarifying that this proposal is only relevant to en.wv and is not binding or even proposed for other editions of Wikiversity. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:54, 15 May 2026 (UTC)
*Note: I am not a regular here, and just visit Wikiversity for the WikiJournal project. Challenges of Wikinews included that it required timely reporting and fact-checking processes which differed greatly from the well-established ones in Wikipedia. Here in Wikiversity, there is the WikiJournal project, and that can take some some forms of journalism, just not breaking news reporting. I am in favor of salvaging parts of Wikinews if helpful. Could it, would it be feasible to adapt Wikijournal to accept some forms of news journalism, but just not the timed news reporting? For example, WikiJournal already is doing conference proceedings, and could likely do related event reports even months after the event ended. It could probably accept long-form investigative reporting, which is a sort of news that is not breaking news. I am not sure what the possibilities are, but I would prefer to build up systems that already work rather than import systems which had problems elsewhere. Thanks. [[User:Bluerasberry|&lt;span style=&quot;background:#cedff2;color:#11e&quot;&gt;''' Blue Rasberry '''&lt;/span&gt;]][[User talk:Bluerasberry|&lt;span style=&quot;cursor:help&quot;&gt;&lt;span style=&quot;background:#cedff2;color:#11e&quot;&gt;(talk)&lt;/span&gt;&lt;/span&gt;]] 19:17, 22 May 2026 (UTC)
*:I agree that there are certain kinds of journalism that are perfectly valid and not time-bound like breaking news reporting, so that won't suffer from the issues noted before. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 21:15, 22 May 2026 (UTC)
*::@[[User:Bluerasberry|Bluerasberry]] WikiJournal is not interested in taking on news journalism. WikiJournal is publishing conference proceedings at the request of some Wikimedian educators, and conference proceedings is what a &quot;regular&quot; journal publishes. News journalism is quite different from this, and if WikiJournal starts to deviate towards publishing news journalism, it will create barrier towards future initiatives like being indexed in Medline or Web of Science, and may risk being delisted from Scopus. [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 22:43, 5 June 2026 (UTC)
*:::Thats a good point. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:09, 9 June 2026 (UTC)

== Create an autopatrolled user group? ==

{{tracked|T428269|resolved}}
I would like to propose creating the user group &lt;code&gt;autopatrolled&lt;/code&gt; (autopatrolled user), in which for non-curators and non-custodians, their page creations and file uploads would be automatically marked as patrolled by the MediaWiki software. Custodians may grant the user group, at their discretion, to users who create good quality pages that do not need frequent patrolling.

On a side note, the term {{tq|autopatroller}} would be used, but because we don't have non-curator/custodian patrollers (as we rely on curators and custodians to patrol), I suggest on using the term {{tq|autopatrolled user}}. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:31, 29 May 2026 (UTC)

:'''Support''' re: the name, I don't really understand the reasoning, so I am '''neutral''' on that. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 15:45, 29 May 2026 (UTC)
:: Regarding the name, this is because as we don't have the patroller user group, we rely on curators and custodians to patrol new pages and file uploads. Does that make sense? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:39, 29 May 2026 (UTC)
:::Not really, but I don't think it's the most important thing. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 16:42, 29 May 2026 (UTC)
:::: We'll decide on the name later. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:48, 30 May 2026 (UTC)
:::::Oh, please don't let me stand in the way. I'm just not very smart, so don't hold up a matter on my account. I didn't want to derail the proposal, which is a fine and sensible one. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 04:16, 30 May 2026 (UTC)
: '''Support''' - sounds like a good idea
:* Suggest adding a draft section about this group to [[Wikiversity:Patrolling]]. There is a statement in the Introduction of the page that I'm not sure if its correct and at least could be improved: &quot;Wikiversity also uses an autopatrol right, meaning trusted users' contributions are automatically marked as checked so patrollers can focus on reviewing newer or anonymous editors.&quot;
:* Regarding autopatroller vs autropatrolled user, what terms are used on similar WMF wiki projects?
: -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:28, 30 May 2026 (UTC)
::# I would create a starting page about the user groups, with experienced editors expanding the page. A summarized part of that page would also be added to [[Wikiversity:Patrolling]].
::# For a similar example, English Wikipedia uses the term {{tq|Autopatrolled}}, just that term only.
:: [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:22, 30 May 2026 (UTC)
: @[[User:Jtneill|Jtneill]] and @[[User:Koavf|Koavf]]: the autopatroller user group has been implemented here. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:14, 8 June 2026 (UTC)
::Thanks. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:13, 9 June 2026 (UTC)</text>
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== Proposal to rehost Wikinews here ==

As many of you know, and mentioned here at the Colloquium, our sister project Wikinews recently closed, with all 31 active editions made read-only. [[User:BigKrow]] has asked about the prospect of writing news stories here and I suggested that since we already have [[School:Journalism]] and some resources related to the [[:Category:Journalism|broader topic of journalism]]. I would like to propose that we have continued and indefinite space for {{w|citizen journalism}} by essentially repurposing Wikinews into a sub-project here. The only special infrastructure that Wikinews required was [[:mw:Extension:DynamicPageList]], which was deactivated and caused issues due to a lack of maintenance.

I will add this proposal to the site banner, but I recognize that that may be a conflict of interest, so if anyone requests that I remove it, I will. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:30, 14 May 2026 (UTC)
:I would like to see this conversation go for at least 30 days to establish a consensus. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:35, 14 May 2026 (UTC)
::A few days shy of 30, it seems obvious that this is not going to pass. So I '''withdraw''' as presumptively '''failed'''. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:14, 9 June 2026 (UTC)

===Votes===
*{{support}} as proposer (with BK's inspiration). I think that an ongoing experiment in citizen journalism is a fit and appropriate use of this site. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:35, 14 May 2026 (UTC)
*{{support}}, hope to seeing ideas about this, and thank you @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 11:08, 14 May 2026 (UTC)
*{{support}} Other than perhaps inflating the total number of pages reported, I see the idea of &quot;practicing journalism&quot; a worthy and relevant activity within the domain of Wikiversity. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 21:41, 14 May 2026 (UTC)
*{{support}} Conditional on development of (a) community guidelines that ensure alignment with Wikiversity's purpose, and (b) clear, nested page-naming structures for projects. More detail below. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:48, 15 May 2026 (UTC)
*{{contra}} This proposal doesn't seem interested in expanding educational materials in journalism, but rather in providing space and protection for Wikinews contributors. But this is contrary to the goals of Wikiversity, and I'm not sure it's a good idea, even with regard to WMF. If WMF decides to close a project and another community lets it run on its domain, that's a bit of an undermining of WMF's and the community's decisions. Given that Wikiversity has had several conflicts with other communities and WMF in its history, I'm against it.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:59, 15 May 2026 (UTC)
*{{contra}} This seems like a proposal to continue the mission of WikiNews, but not a proposal specifically to improve Wikiversity. I concur with Juandev's comments. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 20:29, 30 May 2026 (UTC)
* {{oppose}} per above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:05, 1 June 2026 (UTC)
*{{oppose}} Wikiversity isn’t Wikinews and it also isn’t a dumping ground for anything not covered by other projects. It was already suggested, rather bafflingly, that Wikinews parasitize Wikipedia as a host. If it were allowed to freeload off of Wikiversity it would simply promote a view I and likely many others have— that Wikiversity (as it currently exists) has no standards and mostly just exists to host subpar content that wouldn’t be tolerated on any other Wikimedia site. Wikinews needs a new, non-Wikimedia host, and Wikiversity needs to get its act together by enforcing a minimum scope and standard for what it allows. --[[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:16, 4 June 2026 (UTC)
* {{oppose}} per above. Wikiversity&lt;math&gt;\not=&lt;/math&gt; Wikinews - not a good idea to mix the scope of projects. --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 12:03, 8 June 2026 (UTC)
* {{abstain}} I will abstain since I'm not an active Wikiversity contributor. But I just feel like Wikinews had a very clear and specific goal of providing news, and Wikiversity is just a different project with different goals. For me, it would be odd to rehost Wikinews here. But please do not count my vote, this is only a comment. --[[User:Antimundo|Antimundo]] ([[User talk:Antimundo|discuss]] • [[Special:Contributions/Antimundo|contribs]]) 13:19, 6 June 2026 (UTC)
* {{oppose}} Although I think it's a pity that Wikinews is closed. --[[User:Dick Bos|Dick Bos]] ([[User talk:Dick Bos|discuss]] • [[Special:Contributions/Dick Bos|contribs]]) 19:06, 8 June 2026 (UTC)
*{{support}} In 2018 I initiated [[:Category:Videoconferences on media and democracy]] as a platform for disseminating public affairs events. In 2021 I officially initiated a podcast series on &quot;Media &amp; Democracy&quot; syndicated for the [[w:List of Pacifica Radio stations and affiliates|Pacifica radio network]]. In 2024 I converted it from irregular to fortnightly. I think this is all educational and supports the Wikiversity education mission, and I think that &quot;rehost Wikinews here&quot; would be appropriate. (I had some experience with Wikinews a few years ago. I felt it was too tightly controlled: Article submissions went stale, because I could not get official permission to publish and I could not get the information needed to understand what I was supposed to do to obtain the official permission. I would be opposed to rehosting Wikinews here if the policy similarly made it unreasonably difficult for volunteer contributor to get the information needed to meet the journalistic standards imposed by the overworked editors.) {{unsigned|DavidMCEddy}}

===Comments and questions===
:Definitely worthy of discussion, so I have no problem with the proposal in the sitenotice.
:Initial questions:
:* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages?
:* What are &quot;active editions&quot;?
:* How can Wikiversity navigate the concerns that lead to the closure of Wikinews?
:* Are any changes to the scope of Wikinews proposed?
:* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension?
:** e.g., I'm not sure that a page like [[User:BigKrow/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource.
:-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 05:52, 14 May 2026 (UTC)
:* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages?
::*No, not at this time.
:* What are &quot;active editions&quot;?
::*There were 30 other active editions of Wikinews in addition to English (e.g. [[:n:es:]]) at the time of universal closure (2026-05-04).
:* How can Wikiversity navigate the concerns that lead to the closure of Wikinews?
::*One of the biggest issues was the problems with DPL, which is now irrelevant. Another was the lack of activity, which can be ameliorated by having it be part of an existing project instead of its own domain (e.g. some editions of Wikipedia host their own Wikinews already and those projects were not impacted by the closure).
:* Are any changes to the scope of Wikinews proposed?
::*Not at this juncture. I would also propose as far as implemention goes that we would request a new namespace and that the material be more-or-less sequestered into its own ongoing project, like Wikijournal is or like the Cookbook and Wikijunior are at our sister [[:b:]].
:* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension?
:** e.g., I'm not sure that a page like [[Story/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource.
::*The process of citizen journalists practicing their craft in real-time and collaborating with others to do so is itself an education activity. We would essentially be hosting a real-time experiment in citizen journalism, online communities, and collaborative learning in addition to the prospect of spreading educational information from someone actually reading the news. I would propose that we could also make a more deliberate attempt to engage with learning &lt;em&gt;about&lt;/em&gt; what does and doesn't work with collaborative news writing by experimentation (e.g. audio news, syndicating to other sites, incorporating freely-licensed news from other sources, writing hyper-local news, writing briefs versus longer-term reportage) and also seeing if the problems noted in the Task Force report that recommended closure can be overcome. Note that we have already done some local investigation about and learning about wiki-based journalism on Wikinews here at [[Journalism studies and Wikinews]]. We could continue that learning and refine the process, including incorporating journalism students from universities. As for tensions, Wikinews is the only sister project that must be done with a quick turn-around: if you take a long time to [[:s:|transcribe a book]], that's just how long it takes, but if you take a long time to write news, it ceases to be news entirely. Wikiversity has been a very slow-growing project that has definitely had some successes but has generally come together over a long period with most learning resources being individual passion projects (or sometimes, frankly, crankery) which would not work with collaborative news that requires more than just a single editor writing whatever he feels like.
::Please let me know any other questions/concerns and any other editors feel free to give your own perspective. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 06:13, 14 May 2026 (UTC)
:::Thanks, Justin — it is food for thought.
:::In attempting to understand how we've arrived here, I've summarised some of the background on this page: [[Wikinews]].
:::Perhaps it could be helpful to flesh out more of the vision / ideas / possibilities / challenges on that page? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:49, 14 May 2026 (UTC)

:::*Having given it some thought, in principle, I support hosting [[citizen journalism]] on Wikiversity where it is clearly connected to a learning project and/or constitutes original research, both of which align strongly with [[Wikiversity:Mission|Wikiversity’s educational mission]].
:::*My chief concern is the potential for news content that is not clearly linked to the purpose of Wikiversity. To avoid this, some community-agreed guidelines would be prudent. These need not be overly restrictive; they should support boldness and experimentation while helping ensure alignment with Wikiversity's purpose.
:::*Given the reported low and declining activity on Wikinews, it seems unlikely that English Wikiversity would be overwhelmed by an influx of news-related editing. My impression is that English Wikinews was the most active edition, but even so, many contributors are likely to disperse to other projects or cease editing altogether. A modest migration of interested editors to Wikiversity seems manageable.
:::*At this stage, I do not think a dedicated namespace is necessary. Subpages under [[Wikinews]] or nested pages under relevant learning or research projects, or user-space draft pages should be suitable. I agree that [[Wikijournal]] offers a useful model, as do several existing course structures on Wikiversity.
:::*I support [[User:Koavf]]’s suggestions about framing Wikinews activity explicitly around learning. This would create a distinctive space for experimenting with collaborative news production in ways that are pedagogically meaningful. I agree that the [[journalism studies and Wikinews]] project developed by David and Leigh Blackall through the University of Wollongong is an excellent example of the intersection between Wikiversity and Wikinews. The [[Wikinews]] page could evolve into a hub for such projects.
:::*I've tidied the [[:Category:Wikinews|Wikinews category]] and merged some content into the [[Wikinews]] page. As part of a reinvigoration effort, please review these and related resources such as [[:Category:Journalism]] and [[School:Journalism]].
:::*A further argument in favour of this initiative is that Wikipedia explicitly excludes both news reporting and original research. So, there is value in maintaining spaces within the Wikimedia ecosystem where these forms of knowledge production can be openly developed and curated. Such work can, in turn, generate valuable evidence and source material that may later inform Wikipedia articles.
:::*The closure of WMF-hosted Wikinews does not imply that open wiki-based news curation lacks value. Indeed, the closure documentation appears supportive of experimentation with alternative news models across Wikimedia projects, including through Wikipedia and Wikidata. In that context, Wikiversity seems a natural home for a Wikinews experiment, provided it is clearly grounded in learning and/or research.
:::-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:39, 15 May 2026 (UTC)
My understanding towards Wikinews' failure is that everything takes too long to be approved for the publish status, which means that any breaking news would have already become days-old stale news. Wikinews has a brand recognition (for right or wrong reasons) than Wikiversity and I wonder how effective Wikiversity can attract the &quot;Wikinews refugees&quot; to edit here. And just a quick note on the governance. Since each Wikiversity language operates independently, each language has to vote &amp; adopt this proposal independently. [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 13:47, 15 May 2026 (UTC)

:Your assessment about Wikinews is partially correct. I referenced it earlier, but to be explicit, there is a [[:m:Proposal for Closing Wikinews|report by a task force on sister projects]] that outlines their concerns. There are a few, one of which was the nature of the staleness of news. Thanks also for clarifying that this proposal is only relevant to en.wv and is not binding or even proposed for other editions of Wikiversity. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:54, 15 May 2026 (UTC)
*Note: I am not a regular here, and just visit Wikiversity for the WikiJournal project. Challenges of Wikinews included that it required timely reporting and fact-checking processes which differed greatly from the well-established ones in Wikipedia. Here in Wikiversity, there is the WikiJournal project, and that can take some some forms of journalism, just not breaking news reporting. I am in favor of salvaging parts of Wikinews if helpful. Could it, would it be feasible to adapt Wikijournal to accept some forms of news journalism, but just not the timed news reporting? For example, WikiJournal already is doing conference proceedings, and could likely do related event reports even months after the event ended. It could probably accept long-form investigative reporting, which is a sort of news that is not breaking news. I am not sure what the possibilities are, but I would prefer to build up systems that already work rather than import systems which had problems elsewhere. Thanks. [[User:Bluerasberry|&lt;span style=&quot;background:#cedff2;color:#11e&quot;&gt;''' Blue Rasberry '''&lt;/span&gt;]][[User talk:Bluerasberry|&lt;span style=&quot;cursor:help&quot;&gt;&lt;span style=&quot;background:#cedff2;color:#11e&quot;&gt;(talk)&lt;/span&gt;&lt;/span&gt;]] 19:17, 22 May 2026 (UTC)
*:I agree that there are certain kinds of journalism that are perfectly valid and not time-bound like breaking news reporting, so that won't suffer from the issues noted before. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 21:15, 22 May 2026 (UTC)
*::@[[User:Bluerasberry|Bluerasberry]] WikiJournal is not interested in taking on news journalism. WikiJournal is publishing conference proceedings at the request of some Wikimedian educators, and conference proceedings is what a &quot;regular&quot; journal publishes. News journalism is quite different from this, and if WikiJournal starts to deviate towards publishing news journalism, it will create barrier towards future initiatives like being indexed in Medline or Web of Science, and may risk being delisted from Scopus. [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 22:43, 5 June 2026 (UTC)
*:::Thats a good point. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:09, 9 June 2026 (UTC)

== Create an autopatrolled user group? ==

{{tracked|T428269|resolved}}
I would like to propose creating the user group &lt;code&gt;autopatrolled&lt;/code&gt; (autopatrolled user), in which for non-curators and non-custodians, their page creations and file uploads would be automatically marked as patrolled by the MediaWiki software. Custodians may grant the user group, at their discretion, to users who create good quality pages that do not need frequent patrolling.

On a side note, the term {{tq|autopatroller}} would be used, but because we don't have non-curator/custodian patrollers (as we rely on curators and custodians to patrol), I suggest on using the term {{tq|autopatrolled user}}. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:31, 29 May 2026 (UTC)

:'''Support''' re: the name, I don't really understand the reasoning, so I am '''neutral''' on that. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 15:45, 29 May 2026 (UTC)
:: Regarding the name, this is because as we don't have the patroller user group, we rely on curators and custodians to patrol new pages and file uploads. Does that make sense? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:39, 29 May 2026 (UTC)
:::Not really, but I don't think it's the most important thing. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 16:42, 29 May 2026 (UTC)
:::: We'll decide on the name later. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:48, 30 May 2026 (UTC)
:::::Oh, please don't let me stand in the way. I'm just not very smart, so don't hold up a matter on my account. I didn't want to derail the proposal, which is a fine and sensible one. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 04:16, 30 May 2026 (UTC)
: '''Support''' - sounds like a good idea
:* Suggest adding a draft section about this group to [[Wikiversity:Patrolling]]. There is a statement in the Introduction of the page that I'm not sure if its correct and at least could be improved: &quot;Wikiversity also uses an autopatrol right, meaning trusted users' contributions are automatically marked as checked so patrollers can focus on reviewing newer or anonymous editors.&quot;
:* Regarding autopatroller vs autropatrolled user, what terms are used on similar WMF wiki projects?
: -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:28, 30 May 2026 (UTC)
::# I would create a starting page about the user groups, with experienced editors expanding the page. A summarized part of that page would also be added to [[Wikiversity:Patrolling]].
::# For a similar example, English Wikipedia uses the term {{tq|Autopatrolled}}, just that term only.
:: [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:22, 30 May 2026 (UTC)
: @[[User:Jtneill|Jtneill]] and @[[User:Koavf|Koavf]]: the autopatroller user group has been implemented here. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:14, 8 June 2026 (UTC)
::Thanks. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:13, 9 June 2026 (UTC)

== How much of Wikiversity’s content is LLM slop? ==

Because it seems like a non-trivial amount, along with AI slop images as well. Is there some kind of AI cleanup project established yet? [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:20, 4 June 2026 (UTC)

:We have discussed AI but I don't know of any explicit initiative to find and delete AI-generated noise. Individual modules have been deleted for having been made by AI. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 08:50, 4 June 2026 (UTC)
:Recently agreed [[Wikiversity:Artificial intelligence|policy]] welcome users to tag AI generated pages. Me personally I am not against the use of AI. What is the difference in abstract schematic image created by a human and the same by an AI. If the users does not have finances to pay digital artest and you dont want to let them use AI, would you pay the artest for them? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:07, 8 June 2026 (UTC)
::Wikimedia has a lot of ''volunteer'' artists who can illustrate if asked. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:11, 9 June 2026 (UTC)
:::Interesting! That's good to know. Where can we find the volunteer artists for illustrating? [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 20:11, 9 June 2026 (UTC)
::::Wikimedia commons has [[commons:Commons:Graphic Lab/Illustration workshop]] [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 02:18, 10 June 2026 (UTC)</text>
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      <comment>/* New user what do I do here */ archive from [[Wikiversity:Colloquium]] ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]])</comment>
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== Proposal to rehost Wikinews here ==

As many of you know, and mentioned here at the Colloquium, our sister project Wikinews recently closed, with all 31 active editions made read-only. [[User:BigKrow]] has asked about the prospect of writing news stories here and I suggested that since we already have [[School:Journalism]] and some resources related to the [[:Category:Journalism|broader topic of journalism]]. I would like to propose that we have continued and indefinite space for {{w|citizen journalism}} by essentially repurposing Wikinews into a sub-project here. The only special infrastructure that Wikinews required was [[:mw:Extension:DynamicPageList]], which was deactivated and caused issues due to a lack of maintenance.

I will add this proposal to the site banner, but I recognize that that may be a conflict of interest, so if anyone requests that I remove it, I will. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:30, 14 May 2026 (UTC)
:I would like to see this conversation go for at least 30 days to establish a consensus. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:35, 14 May 2026 (UTC)
::A few days shy of 30, it seems obvious that this is not going to pass. So I '''withdraw''' as presumptively '''failed'''. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:14, 9 June 2026 (UTC)

===Votes===
*{{support}} as proposer (with BK's inspiration). I think that an ongoing experiment in citizen journalism is a fit and appropriate use of this site. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:35, 14 May 2026 (UTC)
*{{support}}, hope to seeing ideas about this, and thank you @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 11:08, 14 May 2026 (UTC)
*{{support}} Other than perhaps inflating the total number of pages reported, I see the idea of &quot;practicing journalism&quot; a worthy and relevant activity within the domain of Wikiversity. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 21:41, 14 May 2026 (UTC)
*{{support}} Conditional on development of (a) community guidelines that ensure alignment with Wikiversity's purpose, and (b) clear, nested page-naming structures for projects. More detail below. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:48, 15 May 2026 (UTC)
*{{contra}} This proposal doesn't seem interested in expanding educational materials in journalism, but rather in providing space and protection for Wikinews contributors. But this is contrary to the goals of Wikiversity, and I'm not sure it's a good idea, even with regard to WMF. If WMF decides to close a project and another community lets it run on its domain, that's a bit of an undermining of WMF's and the community's decisions. Given that Wikiversity has had several conflicts with other communities and WMF in its history, I'm against it.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:59, 15 May 2026 (UTC)
*{{contra}} This seems like a proposal to continue the mission of WikiNews, but not a proposal specifically to improve Wikiversity. I concur with Juandev's comments. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 20:29, 30 May 2026 (UTC)
* {{oppose}} per above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:05, 1 June 2026 (UTC)
*{{oppose}} Wikiversity isn’t Wikinews and it also isn’t a dumping ground for anything not covered by other projects. It was already suggested, rather bafflingly, that Wikinews parasitize Wikipedia as a host. If it were allowed to freeload off of Wikiversity it would simply promote a view I and likely many others have— that Wikiversity (as it currently exists) has no standards and mostly just exists to host subpar content that wouldn’t be tolerated on any other Wikimedia site. Wikinews needs a new, non-Wikimedia host, and Wikiversity needs to get its act together by enforcing a minimum scope and standard for what it allows. --[[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:16, 4 June 2026 (UTC)
* {{oppose}} per above. Wikiversity&lt;math&gt;\not=&lt;/math&gt; Wikinews - not a good idea to mix the scope of projects. --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 12:03, 8 June 2026 (UTC)
* {{abstain}} I will abstain since I'm not an active Wikiversity contributor. But I just feel like Wikinews had a very clear and specific goal of providing news, and Wikiversity is just a different project with different goals. For me, it would be odd to rehost Wikinews here. But please do not count my vote, this is only a comment. --[[User:Antimundo|Antimundo]] ([[User talk:Antimundo|discuss]] • [[Special:Contributions/Antimundo|contribs]]) 13:19, 6 June 2026 (UTC)
* {{oppose}} Although I think it's a pity that Wikinews is closed. --[[User:Dick Bos|Dick Bos]] ([[User talk:Dick Bos|discuss]] • [[Special:Contributions/Dick Bos|contribs]]) 19:06, 8 June 2026 (UTC)
*{{support}} In 2018 I initiated [[:Category:Videoconferences on media and democracy]] as a platform for disseminating public affairs events. In 2021 I officially initiated a podcast series on &quot;Media &amp; Democracy&quot; syndicated for the [[w:List of Pacifica Radio stations and affiliates|Pacifica radio network]]. In 2024 I converted it from irregular to fortnightly. I think this is all educational and supports the Wikiversity education mission, and I think that &quot;rehost Wikinews here&quot; would be appropriate. (I had some experience with Wikinews a few years ago. I felt it was too tightly controlled: Article submissions went stale, because I could not get official permission to publish and I could not get the information needed to understand what I was supposed to do to obtain the official permission. I would be opposed to rehosting Wikinews here if the policy similarly made it unreasonably difficult for volunteer contributor to get the information needed to meet the journalistic standards imposed by the overworked editors.) {{unsigned|DavidMCEddy}}

===Comments and questions===
:Definitely worthy of discussion, so I have no problem with the proposal in the sitenotice.
:Initial questions:
:* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages?
:* What are &quot;active editions&quot;?
:* How can Wikiversity navigate the concerns that lead to the closure of Wikinews?
:* Are any changes to the scope of Wikinews proposed?
:* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension?
:** e.g., I'm not sure that a page like [[User:BigKrow/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource.
:-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 05:52, 14 May 2026 (UTC)
:* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages?
::*No, not at this time.
:* What are &quot;active editions&quot;?
::*There were 30 other active editions of Wikinews in addition to English (e.g. [[:n:es:]]) at the time of universal closure (2026-05-04).
:* How can Wikiversity navigate the concerns that lead to the closure of Wikinews?
::*One of the biggest issues was the problems with DPL, which is now irrelevant. Another was the lack of activity, which can be ameliorated by having it be part of an existing project instead of its own domain (e.g. some editions of Wikipedia host their own Wikinews already and those projects were not impacted by the closure).
:* Are any changes to the scope of Wikinews proposed?
::*Not at this juncture. I would also propose as far as implemention goes that we would request a new namespace and that the material be more-or-less sequestered into its own ongoing project, like Wikijournal is or like the Cookbook and Wikijunior are at our sister [[:b:]].
:* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension?
:** e.g., I'm not sure that a page like [[Story/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource.
::*The process of citizen journalists practicing their craft in real-time and collaborating with others to do so is itself an education activity. We would essentially be hosting a real-time experiment in citizen journalism, online communities, and collaborative learning in addition to the prospect of spreading educational information from someone actually reading the news. I would propose that we could also make a more deliberate attempt to engage with learning &lt;em&gt;about&lt;/em&gt; what does and doesn't work with collaborative news writing by experimentation (e.g. audio news, syndicating to other sites, incorporating freely-licensed news from other sources, writing hyper-local news, writing briefs versus longer-term reportage) and also seeing if the problems noted in the Task Force report that recommended closure can be overcome. Note that we have already done some local investigation about and learning about wiki-based journalism on Wikinews here at [[Journalism studies and Wikinews]]. We could continue that learning and refine the process, including incorporating journalism students from universities. As for tensions, Wikinews is the only sister project that must be done with a quick turn-around: if you take a long time to [[:s:|transcribe a book]], that's just how long it takes, but if you take a long time to write news, it ceases to be news entirely. Wikiversity has been a very slow-growing project that has definitely had some successes but has generally come together over a long period with most learning resources being individual passion projects (or sometimes, frankly, crankery) which would not work with collaborative news that requires more than just a single editor writing whatever he feels like.
::Please let me know any other questions/concerns and any other editors feel free to give your own perspective. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 06:13, 14 May 2026 (UTC)
:::Thanks, Justin — it is food for thought.
:::In attempting to understand how we've arrived here, I've summarised some of the background on this page: [[Wikinews]].
:::Perhaps it could be helpful to flesh out more of the vision / ideas / possibilities / challenges on that page? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:49, 14 May 2026 (UTC)

:::*Having given it some thought, in principle, I support hosting [[citizen journalism]] on Wikiversity where it is clearly connected to a learning project and/or constitutes original research, both of which align strongly with [[Wikiversity:Mission|Wikiversity’s educational mission]].
:::*My chief concern is the potential for news content that is not clearly linked to the purpose of Wikiversity. To avoid this, some community-agreed guidelines would be prudent. These need not be overly restrictive; they should support boldness and experimentation while helping ensure alignment with Wikiversity's purpose.
:::*Given the reported low and declining activity on Wikinews, it seems unlikely that English Wikiversity would be overwhelmed by an influx of news-related editing. My impression is that English Wikinews was the most active edition, but even so, many contributors are likely to disperse to other projects or cease editing altogether. A modest migration of interested editors to Wikiversity seems manageable.
:::*At this stage, I do not think a dedicated namespace is necessary. Subpages under [[Wikinews]] or nested pages under relevant learning or research projects, or user-space draft pages should be suitable. I agree that [[Wikijournal]] offers a useful model, as do several existing course structures on Wikiversity.
:::*I support [[User:Koavf]]’s suggestions about framing Wikinews activity explicitly around learning. This would create a distinctive space for experimenting with collaborative news production in ways that are pedagogically meaningful. I agree that the [[journalism studies and Wikinews]] project developed by David and Leigh Blackall through the University of Wollongong is an excellent example of the intersection between Wikiversity and Wikinews. The [[Wikinews]] page could evolve into a hub for such projects.
:::*I've tidied the [[:Category:Wikinews|Wikinews category]] and merged some content into the [[Wikinews]] page. As part of a reinvigoration effort, please review these and related resources such as [[:Category:Journalism]] and [[School:Journalism]].
:::*A further argument in favour of this initiative is that Wikipedia explicitly excludes both news reporting and original research. So, there is value in maintaining spaces within the Wikimedia ecosystem where these forms of knowledge production can be openly developed and curated. Such work can, in turn, generate valuable evidence and source material that may later inform Wikipedia articles.
:::*The closure of WMF-hosted Wikinews does not imply that open wiki-based news curation lacks value. Indeed, the closure documentation appears supportive of experimentation with alternative news models across Wikimedia projects, including through Wikipedia and Wikidata. In that context, Wikiversity seems a natural home for a Wikinews experiment, provided it is clearly grounded in learning and/or research.
:::-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:39, 15 May 2026 (UTC)
My understanding towards Wikinews' failure is that everything takes too long to be approved for the publish status, which means that any breaking news would have already become days-old stale news. Wikinews has a brand recognition (for right or wrong reasons) than Wikiversity and I wonder how effective Wikiversity can attract the &quot;Wikinews refugees&quot; to edit here. And just a quick note on the governance. Since each Wikiversity language operates independently, each language has to vote &amp; adopt this proposal independently. [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 13:47, 15 May 2026 (UTC)

:Your assessment about Wikinews is partially correct. I referenced it earlier, but to be explicit, there is a [[:m:Proposal for Closing Wikinews|report by a task force on sister projects]] that outlines their concerns. There are a few, one of which was the nature of the staleness of news. Thanks also for clarifying that this proposal is only relevant to en.wv and is not binding or even proposed for other editions of Wikiversity. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:54, 15 May 2026 (UTC)
*Note: I am not a regular here, and just visit Wikiversity for the WikiJournal project. Challenges of Wikinews included that it required timely reporting and fact-checking processes which differed greatly from the well-established ones in Wikipedia. Here in Wikiversity, there is the WikiJournal project, and that can take some some forms of journalism, just not breaking news reporting. I am in favor of salvaging parts of Wikinews if helpful. Could it, would it be feasible to adapt Wikijournal to accept some forms of news journalism, but just not the timed news reporting? For example, WikiJournal already is doing conference proceedings, and could likely do related event reports even months after the event ended. It could probably accept long-form investigative reporting, which is a sort of news that is not breaking news. I am not sure what the possibilities are, but I would prefer to build up systems that already work rather than import systems which had problems elsewhere. Thanks. [[User:Bluerasberry|&lt;span style=&quot;background:#cedff2;color:#11e&quot;&gt;''' Blue Rasberry '''&lt;/span&gt;]][[User talk:Bluerasberry|&lt;span style=&quot;cursor:help&quot;&gt;&lt;span style=&quot;background:#cedff2;color:#11e&quot;&gt;(talk)&lt;/span&gt;&lt;/span&gt;]] 19:17, 22 May 2026 (UTC)
*:I agree that there are certain kinds of journalism that are perfectly valid and not time-bound like breaking news reporting, so that won't suffer from the issues noted before. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 21:15, 22 May 2026 (UTC)
*::@[[User:Bluerasberry|Bluerasberry]] WikiJournal is not interested in taking on news journalism. WikiJournal is publishing conference proceedings at the request of some Wikimedian educators, and conference proceedings is what a &quot;regular&quot; journal publishes. News journalism is quite different from this, and if WikiJournal starts to deviate towards publishing news journalism, it will create barrier towards future initiatives like being indexed in Medline or Web of Science, and may risk being delisted from Scopus. [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 22:43, 5 June 2026 (UTC)
*:::Thats a good point. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:09, 9 June 2026 (UTC)

== Create an autopatrolled user group? ==

{{tracked|T428269|resolved}}
I would like to propose creating the user group &lt;code&gt;autopatrolled&lt;/code&gt; (autopatrolled user), in which for non-curators and non-custodians, their page creations and file uploads would be automatically marked as patrolled by the MediaWiki software. Custodians may grant the user group, at their discretion, to users who create good quality pages that do not need frequent patrolling.

On a side note, the term {{tq|autopatroller}} would be used, but because we don't have non-curator/custodian patrollers (as we rely on curators and custodians to patrol), I suggest on using the term {{tq|autopatrolled user}}. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:31, 29 May 2026 (UTC)

:'''Support''' re: the name, I don't really understand the reasoning, so I am '''neutral''' on that. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 15:45, 29 May 2026 (UTC)
:: Regarding the name, this is because as we don't have the patroller user group, we rely on curators and custodians to patrol new pages and file uploads. Does that make sense? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:39, 29 May 2026 (UTC)
:::Not really, but I don't think it's the most important thing. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 16:42, 29 May 2026 (UTC)
:::: We'll decide on the name later. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:48, 30 May 2026 (UTC)
:::::Oh, please don't let me stand in the way. I'm just not very smart, so don't hold up a matter on my account. I didn't want to derail the proposal, which is a fine and sensible one. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 04:16, 30 May 2026 (UTC)
: '''Support''' - sounds like a good idea
:* Suggest adding a draft section about this group to [[Wikiversity:Patrolling]]. There is a statement in the Introduction of the page that I'm not sure if its correct and at least could be improved: &quot;Wikiversity also uses an autopatrol right, meaning trusted users' contributions are automatically marked as checked so patrollers can focus on reviewing newer or anonymous editors.&quot;
:* Regarding autopatroller vs autropatrolled user, what terms are used on similar WMF wiki projects?
: -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:28, 30 May 2026 (UTC)
::# I would create a starting page about the user groups, with experienced editors expanding the page. A summarized part of that page would also be added to [[Wikiversity:Patrolling]].
::# For a similar example, English Wikipedia uses the term {{tq|Autopatrolled}}, just that term only.
:: [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:22, 30 May 2026 (UTC)
: @[[User:Jtneill|Jtneill]] and @[[User:Koavf|Koavf]]: the autopatroller user group has been implemented here. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:14, 8 June 2026 (UTC)
::Thanks. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:13, 9 June 2026 (UTC)

== How much of Wikiversity’s content is LLM slop? ==

Because it seems like a non-trivial amount, along with AI slop images as well. Is there some kind of AI cleanup project established yet? [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:20, 4 June 2026 (UTC)

:We have discussed AI but I don't know of any explicit initiative to find and delete AI-generated noise. Individual modules have been deleted for having been made by AI. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 08:50, 4 June 2026 (UTC)
:Recently agreed [[Wikiversity:Artificial intelligence|policy]] welcome users to tag AI generated pages. Me personally I am not against the use of AI. What is the difference in abstract schematic image created by a human and the same by an AI. If the users does not have finances to pay digital artest and you dont want to let them use AI, would you pay the artest for them? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:07, 8 June 2026 (UTC)
::Wikimedia has a lot of ''volunteer'' artists who can illustrate if asked. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:11, 9 June 2026 (UTC)
:::Interesting! That's good to know. Where can we find the volunteer artists for illustrating? [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 20:11, 9 June 2026 (UTC)
::::Wikimedia commons has [[commons:Commons:Graphic Lab/Illustration workshop]] [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 02:18, 10 June 2026 (UTC)

== New user what do I do here ==
I love wikipedia and the wikiversity project seems super interesting. However I know very little about wikiversity and would like to know how i can best contribute to the project. Also if there are forums or discord or reddit that would be very helpful.
(One last thing is it normal that my userboxes don't work here) {{unsigned|AUBSTRAWBS}}
:Hey {{ping|AUBSTRAWBS}} Welcome to Wikiversity! I've left a welcome message on your talk page so that should provide you a plethora of useful links for you to look at so you can familiarize yourself with the project. Also, feel free to create the userboxes you need. Wikiversity doesn't have as many userboxes as Wikipedia. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:45, 8 June 2026 (UTC)

:Thank you very much :) hope to contribute a lot.  [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 21:50, 8 June 2026 (UTC)</text>
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== Proposal to rehost Wikinews here ==

As many of you know, and mentioned here at the Colloquium, our sister project Wikinews recently closed, with all 31 active editions made read-only. [[User:BigKrow]] has asked about the prospect of writing news stories here and I suggested that since we already have [[School:Journalism]] and some resources related to the [[:Category:Journalism|broader topic of journalism]]. I would like to propose that we have continued and indefinite space for {{w|citizen journalism}} by essentially repurposing Wikinews into a sub-project here. The only special infrastructure that Wikinews required was [[:mw:Extension:DynamicPageList]], which was deactivated and caused issues due to a lack of maintenance.

I will add this proposal to the site banner, but I recognize that that may be a conflict of interest, so if anyone requests that I remove it, I will. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:30, 14 May 2026 (UTC)
:I would like to see this conversation go for at least 30 days to establish a consensus. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:35, 14 May 2026 (UTC)
::A few days shy of 30, it seems obvious that this is not going to pass. So I '''withdraw''' as presumptively '''failed'''. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:14, 9 June 2026 (UTC)

===Votes===
*{{support}} as proposer (with BK's inspiration). I think that an ongoing experiment in citizen journalism is a fit and appropriate use of this site. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:35, 14 May 2026 (UTC)
*{{support}}, hope to seeing ideas about this, and thank you @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 11:08, 14 May 2026 (UTC)
*{{support}} Other than perhaps inflating the total number of pages reported, I see the idea of &quot;practicing journalism&quot; a worthy and relevant activity within the domain of Wikiversity. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 21:41, 14 May 2026 (UTC)
*{{support}} Conditional on development of (a) community guidelines that ensure alignment with Wikiversity's purpose, and (b) clear, nested page-naming structures for projects. More detail below. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:48, 15 May 2026 (UTC)
*{{contra}} This proposal doesn't seem interested in expanding educational materials in journalism, but rather in providing space and protection for Wikinews contributors. But this is contrary to the goals of Wikiversity, and I'm not sure it's a good idea, even with regard to WMF. If WMF decides to close a project and another community lets it run on its domain, that's a bit of an undermining of WMF's and the community's decisions. Given that Wikiversity has had several conflicts with other communities and WMF in its history, I'm against it.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:59, 15 May 2026 (UTC)
*{{contra}} This seems like a proposal to continue the mission of WikiNews, but not a proposal specifically to improve Wikiversity. I concur with Juandev's comments. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 20:29, 30 May 2026 (UTC)
* {{oppose}} per above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:05, 1 June 2026 (UTC)
*{{oppose}} Wikiversity isn’t Wikinews and it also isn’t a dumping ground for anything not covered by other projects. It was already suggested, rather bafflingly, that Wikinews parasitize Wikipedia as a host. If it were allowed to freeload off of Wikiversity it would simply promote a view I and likely many others have— that Wikiversity (as it currently exists) has no standards and mostly just exists to host subpar content that wouldn’t be tolerated on any other Wikimedia site. Wikinews needs a new, non-Wikimedia host, and Wikiversity needs to get its act together by enforcing a minimum scope and standard for what it allows. --[[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:16, 4 June 2026 (UTC)
* {{oppose}} per above. Wikiversity&lt;math&gt;\not=&lt;/math&gt; Wikinews - not a good idea to mix the scope of projects. --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 12:03, 8 June 2026 (UTC)
* {{abstain}} I will abstain since I'm not an active Wikiversity contributor. But I just feel like Wikinews had a very clear and specific goal of providing news, and Wikiversity is just a different project with different goals. For me, it would be odd to rehost Wikinews here. But please do not count my vote, this is only a comment. --[[User:Antimundo|Antimundo]] ([[User talk:Antimundo|discuss]] • [[Special:Contributions/Antimundo|contribs]]) 13:19, 6 June 2026 (UTC)
* {{oppose}} Although I think it's a pity that Wikinews is closed. --[[User:Dick Bos|Dick Bos]] ([[User talk:Dick Bos|discuss]] • [[Special:Contributions/Dick Bos|contribs]]) 19:06, 8 June 2026 (UTC)
*{{support}} In 2018 I initiated [[:Category:Videoconferences on media and democracy]] as a platform for disseminating public affairs events. In 2021 I officially initiated a podcast series on &quot;Media &amp; Democracy&quot; syndicated for the [[w:List of Pacifica Radio stations and affiliates|Pacifica radio network]]. In 2024 I converted it from irregular to fortnightly. I think this is all educational and supports the Wikiversity education mission, and I think that &quot;rehost Wikinews here&quot; would be appropriate. (I had some experience with Wikinews a few years ago. I felt it was too tightly controlled: Article submissions went stale, because I could not get official permission to publish and I could not get the information needed to understand what I was supposed to do to obtain the official permission. I would be opposed to rehosting Wikinews here if the policy similarly made it unreasonably difficult for volunteer contributor to get the information needed to meet the journalistic standards imposed by the overworked editors.) {{unsigned|DavidMCEddy}}

===Comments and questions===
:Definitely worthy of discussion, so I have no problem with the proposal in the sitenotice.
:Initial questions:
:* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages?
:* What are &quot;active editions&quot;?
:* How can Wikiversity navigate the concerns that lead to the closure of Wikinews?
:* Are any changes to the scope of Wikinews proposed?
:* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension?
:** e.g., I'm not sure that a page like [[User:BigKrow/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource.
:-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 05:52, 14 May 2026 (UTC)
:* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages?
::*No, not at this time.
:* What are &quot;active editions&quot;?
::*There were 30 other active editions of Wikinews in addition to English (e.g. [[:n:es:]]) at the time of universal closure (2026-05-04).
:* How can Wikiversity navigate the concerns that lead to the closure of Wikinews?
::*One of the biggest issues was the problems with DPL, which is now irrelevant. Another was the lack of activity, which can be ameliorated by having it be part of an existing project instead of its own domain (e.g. some editions of Wikipedia host their own Wikinews already and those projects were not impacted by the closure).
:* Are any changes to the scope of Wikinews proposed?
::*Not at this juncture. I would also propose as far as implemention goes that we would request a new namespace and that the material be more-or-less sequestered into its own ongoing project, like Wikijournal is or like the Cookbook and Wikijunior are at our sister [[:b:]].
:* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension?
:** e.g., I'm not sure that a page like [[Story/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource.
::*The process of citizen journalists practicing their craft in real-time and collaborating with others to do so is itself an education activity. We would essentially be hosting a real-time experiment in citizen journalism, online communities, and collaborative learning in addition to the prospect of spreading educational information from someone actually reading the news. I would propose that we could also make a more deliberate attempt to engage with learning &lt;em&gt;about&lt;/em&gt; what does and doesn't work with collaborative news writing by experimentation (e.g. audio news, syndicating to other sites, incorporating freely-licensed news from other sources, writing hyper-local news, writing briefs versus longer-term reportage) and also seeing if the problems noted in the Task Force report that recommended closure can be overcome. Note that we have already done some local investigation about and learning about wiki-based journalism on Wikinews here at [[Journalism studies and Wikinews]]. We could continue that learning and refine the process, including incorporating journalism students from universities. As for tensions, Wikinews is the only sister project that must be done with a quick turn-around: if you take a long time to [[:s:|transcribe a book]], that's just how long it takes, but if you take a long time to write news, it ceases to be news entirely. Wikiversity has been a very slow-growing project that has definitely had some successes but has generally come together over a long period with most learning resources being individual passion projects (or sometimes, frankly, crankery) which would not work with collaborative news that requires more than just a single editor writing whatever he feels like.
::Please let me know any other questions/concerns and any other editors feel free to give your own perspective. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 06:13, 14 May 2026 (UTC)
:::Thanks, Justin — it is food for thought.
:::In attempting to understand how we've arrived here, I've summarised some of the background on this page: [[Wikinews]].
:::Perhaps it could be helpful to flesh out more of the vision / ideas / possibilities / challenges on that page? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:49, 14 May 2026 (UTC)

:::*Having given it some thought, in principle, I support hosting [[citizen journalism]] on Wikiversity where it is clearly connected to a learning project and/or constitutes original research, both of which align strongly with [[Wikiversity:Mission|Wikiversity’s educational mission]].
:::*My chief concern is the potential for news content that is not clearly linked to the purpose of Wikiversity. To avoid this, some community-agreed guidelines would be prudent. These need not be overly restrictive; they should support boldness and experimentation while helping ensure alignment with Wikiversity's purpose.
:::*Given the reported low and declining activity on Wikinews, it seems unlikely that English Wikiversity would be overwhelmed by an influx of news-related editing. My impression is that English Wikinews was the most active edition, but even so, many contributors are likely to disperse to other projects or cease editing altogether. A modest migration of interested editors to Wikiversity seems manageable.
:::*At this stage, I do not think a dedicated namespace is necessary. Subpages under [[Wikinews]] or nested pages under relevant learning or research projects, or user-space draft pages should be suitable. I agree that [[Wikijournal]] offers a useful model, as do several existing course structures on Wikiversity.
:::*I support [[User:Koavf]]’s suggestions about framing Wikinews activity explicitly around learning. This would create a distinctive space for experimenting with collaborative news production in ways that are pedagogically meaningful. I agree that the [[journalism studies and Wikinews]] project developed by David and Leigh Blackall through the University of Wollongong is an excellent example of the intersection between Wikiversity and Wikinews. The [[Wikinews]] page could evolve into a hub for such projects.
:::*I've tidied the [[:Category:Wikinews|Wikinews category]] and merged some content into the [[Wikinews]] page. As part of a reinvigoration effort, please review these and related resources such as [[:Category:Journalism]] and [[School:Journalism]].
:::*A further argument in favour of this initiative is that Wikipedia explicitly excludes both news reporting and original research. So, there is value in maintaining spaces within the Wikimedia ecosystem where these forms of knowledge production can be openly developed and curated. Such work can, in turn, generate valuable evidence and source material that may later inform Wikipedia articles.
:::*The closure of WMF-hosted Wikinews does not imply that open wiki-based news curation lacks value. Indeed, the closure documentation appears supportive of experimentation with alternative news models across Wikimedia projects, including through Wikipedia and Wikidata. In that context, Wikiversity seems a natural home for a Wikinews experiment, provided it is clearly grounded in learning and/or research.
:::-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:39, 15 May 2026 (UTC)
My understanding towards Wikinews' failure is that everything takes too long to be approved for the publish status, which means that any breaking news would have already become days-old stale news. Wikinews has a brand recognition (for right or wrong reasons) than Wikiversity and I wonder how effective Wikiversity can attract the &quot;Wikinews refugees&quot; to edit here. And just a quick note on the governance. Since each Wikiversity language operates independently, each language has to vote &amp; adopt this proposal independently. [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 13:47, 15 May 2026 (UTC)

:Your assessment about Wikinews is partially correct. I referenced it earlier, but to be explicit, there is a [[:m:Proposal for Closing Wikinews|report by a task force on sister projects]] that outlines their concerns. There are a few, one of which was the nature of the staleness of news. Thanks also for clarifying that this proposal is only relevant to en.wv and is not binding or even proposed for other editions of Wikiversity. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:54, 15 May 2026 (UTC)
*Note: I am not a regular here, and just visit Wikiversity for the WikiJournal project. Challenges of Wikinews included that it required timely reporting and fact-checking processes which differed greatly from the well-established ones in Wikipedia. Here in Wikiversity, there is the WikiJournal project, and that can take some some forms of journalism, just not breaking news reporting. I am in favor of salvaging parts of Wikinews if helpful. Could it, would it be feasible to adapt Wikijournal to accept some forms of news journalism, but just not the timed news reporting? For example, WikiJournal already is doing conference proceedings, and could likely do related event reports even months after the event ended. It could probably accept long-form investigative reporting, which is a sort of news that is not breaking news. I am not sure what the possibilities are, but I would prefer to build up systems that already work rather than import systems which had problems elsewhere. Thanks. [[User:Bluerasberry|&lt;span style=&quot;background:#cedff2;color:#11e&quot;&gt;''' Blue Rasberry '''&lt;/span&gt;]][[User talk:Bluerasberry|&lt;span style=&quot;cursor:help&quot;&gt;&lt;span style=&quot;background:#cedff2;color:#11e&quot;&gt;(talk)&lt;/span&gt;&lt;/span&gt;]] 19:17, 22 May 2026 (UTC)
*:I agree that there are certain kinds of journalism that are perfectly valid and not time-bound like breaking news reporting, so that won't suffer from the issues noted before. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 21:15, 22 May 2026 (UTC)
*::@[[User:Bluerasberry|Bluerasberry]] WikiJournal is not interested in taking on news journalism. WikiJournal is publishing conference proceedings at the request of some Wikimedian educators, and conference proceedings is what a &quot;regular&quot; journal publishes. News journalism is quite different from this, and if WikiJournal starts to deviate towards publishing news journalism, it will create barrier towards future initiatives like being indexed in Medline or Web of Science, and may risk being delisted from Scopus. [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 22:43, 5 June 2026 (UTC)
*:::Thats a good point. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:09, 9 June 2026 (UTC)

== Create an autopatrolled user group? ==

{{tracked|T428269|resolved}}
I would like to propose creating the user group &lt;code&gt;autopatrolled&lt;/code&gt; (autopatrolled user), in which for non-curators and non-custodians, their page creations and file uploads would be automatically marked as patrolled by the MediaWiki software. Custodians may grant the user group, at their discretion, to users who create good quality pages that do not need frequent patrolling.

On a side note, the term {{tq|autopatroller}} would be used, but because we don't have non-curator/custodian patrollers (as we rely on curators and custodians to patrol), I suggest on using the term {{tq|autopatrolled user}}. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:31, 29 May 2026 (UTC)

:'''Support''' re: the name, I don't really understand the reasoning, so I am '''neutral''' on that. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 15:45, 29 May 2026 (UTC)
:: Regarding the name, this is because as we don't have the patroller user group, we rely on curators and custodians to patrol new pages and file uploads. Does that make sense? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:39, 29 May 2026 (UTC)
:::Not really, but I don't think it's the most important thing. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 16:42, 29 May 2026 (UTC)
:::: We'll decide on the name later. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:48, 30 May 2026 (UTC)
:::::Oh, please don't let me stand in the way. I'm just not very smart, so don't hold up a matter on my account. I didn't want to derail the proposal, which is a fine and sensible one. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 04:16, 30 May 2026 (UTC)
: '''Support''' - sounds like a good idea
:* Suggest adding a draft section about this group to [[Wikiversity:Patrolling]]. There is a statement in the Introduction of the page that I'm not sure if its correct and at least could be improved: &quot;Wikiversity also uses an autopatrol right, meaning trusted users' contributions are automatically marked as checked so patrollers can focus on reviewing newer or anonymous editors.&quot;
:* Regarding autopatroller vs autropatrolled user, what terms are used on similar WMF wiki projects?
: -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:28, 30 May 2026 (UTC)
::# I would create a starting page about the user groups, with experienced editors expanding the page. A summarized part of that page would also be added to [[Wikiversity:Patrolling]].
::# For a similar example, English Wikipedia uses the term {{tq|Autopatrolled}}, just that term only.
:: [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:22, 30 May 2026 (UTC)
: @[[User:Jtneill|Jtneill]] and @[[User:Koavf|Koavf]]: the autopatroller user group has been implemented here. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:14, 8 June 2026 (UTC)
::Thanks. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:13, 9 June 2026 (UTC)

== How much of Wikiversity’s content is LLM slop? ==

Because it seems like a non-trivial amount, along with AI slop images as well. Is there some kind of AI cleanup project established yet? [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:20, 4 June 2026 (UTC)

:We have discussed AI but I don't know of any explicit initiative to find and delete AI-generated noise. Individual modules have been deleted for having been made by AI. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 08:50, 4 June 2026 (UTC)
:Recently agreed [[Wikiversity:Artificial intelligence|policy]] welcome users to tag AI generated pages. Me personally I am not against the use of AI. What is the difference in abstract schematic image created by a human and the same by an AI. If the users does not have finances to pay digital artest and you dont want to let them use AI, would you pay the artest for them? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:07, 8 June 2026 (UTC)
::Wikimedia has a lot of ''volunteer'' artists who can illustrate if asked. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:11, 9 June 2026 (UTC)
:::Interesting! That's good to know. Where can we find the volunteer artists for illustrating? [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 20:11, 9 June 2026 (UTC)
::::Wikimedia commons has [[commons:Commons:Graphic Lab/Illustration workshop]] [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 02:18, 10 June 2026 (UTC)

== New user what do I do here ==
I love wikipedia and the wikiversity project seems super interesting. However I know very little about wikiversity and would like to know how i can best contribute to the project. Also if there are forums or discord or reddit that would be very helpful.
(One last thing is it normal that my userboxes don't work here) {{unsigned|AUBSTRAWBS}}
:Hey {{ping|AUBSTRAWBS}} Welcome to Wikiversity! I've left a welcome message on your talk page so that should provide you a plethora of useful links for you to look at so you can familiarize yourself with the project. Also, feel free to create the userboxes you need. Wikiversity doesn't have as many userboxes as Wikipedia. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:45, 8 June 2026 (UTC)

:Thank you very much :) hope to contribute a lot.  [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 21:50, 8 June 2026 (UTC)

== Deployment of Legal and Safety Contacts Link in the Footer of Your Wiki ==

Hello community,

The Wikimedia Foundation has provided [[foundation:Legal:Wikimedia Foundation Legal and Safety Contact Information|a single legal and safety contact page]], to be linked in the footer of your wiki, to ensure access to accurate legal information. This is a regulatory requirement.

We have already rolled out links to English, German, Italian, Spanish Wikipedias and other wikis and we will deploy to your wiki soon.

Please [[m:Wikimedia Foundation Legal and Safety Contacts FAQ|read more on the project page]] and leave any comments in this thread or on [[m:Talk:Wikimedia Foundation Legal and Safety Contacts FAQ|the talk page]]. –– [[User:STei (WMF)|STei (WMF)]] ([[User talk:STei (WMF)|discuss]] • [[Special:Contributions/STei (WMF)|contribs]]) 18:12, 9 June 2026 (UTC)

:Thanks for the notice. In case anyone is not clear, we cannot locally change the text at the footer, as it [[:mw:Manual:Footer|requires access to the server settings]]. If we locally needed to change it, we would have to file a ticket at [[:phab:]]. Since the above was sent by someone from the WMF, I think they are on it and it will be updated without any action from anyone here. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:24, 9 June 2026 (UTC)</text>
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      <id>2819786</id>
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== Proposal to rehost Wikinews here ==

As many of you know, and mentioned here at the Colloquium, our sister project Wikinews recently closed, with all 31 active editions made read-only. [[User:BigKrow]] has asked about the prospect of writing news stories here and I suggested that since we already have [[School:Journalism]] and some resources related to the [[:Category:Journalism|broader topic of journalism]]. I would like to propose that we have continued and indefinite space for {{w|citizen journalism}} by essentially repurposing Wikinews into a sub-project here. The only special infrastructure that Wikinews required was [[:mw:Extension:DynamicPageList]], which was deactivated and caused issues due to a lack of maintenance.

I will add this proposal to the site banner, but I recognize that that may be a conflict of interest, so if anyone requests that I remove it, I will. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:30, 14 May 2026 (UTC)
:I would like to see this conversation go for at least 30 days to establish a consensus. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:35, 14 May 2026 (UTC)
::A few days shy of 30, it seems obvious that this is not going to pass. So I '''withdraw''' as presumptively '''failed'''. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:14, 9 June 2026 (UTC)

===Votes===
*{{support}} as proposer (with BK's inspiration). I think that an ongoing experiment in citizen journalism is a fit and appropriate use of this site. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:35, 14 May 2026 (UTC)
*{{support}}, hope to seeing ideas about this, and thank you @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 11:08, 14 May 2026 (UTC)
*{{support}} Other than perhaps inflating the total number of pages reported, I see the idea of &quot;practicing journalism&quot; a worthy and relevant activity within the domain of Wikiversity. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 21:41, 14 May 2026 (UTC)
*{{support}} Conditional on development of (a) community guidelines that ensure alignment with Wikiversity's purpose, and (b) clear, nested page-naming structures for projects. More detail below. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:48, 15 May 2026 (UTC)
*{{contra}} This proposal doesn't seem interested in expanding educational materials in journalism, but rather in providing space and protection for Wikinews contributors. But this is contrary to the goals of Wikiversity, and I'm not sure it's a good idea, even with regard to WMF. If WMF decides to close a project and another community lets it run on its domain, that's a bit of an undermining of WMF's and the community's decisions. Given that Wikiversity has had several conflicts with other communities and WMF in its history, I'm against it.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:59, 15 May 2026 (UTC)
*{{contra}} This seems like a proposal to continue the mission of WikiNews, but not a proposal specifically to improve Wikiversity. I concur with Juandev's comments. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 20:29, 30 May 2026 (UTC)
* {{oppose}} per above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:05, 1 June 2026 (UTC)
*{{oppose}} Wikiversity isn’t Wikinews and it also isn’t a dumping ground for anything not covered by other projects. It was already suggested, rather bafflingly, that Wikinews parasitize Wikipedia as a host. If it were allowed to freeload off of Wikiversity it would simply promote a view I and likely many others have— that Wikiversity (as it currently exists) has no standards and mostly just exists to host subpar content that wouldn’t be tolerated on any other Wikimedia site. Wikinews needs a new, non-Wikimedia host, and Wikiversity needs to get its act together by enforcing a minimum scope and standard for what it allows. --[[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:16, 4 June 2026 (UTC)
* {{oppose}} per above. Wikiversity&lt;math&gt;\not=&lt;/math&gt; Wikinews - not a good idea to mix the scope of projects. --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 12:03, 8 June 2026 (UTC)
* {{abstain}} I will abstain since I'm not an active Wikiversity contributor. But I just feel like Wikinews had a very clear and specific goal of providing news, and Wikiversity is just a different project with different goals. For me, it would be odd to rehost Wikinews here. But please do not count my vote, this is only a comment. --[[User:Antimundo|Antimundo]] ([[User talk:Antimundo|discuss]] • [[Special:Contributions/Antimundo|contribs]]) 13:19, 6 June 2026 (UTC)
* {{oppose}} Although I think it's a pity that Wikinews is closed. --[[User:Dick Bos|Dick Bos]] ([[User talk:Dick Bos|discuss]] • [[Special:Contributions/Dick Bos|contribs]]) 19:06, 8 June 2026 (UTC)
*{{support}} In 2018 I initiated [[:Category:Videoconferences on media and democracy]] as a platform for disseminating public affairs events. In 2021 I officially initiated a podcast series on &quot;Media &amp; Democracy&quot; syndicated for the [[w:List of Pacifica Radio stations and affiliates|Pacifica radio network]]. In 2024 I converted it from irregular to fortnightly. I think this is all educational and supports the Wikiversity education mission, and I think that &quot;rehost Wikinews here&quot; would be appropriate. (I had some experience with Wikinews a few years ago. I felt it was too tightly controlled: Article submissions went stale, because I could not get official permission to publish and I could not get the information needed to understand what I was supposed to do to obtain the official permission. I would be opposed to rehosting Wikinews here if the policy similarly made it unreasonably difficult for volunteer contributor to get the information needed to meet the journalistic standards imposed by the overworked editors.) {{unsigned|DavidMCEddy}}

===Comments and questions===
:Definitely worthy of discussion, so I have no problem with the proposal in the sitenotice.
:Initial questions:
:* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages?
:* What are &quot;active editions&quot;?
:* How can Wikiversity navigate the concerns that lead to the closure of Wikinews?
:* Are any changes to the scope of Wikinews proposed?
:* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension?
:** e.g., I'm not sure that a page like [[User:BigKrow/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource.
:-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 05:52, 14 May 2026 (UTC)
:* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages?
::*No, not at this time.
:* What are &quot;active editions&quot;?
::*There were 30 other active editions of Wikinews in addition to English (e.g. [[:n:es:]]) at the time of universal closure (2026-05-04).
:* How can Wikiversity navigate the concerns that lead to the closure of Wikinews?
::*One of the biggest issues was the problems with DPL, which is now irrelevant. Another was the lack of activity, which can be ameliorated by having it be part of an existing project instead of its own domain (e.g. some editions of Wikipedia host their own Wikinews already and those projects were not impacted by the closure).
:* Are any changes to the scope of Wikinews proposed?
::*Not at this juncture. I would also propose as far as implemention goes that we would request a new namespace and that the material be more-or-less sequestered into its own ongoing project, like Wikijournal is or like the Cookbook and Wikijunior are at our sister [[:b:]].
:* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension?
:** e.g., I'm not sure that a page like [[Story/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource.
::*The process of citizen journalists practicing their craft in real-time and collaborating with others to do so is itself an education activity. We would essentially be hosting a real-time experiment in citizen journalism, online communities, and collaborative learning in addition to the prospect of spreading educational information from someone actually reading the news. I would propose that we could also make a more deliberate attempt to engage with learning &lt;em&gt;about&lt;/em&gt; what does and doesn't work with collaborative news writing by experimentation (e.g. audio news, syndicating to other sites, incorporating freely-licensed news from other sources, writing hyper-local news, writing briefs versus longer-term reportage) and also seeing if the problems noted in the Task Force report that recommended closure can be overcome. Note that we have already done some local investigation about and learning about wiki-based journalism on Wikinews here at [[Journalism studies and Wikinews]]. We could continue that learning and refine the process, including incorporating journalism students from universities. As for tensions, Wikinews is the only sister project that must be done with a quick turn-around: if you take a long time to [[:s:|transcribe a book]], that's just how long it takes, but if you take a long time to write news, it ceases to be news entirely. Wikiversity has been a very slow-growing project that has definitely had some successes but has generally come together over a long period with most learning resources being individual passion projects (or sometimes, frankly, crankery) which would not work with collaborative news that requires more than just a single editor writing whatever he feels like.
::Please let me know any other questions/concerns and any other editors feel free to give your own perspective. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 06:13, 14 May 2026 (UTC)
:::Thanks, Justin — it is food for thought.
:::In attempting to understand how we've arrived here, I've summarised some of the background on this page: [[Wikinews]].
:::Perhaps it could be helpful to flesh out more of the vision / ideas / possibilities / challenges on that page? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:49, 14 May 2026 (UTC)

:::*Having given it some thought, in principle, I support hosting [[citizen journalism]] on Wikiversity where it is clearly connected to a learning project and/or constitutes original research, both of which align strongly with [[Wikiversity:Mission|Wikiversity’s educational mission]].
:::*My chief concern is the potential for news content that is not clearly linked to the purpose of Wikiversity. To avoid this, some community-agreed guidelines would be prudent. These need not be overly restrictive; they should support boldness and experimentation while helping ensure alignment with Wikiversity's purpose.
:::*Given the reported low and declining activity on Wikinews, it seems unlikely that English Wikiversity would be overwhelmed by an influx of news-related editing. My impression is that English Wikinews was the most active edition, but even so, many contributors are likely to disperse to other projects or cease editing altogether. A modest migration of interested editors to Wikiversity seems manageable.
:::*At this stage, I do not think a dedicated namespace is necessary. Subpages under [[Wikinews]] or nested pages under relevant learning or research projects, or user-space draft pages should be suitable. I agree that [[Wikijournal]] offers a useful model, as do several existing course structures on Wikiversity.
:::*I support [[User:Koavf]]’s suggestions about framing Wikinews activity explicitly around learning. This would create a distinctive space for experimenting with collaborative news production in ways that are pedagogically meaningful. I agree that the [[journalism studies and Wikinews]] project developed by David and Leigh Blackall through the University of Wollongong is an excellent example of the intersection between Wikiversity and Wikinews. The [[Wikinews]] page could evolve into a hub for such projects.
:::*I've tidied the [[:Category:Wikinews|Wikinews category]] and merged some content into the [[Wikinews]] page. As part of a reinvigoration effort, please review these and related resources such as [[:Category:Journalism]] and [[School:Journalism]].
:::*A further argument in favour of this initiative is that Wikipedia explicitly excludes both news reporting and original research. So, there is value in maintaining spaces within the Wikimedia ecosystem where these forms of knowledge production can be openly developed and curated. Such work can, in turn, generate valuable evidence and source material that may later inform Wikipedia articles.
:::*The closure of WMF-hosted Wikinews does not imply that open wiki-based news curation lacks value. Indeed, the closure documentation appears supportive of experimentation with alternative news models across Wikimedia projects, including through Wikipedia and Wikidata. In that context, Wikiversity seems a natural home for a Wikinews experiment, provided it is clearly grounded in learning and/or research.
:::-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:39, 15 May 2026 (UTC)
My understanding towards Wikinews' failure is that everything takes too long to be approved for the publish status, which means that any breaking news would have already become days-old stale news. Wikinews has a brand recognition (for right or wrong reasons) than Wikiversity and I wonder how effective Wikiversity can attract the &quot;Wikinews refugees&quot; to edit here. And just a quick note on the governance. Since each Wikiversity language operates independently, each language has to vote &amp; adopt this proposal independently. [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 13:47, 15 May 2026 (UTC)

:Your assessment about Wikinews is partially correct. I referenced it earlier, but to be explicit, there is a [[:m:Proposal for Closing Wikinews|report by a task force on sister projects]] that outlines their concerns. There are a few, one of which was the nature of the staleness of news. Thanks also for clarifying that this proposal is only relevant to en.wv and is not binding or even proposed for other editions of Wikiversity. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:54, 15 May 2026 (UTC)
*Note: I am not a regular here, and just visit Wikiversity for the WikiJournal project. Challenges of Wikinews included that it required timely reporting and fact-checking processes which differed greatly from the well-established ones in Wikipedia. Here in Wikiversity, there is the WikiJournal project, and that can take some some forms of journalism, just not breaking news reporting. I am in favor of salvaging parts of Wikinews if helpful. Could it, would it be feasible to adapt Wikijournal to accept some forms of news journalism, but just not the timed news reporting? For example, WikiJournal already is doing conference proceedings, and could likely do related event reports even months after the event ended. It could probably accept long-form investigative reporting, which is a sort of news that is not breaking news. I am not sure what the possibilities are, but I would prefer to build up systems that already work rather than import systems which had problems elsewhere. Thanks. [[User:Bluerasberry|&lt;span style=&quot;background:#cedff2;color:#11e&quot;&gt;''' Blue Rasberry '''&lt;/span&gt;]][[User talk:Bluerasberry|&lt;span style=&quot;cursor:help&quot;&gt;&lt;span style=&quot;background:#cedff2;color:#11e&quot;&gt;(talk)&lt;/span&gt;&lt;/span&gt;]] 19:17, 22 May 2026 (UTC)
*:I agree that there are certain kinds of journalism that are perfectly valid and not time-bound like breaking news reporting, so that won't suffer from the issues noted before. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 21:15, 22 May 2026 (UTC)
*::@[[User:Bluerasberry|Bluerasberry]] WikiJournal is not interested in taking on news journalism. WikiJournal is publishing conference proceedings at the request of some Wikimedian educators, and conference proceedings is what a &quot;regular&quot; journal publishes. News journalism is quite different from this, and if WikiJournal starts to deviate towards publishing news journalism, it will create barrier towards future initiatives like being indexed in Medline or Web of Science, and may risk being delisted from Scopus. [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 22:43, 5 June 2026 (UTC)
*:::Thats a good point. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:09, 9 June 2026 (UTC)

== Create an autopatrolled user group? ==

{{tracked|T428269|resolved}}
I would like to propose creating the user group &lt;code&gt;autopatrolled&lt;/code&gt; (autopatrolled user), in which for non-curators and non-custodians, their page creations and file uploads would be automatically marked as patrolled by the MediaWiki software. Custodians may grant the user group, at their discretion, to users who create good quality pages that do not need frequent patrolling.

On a side note, the term {{tq|autopatroller}} would be used, but because we don't have non-curator/custodian patrollers (as we rely on curators and custodians to patrol), I suggest on using the term {{tq|autopatrolled user}}. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:31, 29 May 2026 (UTC)

:'''Support''' re: the name, I don't really understand the reasoning, so I am '''neutral''' on that. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 15:45, 29 May 2026 (UTC)
:: Regarding the name, this is because as we don't have the patroller user group, we rely on curators and custodians to patrol new pages and file uploads. Does that make sense? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:39, 29 May 2026 (UTC)
:::Not really, but I don't think it's the most important thing. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 16:42, 29 May 2026 (UTC)
:::: We'll decide on the name later. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:48, 30 May 2026 (UTC)
:::::Oh, please don't let me stand in the way. I'm just not very smart, so don't hold up a matter on my account. I didn't want to derail the proposal, which is a fine and sensible one. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 04:16, 30 May 2026 (UTC)
: '''Support''' - sounds like a good idea
:* Suggest adding a draft section about this group to [[Wikiversity:Patrolling]]. There is a statement in the Introduction of the page that I'm not sure if its correct and at least could be improved: &quot;Wikiversity also uses an autopatrol right, meaning trusted users' contributions are automatically marked as checked so patrollers can focus on reviewing newer or anonymous editors.&quot;
:* Regarding autopatroller vs autropatrolled user, what terms are used on similar WMF wiki projects?
: -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:28, 30 May 2026 (UTC)
::# I would create a starting page about the user groups, with experienced editors expanding the page. A summarized part of that page would also be added to [[Wikiversity:Patrolling]].
::# For a similar example, English Wikipedia uses the term {{tq|Autopatrolled}}, just that term only.
:: [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:22, 30 May 2026 (UTC)
: @[[User:Jtneill|Jtneill]] and @[[User:Koavf|Koavf]]: the autopatroller user group has been implemented here. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:14, 8 June 2026 (UTC)
::Thanks. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:13, 9 June 2026 (UTC)

== How much of Wikiversity’s content is LLM slop? ==

Because it seems like a non-trivial amount, along with AI slop images as well. Is there some kind of AI cleanup project established yet? [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:20, 4 June 2026 (UTC)

:We have discussed AI but I don't know of any explicit initiative to find and delete AI-generated noise. Individual modules have been deleted for having been made by AI. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 08:50, 4 June 2026 (UTC)
:Recently agreed [[Wikiversity:Artificial intelligence|policy]] welcome users to tag AI generated pages. Me personally I am not against the use of AI. What is the difference in abstract schematic image created by a human and the same by an AI. If the users does not have finances to pay digital artest and you dont want to let them use AI, would you pay the artest for them? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:07, 8 June 2026 (UTC)
::Wikimedia has a lot of ''volunteer'' artists who can illustrate if asked. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:11, 9 June 2026 (UTC)
:::Interesting! That's good to know. Where can we find the volunteer artists for illustrating? [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 20:11, 9 June 2026 (UTC)
::::Wikimedia commons has [[commons:Commons:Graphic Lab/Illustration workshop]] [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 02:18, 10 June 2026 (UTC)

== New user what do I do here ==
I love wikipedia and the wikiversity project seems super interesting. However I know very little about wikiversity and would like to know how i can best contribute to the project. Also if there are forums or discord or reddit that would be very helpful.
(One last thing is it normal that my userboxes don't work here) {{unsigned|AUBSTRAWBS}}
:Hey {{ping|AUBSTRAWBS}} Welcome to Wikiversity! I've left a welcome message on your talk page so that should provide you a plethora of useful links for you to look at so you can familiarize yourself with the project. Also, feel free to create the userboxes you need. Wikiversity doesn't have as many userboxes as Wikipedia. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:45, 8 June 2026 (UTC)

:Thank you very much :) hope to contribute a lot.  [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 21:50, 8 June 2026 (UTC)

== Deployment of Legal and Safety Contacts Link in the Footer of Your Wiki ==

Hello community,

The Wikimedia Foundation has provided [[foundation:Legal:Wikimedia Foundation Legal and Safety Contact Information|a single legal and safety contact page]], to be linked in the footer of your wiki, to ensure access to accurate legal information. This is a regulatory requirement.

We have already rolled out links to English, German, Italian, Spanish Wikipedias and other wikis and we will deploy to your wiki soon.

Please [[m:Wikimedia Foundation Legal and Safety Contacts FAQ|read more on the project page]] and leave any comments in this thread or on [[m:Talk:Wikimedia Foundation Legal and Safety Contacts FAQ|the talk page]]. –– [[User:STei (WMF)|STei (WMF)]] ([[User talk:STei (WMF)|discuss]] • [[Special:Contributions/STei (WMF)|contribs]]) 18:12, 9 June 2026 (UTC)

:Thanks for the notice. In case anyone is not clear, we cannot locally change the text at the footer, as it [[:mw:Manual:Footer|requires access to the server settings]]. If we locally needed to change it, we would have to file a ticket at [[:phab:]]. Since the above was sent by someone from the WMF, I think they are on it and it will be updated without any action from anyone here. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:24, 9 June 2026 (UTC)

== Image not displaying ==

Can anyone work out why this image isn't displaying?&lt;br&gt;
[[Educational Media Awareness Campaign/Physics/POTD 10]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:45, 11 June 2026 (UTC)

:Not sure, but it was an issue with the file itself and either way, it should be (and I have since done this) replaced with the SVG [[:File:Telescope-schematic.svg]]. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 13:59, 11 June 2026 (UTC)</text>
      <sha1>jwyugbfrqqi58k542g5rk7mb7jjs36o</sha1>
    </revision>
    <revision>
      <id>2819788</id>
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      <timestamp>2026-07-30T04:49:29Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>/* June 2026 Wikimedia Café meetups regarding the English Wikipedia Editor Reflections project */ archive from [[Wikiversity:Colloquium]] ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]])</comment>
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      <text bytes="29785" sha1="31qjhfa14an24l6qyw6w881ba3skpm7" xml:space="preserve">{{archive}}

== Proposal to rehost Wikinews here ==

As many of you know, and mentioned here at the Colloquium, our sister project Wikinews recently closed, with all 31 active editions made read-only. [[User:BigKrow]] has asked about the prospect of writing news stories here and I suggested that since we already have [[School:Journalism]] and some resources related to the [[:Category:Journalism|broader topic of journalism]]. I would like to propose that we have continued and indefinite space for {{w|citizen journalism}} by essentially repurposing Wikinews into a sub-project here. The only special infrastructure that Wikinews required was [[:mw:Extension:DynamicPageList]], which was deactivated and caused issues due to a lack of maintenance.

I will add this proposal to the site banner, but I recognize that that may be a conflict of interest, so if anyone requests that I remove it, I will. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:30, 14 May 2026 (UTC)
:I would like to see this conversation go for at least 30 days to establish a consensus. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:35, 14 May 2026 (UTC)
::A few days shy of 30, it seems obvious that this is not going to pass. So I '''withdraw''' as presumptively '''failed'''. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:14, 9 June 2026 (UTC)

===Votes===
*{{support}} as proposer (with BK's inspiration). I think that an ongoing experiment in citizen journalism is a fit and appropriate use of this site. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:35, 14 May 2026 (UTC)
*{{support}}, hope to seeing ideas about this, and thank you @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 11:08, 14 May 2026 (UTC)
*{{support}} Other than perhaps inflating the total number of pages reported, I see the idea of &quot;practicing journalism&quot; a worthy and relevant activity within the domain of Wikiversity. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 21:41, 14 May 2026 (UTC)
*{{support}} Conditional on development of (a) community guidelines that ensure alignment with Wikiversity's purpose, and (b) clear, nested page-naming structures for projects. More detail below. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:48, 15 May 2026 (UTC)
*{{contra}} This proposal doesn't seem interested in expanding educational materials in journalism, but rather in providing space and protection for Wikinews contributors. But this is contrary to the goals of Wikiversity, and I'm not sure it's a good idea, even with regard to WMF. If WMF decides to close a project and another community lets it run on its domain, that's a bit of an undermining of WMF's and the community's decisions. Given that Wikiversity has had several conflicts with other communities and WMF in its history, I'm against it.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:59, 15 May 2026 (UTC)
*{{contra}} This seems like a proposal to continue the mission of WikiNews, but not a proposal specifically to improve Wikiversity. I concur with Juandev's comments. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 20:29, 30 May 2026 (UTC)
* {{oppose}} per above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:05, 1 June 2026 (UTC)
*{{oppose}} Wikiversity isn’t Wikinews and it also isn’t a dumping ground for anything not covered by other projects. It was already suggested, rather bafflingly, that Wikinews parasitize Wikipedia as a host. If it were allowed to freeload off of Wikiversity it would simply promote a view I and likely many others have— that Wikiversity (as it currently exists) has no standards and mostly just exists to host subpar content that wouldn’t be tolerated on any other Wikimedia site. Wikinews needs a new, non-Wikimedia host, and Wikiversity needs to get its act together by enforcing a minimum scope and standard for what it allows. --[[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:16, 4 June 2026 (UTC)
* {{oppose}} per above. Wikiversity&lt;math&gt;\not=&lt;/math&gt; Wikinews - not a good idea to mix the scope of projects. --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 12:03, 8 June 2026 (UTC)
* {{abstain}} I will abstain since I'm not an active Wikiversity contributor. But I just feel like Wikinews had a very clear and specific goal of providing news, and Wikiversity is just a different project with different goals. For me, it would be odd to rehost Wikinews here. But please do not count my vote, this is only a comment. --[[User:Antimundo|Antimundo]] ([[User talk:Antimundo|discuss]] • [[Special:Contributions/Antimundo|contribs]]) 13:19, 6 June 2026 (UTC)
* {{oppose}} Although I think it's a pity that Wikinews is closed. --[[User:Dick Bos|Dick Bos]] ([[User talk:Dick Bos|discuss]] • [[Special:Contributions/Dick Bos|contribs]]) 19:06, 8 June 2026 (UTC)
*{{support}} In 2018 I initiated [[:Category:Videoconferences on media and democracy]] as a platform for disseminating public affairs events. In 2021 I officially initiated a podcast series on &quot;Media &amp; Democracy&quot; syndicated for the [[w:List of Pacifica Radio stations and affiliates|Pacifica radio network]]. In 2024 I converted it from irregular to fortnightly. I think this is all educational and supports the Wikiversity education mission, and I think that &quot;rehost Wikinews here&quot; would be appropriate. (I had some experience with Wikinews a few years ago. I felt it was too tightly controlled: Article submissions went stale, because I could not get official permission to publish and I could not get the information needed to understand what I was supposed to do to obtain the official permission. I would be opposed to rehosting Wikinews here if the policy similarly made it unreasonably difficult for volunteer contributor to get the information needed to meet the journalistic standards imposed by the overworked editors.) {{unsigned|DavidMCEddy}}

===Comments and questions===
:Definitely worthy of discussion, so I have no problem with the proposal in the sitenotice.
:Initial questions:
:* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages?
:* What are &quot;active editions&quot;?
:* How can Wikiversity navigate the concerns that lead to the closure of Wikinews?
:* Are any changes to the scope of Wikinews proposed?
:* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension?
:** e.g., I'm not sure that a page like [[User:BigKrow/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource.
:-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 05:52, 14 May 2026 (UTC)
:* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages?
::*No, not at this time.
:* What are &quot;active editions&quot;?
::*There were 30 other active editions of Wikinews in addition to English (e.g. [[:n:es:]]) at the time of universal closure (2026-05-04).
:* How can Wikiversity navigate the concerns that lead to the closure of Wikinews?
::*One of the biggest issues was the problems with DPL, which is now irrelevant. Another was the lack of activity, which can be ameliorated by having it be part of an existing project instead of its own domain (e.g. some editions of Wikipedia host their own Wikinews already and those projects were not impacted by the closure).
:* Are any changes to the scope of Wikinews proposed?
::*Not at this juncture. I would also propose as far as implemention goes that we would request a new namespace and that the material be more-or-less sequestered into its own ongoing project, like Wikijournal is or like the Cookbook and Wikijunior are at our sister [[:b:]].
:* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension?
:** e.g., I'm not sure that a page like [[Story/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource.
::*The process of citizen journalists practicing their craft in real-time and collaborating with others to do so is itself an education activity. We would essentially be hosting a real-time experiment in citizen journalism, online communities, and collaborative learning in addition to the prospect of spreading educational information from someone actually reading the news. I would propose that we could also make a more deliberate attempt to engage with learning &lt;em&gt;about&lt;/em&gt; what does and doesn't work with collaborative news writing by experimentation (e.g. audio news, syndicating to other sites, incorporating freely-licensed news from other sources, writing hyper-local news, writing briefs versus longer-term reportage) and also seeing if the problems noted in the Task Force report that recommended closure can be overcome. Note that we have already done some local investigation about and learning about wiki-based journalism on Wikinews here at [[Journalism studies and Wikinews]]. We could continue that learning and refine the process, including incorporating journalism students from universities. As for tensions, Wikinews is the only sister project that must be done with a quick turn-around: if you take a long time to [[:s:|transcribe a book]], that's just how long it takes, but if you take a long time to write news, it ceases to be news entirely. Wikiversity has been a very slow-growing project that has definitely had some successes but has generally come together over a long period with most learning resources being individual passion projects (or sometimes, frankly, crankery) which would not work with collaborative news that requires more than just a single editor writing whatever he feels like.
::Please let me know any other questions/concerns and any other editors feel free to give your own perspective. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 06:13, 14 May 2026 (UTC)
:::Thanks, Justin — it is food for thought.
:::In attempting to understand how we've arrived here, I've summarised some of the background on this page: [[Wikinews]].
:::Perhaps it could be helpful to flesh out more of the vision / ideas / possibilities / challenges on that page? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:49, 14 May 2026 (UTC)

:::*Having given it some thought, in principle, I support hosting [[citizen journalism]] on Wikiversity where it is clearly connected to a learning project and/or constitutes original research, both of which align strongly with [[Wikiversity:Mission|Wikiversity’s educational mission]].
:::*My chief concern is the potential for news content that is not clearly linked to the purpose of Wikiversity. To avoid this, some community-agreed guidelines would be prudent. These need not be overly restrictive; they should support boldness and experimentation while helping ensure alignment with Wikiversity's purpose.
:::*Given the reported low and declining activity on Wikinews, it seems unlikely that English Wikiversity would be overwhelmed by an influx of news-related editing. My impression is that English Wikinews was the most active edition, but even so, many contributors are likely to disperse to other projects or cease editing altogether. A modest migration of interested editors to Wikiversity seems manageable.
:::*At this stage, I do not think a dedicated namespace is necessary. Subpages under [[Wikinews]] or nested pages under relevant learning or research projects, or user-space draft pages should be suitable. I agree that [[Wikijournal]] offers a useful model, as do several existing course structures on Wikiversity.
:::*I support [[User:Koavf]]’s suggestions about framing Wikinews activity explicitly around learning. This would create a distinctive space for experimenting with collaborative news production in ways that are pedagogically meaningful. I agree that the [[journalism studies and Wikinews]] project developed by David and Leigh Blackall through the University of Wollongong is an excellent example of the intersection between Wikiversity and Wikinews. The [[Wikinews]] page could evolve into a hub for such projects.
:::*I've tidied the [[:Category:Wikinews|Wikinews category]] and merged some content into the [[Wikinews]] page. As part of a reinvigoration effort, please review these and related resources such as [[:Category:Journalism]] and [[School:Journalism]].
:::*A further argument in favour of this initiative is that Wikipedia explicitly excludes both news reporting and original research. So, there is value in maintaining spaces within the Wikimedia ecosystem where these forms of knowledge production can be openly developed and curated. Such work can, in turn, generate valuable evidence and source material that may later inform Wikipedia articles.
:::*The closure of WMF-hosted Wikinews does not imply that open wiki-based news curation lacks value. Indeed, the closure documentation appears supportive of experimentation with alternative news models across Wikimedia projects, including through Wikipedia and Wikidata. In that context, Wikiversity seems a natural home for a Wikinews experiment, provided it is clearly grounded in learning and/or research.
:::-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:39, 15 May 2026 (UTC)
My understanding towards Wikinews' failure is that everything takes too long to be approved for the publish status, which means that any breaking news would have already become days-old stale news. Wikinews has a brand recognition (for right or wrong reasons) than Wikiversity and I wonder how effective Wikiversity can attract the &quot;Wikinews refugees&quot; to edit here. And just a quick note on the governance. Since each Wikiversity language operates independently, each language has to vote &amp; adopt this proposal independently. [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 13:47, 15 May 2026 (UTC)

:Your assessment about Wikinews is partially correct. I referenced it earlier, but to be explicit, there is a [[:m:Proposal for Closing Wikinews|report by a task force on sister projects]] that outlines their concerns. There are a few, one of which was the nature of the staleness of news. Thanks also for clarifying that this proposal is only relevant to en.wv and is not binding or even proposed for other editions of Wikiversity. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:54, 15 May 2026 (UTC)
*Note: I am not a regular here, and just visit Wikiversity for the WikiJournal project. Challenges of Wikinews included that it required timely reporting and fact-checking processes which differed greatly from the well-established ones in Wikipedia. Here in Wikiversity, there is the WikiJournal project, and that can take some some forms of journalism, just not breaking news reporting. I am in favor of salvaging parts of Wikinews if helpful. Could it, would it be feasible to adapt Wikijournal to accept some forms of news journalism, but just not the timed news reporting? For example, WikiJournal already is doing conference proceedings, and could likely do related event reports even months after the event ended. It could probably accept long-form investigative reporting, which is a sort of news that is not breaking news. I am not sure what the possibilities are, but I would prefer to build up systems that already work rather than import systems which had problems elsewhere. Thanks. [[User:Bluerasberry|&lt;span style=&quot;background:#cedff2;color:#11e&quot;&gt;''' Blue Rasberry '''&lt;/span&gt;]][[User talk:Bluerasberry|&lt;span style=&quot;cursor:help&quot;&gt;&lt;span style=&quot;background:#cedff2;color:#11e&quot;&gt;(talk)&lt;/span&gt;&lt;/span&gt;]] 19:17, 22 May 2026 (UTC)
*:I agree that there are certain kinds of journalism that are perfectly valid and not time-bound like breaking news reporting, so that won't suffer from the issues noted before. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 21:15, 22 May 2026 (UTC)
*::@[[User:Bluerasberry|Bluerasberry]] WikiJournal is not interested in taking on news journalism. WikiJournal is publishing conference proceedings at the request of some Wikimedian educators, and conference proceedings is what a &quot;regular&quot; journal publishes. News journalism is quite different from this, and if WikiJournal starts to deviate towards publishing news journalism, it will create barrier towards future initiatives like being indexed in Medline or Web of Science, and may risk being delisted from Scopus. [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 22:43, 5 June 2026 (UTC)
*:::Thats a good point. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:09, 9 June 2026 (UTC)

== Create an autopatrolled user group? ==

{{tracked|T428269|resolved}}
I would like to propose creating the user group &lt;code&gt;autopatrolled&lt;/code&gt; (autopatrolled user), in which for non-curators and non-custodians, their page creations and file uploads would be automatically marked as patrolled by the MediaWiki software. Custodians may grant the user group, at their discretion, to users who create good quality pages that do not need frequent patrolling.

On a side note, the term {{tq|autopatroller}} would be used, but because we don't have non-curator/custodian patrollers (as we rely on curators and custodians to patrol), I suggest on using the term {{tq|autopatrolled user}}. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:31, 29 May 2026 (UTC)

:'''Support''' re: the name, I don't really understand the reasoning, so I am '''neutral''' on that. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 15:45, 29 May 2026 (UTC)
:: Regarding the name, this is because as we don't have the patroller user group, we rely on curators and custodians to patrol new pages and file uploads. Does that make sense? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:39, 29 May 2026 (UTC)
:::Not really, but I don't think it's the most important thing. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 16:42, 29 May 2026 (UTC)
:::: We'll decide on the name later. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:48, 30 May 2026 (UTC)
:::::Oh, please don't let me stand in the way. I'm just not very smart, so don't hold up a matter on my account. I didn't want to derail the proposal, which is a fine and sensible one. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 04:16, 30 May 2026 (UTC)
: '''Support''' - sounds like a good idea
:* Suggest adding a draft section about this group to [[Wikiversity:Patrolling]]. There is a statement in the Introduction of the page that I'm not sure if its correct and at least could be improved: &quot;Wikiversity also uses an autopatrol right, meaning trusted users' contributions are automatically marked as checked so patrollers can focus on reviewing newer or anonymous editors.&quot;
:* Regarding autopatroller vs autropatrolled user, what terms are used on similar WMF wiki projects?
: -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:28, 30 May 2026 (UTC)
::# I would create a starting page about the user groups, with experienced editors expanding the page. A summarized part of that page would also be added to [[Wikiversity:Patrolling]].
::# For a similar example, English Wikipedia uses the term {{tq|Autopatrolled}}, just that term only.
:: [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:22, 30 May 2026 (UTC)
: @[[User:Jtneill|Jtneill]] and @[[User:Koavf|Koavf]]: the autopatroller user group has been implemented here. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:14, 8 June 2026 (UTC)
::Thanks. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:13, 9 June 2026 (UTC)

== How much of Wikiversity’s content is LLM slop? ==

Because it seems like a non-trivial amount, along with AI slop images as well. Is there some kind of AI cleanup project established yet? [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:20, 4 June 2026 (UTC)

:We have discussed AI but I don't know of any explicit initiative to find and delete AI-generated noise. Individual modules have been deleted for having been made by AI. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 08:50, 4 June 2026 (UTC)
:Recently agreed [[Wikiversity:Artificial intelligence|policy]] welcome users to tag AI generated pages. Me personally I am not against the use of AI. What is the difference in abstract schematic image created by a human and the same by an AI. If the users does not have finances to pay digital artest and you dont want to let them use AI, would you pay the artest for them? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:07, 8 June 2026 (UTC)
::Wikimedia has a lot of ''volunteer'' artists who can illustrate if asked. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:11, 9 June 2026 (UTC)
:::Interesting! That's good to know. Where can we find the volunteer artists for illustrating? [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 20:11, 9 June 2026 (UTC)
::::Wikimedia commons has [[commons:Commons:Graphic Lab/Illustration workshop]] [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 02:18, 10 June 2026 (UTC)

== New user what do I do here ==
I love wikipedia and the wikiversity project seems super interesting. However I know very little about wikiversity and would like to know how i can best contribute to the project. Also if there are forums or discord or reddit that would be very helpful.
(One last thing is it normal that my userboxes don't work here) {{unsigned|AUBSTRAWBS}}
:Hey {{ping|AUBSTRAWBS}} Welcome to Wikiversity! I've left a welcome message on your talk page so that should provide you a plethora of useful links for you to look at so you can familiarize yourself with the project. Also, feel free to create the userboxes you need. Wikiversity doesn't have as many userboxes as Wikipedia. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:45, 8 June 2026 (UTC)

:Thank you very much :) hope to contribute a lot.  [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 21:50, 8 June 2026 (UTC)

== Deployment of Legal and Safety Contacts Link in the Footer of Your Wiki ==

Hello community,

The Wikimedia Foundation has provided [[foundation:Legal:Wikimedia Foundation Legal and Safety Contact Information|a single legal and safety contact page]], to be linked in the footer of your wiki, to ensure access to accurate legal information. This is a regulatory requirement.

We have already rolled out links to English, German, Italian, Spanish Wikipedias and other wikis and we will deploy to your wiki soon.

Please [[m:Wikimedia Foundation Legal and Safety Contacts FAQ|read more on the project page]] and leave any comments in this thread or on [[m:Talk:Wikimedia Foundation Legal and Safety Contacts FAQ|the talk page]]. –– [[User:STei (WMF)|STei (WMF)]] ([[User talk:STei (WMF)|discuss]] • [[Special:Contributions/STei (WMF)|contribs]]) 18:12, 9 June 2026 (UTC)

:Thanks for the notice. In case anyone is not clear, we cannot locally change the text at the footer, as it [[:mw:Manual:Footer|requires access to the server settings]]. If we locally needed to change it, we would have to file a ticket at [[:phab:]]. Since the above was sent by someone from the WMF, I think they are on it and it will be updated without any action from anyone here. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:24, 9 June 2026 (UTC)

== Image not displaying ==

Can anyone work out why this image isn't displaying?&lt;br&gt;
[[Educational Media Awareness Campaign/Physics/POTD 10]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:45, 11 June 2026 (UTC)

:Not sure, but it was an issue with the file itself and either way, it should be (and I have since done this) replaced with the SVG [[:File:Telescope-schematic.svg]]. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 13:59, 11 June 2026 (UTC)

== June 2026 Wikimedia Café meetups regarding the English Wikipedia Editor Reflections project ==

&lt;div class=&quot;border-box&quot; style=&quot;background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)&quot;&gt;
&lt;div class=&quot;box&quot; style=&quot;float:left; padding-top: 10px; padding-right: 10px; padding-left: 10px; padding-bottom: 10px;&quot;&gt;[[File:Wikimedia Café logo in plain SVG format.svg|60px|alt=The logo for the Wikimedia Café]]&lt;/div&gt;
Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities during the last weekend of June. Both sessions will focus on the [https://en.wikipedia.org/wiki/Wikipedia:Editor_reflections English Wikipedia Editor Reflections project]. The featured guest in the Café will be [https://en.wikipedia.org/wiki/User:Clovermoss User:Clovermoss]. Participants may attend either or both sessions.

#'''27 June 2026 15:00 UTC''' ([https://zonestamp.toolforge.org/1782572400 timestamp converter]), at a time friendly to the Americas, Africa, and Europe
#'''28 June 2026 03:00 UTC''' ([https://zonestamp.toolforge.org/1782615600 timestamp converter]), at a time friendly to Asia and the Pacific

Please see the Café page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]!
&lt;br /&gt;
[[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]&lt;/div&gt;
&lt;span style=&quot;white-space:nowrap;&quot;&gt;[[User:Pine|&lt;span style=&quot;color:#01796f; text-shadow:#00BFFF 0 0 1.0em&quot;&gt;↠Pine&lt;/span&gt;]] [[User talk:Pine|&lt;span style=&quot;color:DeepSkyBlue&quot;&gt;(&lt;b style=&quot;color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em&quot;&gt;✉&lt;/b&gt;)&lt;/span&gt;]]&lt;/span&gt; 04:00, 15 June 2026 (UTC)</text>
      <sha1>31qjhfa14an24l6qyw6w881ba3skpm7</sha1>
    </revision>
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        <id>10242</id>
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      <comment>/* Mobile friendly main page */ archive from [[Wikiversity:Colloquium]] ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]])</comment>
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== Proposal to rehost Wikinews here ==

As many of you know, and mentioned here at the Colloquium, our sister project Wikinews recently closed, with all 31 active editions made read-only. [[User:BigKrow]] has asked about the prospect of writing news stories here and I suggested that since we already have [[School:Journalism]] and some resources related to the [[:Category:Journalism|broader topic of journalism]]. I would like to propose that we have continued and indefinite space for {{w|citizen journalism}} by essentially repurposing Wikinews into a sub-project here. The only special infrastructure that Wikinews required was [[:mw:Extension:DynamicPageList]], which was deactivated and caused issues due to a lack of maintenance.

I will add this proposal to the site banner, but I recognize that that may be a conflict of interest, so if anyone requests that I remove it, I will. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:30, 14 May 2026 (UTC)
:I would like to see this conversation go for at least 30 days to establish a consensus. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:35, 14 May 2026 (UTC)
::A few days shy of 30, it seems obvious that this is not going to pass. So I '''withdraw''' as presumptively '''failed'''. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:14, 9 June 2026 (UTC)

===Votes===
*{{support}} as proposer (with BK's inspiration). I think that an ongoing experiment in citizen journalism is a fit and appropriate use of this site. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:35, 14 May 2026 (UTC)
*{{support}}, hope to seeing ideas about this, and thank you @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 11:08, 14 May 2026 (UTC)
*{{support}} Other than perhaps inflating the total number of pages reported, I see the idea of &quot;practicing journalism&quot; a worthy and relevant activity within the domain of Wikiversity. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 21:41, 14 May 2026 (UTC)
*{{support}} Conditional on development of (a) community guidelines that ensure alignment with Wikiversity's purpose, and (b) clear, nested page-naming structures for projects. More detail below. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:48, 15 May 2026 (UTC)
*{{contra}} This proposal doesn't seem interested in expanding educational materials in journalism, but rather in providing space and protection for Wikinews contributors. But this is contrary to the goals of Wikiversity, and I'm not sure it's a good idea, even with regard to WMF. If WMF decides to close a project and another community lets it run on its domain, that's a bit of an undermining of WMF's and the community's decisions. Given that Wikiversity has had several conflicts with other communities and WMF in its history, I'm against it.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:59, 15 May 2026 (UTC)
*{{contra}} This seems like a proposal to continue the mission of WikiNews, but not a proposal specifically to improve Wikiversity. I concur with Juandev's comments. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 20:29, 30 May 2026 (UTC)
* {{oppose}} per above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:05, 1 June 2026 (UTC)
*{{oppose}} Wikiversity isn’t Wikinews and it also isn’t a dumping ground for anything not covered by other projects. It was already suggested, rather bafflingly, that Wikinews parasitize Wikipedia as a host. If it were allowed to freeload off of Wikiversity it would simply promote a view I and likely many others have— that Wikiversity (as it currently exists) has no standards and mostly just exists to host subpar content that wouldn’t be tolerated on any other Wikimedia site. Wikinews needs a new, non-Wikimedia host, and Wikiversity needs to get its act together by enforcing a minimum scope and standard for what it allows. --[[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:16, 4 June 2026 (UTC)
* {{oppose}} per above. Wikiversity&lt;math&gt;\not=&lt;/math&gt; Wikinews - not a good idea to mix the scope of projects. --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 12:03, 8 June 2026 (UTC)
* {{abstain}} I will abstain since I'm not an active Wikiversity contributor. But I just feel like Wikinews had a very clear and specific goal of providing news, and Wikiversity is just a different project with different goals. For me, it would be odd to rehost Wikinews here. But please do not count my vote, this is only a comment. --[[User:Antimundo|Antimundo]] ([[User talk:Antimundo|discuss]] • [[Special:Contributions/Antimundo|contribs]]) 13:19, 6 June 2026 (UTC)
* {{oppose}} Although I think it's a pity that Wikinews is closed. --[[User:Dick Bos|Dick Bos]] ([[User talk:Dick Bos|discuss]] • [[Special:Contributions/Dick Bos|contribs]]) 19:06, 8 June 2026 (UTC)
*{{support}} In 2018 I initiated [[:Category:Videoconferences on media and democracy]] as a platform for disseminating public affairs events. In 2021 I officially initiated a podcast series on &quot;Media &amp; Democracy&quot; syndicated for the [[w:List of Pacifica Radio stations and affiliates|Pacifica radio network]]. In 2024 I converted it from irregular to fortnightly. I think this is all educational and supports the Wikiversity education mission, and I think that &quot;rehost Wikinews here&quot; would be appropriate. (I had some experience with Wikinews a few years ago. I felt it was too tightly controlled: Article submissions went stale, because I could not get official permission to publish and I could not get the information needed to understand what I was supposed to do to obtain the official permission. I would be opposed to rehosting Wikinews here if the policy similarly made it unreasonably difficult for volunteer contributor to get the information needed to meet the journalistic standards imposed by the overworked editors.) {{unsigned|DavidMCEddy}}

===Comments and questions===
:Definitely worthy of discussion, so I have no problem with the proposal in the sitenotice.
:Initial questions:
:* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages?
:* What are &quot;active editions&quot;?
:* How can Wikiversity navigate the concerns that lead to the closure of Wikinews?
:* Are any changes to the scope of Wikinews proposed?
:* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension?
:** e.g., I'm not sure that a page like [[User:BigKrow/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource.
:-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 05:52, 14 May 2026 (UTC)
:* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages?
::*No, not at this time.
:* What are &quot;active editions&quot;?
::*There were 30 other active editions of Wikinews in addition to English (e.g. [[:n:es:]]) at the time of universal closure (2026-05-04).
:* How can Wikiversity navigate the concerns that lead to the closure of Wikinews?
::*One of the biggest issues was the problems with DPL, which is now irrelevant. Another was the lack of activity, which can be ameliorated by having it be part of an existing project instead of its own domain (e.g. some editions of Wikipedia host their own Wikinews already and those projects were not impacted by the closure).
:* Are any changes to the scope of Wikinews proposed?
::*Not at this juncture. I would also propose as far as implemention goes that we would request a new namespace and that the material be more-or-less sequestered into its own ongoing project, like Wikijournal is or like the Cookbook and Wikijunior are at our sister [[:b:]].
:* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension?
:** e.g., I'm not sure that a page like [[Story/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource.
::*The process of citizen journalists practicing their craft in real-time and collaborating with others to do so is itself an education activity. We would essentially be hosting a real-time experiment in citizen journalism, online communities, and collaborative learning in addition to the prospect of spreading educational information from someone actually reading the news. I would propose that we could also make a more deliberate attempt to engage with learning &lt;em&gt;about&lt;/em&gt; what does and doesn't work with collaborative news writing by experimentation (e.g. audio news, syndicating to other sites, incorporating freely-licensed news from other sources, writing hyper-local news, writing briefs versus longer-term reportage) and also seeing if the problems noted in the Task Force report that recommended closure can be overcome. Note that we have already done some local investigation about and learning about wiki-based journalism on Wikinews here at [[Journalism studies and Wikinews]]. We could continue that learning and refine the process, including incorporating journalism students from universities. As for tensions, Wikinews is the only sister project that must be done with a quick turn-around: if you take a long time to [[:s:|transcribe a book]], that's just how long it takes, but if you take a long time to write news, it ceases to be news entirely. Wikiversity has been a very slow-growing project that has definitely had some successes but has generally come together over a long period with most learning resources being individual passion projects (or sometimes, frankly, crankery) which would not work with collaborative news that requires more than just a single editor writing whatever he feels like.
::Please let me know any other questions/concerns and any other editors feel free to give your own perspective. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 06:13, 14 May 2026 (UTC)
:::Thanks, Justin — it is food for thought.
:::In attempting to understand how we've arrived here, I've summarised some of the background on this page: [[Wikinews]].
:::Perhaps it could be helpful to flesh out more of the vision / ideas / possibilities / challenges on that page? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:49, 14 May 2026 (UTC)

:::*Having given it some thought, in principle, I support hosting [[citizen journalism]] on Wikiversity where it is clearly connected to a learning project and/or constitutes original research, both of which align strongly with [[Wikiversity:Mission|Wikiversity’s educational mission]].
:::*My chief concern is the potential for news content that is not clearly linked to the purpose of Wikiversity. To avoid this, some community-agreed guidelines would be prudent. These need not be overly restrictive; they should support boldness and experimentation while helping ensure alignment with Wikiversity's purpose.
:::*Given the reported low and declining activity on Wikinews, it seems unlikely that English Wikiversity would be overwhelmed by an influx of news-related editing. My impression is that English Wikinews was the most active edition, but even so, many contributors are likely to disperse to other projects or cease editing altogether. A modest migration of interested editors to Wikiversity seems manageable.
:::*At this stage, I do not think a dedicated namespace is necessary. Subpages under [[Wikinews]] or nested pages under relevant learning or research projects, or user-space draft pages should be suitable. I agree that [[Wikijournal]] offers a useful model, as do several existing course structures on Wikiversity.
:::*I support [[User:Koavf]]’s suggestions about framing Wikinews activity explicitly around learning. This would create a distinctive space for experimenting with collaborative news production in ways that are pedagogically meaningful. I agree that the [[journalism studies and Wikinews]] project developed by David and Leigh Blackall through the University of Wollongong is an excellent example of the intersection between Wikiversity and Wikinews. The [[Wikinews]] page could evolve into a hub for such projects.
:::*I've tidied the [[:Category:Wikinews|Wikinews category]] and merged some content into the [[Wikinews]] page. As part of a reinvigoration effort, please review these and related resources such as [[:Category:Journalism]] and [[School:Journalism]].
:::*A further argument in favour of this initiative is that Wikipedia explicitly excludes both news reporting and original research. So, there is value in maintaining spaces within the Wikimedia ecosystem where these forms of knowledge production can be openly developed and curated. Such work can, in turn, generate valuable evidence and source material that may later inform Wikipedia articles.
:::*The closure of WMF-hosted Wikinews does not imply that open wiki-based news curation lacks value. Indeed, the closure documentation appears supportive of experimentation with alternative news models across Wikimedia projects, including through Wikipedia and Wikidata. In that context, Wikiversity seems a natural home for a Wikinews experiment, provided it is clearly grounded in learning and/or research.
:::-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:39, 15 May 2026 (UTC)
My understanding towards Wikinews' failure is that everything takes too long to be approved for the publish status, which means that any breaking news would have already become days-old stale news. Wikinews has a brand recognition (for right or wrong reasons) than Wikiversity and I wonder how effective Wikiversity can attract the &quot;Wikinews refugees&quot; to edit here. And just a quick note on the governance. Since each Wikiversity language operates independently, each language has to vote &amp; adopt this proposal independently. [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 13:47, 15 May 2026 (UTC)

:Your assessment about Wikinews is partially correct. I referenced it earlier, but to be explicit, there is a [[:m:Proposal for Closing Wikinews|report by a task force on sister projects]] that outlines their concerns. There are a few, one of which was the nature of the staleness of news. Thanks also for clarifying that this proposal is only relevant to en.wv and is not binding or even proposed for other editions of Wikiversity. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:54, 15 May 2026 (UTC)
*Note: I am not a regular here, and just visit Wikiversity for the WikiJournal project. Challenges of Wikinews included that it required timely reporting and fact-checking processes which differed greatly from the well-established ones in Wikipedia. Here in Wikiversity, there is the WikiJournal project, and that can take some some forms of journalism, just not breaking news reporting. I am in favor of salvaging parts of Wikinews if helpful. Could it, would it be feasible to adapt Wikijournal to accept some forms of news journalism, but just not the timed news reporting? For example, WikiJournal already is doing conference proceedings, and could likely do related event reports even months after the event ended. It could probably accept long-form investigative reporting, which is a sort of news that is not breaking news. I am not sure what the possibilities are, but I would prefer to build up systems that already work rather than import systems which had problems elsewhere. Thanks. [[User:Bluerasberry|&lt;span style=&quot;background:#cedff2;color:#11e&quot;&gt;''' Blue Rasberry '''&lt;/span&gt;]][[User talk:Bluerasberry|&lt;span style=&quot;cursor:help&quot;&gt;&lt;span style=&quot;background:#cedff2;color:#11e&quot;&gt;(talk)&lt;/span&gt;&lt;/span&gt;]] 19:17, 22 May 2026 (UTC)
*:I agree that there are certain kinds of journalism that are perfectly valid and not time-bound like breaking news reporting, so that won't suffer from the issues noted before. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 21:15, 22 May 2026 (UTC)
*::@[[User:Bluerasberry|Bluerasberry]] WikiJournal is not interested in taking on news journalism. WikiJournal is publishing conference proceedings at the request of some Wikimedian educators, and conference proceedings is what a &quot;regular&quot; journal publishes. News journalism is quite different from this, and if WikiJournal starts to deviate towards publishing news journalism, it will create barrier towards future initiatives like being indexed in Medline or Web of Science, and may risk being delisted from Scopus. [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 22:43, 5 June 2026 (UTC)
*:::Thats a good point. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:09, 9 June 2026 (UTC)

== Create an autopatrolled user group? ==

{{tracked|T428269|resolved}}
I would like to propose creating the user group &lt;code&gt;autopatrolled&lt;/code&gt; (autopatrolled user), in which for non-curators and non-custodians, their page creations and file uploads would be automatically marked as patrolled by the MediaWiki software. Custodians may grant the user group, at their discretion, to users who create good quality pages that do not need frequent patrolling.

On a side note, the term {{tq|autopatroller}} would be used, but because we don't have non-curator/custodian patrollers (as we rely on curators and custodians to patrol), I suggest on using the term {{tq|autopatrolled user}}. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:31, 29 May 2026 (UTC)

:'''Support''' re: the name, I don't really understand the reasoning, so I am '''neutral''' on that. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 15:45, 29 May 2026 (UTC)
:: Regarding the name, this is because as we don't have the patroller user group, we rely on curators and custodians to patrol new pages and file uploads. Does that make sense? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:39, 29 May 2026 (UTC)
:::Not really, but I don't think it's the most important thing. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 16:42, 29 May 2026 (UTC)
:::: We'll decide on the name later. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:48, 30 May 2026 (UTC)
:::::Oh, please don't let me stand in the way. I'm just not very smart, so don't hold up a matter on my account. I didn't want to derail the proposal, which is a fine and sensible one. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 04:16, 30 May 2026 (UTC)
: '''Support''' - sounds like a good idea
:* Suggest adding a draft section about this group to [[Wikiversity:Patrolling]]. There is a statement in the Introduction of the page that I'm not sure if its correct and at least could be improved: &quot;Wikiversity also uses an autopatrol right, meaning trusted users' contributions are automatically marked as checked so patrollers can focus on reviewing newer or anonymous editors.&quot;
:* Regarding autopatroller vs autropatrolled user, what terms are used on similar WMF wiki projects?
: -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:28, 30 May 2026 (UTC)
::# I would create a starting page about the user groups, with experienced editors expanding the page. A summarized part of that page would also be added to [[Wikiversity:Patrolling]].
::# For a similar example, English Wikipedia uses the term {{tq|Autopatrolled}}, just that term only.
:: [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:22, 30 May 2026 (UTC)
: @[[User:Jtneill|Jtneill]] and @[[User:Koavf|Koavf]]: the autopatroller user group has been implemented here. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:14, 8 June 2026 (UTC)
::Thanks. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:13, 9 June 2026 (UTC)

== How much of Wikiversity’s content is LLM slop? ==

Because it seems like a non-trivial amount, along with AI slop images as well. Is there some kind of AI cleanup project established yet? [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:20, 4 June 2026 (UTC)

:We have discussed AI but I don't know of any explicit initiative to find and delete AI-generated noise. Individual modules have been deleted for having been made by AI. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 08:50, 4 June 2026 (UTC)
:Recently agreed [[Wikiversity:Artificial intelligence|policy]] welcome users to tag AI generated pages. Me personally I am not against the use of AI. What is the difference in abstract schematic image created by a human and the same by an AI. If the users does not have finances to pay digital artest and you dont want to let them use AI, would you pay the artest for them? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:07, 8 June 2026 (UTC)
::Wikimedia has a lot of ''volunteer'' artists who can illustrate if asked. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:11, 9 June 2026 (UTC)
:::Interesting! That's good to know. Where can we find the volunteer artists for illustrating? [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 20:11, 9 June 2026 (UTC)
::::Wikimedia commons has [[commons:Commons:Graphic Lab/Illustration workshop]] [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 02:18, 10 June 2026 (UTC)

== New user what do I do here ==
I love wikipedia and the wikiversity project seems super interesting. However I know very little about wikiversity and would like to know how i can best contribute to the project. Also if there are forums or discord or reddit that would be very helpful.
(One last thing is it normal that my userboxes don't work here) {{unsigned|AUBSTRAWBS}}
:Hey {{ping|AUBSTRAWBS}} Welcome to Wikiversity! I've left a welcome message on your talk page so that should provide you a plethora of useful links for you to look at so you can familiarize yourself with the project. Also, feel free to create the userboxes you need. Wikiversity doesn't have as many userboxes as Wikipedia. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:45, 8 June 2026 (UTC)

:Thank you very much :) hope to contribute a lot.  [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 21:50, 8 June 2026 (UTC)

== Deployment of Legal and Safety Contacts Link in the Footer of Your Wiki ==

Hello community,

The Wikimedia Foundation has provided [[foundation:Legal:Wikimedia Foundation Legal and Safety Contact Information|a single legal and safety contact page]], to be linked in the footer of your wiki, to ensure access to accurate legal information. This is a regulatory requirement.

We have already rolled out links to English, German, Italian, Spanish Wikipedias and other wikis and we will deploy to your wiki soon.

Please [[m:Wikimedia Foundation Legal and Safety Contacts FAQ|read more on the project page]] and leave any comments in this thread or on [[m:Talk:Wikimedia Foundation Legal and Safety Contacts FAQ|the talk page]]. –– [[User:STei (WMF)|STei (WMF)]] ([[User talk:STei (WMF)|discuss]] • [[Special:Contributions/STei (WMF)|contribs]]) 18:12, 9 June 2026 (UTC)

:Thanks for the notice. In case anyone is not clear, we cannot locally change the text at the footer, as it [[:mw:Manual:Footer|requires access to the server settings]]. If we locally needed to change it, we would have to file a ticket at [[:phab:]]. Since the above was sent by someone from the WMF, I think they are on it and it will be updated without any action from anyone here. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:24, 9 June 2026 (UTC)

== Image not displaying ==

Can anyone work out why this image isn't displaying?&lt;br&gt;
[[Educational Media Awareness Campaign/Physics/POTD 10]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:45, 11 June 2026 (UTC)

:Not sure, but it was an issue with the file itself and either way, it should be (and I have since done this) replaced with the SVG [[:File:Telescope-schematic.svg]]. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 13:59, 11 June 2026 (UTC)

== June 2026 Wikimedia Café meetups regarding the English Wikipedia Editor Reflections project ==

&lt;div class=&quot;border-box&quot; style=&quot;background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)&quot;&gt;
&lt;div class=&quot;box&quot; style=&quot;float:left; padding-top: 10px; padding-right: 10px; padding-left: 10px; padding-bottom: 10px;&quot;&gt;[[File:Wikimedia Café logo in plain SVG format.svg|60px|alt=The logo for the Wikimedia Café]]&lt;/div&gt;
Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities during the last weekend of June. Both sessions will focus on the [https://en.wikipedia.org/wiki/Wikipedia:Editor_reflections English Wikipedia Editor Reflections project]. The featured guest in the Café will be [https://en.wikipedia.org/wiki/User:Clovermoss User:Clovermoss]. Participants may attend either or both sessions.

#'''27 June 2026 15:00 UTC''' ([https://zonestamp.toolforge.org/1782572400 timestamp converter]), at a time friendly to the Americas, Africa, and Europe
#'''28 June 2026 03:00 UTC''' ([https://zonestamp.toolforge.org/1782615600 timestamp converter]), at a time friendly to Asia and the Pacific

Please see the Café page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]!
&lt;br /&gt;
[[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]&lt;/div&gt;
&lt;span style=&quot;white-space:nowrap;&quot;&gt;[[User:Pine|&lt;span style=&quot;color:#01796f; text-shadow:#00BFFF 0 0 1.0em&quot;&gt;↠Pine&lt;/span&gt;]] [[User talk:Pine|&lt;span style=&quot;color:DeepSkyBlue&quot;&gt;(&lt;b style=&quot;color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em&quot;&gt;✉&lt;/b&gt;)&lt;/span&gt;]]&lt;/span&gt; 04:00, 15 June 2026 (UTC)

== Mobile friendly main page ==

Hello, I have recently been using wikiversity on mobile and unlike wikipedia some images and boxes stick out instead of all having a set width which means you can scroll a little side to side, which makes the site feel a bit unfinished. Its just a suggestion but I think it will wake the user experience much better {{unsigned|AUBSTRAWBS}}
:{{Ping|AUBSTRAWBS}} I don't use a smartphone. Can you give me more details or even take some screenshots? You can upload them at [[:c:Category:English Wikiversity screenshots]]. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 13:30, 18 June 2026 (UTC)
::Hi i uploaded an image of the problem. Since some of the images are larger than the screen and not adjusted to fit they stick out and makes the page larger which lets you scroll right and have a big white rectangle on the side [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 14:03, 18 June 2026 (UTC)
:::Thanks. I agree that this is an issue, but it's a pretty minor-to-moderate one to me and I don't think I will be able to dedicate time to fix it myself. Showing it to others here is useful in case someone else wants to tinker with the CSS to resolve it. Thanks for bringing it to the community's attention. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 15:42, 18 June 2026 (UTC)
::::I do know CSS as I like to maintain a blog online so I could try and fix it but I don't know if I have the access to do that, would i need to be a curator/ custodian. Alternatively i could edit a sandbox version of the main page and then send it to someone. [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 20:00, 18 June 2026 (UTC)
:::::Oh great. There are a lot of draft versions of the main page like [[Wikiversity:Main Page/Draft version 0.2]], so you can make [[Wikiversity:Main Page/Sandbox]] if you want and edit there. If you can tinker it to your liking, I can edit the main page. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 20:14, 18 June 2026 (UTC)
::::::thank you, i'll check it out [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 22:16, 18 June 2026 (UTC)</text>
      <sha1>fzmwqybkaq2irm2swcyflu2rqicfvug</sha1>
    </revision>
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      <comment>/* Main page titles */ archive from [[Wikiversity:Colloquium]] ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]])</comment>
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== Proposal to rehost Wikinews here ==

As many of you know, and mentioned here at the Colloquium, our sister project Wikinews recently closed, with all 31 active editions made read-only. [[User:BigKrow]] has asked about the prospect of writing news stories here and I suggested that since we already have [[School:Journalism]] and some resources related to the [[:Category:Journalism|broader topic of journalism]]. I would like to propose that we have continued and indefinite space for {{w|citizen journalism}} by essentially repurposing Wikinews into a sub-project here. The only special infrastructure that Wikinews required was [[:mw:Extension:DynamicPageList]], which was deactivated and caused issues due to a lack of maintenance.

I will add this proposal to the site banner, but I recognize that that may be a conflict of interest, so if anyone requests that I remove it, I will. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:30, 14 May 2026 (UTC)
:I would like to see this conversation go for at least 30 days to establish a consensus. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:35, 14 May 2026 (UTC)
::A few days shy of 30, it seems obvious that this is not going to pass. So I '''withdraw''' as presumptively '''failed'''. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:14, 9 June 2026 (UTC)

===Votes===
*{{support}} as proposer (with BK's inspiration). I think that an ongoing experiment in citizen journalism is a fit and appropriate use of this site. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:35, 14 May 2026 (UTC)
*{{support}}, hope to seeing ideas about this, and thank you @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 11:08, 14 May 2026 (UTC)
*{{support}} Other than perhaps inflating the total number of pages reported, I see the idea of &quot;practicing journalism&quot; a worthy and relevant activity within the domain of Wikiversity. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 21:41, 14 May 2026 (UTC)
*{{support}} Conditional on development of (a) community guidelines that ensure alignment with Wikiversity's purpose, and (b) clear, nested page-naming structures for projects. More detail below. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:48, 15 May 2026 (UTC)
*{{contra}} This proposal doesn't seem interested in expanding educational materials in journalism, but rather in providing space and protection for Wikinews contributors. But this is contrary to the goals of Wikiversity, and I'm not sure it's a good idea, even with regard to WMF. If WMF decides to close a project and another community lets it run on its domain, that's a bit of an undermining of WMF's and the community's decisions. Given that Wikiversity has had several conflicts with other communities and WMF in its history, I'm against it.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:59, 15 May 2026 (UTC)
*{{contra}} This seems like a proposal to continue the mission of WikiNews, but not a proposal specifically to improve Wikiversity. I concur with Juandev's comments. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 20:29, 30 May 2026 (UTC)
* {{oppose}} per above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:05, 1 June 2026 (UTC)
*{{oppose}} Wikiversity isn’t Wikinews and it also isn’t a dumping ground for anything not covered by other projects. It was already suggested, rather bafflingly, that Wikinews parasitize Wikipedia as a host. If it were allowed to freeload off of Wikiversity it would simply promote a view I and likely many others have— that Wikiversity (as it currently exists) has no standards and mostly just exists to host subpar content that wouldn’t be tolerated on any other Wikimedia site. Wikinews needs a new, non-Wikimedia host, and Wikiversity needs to get its act together by enforcing a minimum scope and standard for what it allows. --[[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:16, 4 June 2026 (UTC)
* {{oppose}} per above. Wikiversity&lt;math&gt;\not=&lt;/math&gt; Wikinews - not a good idea to mix the scope of projects. --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 12:03, 8 June 2026 (UTC)
* {{abstain}} I will abstain since I'm not an active Wikiversity contributor. But I just feel like Wikinews had a very clear and specific goal of providing news, and Wikiversity is just a different project with different goals. For me, it would be odd to rehost Wikinews here. But please do not count my vote, this is only a comment. --[[User:Antimundo|Antimundo]] ([[User talk:Antimundo|discuss]] • [[Special:Contributions/Antimundo|contribs]]) 13:19, 6 June 2026 (UTC)
* {{oppose}} Although I think it's a pity that Wikinews is closed. --[[User:Dick Bos|Dick Bos]] ([[User talk:Dick Bos|discuss]] • [[Special:Contributions/Dick Bos|contribs]]) 19:06, 8 June 2026 (UTC)
*{{support}} In 2018 I initiated [[:Category:Videoconferences on media and democracy]] as a platform for disseminating public affairs events. In 2021 I officially initiated a podcast series on &quot;Media &amp; Democracy&quot; syndicated for the [[w:List of Pacifica Radio stations and affiliates|Pacifica radio network]]. In 2024 I converted it from irregular to fortnightly. I think this is all educational and supports the Wikiversity education mission, and I think that &quot;rehost Wikinews here&quot; would be appropriate. (I had some experience with Wikinews a few years ago. I felt it was too tightly controlled: Article submissions went stale, because I could not get official permission to publish and I could not get the information needed to understand what I was supposed to do to obtain the official permission. I would be opposed to rehosting Wikinews here if the policy similarly made it unreasonably difficult for volunteer contributor to get the information needed to meet the journalistic standards imposed by the overworked editors.) {{unsigned|DavidMCEddy}}

===Comments and questions===
:Definitely worthy of discussion, so I have no problem with the proposal in the sitenotice.
:Initial questions:
:* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages?
:* What are &quot;active editions&quot;?
:* How can Wikiversity navigate the concerns that lead to the closure of Wikinews?
:* Are any changes to the scope of Wikinews proposed?
:* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension?
:** e.g., I'm not sure that a page like [[User:BigKrow/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource.
:-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 05:52, 14 May 2026 (UTC)
:* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages?
::*No, not at this time.
:* What are &quot;active editions&quot;?
::*There were 30 other active editions of Wikinews in addition to English (e.g. [[:n:es:]]) at the time of universal closure (2026-05-04).
:* How can Wikiversity navigate the concerns that lead to the closure of Wikinews?
::*One of the biggest issues was the problems with DPL, which is now irrelevant. Another was the lack of activity, which can be ameliorated by having it be part of an existing project instead of its own domain (e.g. some editions of Wikipedia host their own Wikinews already and those projects were not impacted by the closure).
:* Are any changes to the scope of Wikinews proposed?
::*Not at this juncture. I would also propose as far as implemention goes that we would request a new namespace and that the material be more-or-less sequestered into its own ongoing project, like Wikijournal is or like the Cookbook and Wikijunior are at our sister [[:b:]].
:* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension?
:** e.g., I'm not sure that a page like [[Story/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource.
::*The process of citizen journalists practicing their craft in real-time and collaborating with others to do so is itself an education activity. We would essentially be hosting a real-time experiment in citizen journalism, online communities, and collaborative learning in addition to the prospect of spreading educational information from someone actually reading the news. I would propose that we could also make a more deliberate attempt to engage with learning &lt;em&gt;about&lt;/em&gt; what does and doesn't work with collaborative news writing by experimentation (e.g. audio news, syndicating to other sites, incorporating freely-licensed news from other sources, writing hyper-local news, writing briefs versus longer-term reportage) and also seeing if the problems noted in the Task Force report that recommended closure can be overcome. Note that we have already done some local investigation about and learning about wiki-based journalism on Wikinews here at [[Journalism studies and Wikinews]]. We could continue that learning and refine the process, including incorporating journalism students from universities. As for tensions, Wikinews is the only sister project that must be done with a quick turn-around: if you take a long time to [[:s:|transcribe a book]], that's just how long it takes, but if you take a long time to write news, it ceases to be news entirely. Wikiversity has been a very slow-growing project that has definitely had some successes but has generally come together over a long period with most learning resources being individual passion projects (or sometimes, frankly, crankery) which would not work with collaborative news that requires more than just a single editor writing whatever he feels like.
::Please let me know any other questions/concerns and any other editors feel free to give your own perspective. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 06:13, 14 May 2026 (UTC)
:::Thanks, Justin — it is food for thought.
:::In attempting to understand how we've arrived here, I've summarised some of the background on this page: [[Wikinews]].
:::Perhaps it could be helpful to flesh out more of the vision / ideas / possibilities / challenges on that page? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:49, 14 May 2026 (UTC)

:::*Having given it some thought, in principle, I support hosting [[citizen journalism]] on Wikiversity where it is clearly connected to a learning project and/or constitutes original research, both of which align strongly with [[Wikiversity:Mission|Wikiversity’s educational mission]].
:::*My chief concern is the potential for news content that is not clearly linked to the purpose of Wikiversity. To avoid this, some community-agreed guidelines would be prudent. These need not be overly restrictive; they should support boldness and experimentation while helping ensure alignment with Wikiversity's purpose.
:::*Given the reported low and declining activity on Wikinews, it seems unlikely that English Wikiversity would be overwhelmed by an influx of news-related editing. My impression is that English Wikinews was the most active edition, but even so, many contributors are likely to disperse to other projects or cease editing altogether. A modest migration of interested editors to Wikiversity seems manageable.
:::*At this stage, I do not think a dedicated namespace is necessary. Subpages under [[Wikinews]] or nested pages under relevant learning or research projects, or user-space draft pages should be suitable. I agree that [[Wikijournal]] offers a useful model, as do several existing course structures on Wikiversity.
:::*I support [[User:Koavf]]’s suggestions about framing Wikinews activity explicitly around learning. This would create a distinctive space for experimenting with collaborative news production in ways that are pedagogically meaningful. I agree that the [[journalism studies and Wikinews]] project developed by David and Leigh Blackall through the University of Wollongong is an excellent example of the intersection between Wikiversity and Wikinews. The [[Wikinews]] page could evolve into a hub for such projects.
:::*I've tidied the [[:Category:Wikinews|Wikinews category]] and merged some content into the [[Wikinews]] page. As part of a reinvigoration effort, please review these and related resources such as [[:Category:Journalism]] and [[School:Journalism]].
:::*A further argument in favour of this initiative is that Wikipedia explicitly excludes both news reporting and original research. So, there is value in maintaining spaces within the Wikimedia ecosystem where these forms of knowledge production can be openly developed and curated. Such work can, in turn, generate valuable evidence and source material that may later inform Wikipedia articles.
:::*The closure of WMF-hosted Wikinews does not imply that open wiki-based news curation lacks value. Indeed, the closure documentation appears supportive of experimentation with alternative news models across Wikimedia projects, including through Wikipedia and Wikidata. In that context, Wikiversity seems a natural home for a Wikinews experiment, provided it is clearly grounded in learning and/or research.
:::-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:39, 15 May 2026 (UTC)
My understanding towards Wikinews' failure is that everything takes too long to be approved for the publish status, which means that any breaking news would have already become days-old stale news. Wikinews has a brand recognition (for right or wrong reasons) than Wikiversity and I wonder how effective Wikiversity can attract the &quot;Wikinews refugees&quot; to edit here. And just a quick note on the governance. Since each Wikiversity language operates independently, each language has to vote &amp; adopt this proposal independently. [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 13:47, 15 May 2026 (UTC)

:Your assessment about Wikinews is partially correct. I referenced it earlier, but to be explicit, there is a [[:m:Proposal for Closing Wikinews|report by a task force on sister projects]] that outlines their concerns. There are a few, one of which was the nature of the staleness of news. Thanks also for clarifying that this proposal is only relevant to en.wv and is not binding or even proposed for other editions of Wikiversity. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:54, 15 May 2026 (UTC)
*Note: I am not a regular here, and just visit Wikiversity for the WikiJournal project. Challenges of Wikinews included that it required timely reporting and fact-checking processes which differed greatly from the well-established ones in Wikipedia. Here in Wikiversity, there is the WikiJournal project, and that can take some some forms of journalism, just not breaking news reporting. I am in favor of salvaging parts of Wikinews if helpful. Could it, would it be feasible to adapt Wikijournal to accept some forms of news journalism, but just not the timed news reporting? For example, WikiJournal already is doing conference proceedings, and could likely do related event reports even months after the event ended. It could probably accept long-form investigative reporting, which is a sort of news that is not breaking news. I am not sure what the possibilities are, but I would prefer to build up systems that already work rather than import systems which had problems elsewhere. Thanks. [[User:Bluerasberry|&lt;span style=&quot;background:#cedff2;color:#11e&quot;&gt;''' Blue Rasberry '''&lt;/span&gt;]][[User talk:Bluerasberry|&lt;span style=&quot;cursor:help&quot;&gt;&lt;span style=&quot;background:#cedff2;color:#11e&quot;&gt;(talk)&lt;/span&gt;&lt;/span&gt;]] 19:17, 22 May 2026 (UTC)
*:I agree that there are certain kinds of journalism that are perfectly valid and not time-bound like breaking news reporting, so that won't suffer from the issues noted before. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 21:15, 22 May 2026 (UTC)
*::@[[User:Bluerasberry|Bluerasberry]] WikiJournal is not interested in taking on news journalism. WikiJournal is publishing conference proceedings at the request of some Wikimedian educators, and conference proceedings is what a &quot;regular&quot; journal publishes. News journalism is quite different from this, and if WikiJournal starts to deviate towards publishing news journalism, it will create barrier towards future initiatives like being indexed in Medline or Web of Science, and may risk being delisted from Scopus. [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 22:43, 5 June 2026 (UTC)
*:::Thats a good point. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:09, 9 June 2026 (UTC)

== Create an autopatrolled user group? ==

{{tracked|T428269|resolved}}
I would like to propose creating the user group &lt;code&gt;autopatrolled&lt;/code&gt; (autopatrolled user), in which for non-curators and non-custodians, their page creations and file uploads would be automatically marked as patrolled by the MediaWiki software. Custodians may grant the user group, at their discretion, to users who create good quality pages that do not need frequent patrolling.

On a side note, the term {{tq|autopatroller}} would be used, but because we don't have non-curator/custodian patrollers (as we rely on curators and custodians to patrol), I suggest on using the term {{tq|autopatrolled user}}. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:31, 29 May 2026 (UTC)

:'''Support''' re: the name, I don't really understand the reasoning, so I am '''neutral''' on that. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 15:45, 29 May 2026 (UTC)
:: Regarding the name, this is because as we don't have the patroller user group, we rely on curators and custodians to patrol new pages and file uploads. Does that make sense? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:39, 29 May 2026 (UTC)
:::Not really, but I don't think it's the most important thing. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 16:42, 29 May 2026 (UTC)
:::: We'll decide on the name later. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:48, 30 May 2026 (UTC)
:::::Oh, please don't let me stand in the way. I'm just not very smart, so don't hold up a matter on my account. I didn't want to derail the proposal, which is a fine and sensible one. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 04:16, 30 May 2026 (UTC)
: '''Support''' - sounds like a good idea
:* Suggest adding a draft section about this group to [[Wikiversity:Patrolling]]. There is a statement in the Introduction of the page that I'm not sure if its correct and at least could be improved: &quot;Wikiversity also uses an autopatrol right, meaning trusted users' contributions are automatically marked as checked so patrollers can focus on reviewing newer or anonymous editors.&quot;
:* Regarding autopatroller vs autropatrolled user, what terms are used on similar WMF wiki projects?
: -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:28, 30 May 2026 (UTC)
::# I would create a starting page about the user groups, with experienced editors expanding the page. A summarized part of that page would also be added to [[Wikiversity:Patrolling]].
::# For a similar example, English Wikipedia uses the term {{tq|Autopatrolled}}, just that term only.
:: [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:22, 30 May 2026 (UTC)
: @[[User:Jtneill|Jtneill]] and @[[User:Koavf|Koavf]]: the autopatroller user group has been implemented here. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:14, 8 June 2026 (UTC)
::Thanks. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:13, 9 June 2026 (UTC)

== How much of Wikiversity’s content is LLM slop? ==

Because it seems like a non-trivial amount, along with AI slop images as well. Is there some kind of AI cleanup project established yet? [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:20, 4 June 2026 (UTC)

:We have discussed AI but I don't know of any explicit initiative to find and delete AI-generated noise. Individual modules have been deleted for having been made by AI. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 08:50, 4 June 2026 (UTC)
:Recently agreed [[Wikiversity:Artificial intelligence|policy]] welcome users to tag AI generated pages. Me personally I am not against the use of AI. What is the difference in abstract schematic image created by a human and the same by an AI. If the users does not have finances to pay digital artest and you dont want to let them use AI, would you pay the artest for them? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:07, 8 June 2026 (UTC)
::Wikimedia has a lot of ''volunteer'' artists who can illustrate if asked. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:11, 9 June 2026 (UTC)
:::Interesting! That's good to know. Where can we find the volunteer artists for illustrating? [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 20:11, 9 June 2026 (UTC)
::::Wikimedia commons has [[commons:Commons:Graphic Lab/Illustration workshop]] [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 02:18, 10 June 2026 (UTC)

== New user what do I do here ==
I love wikipedia and the wikiversity project seems super interesting. However I know very little about wikiversity and would like to know how i can best contribute to the project. Also if there are forums or discord or reddit that would be very helpful.
(One last thing is it normal that my userboxes don't work here) {{unsigned|AUBSTRAWBS}}
:Hey {{ping|AUBSTRAWBS}} Welcome to Wikiversity! I've left a welcome message on your talk page so that should provide you a plethora of useful links for you to look at so you can familiarize yourself with the project. Also, feel free to create the userboxes you need. Wikiversity doesn't have as many userboxes as Wikipedia. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:45, 8 June 2026 (UTC)

:Thank you very much :) hope to contribute a lot.  [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 21:50, 8 June 2026 (UTC)

== Deployment of Legal and Safety Contacts Link in the Footer of Your Wiki ==

Hello community,

The Wikimedia Foundation has provided [[foundation:Legal:Wikimedia Foundation Legal and Safety Contact Information|a single legal and safety contact page]], to be linked in the footer of your wiki, to ensure access to accurate legal information. This is a regulatory requirement.

We have already rolled out links to English, German, Italian, Spanish Wikipedias and other wikis and we will deploy to your wiki soon.

Please [[m:Wikimedia Foundation Legal and Safety Contacts FAQ|read more on the project page]] and leave any comments in this thread or on [[m:Talk:Wikimedia Foundation Legal and Safety Contacts FAQ|the talk page]]. –– [[User:STei (WMF)|STei (WMF)]] ([[User talk:STei (WMF)|discuss]] • [[Special:Contributions/STei (WMF)|contribs]]) 18:12, 9 June 2026 (UTC)

:Thanks for the notice. In case anyone is not clear, we cannot locally change the text at the footer, as it [[:mw:Manual:Footer|requires access to the server settings]]. If we locally needed to change it, we would have to file a ticket at [[:phab:]]. Since the above was sent by someone from the WMF, I think they are on it and it will be updated without any action from anyone here. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:24, 9 June 2026 (UTC)

== Image not displaying ==

Can anyone work out why this image isn't displaying?&lt;br&gt;
[[Educational Media Awareness Campaign/Physics/POTD 10]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:45, 11 June 2026 (UTC)

:Not sure, but it was an issue with the file itself and either way, it should be (and I have since done this) replaced with the SVG [[:File:Telescope-schematic.svg]]. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 13:59, 11 June 2026 (UTC)

== June 2026 Wikimedia Café meetups regarding the English Wikipedia Editor Reflections project ==

&lt;div class=&quot;border-box&quot; style=&quot;background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)&quot;&gt;
&lt;div class=&quot;box&quot; style=&quot;float:left; padding-top: 10px; padding-right: 10px; padding-left: 10px; padding-bottom: 10px;&quot;&gt;[[File:Wikimedia Café logo in plain SVG format.svg|60px|alt=The logo for the Wikimedia Café]]&lt;/div&gt;
Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities during the last weekend of June. Both sessions will focus on the [https://en.wikipedia.org/wiki/Wikipedia:Editor_reflections English Wikipedia Editor Reflections project]. The featured guest in the Café will be [https://en.wikipedia.org/wiki/User:Clovermoss User:Clovermoss]. Participants may attend either or both sessions.

#'''27 June 2026 15:00 UTC''' ([https://zonestamp.toolforge.org/1782572400 timestamp converter]), at a time friendly to the Americas, Africa, and Europe
#'''28 June 2026 03:00 UTC''' ([https://zonestamp.toolforge.org/1782615600 timestamp converter]), at a time friendly to Asia and the Pacific

Please see the Café page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]!
&lt;br /&gt;
[[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]&lt;/div&gt;
&lt;span style=&quot;white-space:nowrap;&quot;&gt;[[User:Pine|&lt;span style=&quot;color:#01796f; text-shadow:#00BFFF 0 0 1.0em&quot;&gt;↠Pine&lt;/span&gt;]] [[User talk:Pine|&lt;span style=&quot;color:DeepSkyBlue&quot;&gt;(&lt;b style=&quot;color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em&quot;&gt;✉&lt;/b&gt;)&lt;/span&gt;]]&lt;/span&gt; 04:00, 15 June 2026 (UTC)

== Mobile friendly main page ==

Hello, I have recently been using wikiversity on mobile and unlike wikipedia some images and boxes stick out instead of all having a set width which means you can scroll a little side to side, which makes the site feel a bit unfinished. Its just a suggestion but I think it will wake the user experience much better {{unsigned|AUBSTRAWBS}}
:{{Ping|AUBSTRAWBS}} I don't use a smartphone. Can you give me more details or even take some screenshots? You can upload them at [[:c:Category:English Wikiversity screenshots]]. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 13:30, 18 June 2026 (UTC)
::Hi i uploaded an image of the problem. Since some of the images are larger than the screen and not adjusted to fit they stick out and makes the page larger which lets you scroll right and have a big white rectangle on the side [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 14:03, 18 June 2026 (UTC)
:::Thanks. I agree that this is an issue, but it's a pretty minor-to-moderate one to me and I don't think I will be able to dedicate time to fix it myself. Showing it to others here is useful in case someone else wants to tinker with the CSS to resolve it. Thanks for bringing it to the community's attention. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 15:42, 18 June 2026 (UTC)
::::I do know CSS as I like to maintain a blog online so I could try and fix it but I don't know if I have the access to do that, would i need to be a curator/ custodian. Alternatively i could edit a sandbox version of the main page and then send it to someone. [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 20:00, 18 June 2026 (UTC)
:::::Oh great. There are a lot of draft versions of the main page like [[Wikiversity:Main Page/Draft version 0.2]], so you can make [[Wikiversity:Main Page/Sandbox]] if you want and edit there. If you can tinker it to your liking, I can edit the main page. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 20:14, 18 June 2026 (UTC)
::::::thank you, i'll check it out [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 22:16, 18 June 2026 (UTC)

== Main page titles ==

Currently, the title says &quot;Wikiversity:Main Page&quot;, but in my opinion, it's too basic. I would like to propose changing it with the following options (you may only pick one):
# Option 1: Set both [[MediaWiki:Mainpage-title]] and [[MediaWiki:Mainpage-title-loggedin]] to blank, giving the main page a portal-like design (as with English Wikipedia, English Wikibooks, etc.)
# Option 2: Modify [[MediaWiki:Mainpage-title]] to &lt;code&gt;Welcome to Wikiversity&lt;/code&gt; (for unregistered users), and [[MediaWiki:Mainpage-title-loggedin]] to &lt;code&gt;&lt;nowiki&gt;Welcome to Wikiversity, $1!&lt;/nowiki&gt;&lt;/code&gt;; the latter would display to me as &lt;code&gt;Welcome to Wikiversity, Codename Noreste!&lt;/code&gt;
Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:34, 18 June 2026 (UTC)
: Pinging @[[User:Jtneill|Jtneill]] and @[[User:Koavf|Koavf]] for input above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:03, 24 June 2026 (UTC)
:I'm afraid that I don't have strong feelings on this. Changing to either or staying with the status quo are all fine to me. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 16:10, 24 June 2026 (UTC)
: I like the option of being consistent with Wikipedia and Wikibooks -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:04, 25 June 2026 (UTC)
: {{done}}. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:34, 30 June 2026 (UTC)
: Thankyou - looks good. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:29, 2 July 2026 (UTC)</text>
      <sha1>49vbw6jvymego2btrcdsjbfomwjf6px</sha1>
    </revision>
    <revision>
      <id>2819794</id>
      <parentid>2819792</parentid>
      <timestamp>2026-07-30T04:51:48Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>/* Wiki x AI preconference day @ Wikimania */ archive from [[Wikiversity:Colloquium]] ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]])</comment>
      <origin>2819794</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="35603" sha1="8lzdbnp90trseufckgv0e3jyy4wx4h8" xml:space="preserve">{{archive}}

== Proposal to rehost Wikinews here ==

As many of you know, and mentioned here at the Colloquium, our sister project Wikinews recently closed, with all 31 active editions made read-only. [[User:BigKrow]] has asked about the prospect of writing news stories here and I suggested that since we already have [[School:Journalism]] and some resources related to the [[:Category:Journalism|broader topic of journalism]]. I would like to propose that we have continued and indefinite space for {{w|citizen journalism}} by essentially repurposing Wikinews into a sub-project here. The only special infrastructure that Wikinews required was [[:mw:Extension:DynamicPageList]], which was deactivated and caused issues due to a lack of maintenance.

I will add this proposal to the site banner, but I recognize that that may be a conflict of interest, so if anyone requests that I remove it, I will. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:30, 14 May 2026 (UTC)
:I would like to see this conversation go for at least 30 days to establish a consensus. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:35, 14 May 2026 (UTC)
::A few days shy of 30, it seems obvious that this is not going to pass. So I '''withdraw''' as presumptively '''failed'''. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:14, 9 June 2026 (UTC)

===Votes===
*{{support}} as proposer (with BK's inspiration). I think that an ongoing experiment in citizen journalism is a fit and appropriate use of this site. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:35, 14 May 2026 (UTC)
*{{support}}, hope to seeing ideas about this, and thank you @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 11:08, 14 May 2026 (UTC)
*{{support}} Other than perhaps inflating the total number of pages reported, I see the idea of &quot;practicing journalism&quot; a worthy and relevant activity within the domain of Wikiversity. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 21:41, 14 May 2026 (UTC)
*{{support}} Conditional on development of (a) community guidelines that ensure alignment with Wikiversity's purpose, and (b) clear, nested page-naming structures for projects. More detail below. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:48, 15 May 2026 (UTC)
*{{contra}} This proposal doesn't seem interested in expanding educational materials in journalism, but rather in providing space and protection for Wikinews contributors. But this is contrary to the goals of Wikiversity, and I'm not sure it's a good idea, even with regard to WMF. If WMF decides to close a project and another community lets it run on its domain, that's a bit of an undermining of WMF's and the community's decisions. Given that Wikiversity has had several conflicts with other communities and WMF in its history, I'm against it.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:59, 15 May 2026 (UTC)
*{{contra}} This seems like a proposal to continue the mission of WikiNews, but not a proposal specifically to improve Wikiversity. I concur with Juandev's comments. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 20:29, 30 May 2026 (UTC)
* {{oppose}} per above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:05, 1 June 2026 (UTC)
*{{oppose}} Wikiversity isn’t Wikinews and it also isn’t a dumping ground for anything not covered by other projects. It was already suggested, rather bafflingly, that Wikinews parasitize Wikipedia as a host. If it were allowed to freeload off of Wikiversity it would simply promote a view I and likely many others have— that Wikiversity (as it currently exists) has no standards and mostly just exists to host subpar content that wouldn’t be tolerated on any other Wikimedia site. Wikinews needs a new, non-Wikimedia host, and Wikiversity needs to get its act together by enforcing a minimum scope and standard for what it allows. --[[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:16, 4 June 2026 (UTC)
* {{oppose}} per above. Wikiversity&lt;math&gt;\not=&lt;/math&gt; Wikinews - not a good idea to mix the scope of projects. --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 12:03, 8 June 2026 (UTC)
* {{abstain}} I will abstain since I'm not an active Wikiversity contributor. But I just feel like Wikinews had a very clear and specific goal of providing news, and Wikiversity is just a different project with different goals. For me, it would be odd to rehost Wikinews here. But please do not count my vote, this is only a comment. --[[User:Antimundo|Antimundo]] ([[User talk:Antimundo|discuss]] • [[Special:Contributions/Antimundo|contribs]]) 13:19, 6 June 2026 (UTC)
* {{oppose}} Although I think it's a pity that Wikinews is closed. --[[User:Dick Bos|Dick Bos]] ([[User talk:Dick Bos|discuss]] • [[Special:Contributions/Dick Bos|contribs]]) 19:06, 8 June 2026 (UTC)
*{{support}} In 2018 I initiated [[:Category:Videoconferences on media and democracy]] as a platform for disseminating public affairs events. In 2021 I officially initiated a podcast series on &quot;Media &amp; Democracy&quot; syndicated for the [[w:List of Pacifica Radio stations and affiliates|Pacifica radio network]]. In 2024 I converted it from irregular to fortnightly. I think this is all educational and supports the Wikiversity education mission, and I think that &quot;rehost Wikinews here&quot; would be appropriate. (I had some experience with Wikinews a few years ago. I felt it was too tightly controlled: Article submissions went stale, because I could not get official permission to publish and I could not get the information needed to understand what I was supposed to do to obtain the official permission. I would be opposed to rehosting Wikinews here if the policy similarly made it unreasonably difficult for volunteer contributor to get the information needed to meet the journalistic standards imposed by the overworked editors.) {{unsigned|DavidMCEddy}}

===Comments and questions===
:Definitely worthy of discussion, so I have no problem with the proposal in the sitenotice.
:Initial questions:
:* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages?
:* What are &quot;active editions&quot;?
:* How can Wikiversity navigate the concerns that lead to the closure of Wikinews?
:* Are any changes to the scope of Wikinews proposed?
:* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension?
:** e.g., I'm not sure that a page like [[User:BigKrow/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource.
:-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 05:52, 14 May 2026 (UTC)
:* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages?
::*No, not at this time.
:* What are &quot;active editions&quot;?
::*There were 30 other active editions of Wikinews in addition to English (e.g. [[:n:es:]]) at the time of universal closure (2026-05-04).
:* How can Wikiversity navigate the concerns that lead to the closure of Wikinews?
::*One of the biggest issues was the problems with DPL, which is now irrelevant. Another was the lack of activity, which can be ameliorated by having it be part of an existing project instead of its own domain (e.g. some editions of Wikipedia host their own Wikinews already and those projects were not impacted by the closure).
:* Are any changes to the scope of Wikinews proposed?
::*Not at this juncture. I would also propose as far as implemention goes that we would request a new namespace and that the material be more-or-less sequestered into its own ongoing project, like Wikijournal is or like the Cookbook and Wikijunior are at our sister [[:b:]].
:* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension?
:** e.g., I'm not sure that a page like [[Story/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource.
::*The process of citizen journalists practicing their craft in real-time and collaborating with others to do so is itself an education activity. We would essentially be hosting a real-time experiment in citizen journalism, online communities, and collaborative learning in addition to the prospect of spreading educational information from someone actually reading the news. I would propose that we could also make a more deliberate attempt to engage with learning &lt;em&gt;about&lt;/em&gt; what does and doesn't work with collaborative news writing by experimentation (e.g. audio news, syndicating to other sites, incorporating freely-licensed news from other sources, writing hyper-local news, writing briefs versus longer-term reportage) and also seeing if the problems noted in the Task Force report that recommended closure can be overcome. Note that we have already done some local investigation about and learning about wiki-based journalism on Wikinews here at [[Journalism studies and Wikinews]]. We could continue that learning and refine the process, including incorporating journalism students from universities. As for tensions, Wikinews is the only sister project that must be done with a quick turn-around: if you take a long time to [[:s:|transcribe a book]], that's just how long it takes, but if you take a long time to write news, it ceases to be news entirely. Wikiversity has been a very slow-growing project that has definitely had some successes but has generally come together over a long period with most learning resources being individual passion projects (or sometimes, frankly, crankery) which would not work with collaborative news that requires more than just a single editor writing whatever he feels like.
::Please let me know any other questions/concerns and any other editors feel free to give your own perspective. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 06:13, 14 May 2026 (UTC)
:::Thanks, Justin — it is food for thought.
:::In attempting to understand how we've arrived here, I've summarised some of the background on this page: [[Wikinews]].
:::Perhaps it could be helpful to flesh out more of the vision / ideas / possibilities / challenges on that page? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:49, 14 May 2026 (UTC)

:::*Having given it some thought, in principle, I support hosting [[citizen journalism]] on Wikiversity where it is clearly connected to a learning project and/or constitutes original research, both of which align strongly with [[Wikiversity:Mission|Wikiversity’s educational mission]].
:::*My chief concern is the potential for news content that is not clearly linked to the purpose of Wikiversity. To avoid this, some community-agreed guidelines would be prudent. These need not be overly restrictive; they should support boldness and experimentation while helping ensure alignment with Wikiversity's purpose.
:::*Given the reported low and declining activity on Wikinews, it seems unlikely that English Wikiversity would be overwhelmed by an influx of news-related editing. My impression is that English Wikinews was the most active edition, but even so, many contributors are likely to disperse to other projects or cease editing altogether. A modest migration of interested editors to Wikiversity seems manageable.
:::*At this stage, I do not think a dedicated namespace is necessary. Subpages under [[Wikinews]] or nested pages under relevant learning or research projects, or user-space draft pages should be suitable. I agree that [[Wikijournal]] offers a useful model, as do several existing course structures on Wikiversity.
:::*I support [[User:Koavf]]’s suggestions about framing Wikinews activity explicitly around learning. This would create a distinctive space for experimenting with collaborative news production in ways that are pedagogically meaningful. I agree that the [[journalism studies and Wikinews]] project developed by David and Leigh Blackall through the University of Wollongong is an excellent example of the intersection between Wikiversity and Wikinews. The [[Wikinews]] page could evolve into a hub for such projects.
:::*I've tidied the [[:Category:Wikinews|Wikinews category]] and merged some content into the [[Wikinews]] page. As part of a reinvigoration effort, please review these and related resources such as [[:Category:Journalism]] and [[School:Journalism]].
:::*A further argument in favour of this initiative is that Wikipedia explicitly excludes both news reporting and original research. So, there is value in maintaining spaces within the Wikimedia ecosystem where these forms of knowledge production can be openly developed and curated. Such work can, in turn, generate valuable evidence and source material that may later inform Wikipedia articles.
:::*The closure of WMF-hosted Wikinews does not imply that open wiki-based news curation lacks value. Indeed, the closure documentation appears supportive of experimentation with alternative news models across Wikimedia projects, including through Wikipedia and Wikidata. In that context, Wikiversity seems a natural home for a Wikinews experiment, provided it is clearly grounded in learning and/or research.
:::-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:39, 15 May 2026 (UTC)
My understanding towards Wikinews' failure is that everything takes too long to be approved for the publish status, which means that any breaking news would have already become days-old stale news. Wikinews has a brand recognition (for right or wrong reasons) than Wikiversity and I wonder how effective Wikiversity can attract the &quot;Wikinews refugees&quot; to edit here. And just a quick note on the governance. Since each Wikiversity language operates independently, each language has to vote &amp; adopt this proposal independently. [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 13:47, 15 May 2026 (UTC)

:Your assessment about Wikinews is partially correct. I referenced it earlier, but to be explicit, there is a [[:m:Proposal for Closing Wikinews|report by a task force on sister projects]] that outlines their concerns. There are a few, one of which was the nature of the staleness of news. Thanks also for clarifying that this proposal is only relevant to en.wv and is not binding or even proposed for other editions of Wikiversity. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:54, 15 May 2026 (UTC)
*Note: I am not a regular here, and just visit Wikiversity for the WikiJournal project. Challenges of Wikinews included that it required timely reporting and fact-checking processes which differed greatly from the well-established ones in Wikipedia. Here in Wikiversity, there is the WikiJournal project, and that can take some some forms of journalism, just not breaking news reporting. I am in favor of salvaging parts of Wikinews if helpful. Could it, would it be feasible to adapt Wikijournal to accept some forms of news journalism, but just not the timed news reporting? For example, WikiJournal already is doing conference proceedings, and could likely do related event reports even months after the event ended. It could probably accept long-form investigative reporting, which is a sort of news that is not breaking news. I am not sure what the possibilities are, but I would prefer to build up systems that already work rather than import systems which had problems elsewhere. Thanks. [[User:Bluerasberry|&lt;span style=&quot;background:#cedff2;color:#11e&quot;&gt;''' Blue Rasberry '''&lt;/span&gt;]][[User talk:Bluerasberry|&lt;span style=&quot;cursor:help&quot;&gt;&lt;span style=&quot;background:#cedff2;color:#11e&quot;&gt;(talk)&lt;/span&gt;&lt;/span&gt;]] 19:17, 22 May 2026 (UTC)
*:I agree that there are certain kinds of journalism that are perfectly valid and not time-bound like breaking news reporting, so that won't suffer from the issues noted before. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 21:15, 22 May 2026 (UTC)
*::@[[User:Bluerasberry|Bluerasberry]] WikiJournal is not interested in taking on news journalism. WikiJournal is publishing conference proceedings at the request of some Wikimedian educators, and conference proceedings is what a &quot;regular&quot; journal publishes. News journalism is quite different from this, and if WikiJournal starts to deviate towards publishing news journalism, it will create barrier towards future initiatives like being indexed in Medline or Web of Science, and may risk being delisted from Scopus. [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 22:43, 5 June 2026 (UTC)
*:::Thats a good point. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:09, 9 June 2026 (UTC)

== Create an autopatrolled user group? ==

{{tracked|T428269|resolved}}
I would like to propose creating the user group &lt;code&gt;autopatrolled&lt;/code&gt; (autopatrolled user), in which for non-curators and non-custodians, their page creations and file uploads would be automatically marked as patrolled by the MediaWiki software. Custodians may grant the user group, at their discretion, to users who create good quality pages that do not need frequent patrolling.

On a side note, the term {{tq|autopatroller}} would be used, but because we don't have non-curator/custodian patrollers (as we rely on curators and custodians to patrol), I suggest on using the term {{tq|autopatrolled user}}. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:31, 29 May 2026 (UTC)

:'''Support''' re: the name, I don't really understand the reasoning, so I am '''neutral''' on that. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 15:45, 29 May 2026 (UTC)
:: Regarding the name, this is because as we don't have the patroller user group, we rely on curators and custodians to patrol new pages and file uploads. Does that make sense? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:39, 29 May 2026 (UTC)
:::Not really, but I don't think it's the most important thing. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 16:42, 29 May 2026 (UTC)
:::: We'll decide on the name later. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:48, 30 May 2026 (UTC)
:::::Oh, please don't let me stand in the way. I'm just not very smart, so don't hold up a matter on my account. I didn't want to derail the proposal, which is a fine and sensible one. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 04:16, 30 May 2026 (UTC)
: '''Support''' - sounds like a good idea
:* Suggest adding a draft section about this group to [[Wikiversity:Patrolling]]. There is a statement in the Introduction of the page that I'm not sure if its correct and at least could be improved: &quot;Wikiversity also uses an autopatrol right, meaning trusted users' contributions are automatically marked as checked so patrollers can focus on reviewing newer or anonymous editors.&quot;
:* Regarding autopatroller vs autropatrolled user, what terms are used on similar WMF wiki projects?
: -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:28, 30 May 2026 (UTC)
::# I would create a starting page about the user groups, with experienced editors expanding the page. A summarized part of that page would also be added to [[Wikiversity:Patrolling]].
::# For a similar example, English Wikipedia uses the term {{tq|Autopatrolled}}, just that term only.
:: [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:22, 30 May 2026 (UTC)
: @[[User:Jtneill|Jtneill]] and @[[User:Koavf|Koavf]]: the autopatroller user group has been implemented here. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:14, 8 June 2026 (UTC)
::Thanks. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:13, 9 June 2026 (UTC)

== How much of Wikiversity’s content is LLM slop? ==

Because it seems like a non-trivial amount, along with AI slop images as well. Is there some kind of AI cleanup project established yet? [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:20, 4 June 2026 (UTC)

:We have discussed AI but I don't know of any explicit initiative to find and delete AI-generated noise. Individual modules have been deleted for having been made by AI. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 08:50, 4 June 2026 (UTC)
:Recently agreed [[Wikiversity:Artificial intelligence|policy]] welcome users to tag AI generated pages. Me personally I am not against the use of AI. What is the difference in abstract schematic image created by a human and the same by an AI. If the users does not have finances to pay digital artest and you dont want to let them use AI, would you pay the artest for them? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:07, 8 June 2026 (UTC)
::Wikimedia has a lot of ''volunteer'' artists who can illustrate if asked. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:11, 9 June 2026 (UTC)
:::Interesting! That's good to know. Where can we find the volunteer artists for illustrating? [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 20:11, 9 June 2026 (UTC)
::::Wikimedia commons has [[commons:Commons:Graphic Lab/Illustration workshop]] [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 02:18, 10 June 2026 (UTC)

== New user what do I do here ==
I love wikipedia and the wikiversity project seems super interesting. However I know very little about wikiversity and would like to know how i can best contribute to the project. Also if there are forums or discord or reddit that would be very helpful.
(One last thing is it normal that my userboxes don't work here) {{unsigned|AUBSTRAWBS}}
:Hey {{ping|AUBSTRAWBS}} Welcome to Wikiversity! I've left a welcome message on your talk page so that should provide you a plethora of useful links for you to look at so you can familiarize yourself with the project. Also, feel free to create the userboxes you need. Wikiversity doesn't have as many userboxes as Wikipedia. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:45, 8 June 2026 (UTC)

:Thank you very much :) hope to contribute a lot.  [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 21:50, 8 June 2026 (UTC)

== Deployment of Legal and Safety Contacts Link in the Footer of Your Wiki ==

Hello community,

The Wikimedia Foundation has provided [[foundation:Legal:Wikimedia Foundation Legal and Safety Contact Information|a single legal and safety contact page]], to be linked in the footer of your wiki, to ensure access to accurate legal information. This is a regulatory requirement.

We have already rolled out links to English, German, Italian, Spanish Wikipedias and other wikis and we will deploy to your wiki soon.

Please [[m:Wikimedia Foundation Legal and Safety Contacts FAQ|read more on the project page]] and leave any comments in this thread or on [[m:Talk:Wikimedia Foundation Legal and Safety Contacts FAQ|the talk page]]. –– [[User:STei (WMF)|STei (WMF)]] ([[User talk:STei (WMF)|discuss]] • [[Special:Contributions/STei (WMF)|contribs]]) 18:12, 9 June 2026 (UTC)

:Thanks for the notice. In case anyone is not clear, we cannot locally change the text at the footer, as it [[:mw:Manual:Footer|requires access to the server settings]]. If we locally needed to change it, we would have to file a ticket at [[:phab:]]. Since the above was sent by someone from the WMF, I think they are on it and it will be updated without any action from anyone here. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:24, 9 June 2026 (UTC)

== Image not displaying ==

Can anyone work out why this image isn't displaying?&lt;br&gt;
[[Educational Media Awareness Campaign/Physics/POTD 10]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:45, 11 June 2026 (UTC)

:Not sure, but it was an issue with the file itself and either way, it should be (and I have since done this) replaced with the SVG [[:File:Telescope-schematic.svg]]. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 13:59, 11 June 2026 (UTC)

== June 2026 Wikimedia Café meetups regarding the English Wikipedia Editor Reflections project ==

&lt;div class=&quot;border-box&quot; style=&quot;background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)&quot;&gt;
&lt;div class=&quot;box&quot; style=&quot;float:left; padding-top: 10px; padding-right: 10px; padding-left: 10px; padding-bottom: 10px;&quot;&gt;[[File:Wikimedia Café logo in plain SVG format.svg|60px|alt=The logo for the Wikimedia Café]]&lt;/div&gt;
Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities during the last weekend of June. Both sessions will focus on the [https://en.wikipedia.org/wiki/Wikipedia:Editor_reflections English Wikipedia Editor Reflections project]. The featured guest in the Café will be [https://en.wikipedia.org/wiki/User:Clovermoss User:Clovermoss]. Participants may attend either or both sessions.

#'''27 June 2026 15:00 UTC''' ([https://zonestamp.toolforge.org/1782572400 timestamp converter]), at a time friendly to the Americas, Africa, and Europe
#'''28 June 2026 03:00 UTC''' ([https://zonestamp.toolforge.org/1782615600 timestamp converter]), at a time friendly to Asia and the Pacific

Please see the Café page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]!
&lt;br /&gt;
[[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]&lt;/div&gt;
&lt;span style=&quot;white-space:nowrap;&quot;&gt;[[User:Pine|&lt;span style=&quot;color:#01796f; text-shadow:#00BFFF 0 0 1.0em&quot;&gt;↠Pine&lt;/span&gt;]] [[User talk:Pine|&lt;span style=&quot;color:DeepSkyBlue&quot;&gt;(&lt;b style=&quot;color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em&quot;&gt;✉&lt;/b&gt;)&lt;/span&gt;]]&lt;/span&gt; 04:00, 15 June 2026 (UTC)

== Mobile friendly main page ==

Hello, I have recently been using wikiversity on mobile and unlike wikipedia some images and boxes stick out instead of all having a set width which means you can scroll a little side to side, which makes the site feel a bit unfinished. Its just a suggestion but I think it will wake the user experience much better {{unsigned|AUBSTRAWBS}}
:{{Ping|AUBSTRAWBS}} I don't use a smartphone. Can you give me more details or even take some screenshots? You can upload them at [[:c:Category:English Wikiversity screenshots]]. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 13:30, 18 June 2026 (UTC)
::Hi i uploaded an image of the problem. Since some of the images are larger than the screen and not adjusted to fit they stick out and makes the page larger which lets you scroll right and have a big white rectangle on the side [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 14:03, 18 June 2026 (UTC)
:::Thanks. I agree that this is an issue, but it's a pretty minor-to-moderate one to me and I don't think I will be able to dedicate time to fix it myself. Showing it to others here is useful in case someone else wants to tinker with the CSS to resolve it. Thanks for bringing it to the community's attention. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 15:42, 18 June 2026 (UTC)
::::I do know CSS as I like to maintain a blog online so I could try and fix it but I don't know if I have the access to do that, would i need to be a curator/ custodian. Alternatively i could edit a sandbox version of the main page and then send it to someone. [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 20:00, 18 June 2026 (UTC)
:::::Oh great. There are a lot of draft versions of the main page like [[Wikiversity:Main Page/Draft version 0.2]], so you can make [[Wikiversity:Main Page/Sandbox]] if you want and edit there. If you can tinker it to your liking, I can edit the main page. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 20:14, 18 June 2026 (UTC)
::::::thank you, i'll check it out [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 22:16, 18 June 2026 (UTC)

== Main page titles ==

Currently, the title says &quot;Wikiversity:Main Page&quot;, but in my opinion, it's too basic. I would like to propose changing it with the following options (you may only pick one):
# Option 1: Set both [[MediaWiki:Mainpage-title]] and [[MediaWiki:Mainpage-title-loggedin]] to blank, giving the main page a portal-like design (as with English Wikipedia, English Wikibooks, etc.)
# Option 2: Modify [[MediaWiki:Mainpage-title]] to &lt;code&gt;Welcome to Wikiversity&lt;/code&gt; (for unregistered users), and [[MediaWiki:Mainpage-title-loggedin]] to &lt;code&gt;&lt;nowiki&gt;Welcome to Wikiversity, $1!&lt;/nowiki&gt;&lt;/code&gt;; the latter would display to me as &lt;code&gt;Welcome to Wikiversity, Codename Noreste!&lt;/code&gt;
Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:34, 18 June 2026 (UTC)
: Pinging @[[User:Jtneill|Jtneill]] and @[[User:Koavf|Koavf]] for input above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:03, 24 June 2026 (UTC)
:I'm afraid that I don't have strong feelings on this. Changing to either or staying with the status quo are all fine to me. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 16:10, 24 June 2026 (UTC)
: I like the option of being consistent with Wikipedia and Wikibooks -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:04, 25 June 2026 (UTC)
: {{done}}. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:34, 30 June 2026 (UTC)
: Thankyou - looks good. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:29, 2 July 2026 (UTC)

== Wiki x AI preconference day @ Wikimania ==
There will be a preconference day at Wikimania about [[meta:Artificial_intelligence/2026_Wiki_AI | Wiki AI]]. It will be mostly offline, but there will be at least one hybrid session for demos of community-developed AI tools and workflows. 

* If you've built something cool, that is a chance to show it off, list it on the gallery of tools in progress, and get feedback. 

* If you could ask the people shaping AI on the wikis (WMF, tool builders, model trainers, GLAM and policy folks) a question, what would it be? 

Cheers, &lt;span style=&quot;padding:0 2px 0 2px;background-color:white;color:#bbb;&quot;&gt;&amp;ndash;[[User:Sj|SJ]][[User Talk:Sj|&lt;span style=&quot;color:#ff9900;&quot;&gt;+&lt;/span&gt;]]&lt;/span&gt; 23:12, 20 June 2026 (UTC) and Alaexis&lt;br&gt;{{comment|1=Copied from https://en.wikiversity.org/w/index.php?title=Talk%3AMotivation_and_emotion%2FAssessment%2FUsing_generative_AI&amp;diff=2816357&amp;oldid=2807052}}</text>
      <sha1>8lzdbnp90trseufckgv0e3jyy4wx4h8</sha1>
    </revision>
  </page>
  <page>
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        <id>10242</id>
      </contributor>
      <comment>/* July 2026 Wikimedia Café meetups regarding Wikimedia governance and options for reform */ archive from [[Wikiversity:Colloquium]] ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]])</comment>
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      <text bytes="3112" sha1="2o4gyl4g4p15gi1pkwo0f3lgv1hzu1f" xml:space="preserve">{{archive}}

== July 2026 Wikimedia Café meetups regarding Wikimedia governance and options for reform ==

&lt;div class=&quot;border-box&quot; style=&quot;background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)&quot;&gt;
&lt;div class=&quot;box&quot; style=&quot;float:left; padding-top: 10px; padding-right: 10px; padding-left: 10px; padding-bottom: 10px;&quot;&gt;[[File:Wikimedia Café logo in plain SVG format.svg|60px|alt=The logo for the Wikimedia Café]]&lt;/div&gt;
Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities in July. Both sessions will focus on Wikimedia governance, including possible follow-ups to the [https://meta.wikimedia.org/wiki/Movement_Charter Movement Charter] and options for reform. Participants may attend either or both Café sessions. 

This month, to deconflict the Café meetups from Wikimania, the meetups will be held one day later than usual.

#'''26 July 2026 15:00 UTC''' ([https://zonestamp.toolforge.org/1785078000 timestamp converter]), at a time friendly to the Americas, Africa, and Europe
#'''27 July 2026 03:00 UTC''' ([https://zonestamp.toolforge.org/1785121200 timestamp converter]), at a time friendly to Asia and the Pacific

Please see the Café page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]!
&lt;br /&gt;
[[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]&lt;/div&gt;
&lt;span style=&quot;white-space:nowrap;&quot;&gt;[[User:Pine|&lt;span style=&quot;color:#01796f; text-shadow:#00BFFF 0 0 1.0em&quot;&gt;↠Pine&lt;/span&gt;]] [[User talk:Pine|&lt;span style=&quot;color:DeepSkyBlue&quot;&gt;(&lt;b style=&quot;color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em&quot;&gt;✉&lt;/b&gt;)&lt;/span&gt;]]&lt;/span&gt; 03:51, 13 July 2026 (UTC)

:@[[User:Pine|Pine]], Some wikimedians who may contribute less popular, but hopefully productive, comments,  are those who are currently blocked on Meta. I myself was blocked infinitely on Meta several years ago, but was allowed to contInue contributing on other wikimedia projects, such as this one. There are many long term contributors, some who have contrIbuted tens of thousands of edits  to wikimedia projects who are are now blocked on specific projects. 
:Those who happen to be blocked on Meta have,  to all intents and purposes, been removed from the wikimedia movement.
:Ottawahitech [[Special:Contributions/&amp;#126;2026-40514-81|&amp;#126;2026-40514-81]] ([[User talk:&amp;#126;2026-40514-81|talk]]) 21:41, 18 July 2026 (UTC)
:*Hi Ottawahitech, thanks for the comment. Reviewing local bans on Meta is outside of my current scope, but if you have comments that you'd like to share here in the Colloquium relating to Wikiversity, I can consider the comments in the scope of the Café discussions. &lt;span style=&quot;white-space:nowrap;&quot;&gt;[[User:Pine|&lt;span style=&quot;color:#01796f; text-shadow:#00BFFF 0 0 1.0em&quot;&gt;↠Pine&lt;/span&gt;]] [[User talk:Pine|&lt;span style=&quot;color:DeepSkyBlue&quot;&gt;(&lt;b style=&quot;color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em&quot;&gt;✉&lt;/b&gt;)&lt;/span&gt;]]&lt;/span&gt; 20:05, 25 July 2026 (UTC)</text>
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'''Athena problem''' is an [[:w:List of unsolved problems in mathematics|unsolved problem]] in [[:w:Number theory|number theory]] and [[:w:Formal language theory|formal language theory]] and [[:w:Order theory|order theory]], this problem is named after the ancient Greek goddess [[:w:Athena|Athena]] (which is associated with [[:w:Wisdom|wisdom]]). Athena problem is: Give a [[:w:Natural number|natural number]] ''b'' &gt; 1, find the [[:w:Set (mathematics)|set]] of the [[:w:Minimal element|minimal element]]s of the set of the &quot;[[:w:Prime number|prime number]] [[:w:Greater than|&gt;]] ''b''&quot; [[:w:Numerical digit|digit]] [[:w:String (computer science)|string]]s in the [[:w:Positional numeral system|positional numeral system]] with [[:w:Radix|base]] ''b'' for the [[:w:Subsequence|subsequence]] [[:w:Partially ordered set|ordering]]. (A string ''x'' is a subsequence of another string ''y'', if ''x'' can be obtained from ''y'' by deleting zero or more of the [[:w:Character (computing)|character]]s in ''y''. For example, 514 is a subsequence of 352148, &quot;string&quot; is a subsequence of &quot;meistersinger&quot;. In contrast, 758 is not a subsequence of 378259, &quot;abc&quot; is not a subsequence of &quot;cbacacba&quot;, since the characters must be in the same order) (Unlike [[:w:Substring|substring]], subsequence is not required to occupy consecutive positions within the original sequences, e.g. the [[:w:Longest common subsequence|longest common subsequence problem]] is different from the [[:w:Longest common substring|longest common substring problem]])

Using [[:w:Formal language theory|formal language theory]] terminology, Athena problem is finding the [[:w:Set (mathematics)|set]] of the [[:w:Minimal element|minimal element]]s of the [[:w:Formal language|language]] of base-''b'' [[:w:Representation (mathematics)|representation]]s of the [[:w:Prime number|prime number]]s [[:w:Greater than|&gt;]] ''b'' (which is a set of [[:w:String (computer science)|string]]s of [[:w:Symbol|symbol]]s over the [[:w:Alphabet (formal languages)|alphabet]] ''Σ''&lt;sub&gt;''b''&lt;/sub&gt; := {0, 1, ..., ''b''−1}), under the subsequence ordering (i.e. the [[:w:Binary relation|binary relation]] &quot;is a subsequence of&quot;, which is a [[:w:Partially ordered set|partial ordering]]), for a given natural number ''b'' &gt; 1 (You can draw this partial ordering as a [[:w:Hasse diagram|Hasse diagram]] to find all [[:w:Minimal element|minimal element]]s), this set is called '''Athena set base ''b''''', and the prime numbers in this set is called '''Athena primes base ''b'''''.

By [[:w:Higman's lemma|Higman's lemma]], there are no [[:w:Infinite set|infinite]] [[:w:Antichain|antichain]]s for the subsequence ordering (i.e. the subsequence ordering is always a [[:w:Well-quasi-ordering|well quasi order]]) (i.e. under the subsequence ordering (i.e. the [[:w:Binary relation|binary relation]] &quot;is a subsequence of&quot;, which is a [[:w:Partially ordered set|partial ordering]]), every set of pairwise incomparable (i.e. not [[:w:Comparability|comparable]]) strings is finite), thus there must be only finitely many such minimal elements. In other words, the Athena set in every base ''b'' must be a [[:w:Finite set|finite set]], and every base ''b'' ≥ 2 has only finitely many Athena primes, e.g. in [[:w:Decimal|decimal]] (base ''b'' = 10), the Athena set has exactly 77 [[:w:Element of a set|element]]s (the Athena primes in decimal (base ''b'' = 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}.

For bases 2 ≤ ''b'' ≤ 36, Athena problem is fully solved in bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 24, and also solved in bases ''b'' = 11, 13, 16, 22, 30 if [[:w:Probable prime|probable prime]]s are allowed. For the unsolved bases ''b'' = 17, 19, 21, 23, 25, 26, 27, 28, 29, 31, 32, 34, 35, 36, Athena problem is solved (if probable primes are allowed) except 771 [[:w:Indexed family|families]] of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be [[:w:Empty string|empty]]) of digits in base ''b'', ''y'' is a digit in base ''b'') = sequence {''xz'', ''xyz'', ''xyyz'', ''xyyyz'', ''xyyyyz'', ''xyyyyyz'', ...} (i.e. &quot;''xy''&lt;sup&gt;+&lt;/sup&gt;''z''&quot; in [[:w:Regular expression|regular expression]]), all of these 771 families contain no primes &gt; ''b'' or probable primes &gt; ''b'' with length ≤ 100000.

== Solve the problem ==

To solve the Athena problem for a given base ''b'', we must [[:w:Computing|compute]] the elements up to families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''), and find the smallest prime &gt; ''b'' in all such families.

We call families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') &quot;linear&quot; families, and we reduce these families by removing all trailing digits ''y'' from ''x'', and removing all leading digits ''y'' from ''z'', to make the families be easier, e.g. family 12333{3}33345 in base ''b'' is reduced to family 12{3}45 in base ''b'', since they are in fact the same family. Our [[:w:Algorithm|algorithm]] then proceeds as follows:

* 1. ''M'' := {minimal primes in base ''b'' of length 2 or 3}, ''L'' := union of all ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'') such that ''x'' ≠ 0 and ''gcd''(''z'', ''b'') = 1 and ''Y'' is the set of digits ''y'' in base ''b'' such that ''xyz'' has no subsequence in ''M''.
* 2. While ''L'' contains nonlinear families (families which are not linear families): Explore each family of ''L'', and update ''L''. Examine each family of ''L'' by:
* 2.1. Let ''w'' be the shortest string in the family. If ''w'' has a subsequence in ''M'', then remove the family from ''L''. If ''w'' represents a prime, then add ''w'' to ''M'' and remove the family from ''L''.
* 2.2. If possible, simplify the family.
* 2.3. Using the techniques below (covering congruence, algebraic factorization, or combine of them), check if the family can be proven to only contain composites (only count the numbers &gt; ''b''), and if so then remove the family from ''L''.
* 3. Update ''L'', after each split examine the new families as in step 2.

e.g. in decimal (base ''b'' = 10):

''M'' := {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}

''L'' := {2{0,2}1, 2{0,8}7, 3{0,3,6,9}3, 3{0,3,6,9}9, 4{6}9, 5{0,5,8}1, 5{0,2}7, 6{0,3,6,9}3, 6{0,3,4,6,9}9, 7{0,7}7, 8{0,5}1, 8{0}7, 9{0,2,5,8}1, 9{0,3,6,9}3, 9{0,3,4,6,9}9}

and since 2221 is prime, it follows that the family 2{0,2}1 splits into the families 2{0}1 and 2{0}2{0}1

and since the family 2{0}1 can be proven to contain no primes &gt; base (since all numbers in this family are divisible by 3), it can be removed

and since 20201 is prime, it follows that the family 2{0}2{0}1 splits into the families 2{0}21 and 22{0}1

221 and 2021 are composites, but 20021 is prime, thus add 20021 to ''L''

none of 221, 2201, 22001, 220001, 2200001 are primes, but 22000001 is prime, thus add 22000001 to ''L''

and since the family 3{0,3,6,9}3 can be proven to contain no primes &gt; base (since all numbers in this family are divisible by 3), it can be removed

etc.

Since the number of possible (first digit,last digit) (also called (initial digit,final digit)) combos ([[:w:Ordered pair|ordered pair]]s) of a prime &gt; ''b'' in base ''b'' is (''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(''b'') (all digits except 0 can be the first digit of a prime &gt; ''b'' in base ''b'' (thus ''b''−1 possible digits), but only the digits coprime to ''b'' can be the last digit of a prime &gt; ''b'' in base ''b'' (thus ''eulerphi''(''b'') possible digits), and by the [[:w:Rule of product|rule of product]], there are (''b''−1)×''eulerphi''(''b'') choices of the (first digit,last digit) combo, also, both &quot;numbers of primes in the set of the Athena problem in base ''b''&quot; and &quot;length of the largest prime in the set of the Athena problem in base ''b''&quot; are [[:w:Asymptotic analysis|roughly]] ''[[:w:E (mathematical_constant)|e]]''&lt;sup&gt;''[[:w:Euler's constant|γ]]''×(''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(*b*)&lt;/sup&gt;.

Shrinking the family ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''Y'' is a set of digits in base ''b'')

* If ''y'' ∈ ''Y'' and the string ''xyyz'' represents a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or has a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''}''z'' ∪ ''x''{''Y'' \ ''y''}''y''{''Y'' \ ''y''}''z''.
* If ''y''&lt;sub&gt;1&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;2&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;1&lt;/sub&gt; ≠ ''y''&lt;sub&gt;2&lt;/sub&gt; and the string ''xy''&lt;sub&gt;1&lt;/sub&gt;''y''&lt;sub&gt;2&lt;/sub&gt;''z'' represents a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or has a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''&lt;sub&gt;1&lt;/sub&gt;}{''Y'' \ ''y''&lt;sub&gt;2&lt;/sub&gt;}''z''.
* If ''y''&lt;sub&gt;1&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;2&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;1&lt;/sub&gt; ≠ ''y''&lt;sub&gt;2&lt;/sub&gt; and both the strings ''xy''&lt;sub&gt;1&lt;/sub&gt;''y''&lt;sub&gt;2&lt;/sub&gt;''z'' and ''xy''&lt;sub&gt;2&lt;/sub&gt;''y''&lt;sub&gt;1&lt;/sub&gt;''z'' represent a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or have a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''&lt;sub&gt;1&lt;/sub&gt;}''z'' ∪ ''x''{''Y'' \ ''y''&lt;sub&gt;2&lt;/sub&gt;}''z''.

e.g. in decimal (base ''b'' = 10):

* 2221 is a prime &gt; 10, thus the family 2{0,2}1 splits into the two families 2{0}1 and 2{0}2{0}1.
* 227 is a prime &gt; 10, and it is a subsequence of 5227, thus the family 5{0,2}7 splits into the two families 5{0}7 and 5{0}2{0}7.
* 449 is a prime &gt; 10, and it is a subsequence of 6449, thus the family 6{0,3,4,6,9}9 splits into the two families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9.
* Both 5051 and 5501 are primes &gt; 10, thus the family 5{0,5}1 splits into the two families 5{0}1 and 5{5}1 = {5}1.
* 8501 is a prime &gt; 10, thus the family 8{0,5}1 splits into the family 8{0}{5}1.
* 887 is a prime &gt; 10, and it is a subsequence of 2887, also 2087 is a prime &gt; 10, thus the family 2{0,8}7 splits into the two families 2{0}7 and 28{0}7.
* 349 and 449 are primes &gt; 10, and they are subsequences of 9349 and 9449, respectively, also 9049, 9649, 9949 are primes &gt; 10, thus the family 9{0,3,4,6,9}9 splits into the two families 9{0,3,6,9}9 and 94{0,3,6,9}9.
* 251, 281, 521, 821, 881 are primes &gt; 10, and they are subsequences of 9251, 9281, 9521, 9821, 9881, respectively, also 9001, 9221, 9551, 9851 are primes &gt; 10, thus the family 9{0,2,5,8}1 splits into the numbers {91, 901, 921, 951, 981, 9021, 9051, 9081, 9201, 9501, 9581, 9801, 90581, 95081, 95801}.

If the methods we have discussed cannot be used to rule out or shrink ''x''{''Y''}''z'' where ''Y'' = {''y''&lt;sub&gt;1&lt;/sub&gt;, ''y''&lt;sub&gt;2&lt;/sub&gt;, ..., ''y''&lt;sub&gt;''n''&lt;/sub&gt;}, then we can replace ''x''{''Y''}''z'' by ''xy''&lt;sub&gt;1&lt;/sub&gt;{''Y''}''z'' ∪ ''xy''&lt;sub&gt;2&lt;/sub&gt;{''Y''}''z'' ∪ ... ∪ ''xy''&lt;sub&gt;''n''&lt;/sub&gt;{''Y''}''z'' and re-run the methods on this new [[:w:Formal language|language]].

If all remain families are linear families (i.e. of the form ''x''{''y''}''z'', where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''), then we search the smallest (probable) primes in these families and add these primes to the list.

e.g. in decimal (base ''b'' = 10):

* The smallest prime in the family 5{0}27 is 5000000000000000000000000000027.
* The smallest prime in the family {5}1 is 555555555551.
* The smallest prime in the family 8{5}1 is 8555555555555555555551, but 8555555555555555555551 is not a minimal element since 555555555551 is a subsequence of 8555555555555555555551.

There is no guarantee that the techniques discussed will ever terminate, but in practice they often do. They are able to determine the set of the minimal elements in base ''b'' for 2 ≤ ''b'' ≤ 16 and ''b'' = 18, 20, 22, 24, 30. The bases ''b'' = 17, 19, 21, 23, 25 ≤ ''b'' ≤ 29, 31 ≤ ''b'' ≤ 36 are solved with the exception of 771 families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'').

The following is a &quot;[[:w:Semi-algorithm|semi-algorithm]]&quot; that is guaranteed to solve the Athena problem for a given base ''b'', but it is not so easy to implement:

# ''M'' = ''[[:w:Empty string|∅]]''
# while (''L'' ≠ ''∅'') do
# choose ''x'', a shortest string in ''L''
# ''M'' := ''M'' ∪ {''x''}
# ''L'' := ''L'' − ''sup''({''x''})

In practice, for arbitrary ''L'', we cannot feasibly carry out step 5. Instead, we work with ''L''&amp;#39;, some regular overapproximation to ''L'', until we can show ''L''&amp;#39; = ''∅'' (which implies ''L'' = ''∅''). In practice, ''L''&amp;#39; is usually chosen to be a finite [[:w:Union (set theory)|union]] of sets of the form ''L''&lt;sub&gt;1&lt;/sub&gt;{''L''&lt;sub&gt;2&lt;/sub&gt;}''L''&lt;sub&gt;3&lt;/sub&gt;, where each of ''L''&lt;sub&gt;1&lt;/sub&gt;, ''L''&lt;sub&gt;2&lt;/sub&gt;, ''L''&lt;sub&gt;3&lt;/sub&gt; is finite. In the case we consider in this project, we then have to determine whether such a family contains a prime or not.

Thus, the [[:w:Time complexity|time complexity]] of the Athena problem in base ''b'' may be ''[[:w:Big O notation|O]]''(''[[:w:E (mathematical_constant)|e]]''&lt;sup&gt;''[[:w:Euler's constant|γ]]''×(''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(*b*)&lt;/sup&gt;), and the [[:w:CPU time|CPU time]] of the Athena problem in base ''b'' may be longer than [[:w:Age of the universe|the age of the universe]] for bases ''b'' = 19, 23, 25, 27, 29, 31, 32, 33, 34, 35, also, Athena problem in bases ''b'' around 500 may be [[:w:NP-complete|NP-complete]] or [[:w:NP-hard|NP-hard]], or an [[:w:Undecidable problem|undecidable problem]], or an example of [[:w:Gödel's incompleteness theorems|Gödel's incompleteness theorems]] (like the [[:w:Continuum hypothesis|continuum hypothesis]] and the [[:w:Halting problem|halting problem]]).

To solve the Athena problem, we need to determine whether a given family contains a prime. In practice, if family ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''Y'' is a set of digits in base ''b'') could not be ruled out as only containing composites and ''Y'' contains two or more digits, then a relatively small prime &gt; ''b'' could always be found in this family. Intuitively, this is because there are a large number of small strings in such a family, and at least one is likely to be prime (e.g. there are 2&lt;sup&gt;''n''−2&lt;/sup&gt; strings of length ''n'' in the family 1{3,7}9, and there are over a thousand strings of length 12 in the family 1{3,7}9, thus it is very impossible that these numbers are all composite). In the case ''Y'' contains only one digit, this family is of the form ''x''{''y''}''z'', and there is only a single string of each length &gt; (the length of ''x'' + the length of ''z''), and it is not known if the following [[:w:Decision problem|decision problem]] is recursively solvable (just like [[:w:Sierpiński number|Sierpiński problem]] and [[:w:Riesel number|Riesel problem]], Sierpiński problem and Riesel problem can be generalized to other bases ''b'' (references: http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm), in fact, Athena problem base ''b'' covers the Sierpiński problem base ''b'' and the Riesel problem base ''b'' with ''k'' &lt; ''b'', i.e. finding the smallest prime of the form ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 and ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (or prove such prime does not exist) with ''k'' &lt; ''b'' (specially, for bases ''b'' such that the conjectured smallest Sierpiński number or the conjectured smallest Riesel number is &lt; ''b'', Athena problem base ''b'' covers the Sierpiński problem base ''b'' or the Riesel problem base ''b'', respectively), since the smallest prime of the form ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 and ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (if exists) must be a minimal element in base ''b'', also, Athena problem base ''b'' covers finding the smallest prime of these forms in base ''b'' (or proving that such prime does not exist): (''b''&lt;sup&gt;''n''&lt;/sup&gt;−1)/(''b''−1) (for this form, ''n'' must be prime, and we want ''n'' ≥ 2) (references of this form: http://www.fermatquotient.com/PrimSerien/GenRepu.txt, https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html, http://www.primenumbers.net/Henri/us/MersFermus.htm, http://www.bitman.name/math/table/379, https://pzktupel.de/Primetables/TableRepunitGen.php, https://oeis.org/A084740, https://oeis.org/A084738, https://oeis.org/A128164, https://oeis.org/A285642; or for prime bases ''b'': https://oeis.org/A065854, https://oeis.org/A279068), ''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 (for this form, ''n'' must be power of 2, and we want ''n'' ≥ 1) (references of this form: http://jeppesn.dk/generalized-fermat.html, http://www.noprimeleftbehind.net/crus/GFN-primes.htm, https://web.archive.org/web/20231002190634/http://yves.gallot.pagesperso-orange.fr/primes/index.html, https://pzktupel.de/Primetables/TableFermatGFBB.php, https://oeis.org/A079706, https://oeis.org/A084712, https://oeis.org/A228101), (''b''&lt;sup&gt;''n''&lt;/sup&gt;+1)/2 (for odd ''b'') (for this form, ''n'' must be power of 2, and we want ''n'' ≥ 2) (reference of this form: http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt), (''sqrt''(''b'')×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1)/(''sqrt''(''b'')+1) (for square ''b'') (for this form, 2×''n''+1 must be prime, and we want ''n'' ≥ 2) (references of this form: http://www.fermatquotient.com/PrimSerien/GenRepuP.txt, http://www.primenumbers.net/Henri/us/MersFermus.htm, http://www.bitman.name/math/table/488, https://pzktupel.de/Primetables/TableWagstaffGen.php, https://oeis.org/A084742, https://oeis.org/A084741; or for bases ''b'' with ''sqrt''(''b'') prime: https://oeis.org/A065507), ((''b''−2)×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1)/(''b''−1) (''n'' ≥ 2) (reference of this form: https://oeis.org/A243404), 2×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 (''n'' ≥ 1) (references of this form: https://www.mersenneforum.org/showthread.php?t=6918, https://www.mersenneforum.org/showthread.php?t=19725, https://oeis.org/A119624), 2×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (''n'' ≥ 1) (references of this form: https://www.mersenneforum.org/showthread.php?t=24576, https://www.mersenneforum.org/attachment.php?attachmentid=20976&amp;d=1567314217, https://oeis.org/A119591), ''b''&lt;sup&gt;''n''&lt;/sup&gt;+2 (''n'' ≥ 1) (references of this form: https://oeis.org/A138066, https://oeis.org/A084713, https://oeis.org/A138067), ''b''&lt;sup&gt;''n''&lt;/sup&gt;−2 (''n'' ≥ 2) (references of this form: https://www.primepuzzles.net/puzzles/puzz_887.htm, https://oeis.org/A250200, https://oeis.org/A255707, https://oeis.org/A084714; or for prime bases ''b'': https://oeis.org/A292201), (''b''−1)×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 (''n'' ≥ 1) (references of this form: http://www.noprimeleftbehind.net/Williams-primes-MP.htm, http://www.bitman.name/math/table/477, https://pzktupel.de/Primetables/TableWilliams2.php, https://oeis.org/A305531; or for prime bases ''b'': https://oeis.org/A087139), (''b''−1)×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (''n'' ≥ 1) (references of this form: https://harvey563.tripod.com/wills.txt, http://www.noprimeleftbehind.net/Williams-primes-MM.htm, http://www.bitman.name/math/table/484, https://pzktupel.de/Primetables/TableWilliams1.php; or for prime bases ''b'': https://oeis.org/A122396), ''b''&lt;sup&gt;''n''&lt;/sup&gt;+(''b''−1) (''n'' ≥ 1) (references of this form: http://www.bitman.name/math/table/795, https://pzktupel.de/Primetables/TableWilliams6.php, https://oeis.org/A076845, https://oeis.org/A076846, https://oeis.org/A078178, https://oeis.org/A078179), ''b''&lt;sup&gt;''n''&lt;/sup&gt;−(''b''−1) (''n'' ≥ 2) (references of this form: http://www.bitman.name/math/table/792, https://pzktupel.de/Primetables/TableWilliams5.php, https://oeis.org/A113516, https://oeis.org/A343589; or for prime bases ''b'': https://cs.uwaterloo.ca/journals/JIS/VOL3/mccranie.html, http://www.bitman.name/math/table/435)):

Problem: Given strings ''x'', ''z'' (may be empty), a digit ''y'', and a base ''b'' (''x'' does not [[:w:Leading zero|start with the digit 0]], ''z'' ends with a digit which [[:w:Coprime integers|coprime]] to ''b'', ''y'' is not 0 if ''x'' is empty, ''y'' is coprime to ''b'' if ''z'' is empty), does there exist a prime number whose base-''b'' expansion is of the form ''xy''&lt;sub&gt;''n''&lt;/sub&gt;''z'' for some ''n'' ≥ 0?

Some families can be ruled out to contain no prime &gt; ''b'' by [[:w:Covering set|covering congruence]], [[:w:Factorization of polynomials|algebraic factorization]] (e.g. [[:w:Difference of two squares|difference of two squares]], [[:w:Sum of two cubes|sum of two cubes]], [[:w:Sophie Germain's identity|Sophie Germain's identity of ''x''&lt;sup&gt;4&lt;/sup&gt;+4×''y''&lt;sup&gt;4&lt;/sup&gt;]]), or combine of them, e.g.

* The base 9 family 2{7}: Always divisible by 2 or 5
* The base 11 family 2{5}: Always divisible by 2 or 3
* The base 14 family B{0}1: Always divisible by 3 or 5
* The base 13 family 95{0}3: Always divisible by 5, 7, or 17
* The base 16 family {4}D: Always divisible by 3, 7, or 13
* The base 16 family {8}F: Always divisible by 3, 7, or 13
* The base 21 family {7}D: Always divisible by 2, 13, or 17
* The base 23 family {D}GA: Always divisible by 2, 5, 7, 37, or 79
* The base 9 family {1}: Can be written as (9&lt;sup&gt;''n''&lt;/sup&gt;−1)/8 and can be factored as (3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (3&lt;sup&gt;''n''&lt;/sup&gt;+1) / 8
* The base 8 family 1{0}1: Can be written as 8&lt;sup&gt;''n''&lt;/sup&gt;+1 and can be factored as (2&lt;sup&gt;''n''&lt;/sup&gt;+1) × (4&lt;sup&gt;''n''&lt;/sup&gt;−2&lt;sup&gt;''n''&lt;/sup&gt;+1)
* The base 9 family 3{8}: Can be written as 4×9&lt;sup&gt;''n''&lt;/sup&gt;−1 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (2×3&lt;sup&gt;''n''&lt;/sup&gt;+1)
* The base 16 family 1{5}: Can be written as (4×16&lt;sup&gt;''n''&lt;/sup&gt;−1)/3 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (2×3&lt;sup&gt;''n''&lt;/sup&gt;+1) / 3
* The base 16 family {4}1: Can be written as (4×16&lt;sup&gt;''n''&lt;/sup&gt;−49)/15 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−7) × (2×3&lt;sup&gt;''n''&lt;/sup&gt;+7) / 15
* The base 27 family 7{Q}: Can be written as 8×27&lt;sup&gt;''n''&lt;/sup&gt;−1 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (4×9&lt;sup&gt;''n''&lt;/sup&gt;+2×3&lt;sup&gt;''n''&lt;/sup&gt;+1)
* The base 27 family 9{G}: Can be written as (125×27&lt;sup&gt;''n''&lt;/sup&gt;−8)/13 and can be factored as (5×3&lt;sup&gt;''n''&lt;/sup&gt;−2) × (25×9&lt;sup&gt;''n''&lt;/sup&gt;+10×3&lt;sup&gt;''n''&lt;/sup&gt;+4)
* The base 16 family {C}D: Can be written as (4×16&lt;sup&gt;''n''&lt;/sup&gt;+1)/5 and can be factored as (2×4&lt;sup&gt;''n''&lt;/sup&gt;−2×2&lt;sup&gt;''n''&lt;/sup&gt;+1) × (2×4&lt;sup&gt;''n''&lt;/sup&gt;+2×2&lt;sup&gt;''n''&lt;/sup&gt;+1) / 5
* The base 14 family 8{D}: Can be written as 9×14&lt;sup&gt;''n''&lt;/sup&gt;−1, it is divisible by 5 if ''n'' is odd and can be factored as (3×14&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (3×14&lt;sup&gt;''n''/2&lt;/sup&gt;+1) if ''n'' is even
* The base 12 family {B}9B: Can be written as 12&lt;sup&gt;''n''&lt;/sup&gt;−25, it is divisible by 13 if ''n'' is odd and can be factored as (12&lt;sup&gt;''n''/2&lt;/sup&gt;−5) × (12&lt;sup&gt;''n''/2&lt;/sup&gt;+5) if ''n'' is even
* The base 14 family {D}5: Can be written as 14&lt;sup&gt;''n''&lt;/sup&gt;−9, it is divisible by 5 if ''n'' is odd and can be factored as (14&lt;sup&gt;''n''/2&lt;/sup&gt;−3) × (14&lt;sup&gt;''n''/2&lt;/sup&gt;+3) if ''n'' is even
* The base 17 family 1{9}: Can be written as (25×17&lt;sup&gt;''n''&lt;/sup&gt;−9)/16, it is divisible by 2 if ''n'' is odd and can be factored as (5×17&lt;sup&gt;''n''/2&lt;/sup&gt;−3) × (5×17&lt;sup&gt;''n''/2&lt;/sup&gt;+3) / 16 if ''n'' is even
* The base 17 family 7{9}: Can be written as (121×17&lt;sup&gt;''n''&lt;/sup&gt;−9)/16, it is divisible by 2 if ''n'' is odd and can be factored as (11×17&lt;sup&gt;''n''/2&lt;/sup&gt;−3) × (11×17&lt;sup&gt;''n''/2&lt;/sup&gt;+3) / 16 if ''n'' is even
* The base 19 family 1{6}: Can be written as (4×19&lt;sup&gt;''n''&lt;/sup&gt;−1)/3, it is divisible by 5 if ''n'' is odd and can be factored as (2×19&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (2×19&lt;sup&gt;''n''/2&lt;/sup&gt;+1) / 3 if ''n'' is even
* The base 19 family 7{2}: Can be written as (64×19&lt;sup&gt;''n''&lt;/sup&gt;−1)/9, it is divisible by 5 if ''n'' is odd and can be factored as (8×19&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (8×19&lt;sup&gt;''n''/2&lt;/sup&gt;+1) / 9 if ''n'' is even
* The base 24 family 3{N}: Can be written as 4×24&lt;sup&gt;''n''&lt;/sup&gt;−1, it is divisible by 5 if ''n'' is odd and can be factored as (2×24&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (2×24&lt;sup&gt;''n''/2&lt;/sup&gt;+1) if ''n'' is even

By the [[:w:Prime number theorem|prime number theorem]], the [[:w:Probability|chance]] that a [[:w:Random number|random]] ''n''-digit base ''b'' number is prime is [[:w:Asymptotic analysis|approximately]] 1/''n'' (more accurately, the chance is approximately 1/(''n''×''ln''(''b'')), where ''ln'' is the [[:w:Natural logarithm|natural logarithm]]). If one conjectures the numbers ''x''{''y''}''z'' behave similarly (i.e. the numbers ''x''{''y''}''z'' is a [[:w:Pseudorandomness|pseudorandom sequence]]) you would expect [[:w:Harmonic_series (mathematics)|1/1 + 1/2 + 1/3 + 1/4 + ... = ∞]] primes of the form ''x''{''y''}''z'' (of course, this does not always happen, since some ''x''{''y''}''z'' families can be ruled out to contain no prime &gt; ''b'' (by covering congruence, algebraic factorization, or combine of them), but it is at least a reasonable conjecture in the absence of evidence to the contrary. Hence, the [[:w:Heuristic argument|heuristic argument]] suggests there are always infinitely many primes in family ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') if it cannot be ruled out to contain no prime or only contain finitely many primes, by covering congruence, algebraic factorization, or combine of them. However, some families ''x''{''y''}''z'' could not be proven to contain no primes &gt; ''b'' (by covering congruence, algebraic factorization, or combine of them) but no primes &gt; ''b'' could be found in the family, even after searching through numbers with over 100000 digits. In such a case, the only way to proceed is to [[:w:Primality test|test the primality]] of larger and larger numbers of such form and hope a prime is eventually discovered. e.g. the smallest (probable) prime in the family A{3}A in base ''b'' = 13 is A3&lt;sub&gt;592197&lt;/sub&gt;A, its algebraic form is (41×13&lt;sup&gt;592198&lt;/sup&gt;+27)/4, when written in decimal contains 659677 digits (it is only probable prime, i.e. not definitely prime, since technically, probable primality tests were used to show this (which have a ''very'' small chance of making an error, see https://t5k.org/notes/prp_prob.html) because all known primality tests run far too slowly to run on numbers of this size unless either [https://t5k.org/prove/prove3_1.html ''N''−1] or [https://t5k.org/prove/prove3_2.html ''N''+1] (or both) can be ≥ 1/3 factored).

The numbers in family ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') are of the form (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1) for some fixed ''a'', ''b'', ''c'' such that ''a'' ≥ 1, ''b'' ≥ 2 (''b'' is the base), ''c'' ≠ 0, ''gcd''(''a'',''c'') = 1, ''gcd''(''b'',''c'') = 1. Except in the [[:w:Special case|special case]] ''c'' = ±1 and ''gcd''(''a''+''c'',''b''−1) = 1, when ''n'' is large the known [[:w:Primality test|primality test]]s for such a number are too inefficient to run. In this case one must resort to a [[:w:Probabilistic algorithm|probable]] primality test such as a [[:w:Miller–Rabin primality test|Miller–Rabin primality test]] or a [[:w:Baillie–PSW primality test|Baillie–PSW primality test]], unless a divisor of the number can be found. Since we are testing many numbers in an [[:w:Exponential growth|exponential sequence]], it is possible to use a sieving process to find divisors rather than using [[:w:Trial division|trial division]].

To do this, we made use of Geoffrey Reynolds' ''srsieve'' software (download: https://pzktupel.de/Software/srsieve_1.1.4.7z). This program uses the [[:w:Baby-step giant-step|baby-step giant-step]] [[:w:Algorithm|algorithm]] to find all primes ''p'' which divide ''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'' where ''p'' and ''n'' lie in a [[:w:Interval_(mathematics)|specified range]]. Since this program cannot handle the general case (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1) when ''gcd''(''a''+''c'',''b''−1) &gt; 1 we only used it to sieve the sequence ''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'' for primes ''p'' not dividing ''gcd''(''a''+''c'',''b''−1), and initialized the list of candidates to not include ''n'' for which there is some prime ''p'' dividing ''gcd''(''a''+''c'',''b''−1) for which ''p'' dividing (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1). The program had to be modified slightly to remove a check which would prevent it from running in the case when ''a'', ''b'', and ''c'' were all odd (since then 2 divides ''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'', but 2 may not divide (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1)).

Once the numbers with small divisors had been removed, it remained to test the remaining numbers using a probable primality test. For this we used the software ''LLR'' by Jean Penné. (download: http://jpenne.free.fr/index2.html). Although undocumented, it is possible to run this program on numbers of the form (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1) when ''gcd''(''a''+''c'',''b''−1) &gt; 1, so this program required no modifications. A script was also written which allowed one to run ''srsieve'' while ''LLR'' was testing the remaining candidates, so that when a divisor was found by srsieve on a number which had not yet been tested by ''LLR'' it would be removed from the list of candidates.

For the primes &lt; 10&lt;sup&gt;25000&lt;/sup&gt; for the &quot;easy&quot; bases (bases ''b'' with ≤ 150 primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; (base ''b'' = 26 has 83 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 3 unsolved families, base ''b'' = 36 has 75 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 4 unsolved families, base ''b'' = 17 has 99 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 18 unsolved families, base ''b'' = 21 has 80 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 12 unsolved families, base ''b'' = 19 has 201 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 23 unsolved families), i.e. bases *b* = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 26, 28, 30, 36), we employed ''CM'' by Andreas Enge (download: https://www.multiprecision.org/cm/download.html), an elliptic curve primality proving implementation.

== Data ==

These are the results of the Athena problem in bases 2 ≤ ''b'' ≤ 36 (we stop at base 36 since this base is the maximum base for which it is possible to write the numbers with the [[:w:Symbol|symbol]]s 0, 1, 2, ..., 9 and A, B, C, ..., Z (i.e. the 10 [[:w:Arabic numerals|Arabic numerals]] and the 26 [[:w:Latin script|Latin letters]]): (some large primes are only probable primes, i.e. not definitely primes, since they are too large to be [[:w:Elliptic curve primality|ECPP proved]] and [[:w:Pocklington primality test#Extensions and variants|neither ''N''−1 nor ''N''+1 can be ≥ 1/3 factored]], all of them pass the [[:w:Baillie–PSW primality test|Baillie–PSW primality test]] and the [[:w:Strong pseudoprime|strong primality test]] (i.e. the [[:w:Miller–Rabin primality test|Miller–Rabin primality test]]) with all prime bases ''p'' ≤ 61, however, all primes &lt; 10&lt;sup&gt;25000&lt;/sup&gt; for bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 36 are definitely primes, most of them &gt; 10&lt;sup&gt;299&lt;/sup&gt; are proven primes with [[:w:Elliptic curve primality|ECPP proving]], others &gt; 10&lt;sup&gt;299&lt;/sup&gt; are proven primes with [[:w:Pocklington primality test#Extensions and variants|''N''−1 or ''N''+1 proving]])

All numbers are written in base ''b'', [[:w:Senary#Base 36 as senary compression|using A to Z to represent digit values 10 to 35]], &quot;{}&quot; means repeating, e.g. family 12{3}45 means the sequence {1245, 12345, 123345, 1233345, 12333345, 123333345, ...} (where the members are expressed as base ''b'' strings), subscripts are used to indicate repetitions of digits, e.g. 123&lt;sub&gt;4&lt;/sub&gt;567 means 123333567 (all subscripts are written in decimal).

Base 2: 1 Athena prime (the largest of which has 2 digits (it is 11, and its value is 3 in decimal)): {11}

Base 3: 3 Athena primes (the largest of which has 3 digits (it is 111, and its value is 13 in decimal)): {12, 21, 111}

Base 4: 5 Athena primes (the largest of which has 3 digits (it is 221, and its value is 41 in decimal)): {11, 13, 23, 31, 221}

Base 5: 22 Athena primes (the largest of which has 96 digits (it is 10&lt;sub&gt;93&lt;/sub&gt;13, and its algebraic form is 5&lt;sup&gt;95&lt;/sup&gt;+8)): {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 Athena primes (the largest of which has 5 digits (it is 40041, and its value is 5209 in decimal)): {11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041}

Base 7: 71 Athena primes (the largest of which has 17 digits (it is 3&lt;sub&gt;16&lt;/sub&gt;1, and its algebraic form is (7&lt;sup&gt;17&lt;/sup&gt;−5)/2)): {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: 75 Athena primes (the largest of which has 221 digits (it is 4&lt;sub&gt;220&lt;/sub&gt;7, and its algebraic form is (4×8&lt;sup&gt;221&lt;/sup&gt;+17)/7)): {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: 151 Athena primes (the largest of which has 1161 digits (it is 30&lt;sub&gt;1158&lt;/sub&gt;11, and its algebraic form is 3×9&lt;sup&gt;1160&lt;/sup&gt;+10)): {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: 77 Athena primes (the largest of which has 31 digits (it is 50&lt;sub&gt;28&lt;/sub&gt;27, and its algebraic form is 5×10&lt;sup&gt;30&lt;/sup&gt;+27)): {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: 1068 Athena primes (including 1 unproven probable prime: 57&lt;sub&gt;62668&lt;/sub&gt;), the largest of which has 62669 digits (it is 57&lt;sub&gt;62668&lt;/sub&gt;, and its algebraic form is (57×11&lt;sup&gt;62668&lt;/sup&gt;−7)/10), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel11 Data of Athena (probable) primes base 11]

Base 12: 106 Athena primes (the largest of which has 42 digits (it is 40&lt;sub&gt;39&lt;/sub&gt;77, and its algebraic form is 4×12&lt;sup&gt;41&lt;/sup&gt;+91)): {11, 15, 17, 1B, 25, 27, 31, 35, 37, 3B, 45, 4B, 51, 57, 5B, 61, 67, 6B, 75, 81, 85, 87, 8B, 91, 95, A7, AB, B5, B7, 221, 241, 2A1, 2B1, 2BB, 401, 421, 447, 471, 497, 565, 655, 665, 701, 70B, 721, 747, 771, 77B, 797, 7A1, 7BB, 907, 90B, 9BB, A41, B21, B2B, 2001, 200B, 202B, 222B, 229B, 292B, 299B, 4441, 4707, 4777, 6A05, 6AA5, 729B, 7441, 7B41, 929B, 9777, 992B, 9947, 997B, 9997, A0A1, A201, A605, A6A5, AA65, B001, B0B1, BB01, BB41, 600A5, 7999B, 9999B, AAAA1, B04A1, B0B9B, BAA01, BAAA1, BB09B, BBBB1, 44AAA1, A00065, BBBAA1, AAA0001, B00099B, AA000001, BBBBBB99B, B0000000000000000000000000009B, 400000000000000000000000000000000000000077}

Base 13: 3197 Athena primes (including 4 unproven probable primes: C5&lt;sub&gt;23755&lt;/sub&gt;C, 80&lt;sub&gt;32017&lt;/sub&gt;111, 95&lt;sub&gt;197420&lt;/sub&gt;, A3&lt;sub&gt;592197&lt;/sub&gt;A), the largest of which has 592199 digits (it is A3&lt;sub&gt;592197&lt;/sub&gt;A, and its algebraic form is (41×13&lt;sup&gt;592198&lt;/sup&gt;+27)/4), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel13 Data of Athena (probable) primes base 13]

Base 14: 650 Athena primes, the largest of which has 19699 digits (it is 4D&lt;sub&gt;19698&lt;/sub&gt;, and its algebraic form is 5×14&lt;sup&gt;19698&lt;/sup&gt;−1), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel14 Data of Athena primes base 14]

Base 15: 1284 Athena primes, the largest of which has 157 digits (it is 7&lt;sub&gt;155&lt;/sub&gt;97, and its algebraic form is (15&lt;sup&gt;157&lt;/sup&gt;+59)/2), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel15 Data of Athena primes base 15]

Base 16: 2347 Athena primes (including 3 unproven probable primes: DB&lt;sub&gt;32234&lt;/sub&gt;, 4&lt;sub&gt;72785&lt;/sub&gt;DD, 3&lt;sub&gt;116137&lt;/sub&gt;AF), the largest of which has 116139 digits (it is 3&lt;sub&gt;116137&lt;/sub&gt;AF, and its algebraic form is (16&lt;sup&gt;116139&lt;/sup&gt;+619)/5), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel16 Data of Athena (probable) primes base 16]

Base 17: 10415 known Athena primes (including many unproven probable primes) and 12 unsolved families (1{7}, 1F{0}7, 4{7}A, 70F{0}D, 8{B}9, 9{5}9, A{D}F, B{0}B3, {B}E9, {B}EE, F1{9}, FD0{D}, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel17 Data of known Athena (probable) primes base 17]

Base 18: 549 Athena primes, the largest of which has 6271 digits (it is C0&lt;sub&gt;6268&lt;/sub&gt;C5, and its algebraic form is 12×18&lt;sup&gt;6270&lt;/sup&gt;+221), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel18 Data of Athena primes base 18]

Base 19: 31417 known Athena primes (including many unproven probable primes) and 17 unsolved families (4B5{0}H, {5}3, 5{H}05, 5{H}0H, 5{H}5, 66{B}, 71{0}177, 7AF{0}H, 97{0}3, C{H}C, EE1{6}, F{7}5, F{B}G, F{D}F, H0F{0}7A, HB{0}5B5, II{D}, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel19 Data of known Athena (probable) primes base 19]

Base 20: 3314 Athena primes, the largest of which has 6271 digits (it is G0&lt;sub&gt;6269&lt;/sub&gt;D, and its algebraic form is 16×20&lt;sup&gt;6270&lt;/sup&gt;+13), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel20 Data of Athena primes base 20]

Base 21: 13386 known Athena primes (including many unproven probable primes) and 8 unsolved families (5{0}DJ, {9}D, B3{0}EB, B{H}6H, C{F}0K, {F}35, G{0}FK, H{0}7771, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel21 Data of known Athena (probable) primes base 21]

Base 22: 8003 Athena primes (including 1 unproven probable prime: BK&lt;sub&gt;22001&lt;/sub&gt;5), the largest of which has 22003 digits (it is BK&lt;sub&gt;22001&lt;/sub&gt;5, and its algebraic form is (251×22&lt;sup&gt;22002&lt;/sup&gt;−335)/21), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel22 Data of Athena (probable) primes base 22]

Base 23: 65178 known Athena primes (including many unproven probable primes) and 87 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel23 Data of known Athena (probable) primes base 23] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left23 Data of unsolved families for Athena problem base 23]

Base 24: 3409 Athena primes, the largest of which has 8134 digits (it is N00N&lt;sub&gt;8129&lt;/sub&gt;LN, and its algebraic form is 13249×24&lt;sup&gt;8131&lt;/sup&gt;−49), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel24 Data of Athena primes base 24]

Base 25: 133639 known Athena primes (including many unproven probable primes) and 85 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel25 Data of known Athena (probable) primes base 25] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left25 Data of unsolved families for Athena problem base 25]

Base 26: 25256 known Athena primes (including 7 unproven probable primes: 5&lt;sub&gt;19391&lt;/sub&gt;6F, 7&lt;sub&gt;20279&lt;/sub&gt;OL, LD0&lt;sub&gt;20975&lt;/sub&gt;7, 6K&lt;sub&gt;23300&lt;/sub&gt;5, J0&lt;sub&gt;44303&lt;/sub&gt;KCB, M0&lt;sub&gt;61186&lt;/sub&gt;2BB, 85M&lt;sub&gt;197060&lt;/sub&gt;B) and 3 unsolved families ({A}6F, {H}MH, {I}GL, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel26 Data of known Athena (probable) primes base 26]

Base 27: 102852 known Athena primes (including many unproven probable primes) and 44 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel27 Data of known Athena (probable) primes base 27] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left27 Data of unsolved families for Athena problem base 27]

Base 28: 25528 known Athena primes (including 3 unproven probable primes: N6&lt;sub&gt;24051&lt;/sub&gt;LR, 5OA&lt;sub&gt;31238&lt;/sub&gt;F, O4O&lt;sub&gt;94535&lt;/sub&gt;9) and 1 unsolved family (O{A}F, no primes or probable primes with length ≤ 900000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel28 Data of known Athena (probable) primes base 28]

Base 29: 355242 known Athena primes (including many unproven probable primes) and 125 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel29 Data of known Athena (probable) primes base 29] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left29 Data of unsolved families for Athena problem base 29]

Base 30: 2619 Athena primes (including 1 unproven probable prime: I0&lt;sub&gt;24608&lt;/sub&gt;D), the largest of which has 34206 digits (it is OT&lt;sub&gt;34205&lt;/sub&gt;, and its algebraic form is 25×30&lt;sup&gt;34205&lt;/sup&gt;−1), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel30 Data of Athena (probable) primes base 30]

Base 31: 569323 known Athena primes (including many unproven probable primes) and 77 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel31 Data of known Athena (probable) primes base 31] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left31 Data of unsolved families for Athena problem base 31]

Base 32: 168882 known Athena primes (including many unproven probable primes) and 120 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel32 Data of known Athena (probable) primes base 32] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left32 Data of unsolved families for Athena problem base 32]

Base 33: 280012 known Athena primes (including many unproven probable primes) and 81 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel33 Data of known Athena (probable) primes base 33] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left33 Data of unsolved families for Athena problem base 33]

Base 34: 184785 known Athena primes (including many unproven probable primes) and 47 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel34 Data of known Athena (probable) primes base 34] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left34 Data of unsolved families for Athena problem base 34]

Base 35: 720002 known Athena primes (including many unproven probable primes) and 60 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel35 Data of known Athena (probable) primes base 35] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left35 Data of unsolved families for Athena problem base 35]

Base 36: 35286 known Athena primes (including 3 unproven probable primes: 7K&lt;sub&gt;26567&lt;/sub&gt;Z, S0&lt;sub&gt;75007&lt;/sub&gt;8H, P&lt;sub&gt;81993&lt;/sub&gt;SZ) and 4 unsolved families (B{0}EUV, HM{0}N, N{0}YYN, O{L}Z, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel36 Data of known Athena (probable) primes base 36]

== Condensed table for bases 2 ≤ ''b'' ≤ 36 ==

{|class=&quot;wikitable&quot;
||''b''||number of primes (or probable prime, which is a minimal prime assuming its primality) in the Athena problem base ''b''||base-''b'' form of the top 10 known primes (or probable prime, which is a minimal prime assuming its primality) in the Athena problem base ''b'' (write &quot;''d''&lt;sub&gt;''n''&lt;/sub&gt;&quot; if there are 5 or more (''n'') consecutive same digits ''d'')||length of the top 10 known primes (or probable prime, which is a minimal prime assuming its primality) in the Athena problem base ''b''||length of the top 10 known primes (or probable prime, which is a minimal prime assuming its primality) in the Athena problem base ''b'' in decimal||algebraic ((''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1)) form of the top 10 known primes (or probable prime, which is a minimal prime assuming its primality) in the Athena problem base ''b''||''factordb'' entry of the top 10 known primes (or probable prime, which is a minimal prime assuming its primality) in the Athena problem base ''b''||the top 10 known primes (or probable prime, which is a minimal prime assuming its primality) in the Athena problem base ''b'' written in base ''b'' (use lower case letters instead of upper case letters)||number of left families in the Athena problem base ''b'' (all of these left families are linear families)||searching limit of length for the left families in the Athena problem base ''b'' (if there are different searching limits for the left families in base ''b'', choose the lowest searching limit)||
|-
||2||1||11||2||1||3||http://factordb.com/index.php?id=3&amp;open=ecm||http://factordb.com/index.php?showid=3&amp;base=2||0||–||
|-
||3||3||111&lt;br&gt;21&lt;br&gt;12||3&lt;br&gt;2&lt;br&gt;2||2&lt;br&gt;1&lt;br&gt;1||13&lt;br&gt;7&lt;br&gt;5||http://factordb.com/index.php?id=13&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=7&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=5&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=13&amp;base=3&lt;br&gt;http://factordb.com/index.php?showid=7&amp;base=3&lt;br&gt;http://factordb.com/index.php?showid=5&amp;base=3&lt;nowiki/&gt;||0||–||
|-
||4||5||221&lt;br&gt;31&lt;br&gt;23&lt;br&gt;13&lt;br&gt;11||3&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2||2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;1&lt;br&gt;1||41&lt;br&gt;13&lt;br&gt;11&lt;br&gt;7&lt;br&gt;5||http://factordb.com/index.php?id=41&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=13&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=11&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=7&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=5&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=41&amp;base=4&lt;br&gt;http://factordb.com/index.php?showid=13&amp;base=4&lt;br&gt;http://factordb.com/index.php?showid=11&amp;base=4&lt;br&gt;http://factordb.com/index.php?showid=7&amp;base=4&lt;br&gt;http://factordb.com/index.php?showid=5&amp;base=4&lt;nowiki/&gt;||0||–||
|-
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||7||71||3&lt;sub&gt;16&lt;/sub&gt;1&lt;br&gt;510&lt;sub&gt;7&lt;/sub&gt;1&lt;br&gt;3&lt;sub&gt;6&lt;/sub&gt;01&lt;br&gt;1100021&lt;br&gt;531101&lt;br&gt;351101&lt;br&gt;300053&lt;br&gt;150001&lt;br&gt;100121&lt;br&gt;40054||17&lt;br&gt;10&lt;br&gt;8&lt;br&gt;7&lt;br&gt;6&lt;br&gt;6&lt;br&gt;6&lt;br&gt;6&lt;br&gt;6&lt;br&gt;5||15&lt;br&gt;9&lt;br&gt;7&lt;br&gt;6&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;4||(7&lt;sup&gt;17&lt;/sup&gt;−5)/2&lt;br&gt;36×7&lt;sup&gt;8&lt;/sup&gt;+1&lt;br&gt;(7&lt;sup&gt;8&lt;/sup&gt;−47)/2&lt;br&gt;134471&lt;br&gt;91631&lt;br&gt;62819&lt;br&gt;50459&lt;br&gt;28813&lt;br&gt;16871&lt;br&gt;9643||http://factordb.com/index.php?id=116315256993601&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=207532837&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=2882377&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=134471&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=91631&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=62819&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=50459&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=28813&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=16871&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=9643&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=116315256993601&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=207532837&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=2882377&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=134471&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=91631&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=62819&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=50459&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=28813&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=16871&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=9643&amp;base=7&lt;nowiki/&gt;||0||–||
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||8||75||4&lt;sub&gt;220&lt;/sub&gt;7&lt;br&gt;5&lt;sub&gt;13&lt;/sub&gt;25&lt;br&gt;7&lt;sub&gt;12&lt;/sub&gt;1&lt;br&gt;77774&lt;sub&gt;6&lt;/sub&gt;1&lt;br&gt;74&lt;sub&gt;7&lt;/sub&gt;1&lt;br&gt;4&lt;sub&gt;8&lt;/sub&gt;1&lt;br&gt;5&lt;sub&gt;5&lt;/sub&gt;025&lt;br&gt;5550525&lt;br&gt;5500525&lt;br&gt;4&lt;sub&gt;5&lt;/sub&gt;77||221&lt;br&gt;15&lt;br&gt;13&lt;br&gt;11&lt;br&gt;9&lt;br&gt;9&lt;br&gt;8&lt;br&gt;7&lt;br&gt;7&lt;br&gt;7||200&lt;br&gt;14&lt;br&gt;12&lt;br&gt;10&lt;br&gt;9&lt;br&gt;8&lt;br&gt;8&lt;br&gt;7&lt;br&gt;7&lt;br&gt;7||(4×8&lt;sup&gt;221&lt;/sup&gt;+17)/7&lt;br&gt;(5×8&lt;sup&gt;15&lt;/sup&gt;−173)/7&lt;br&gt;8&lt;sup&gt;13&lt;/sup&gt;−7&lt;br&gt;(28669×8&lt;sup&gt;7&lt;/sup&gt;−25)/7&lt;br&gt;(53×8&lt;sup&gt;8&lt;/sup&gt;−25)/7&lt;br&gt;(4×8&lt;sup&gt;9&lt;/sup&gt;−25)/7&lt;br&gt;(5×8&lt;sup&gt;8&lt;/sup&gt;−2413)/7&lt;br&gt;1495381&lt;br&gt;1474901&lt;br&gt;(4×8&lt;sup&gt;7&lt;/sup&gt;+185)/7||http://factordb.com/index.php?id=1100000000416605822&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=25131694349141&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=549755813881&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=8589035809&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=127027489&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=76695841&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=11983381&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1495381&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1474901&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1198399&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000000416605822&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=25131694349141&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=549755813881&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=8589035809&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=127027489&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=76695841&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=11983381&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=1495381&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=1474901&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=1198399&amp;base=8&lt;nowiki/&gt;||0||–||
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||9||151||30&lt;sub&gt;1158&lt;/sub&gt;11&lt;br&gt;27&lt;sub&gt;686&lt;/sub&gt;07&lt;br&gt;76&lt;sub&gt;329&lt;/sub&gt;2&lt;br&gt;561&lt;sub&gt;36&lt;/sub&gt;&lt;br&gt;10&lt;sub&gt;25&lt;/sub&gt;57&lt;br&gt;30&lt;sub&gt;20&lt;/sub&gt;51&lt;br&gt;8&lt;sub&gt;19&lt;/sub&gt;335&lt;br&gt;727&lt;sub&gt;15&lt;/sub&gt;07&lt;br&gt;51&lt;sub&gt;13&lt;/sub&gt;61&lt;br&gt;10&lt;sub&gt;11&lt;/sub&gt;507||1161&lt;br&gt;689&lt;br&gt;331&lt;br&gt;38&lt;br&gt;28&lt;br&gt;23&lt;br&gt;22&lt;br&gt;19&lt;br&gt;16&lt;br&gt;15||1108&lt;br&gt;657&lt;br&gt;316&lt;br&gt;37&lt;br&gt;26&lt;br&gt;22&lt;br&gt;21&lt;br&gt;19&lt;br&gt;16&lt;br&gt;14||3×9&lt;sup&gt;1160&lt;/sup&gt;+10&lt;br&gt;(23×9&lt;sup&gt;688&lt;/sup&gt;−511)/8&lt;br&gt;(31×9&lt;sup&gt;330&lt;/sup&gt;−19)/4&lt;br&gt;(409×9&lt;sup&gt;36&lt;/sup&gt;−1)/8&lt;br&gt;9&lt;sup&gt;27&lt;/sup&gt;+52&lt;br&gt;3×9&lt;sup&gt;22&lt;/sup&gt;+46&lt;br&gt;9&lt;sup&gt;22&lt;/sup&gt;−454&lt;br&gt;(527×9&lt;sup&gt;17&lt;/sup&gt;−511)/8&lt;br&gt;(41×9&lt;sup&gt;15&lt;/sup&gt;+359)/8&lt;br&gt;9&lt;sup&gt;14&lt;/sup&gt;+412||http://factordb.com/index.php?id=1100000002376318423&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000002495467486&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000002359003642&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000001554010824&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1100000002512830927&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1100000000032261811&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1100000002495736583&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1100000003446800389&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1055192051985121&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=22876792455373&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000002376318423&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=1100000002495467486&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=1100000002359003642&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=1100000001554010824&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=1100000002512830927&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=1100000000032261811&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=1100000002495736583&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=1100000003446800389&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=1055192051985121&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=22876792455373&amp;base=9&lt;nowiki/&gt;||0||–||
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||24||3409||N00N&lt;sub&gt;8129&lt;/sub&gt;LN&lt;br&gt;88N&lt;sub&gt;5951&lt;/sub&gt;&lt;br&gt;A0&lt;sub&gt;2951&lt;/sub&gt;8ID&lt;br&gt;D&lt;sub&gt;2698&lt;/sub&gt;LD&lt;br&gt;N&lt;sub&gt;2644&lt;/sub&gt;LLN&lt;br&gt;BC0&lt;sub&gt;331&lt;/sub&gt;B&lt;br&gt;20&lt;sub&gt;313&lt;/sub&gt;7&lt;br&gt;C7&lt;sub&gt;298&lt;/sub&gt;&lt;br&gt;D0&lt;sub&gt;259&lt;/sub&gt;KKD&lt;br&gt;I0&lt;sub&gt;241&lt;/sub&gt;I5||8134&lt;br&gt;5953&lt;br&gt;2955&lt;br&gt;2700&lt;br&gt;2647&lt;br&gt;334&lt;br&gt;315&lt;br&gt;299&lt;br&gt;263&lt;br&gt;244||11227&lt;br&gt;8216&lt;br&gt;4079&lt;br&gt;3727&lt;br&gt;3654&lt;br&gt;461&lt;br&gt;434&lt;br&gt;413&lt;br&gt;363&lt;br&gt;337||13249×24&lt;sup&gt;8131&lt;/sup&gt;−49&lt;br&gt;201×24&lt;sup&gt;5951&lt;/sup&gt;−1&lt;br&gt;10×24&lt;sup&gt;2954&lt;/sup&gt;+5053&lt;br&gt;(13×24&lt;sup&gt;2700&lt;/sup&gt;+4403)/23&lt;br&gt;24&lt;sup&gt;2647&lt;/sup&gt;−1201&lt;br&gt;276×24&lt;sup&gt;332&lt;/sup&gt;+11&lt;br&gt;2×24&lt;sup&gt;314&lt;/sup&gt;+7&lt;br&gt;(283×24&lt;sup&gt;298&lt;/sup&gt;−7)/23&lt;br&gt;13×24&lt;sup&gt;262&lt;/sup&gt;+12013&lt;br&gt;18×24&lt;sup&gt;243&lt;/sup&gt;+437||http://factordb.com/index.php?id=1100000003593391606&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003593275880&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003593269654&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003593269876&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003593270089&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000002633359842&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000002355610241&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000002326181235&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003593270725&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000002633360037&amp;open=prime&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000003593391606&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000003593275880&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000003593269654&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000003593269876&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000003593270089&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000002633359842&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000002355610241&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000002326181235&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000003593270725&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000002633360037&amp;base=24&lt;nowiki/&gt;||0||–||
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||36||35286~35290||P&lt;sub&gt;81993&lt;/sub&gt;SZ&lt;br&gt;S0&lt;sub&gt;75007&lt;/sub&gt;8H&lt;br&gt;7K&lt;sub&gt;26567&lt;/sub&gt;Z&lt;br&gt;J&lt;sub&gt;10117&lt;/sub&gt;LJ&lt;br&gt;VL0&lt;sub&gt;7258&lt;/sub&gt;J&lt;br&gt;EO0&lt;sub&gt;6177&lt;/sub&gt;V&lt;br&gt;FZ&lt;sub&gt;5777&lt;/sub&gt;3P&lt;br&gt;T09&lt;sub&gt;4618&lt;/sub&gt;1&lt;br&gt;RY&lt;sub&gt;4562&lt;/sub&gt;H&lt;br&gt;OZ&lt;sub&gt;3932&lt;/sub&gt;AZ||81995&lt;br&gt;75010&lt;br&gt;26569&lt;br&gt;10119&lt;br&gt;7261&lt;br&gt;6180&lt;br&gt;5780&lt;br&gt;4621&lt;br&gt;4564&lt;br&gt;3935||127609&lt;br&gt;116739&lt;br&gt;41349&lt;br&gt;15748&lt;br&gt;11301&lt;br&gt;9618&lt;br&gt;8996&lt;br&gt;7192&lt;br&gt;7103&lt;br&gt;6124||(5×36&lt;sup&gt;81995&lt;/sup&gt;+821)/7&lt;br&gt;28×36&lt;sup&gt;75009&lt;/sup&gt;+305&lt;br&gt;(53×36&lt;sup&gt;26568&lt;/sup&gt;+101)/7&lt;br&gt;(19×36&lt;sup&gt;10119&lt;/sup&gt;+2501)/35&lt;br&gt;1137×36&lt;sup&gt;7259&lt;/sup&gt;+19&lt;br&gt;528×36&lt;sup&gt;6178&lt;/sup&gt;+31&lt;br&gt;16×36&lt;sup&gt;5779&lt;/sup&gt;−1163&lt;br&gt;(36549×36&lt;sup&gt;4619&lt;/sup&gt;−289)/35&lt;br&gt;(979×36&lt;sup&gt;4563&lt;/sup&gt;−629)/35&lt;br&gt;25×36&lt;sup&gt;3934&lt;/sup&gt;−901||http://factordb.com/index.php?id=1100000002394962083&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004020085177&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003896952461&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362491&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362489&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362488&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362487&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362486&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362485&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000000840634476&amp;open=prime&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000002394962083&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000004020085177&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003896952461&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362491&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362489&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362488&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362487&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362486&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362485&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000000840634476&amp;base=36&lt;nowiki/&gt;||4||200000||
|}

== The fully proof of Athena problem in decimal (base ''b'' = 10) ==

'''Bold''' for the minimal elements, ''x'' ◁ ''y'' means ''x'' is a subsequence of ''y''.

Assume ''p'' is a prime &gt; 10, and the last digit of ''p'' must lie in {1,3,7,9}.

Case 1: ''p'' ends with 1.

In this case we can write ''p'' = ''x''1. If ''x'' contains 1, 3, 4, 6, or 7, then (respectively) '''11''' ◁ ''p'', '''31''' ◁ ''p'', '''41''' ◁ ''p'', '''61''' ◁ ''p'', or '''71''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 2, 5, 8, or 9.

Case 1.1: ''p'' begins with 2.

In this case we can write ''p'' = 2''y''1. If 5 ◁ ''y'', then '''251''' ◁ ''p''. If 8 ◁ ''y'', then '''281''' ◁ ''p''. If 9 ◁ ''y'', then 29 ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 2.

If 22 ◁ ''y'', then '''2221''' ◁ ''p''. Hence we may assume ''y'' contains zero or one 2's.

If ''y'' contains no 2's, then ''p'' ∈ 2{0}1. But then, since the sum of the digits of ''p'' is 3, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' contains exactly one 2, then we can write ''p'' = 2''z''2''w''1, where ''z'',''w'' ∈ {0}. If 0 ◁ ''z'' and 0 ◁ ''w'', then '''20201''' ◁ ''p''. Hence we may assume either ''z'' or ''w'' is empty.

If ''z'' is empty, then ''p'' ∈ 22{0}1, and the smallest prime ''p'' ∈ 22{0}1 is '''22000001'''.

If ''w'' is empty, then ''p'' ∈ 2{0}21, and the smallest prime ''p'' ∈ 2{0}21 is '''20021'''.

Case 1.2: ''p'' begins with 5.

In this case we can write ''p'' = 5''y''1. If 2 ◁ ''y'', then '''521''' ◁ ''p''. If 9 ◁ ''y'', then 59 ◁ ''p''. Hence we may assume all digits of ''y'' are 0, 5, or 8.

If 05 ◁ ''y'', then '''5051''' ◁ ''p''. If 08 ◁ ''y'', then '''5081''' ◁ ''p''. If 50 ◁ ''y'', then '''5501''' ◁ ''p''. If 58 ◁ ''y'', then '''5581''' ◁ ''p''. If 80 ◁ ''y'', then '''5801''' ◁ ''p''. If 85 ◁ ''y'', then '''5851''' ◁ ''p''. Hence we may assume ''y'' ∈ {0} ∪ {5} ∪ {8}.

If ''y'' ∈ {0}, then ''p'' ∈ 5{0}1. But then, since the sum of the digits of ''p'' is 6, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' ∈ {5}, then ''p'' ∈ 5{5}1, and the smallest prime ''p'' ∈ 5{5}1 is '''555555555551'''.

If ''y'' ∈ {8}, since if 88 ◁ ''y'', then 881 ◁ ''p'', hence we may assume ''y'' ∈ {''𝜆'',8}, and thus ''p'' ∈ {51,581}, but 51 and 581 are both composite.

Case 1.3: ''p'' begins with 8.

In this case we can write p = 8''y''1. If 2 ◁ ''y'', then '''821''' ◁ ''p''. If 8 ◁ ''y'', then '''881''' ◁ ''p''. If 9 ◁ ''y'', then 89 ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 5.

If 50 ◁ ''y'', then '''8501''' ◁ ''p''. Hence we may assume y ∈ {0}{5}.

If 005 ◁ ''y'', then '''80051''' ◁ p. Hence we may assume y ∈ {0} ∪ {5} ∪ 0{5}.

If y ∈ {0}, then ''p'' ∈ 8{0}1. But then, since the sum of the digits of ''p'' is 9, ''p'' is divisible by 3, so ''p'' cannot be prime.

If y ∈ {5}, since if 55555555555 ◁ ''y'', then 555555555551 ◁ ''p'', hence we may assume ''y'' ∈ {''𝜆'', 5, 55, 555, 5555, 55555, 555555, 5555555, 55555555, 555555555, 5555555555}, and thus ''p'' ∈ {81, 851, 8551, 85551, 855551, 8555551, 85555551, 855555551, 8555555551, 85555555551, 855555555551}, but all of these numbers are composite.

If y ∈ 0{5}, since if 55555555555 ◁ ''y'', then 555555555551 ◁ ''p'', hence we may assume ''y'' ∈ {0, 05, 055, 0555, 05555, 055555, 0555555, 05555555, 055555555, 0555555555, 05555555555}, and thus ''p'' ∈ {801, 8051, 80551, 805551, 8055551, 80555551, 805555551, 8055555551, 80555555551, 805555555551, 8055555555551}, and of these numbers only 80555551 and 8055555551 are primes, but 80555551 ◁ 8055555551, thus only '''80555551''' is a minimal element.

Case 1.4: ''p'' begins with 9.

In this case we can write p = 9''y''1. If 9 ◁ ''y'', then '''991''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0, 2, 5, or 8.

If 00 ◁ ''y'', then '''9001''' ◁ ''p''. If 22 ◁ ''y'', then '''9221''' ◁ ''p''. If 55 ◁ ''y'', then '''9551''' ◁ ''p''. If 88 ◁ ''y'', then 881 ◁ ''p''. Hence we may assume ''y'' contains at most one 0, at most one 2, at most one 5, and at most one 8.

If ''y'' only contains at most one 0 and does not contain any of {2,5,8}, then ''y'' ∈ {''𝜆'',0}, and thus ''p'' ∈ {91,901}, but 91 and 901 are both composite. If ''y'' only contains at most one 0 and only one of {2,5,8}, then the sum of the digits of ''p'' is divisible by 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Hence we may assume ''y'' contains at least two of {2,5,8}.

If 25 ◁ ''y'', then 251 ◁ ''p''. If 28 ◁ ''y'', then 281 ◁ ''p''. If 52 ◁ ''y'', then 521 ◁ ''p''. If 82 ◁ ''y'', then 821 ◁ ''p''. Hence we may assume ''y'' contains no 2's (since if ''y'' contains 2, then ''y'' cannot contain either 5's or 8's, which is a contradiction).

If 85 ◁ ''y'', then '''9851''' ◁ ''p''. Hence we may assume ''y'' ∈ {58,580,508,058}, and thus ''p'' ∈ {9581,95801,95081,90581}, and of these numbers only 95801 is prime, but 95801 is not a minimal element since 5801 ◁ 95801.

Case 2: ''p'' ends with 3.

In this case we can write p = ''x''3. If ''x'' contains 1, 2, 4, 5, 7, or 8, then (respectively) '''13''' ◁ ''p'', '''23''' ◁ ''p'', '''43''' ◁ ''p'', '''53''' ◁ ''p'', '''73''' ◁ ''p'', or '''83''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 3, 6, or 9, and thus all digits of ''p'' are 0, 3, 6, or 9. But then, since the digits of ''p'' all have a common factor 3, ''p'' is divisible by 3, so ''p'' cannot be prime.

Case 3: ''p'' ends with 7.

In this case we can write ''p'' = ''x''7. If ''x'' contains 1, 3, 4, 6, or 9, then (respectively) '''17''' ◁ ''p'', '''37''' ◁ ''p'', '''47''' ◁ ''p'', '''67''' ◁ ''p'', or '''97''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 2, 5, 7, or 8.

Case 3.1: ''p'' begins with 2.

In this case we can write ''p'' = 2''y''7. If 2 ◁ ''y'', then '''227''' ◁ ''p''. If 5 ◁ ''y'', then '''257''' ◁ ''p''. If 7 ◁ ''y'', then '''277''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 8.

If 08 ◁ ''y'', then '''2087''' ◁ ''p''. If 88 ◁ ''y'', then 887 ◁ ''p''. Hence we may assume ''y'' ∈ {0} ∪ 8{0}.

If ''y'' ∈ {0}, then ''p'' ∈ 2{0}7. But then, since the sum of the digits of ''p'' is 9, ''p'' is divisible by 3, so ''p'' cannot be prime.

If y ∈ 8{0}, then ''p'' ∈ 28{0}7. But then ''p'' is divisible by 7, since for ''n'' ≥ 0 we have 7 × 40&lt;sub&gt;''n''&lt;/sub&gt;1 = 280&lt;sub&gt;''n''&lt;/sub&gt;7.

Case 3.2: ''p'' begins with 5.

In this case we can write ''p'' = 5''y''7. If 5 ◁ ''y'', then '''557''' ◁ ''p''. If 7 ◁ ''y'', then '''577''' ◁ ''p''. If 8 ◁ ''y'', then '''587''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 2.

If 22 ◁ ''y'', then 227 ◁ ''p''. Hence we may assume ''y'' contains zero or one 2's.

If ''y'' contains no 2's, then ''p'' ∈ 5{0}7. But then, since the sum of the digits of ''p'' is 12, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' contains exactly one 2, then we can write ''p'' = 5''z''2''w''7, where ''z'',''w'' ∈ {0}. If 0 ◁ ''z'' and 0 ◁ ''w'', then '''50207''' ◁ ''p''. Hence we may assume either ''z'' or ''w'' is empty.

If ''z'' is empty, then ''p'' ∈ 52{0}7, and the smallest prime ''p'' ∈ 52{0}7 is '''5200007'''.

If ''w'' is empty, then ''p'' ∈ 5{0}27, and the smallest prime ''p'' ∈ 5{0}27 is '''5000000000000000000000000000027'''.

Case 3.3: ''p'' begins with 7.

In this case we can write ''p'' = 7''y''7. If 2 ◁ ''y'', then '''727''' ◁ ''p''. If 5 ◁ ''y'', then '''757''' ◁ ''p''. If 8 ◁ ''y'', then '''787''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 7, and thus all digits of ''p'' are 0 or 7. But then, since the digits of ''p'' all have a common factor 7, ''p'' is divisible by 7, so ''p'' cannot be prime.

Case 3.4: ''p'' begins with 8.

In this case we can write ''p'' = 8''y''7. If 2 ◁ ''y'', then '''827''' ◁ ''p''. If 5 ◁ ''y'', then '''857''' ◁ ''p''. If 7 ◁ ''y'', then '''877''' ◁ ''p''. If 8 ◁ ''y'', then '''887''' ◁ ''p''. Hence we may assume ''y'' ∈ {0}, and thus ''p'' ∈ 8{0}7. But then, since the sum of the digits of ''p'' is 15, ''p'' is divisible by 3, so ''p'' cannot be prime.

Case 4: ''p'' ends with 9.

In this case we can write ''p'' = ''x''9. If ''x'' contains 1, 2, 5, 7, or 8, then (respectively) '''19''' ◁ ''p'', '''29''' ◁ ''p'', '''59''' ◁ ''p'', '''79''' ◁ ''p'', or '''89''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 3, 4, 6, or 9.

If 44 ◁ ''x'', then '''449''' ◁ ''p''. Hence we may assume ''x'' contains zero or one 4's.

If x contains no 4's, then all digits of ''x'' are 0, 3, 6, or 9, and thus all digits of ''p'' are 0, 3, 6, or 9. But then, since the digits of ''p'' all have a common factor 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Hence we may assume that ''x'' contains exactly one 4.

Case 4.1: ''p'' begins with 3.

In this case we can write ''p'' = 3''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. We must have '''349''' ◁ ''p''.

Case 4.2: ''p'' begins with 4.

In this case we can write ''p'' = 4''y''9, where all digits of ''y'' are 0, 3, 6, or 9. If 0 ◁ ''y'', then '''409''' ◁ ''p''. If 3 ◁ ''y'', then 43 ◁ ''p''. If 9 ◁ ''y'', then '''499''' ◁ ''p''. Hence we may assume ''y'' ∈ {6}, and thus ''p'' ∈ 4{6}9. But then ''p'' is divisible by 7, since for ''n'' ≥ 0 we have 7 × 6&lt;sub&gt;''n''&lt;/sub&gt;7 = 46&lt;sub&gt;''n''&lt;/sub&gt;9.

Case 4.3: ''p'' begins with 6.

In this case we can write p = 6''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. If 0 ◁ ''z'', then 409 ◁ ''p''. If 3 ◁ ''z'', then 43 ◁ ''p''. If 6 ◁ ''z'', then '''6469''' ◁ ''p''. If 9 ◁ ''z'', then 499 ◁ ''p''. Hence we may assume ''z'' is empty.

If 3 ◁ ''y'', then 349 ◁ ''p''. If 9 ◁ ''y'', then '''6949''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 6.

If 06 ◁ ''y'', then '''60649''' ◁ ''p''. Hence we may assume ''y'' ∈ {6}{0}.

If 666 ◁ ''y'', then '''666649''' ◁ ''p''. If 00000 ◁ ''y'', then '''60000049''' ◁ ''p''. Hence we may assume ''y'' ∈ {''𝜆'', 0, 00, 000, 0000, 6, 60, 600, 6000, 60000, 66, 660, 6600, 66000, 660000}, and thus ''p'' ∈ {649, 6049, 60049, 600049, 6000049, 6649, 66049, 660049, 6600049, 66000049, 66649, 666049, 6660049, 66600049, 666000049}, and of these numbers only '''66000049''' and '''66600049''' are primes.

Case 4.4: ''p'' begins with 9.

In this case we can write p = 9''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. If 0 ◁ ''y'', then '''9049''' ◁ ''p''. If 3 ◁ ''y'', then 349 ◁ ''p''. If 6 ◁ ''y'', then '''9649''' ◁ ''p''. If 9 ◁ ''y'', then '''9949''' ◁ ''p''. Hence we may assume ''y'' is empty.

If 0 ◁ ''z'', then 409 ◁ ''p''. If 3 ◁ ''z'', then 43 ◁ ''p''. If 9 ◁ ''z'', then 499 ◁ ''p''. Hence we may assume ''z'' ∈ {6}, and thus ''p'' ∈ 94{6}9, and the smallest prime ''p'' ∈ 94{6}9 is 946669.

[[Category:Number theory]]</text>
      <sha1>tfalcy0f7ikx81l3n9flqpp3dvjquiw</sha1>
    </revision>
    <revision>
      <id>2819767</id>
      <parentid>2819766</parentid>
      <timestamp>2026-07-29T19:26:51Z</timestamp>
      <contributor>
        <username>Athene241</username>
        <id>3100061</id>
      </contributor>
      <comment>/* The fully proof of Athena problem in decimal (base b = 10) */</comment>
      <origin>2819767</origin>
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      <text bytes="128875" sha1="bjecmf2rhslpw9jmyeradp4kaj0xsm5" xml:space="preserve">{{mathematics}}
'''Athena problem''' is an [[:w:List of unsolved problems in mathematics|unsolved problem]] in [[:w:Number theory|number theory]] and [[:w:Formal language theory|formal language theory]] and [[:w:Order theory|order theory]], this problem is named after the ancient Greek goddess [[:w:Athena|Athena]] (which is associated with [[:w:Wisdom|wisdom]]). Athena problem is: Give a [[:w:Natural number|natural number]] ''b'' &gt; 1, find the [[:w:Set (mathematics)|set]] of the [[:w:Minimal element|minimal element]]s of the set of the &quot;[[:w:Prime number|prime number]] [[:w:Greater than|&gt;]] ''b''&quot; [[:w:Numerical digit|digit]] [[:w:String (computer science)|string]]s in the [[:w:Positional numeral system|positional numeral system]] with [[:w:Radix|base]] ''b'' for the [[:w:Subsequence|subsequence]] [[:w:Partially ordered set|ordering]]. (A string ''x'' is a subsequence of another string ''y'', if ''x'' can be obtained from ''y'' by deleting zero or more of the [[:w:Character (computing)|character]]s in ''y''. For example, 514 is a subsequence of 352148, &quot;string&quot; is a subsequence of &quot;meistersinger&quot;. In contrast, 758 is not a subsequence of 378259, &quot;abc&quot; is not a subsequence of &quot;cbacacba&quot;, since the characters must be in the same order) (Unlike [[:w:Substring|substring]], subsequence is not required to occupy consecutive positions within the original sequences, e.g. the [[:w:Longest common subsequence|longest common subsequence problem]] is different from the [[:w:Longest common substring|longest common substring problem]])

Using [[:w:Formal language theory|formal language theory]] terminology, Athena problem is finding the [[:w:Set (mathematics)|set]] of the [[:w:Minimal element|minimal element]]s of the [[:w:Formal language|language]] of base-''b'' [[:w:Representation (mathematics)|representation]]s of the [[:w:Prime number|prime number]]s [[:w:Greater than|&gt;]] ''b'' (which is a set of [[:w:String (computer science)|string]]s of [[:w:Symbol|symbol]]s over the [[:w:Alphabet (formal languages)|alphabet]] ''Σ''&lt;sub&gt;''b''&lt;/sub&gt; := {0, 1, ..., ''b''−1}), under the subsequence ordering (i.e. the [[:w:Binary relation|binary relation]] &quot;is a subsequence of&quot;, which is a [[:w:Partially ordered set|partial ordering]]), for a given natural number ''b'' &gt; 1 (You can draw this partial ordering as a [[:w:Hasse diagram|Hasse diagram]] to find all [[:w:Minimal element|minimal element]]s), this set is called '''Athena set base ''b''''', and the prime numbers in this set is called '''Athena primes base ''b'''''.

By [[:w:Higman's lemma|Higman's lemma]], there are no [[:w:Infinite set|infinite]] [[:w:Antichain|antichain]]s for the subsequence ordering (i.e. the subsequence ordering is always a [[:w:Well-quasi-ordering|well quasi order]]) (i.e. under the subsequence ordering (i.e. the [[:w:Binary relation|binary relation]] &quot;is a subsequence of&quot;, which is a [[:w:Partially ordered set|partial ordering]]), every set of pairwise incomparable (i.e. not [[:w:Comparability|comparable]]) strings is finite), thus there must be only finitely many such minimal elements. In other words, the Athena set in every base ''b'' must be a [[:w:Finite set|finite set]], and every base ''b'' ≥ 2 has only finitely many Athena primes, e.g. in [[:w:Decimal|decimal]] (base ''b'' = 10), the Athena set has exactly 77 [[:w:Element of a set|element]]s (the Athena primes in decimal (base ''b'' = 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}.

For bases 2 ≤ ''b'' ≤ 36, Athena problem is fully solved in bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 24, and also solved in bases ''b'' = 11, 13, 16, 22, 30 if [[:w:Probable prime|probable prime]]s are allowed. For the unsolved bases ''b'' = 17, 19, 21, 23, 25, 26, 27, 28, 29, 31, 32, 34, 35, 36, Athena problem is solved (if probable primes are allowed) except 771 [[:w:Indexed family|families]] of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be [[:w:Empty string|empty]]) of digits in base ''b'', ''y'' is a digit in base ''b'') = sequence {''xz'', ''xyz'', ''xyyz'', ''xyyyz'', ''xyyyyz'', ''xyyyyyz'', ...} (i.e. &quot;''xy''&lt;sup&gt;+&lt;/sup&gt;''z''&quot; in [[:w:Regular expression|regular expression]]), all of these 771 families contain no primes &gt; ''b'' or probable primes &gt; ''b'' with length ≤ 100000.

== Solve the problem ==

To solve the Athena problem for a given base ''b'', we must [[:w:Computing|compute]] the elements up to families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''), and find the smallest prime &gt; ''b'' in all such families.

We call families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') &quot;linear&quot; families, and we reduce these families by removing all trailing digits ''y'' from ''x'', and removing all leading digits ''y'' from ''z'', to make the families be easier, e.g. family 12333{3}33345 in base ''b'' is reduced to family 12{3}45 in base ''b'', since they are in fact the same family. Our [[:w:Algorithm|algorithm]] then proceeds as follows:

* 1. ''M'' := {minimal primes in base ''b'' of length 2 or 3}, ''L'' := union of all ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'') such that ''x'' ≠ 0 and ''gcd''(''z'', ''b'') = 1 and ''Y'' is the set of digits ''y'' in base ''b'' such that ''xyz'' has no subsequence in ''M''.
* 2. While ''L'' contains nonlinear families (families which are not linear families): Explore each family of ''L'', and update ''L''. Examine each family of ''L'' by:
* 2.1. Let ''w'' be the shortest string in the family. If ''w'' has a subsequence in ''M'', then remove the family from ''L''. If ''w'' represents a prime, then add ''w'' to ''M'' and remove the family from ''L''.
* 2.2. If possible, simplify the family.
* 2.3. Using the techniques below (covering congruence, algebraic factorization, or combine of them), check if the family can be proven to only contain composites (only count the numbers &gt; ''b''), and if so then remove the family from ''L''.
* 3. Update ''L'', after each split examine the new families as in step 2.

e.g. in decimal (base ''b'' = 10):

''M'' := {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}

''L'' := {2{0,2}1, 2{0,8}7, 3{0,3,6,9}3, 3{0,3,6,9}9, 4{6}9, 5{0,5,8}1, 5{0,2}7, 6{0,3,6,9}3, 6{0,3,4,6,9}9, 7{0,7}7, 8{0,5}1, 8{0}7, 9{0,2,5,8}1, 9{0,3,6,9}3, 9{0,3,4,6,9}9}

and since 2221 is prime, it follows that the family 2{0,2}1 splits into the families 2{0}1 and 2{0}2{0}1

and since the family 2{0}1 can be proven to contain no primes &gt; base (since all numbers in this family are divisible by 3), it can be removed

and since 20201 is prime, it follows that the family 2{0}2{0}1 splits into the families 2{0}21 and 22{0}1

221 and 2021 are composites, but 20021 is prime, thus add 20021 to ''L''

none of 221, 2201, 22001, 220001, 2200001 are primes, but 22000001 is prime, thus add 22000001 to ''L''

and since the family 3{0,3,6,9}3 can be proven to contain no primes &gt; base (since all numbers in this family are divisible by 3), it can be removed

etc.

Since the number of possible (first digit,last digit) (also called (initial digit,final digit)) combos ([[:w:Ordered pair|ordered pair]]s) of a prime &gt; ''b'' in base ''b'' is (''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(''b'') (all digits except 0 can be the first digit of a prime &gt; ''b'' in base ''b'' (thus ''b''−1 possible digits), but only the digits coprime to ''b'' can be the last digit of a prime &gt; ''b'' in base ''b'' (thus ''eulerphi''(''b'') possible digits), and by the [[:w:Rule of product|rule of product]], there are (''b''−1)×''eulerphi''(''b'') choices of the (first digit,last digit) combo, also, both &quot;numbers of primes in the set of the Athena problem in base ''b''&quot; and &quot;length of the largest prime in the set of the Athena problem in base ''b''&quot; are [[:w:Asymptotic analysis|roughly]] ''[[:w:E (mathematical_constant)|e]]''&lt;sup&gt;''[[:w:Euler's constant|γ]]''×(''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(*b*)&lt;/sup&gt;.

Shrinking the family ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''Y'' is a set of digits in base ''b'')

* If ''y'' ∈ ''Y'' and the string ''xyyz'' represents a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or has a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''}''z'' ∪ ''x''{''Y'' \ ''y''}''y''{''Y'' \ ''y''}''z''.
* If ''y''&lt;sub&gt;1&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;2&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;1&lt;/sub&gt; ≠ ''y''&lt;sub&gt;2&lt;/sub&gt; and the string ''xy''&lt;sub&gt;1&lt;/sub&gt;''y''&lt;sub&gt;2&lt;/sub&gt;''z'' represents a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or has a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''&lt;sub&gt;1&lt;/sub&gt;}{''Y'' \ ''y''&lt;sub&gt;2&lt;/sub&gt;}''z''.
* If ''y''&lt;sub&gt;1&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;2&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;1&lt;/sub&gt; ≠ ''y''&lt;sub&gt;2&lt;/sub&gt; and both the strings ''xy''&lt;sub&gt;1&lt;/sub&gt;''y''&lt;sub&gt;2&lt;/sub&gt;''z'' and ''xy''&lt;sub&gt;2&lt;/sub&gt;''y''&lt;sub&gt;1&lt;/sub&gt;''z'' represent a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or have a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''&lt;sub&gt;1&lt;/sub&gt;}''z'' ∪ ''x''{''Y'' \ ''y''&lt;sub&gt;2&lt;/sub&gt;}''z''.

e.g. in decimal (base ''b'' = 10):

* 2221 is a prime &gt; 10, thus the family 2{0,2}1 splits into the two families 2{0}1 and 2{0}2{0}1.
* 227 is a prime &gt; 10, and it is a subsequence of 5227, thus the family 5{0,2}7 splits into the two families 5{0}7 and 5{0}2{0}7.
* 449 is a prime &gt; 10, and it is a subsequence of 6449, thus the family 6{0,3,4,6,9}9 splits into the two families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9.
* Both 5051 and 5501 are primes &gt; 10, thus the family 5{0,5}1 splits into the two families 5{0}1 and 5{5}1 = {5}1.
* 8501 is a prime &gt; 10, thus the family 8{0,5}1 splits into the family 8{0}{5}1.
* 887 is a prime &gt; 10, and it is a subsequence of 2887, also 2087 is a prime &gt; 10, thus the family 2{0,8}7 splits into the two families 2{0}7 and 28{0}7.
* 349 and 449 are primes &gt; 10, and they are subsequences of 9349 and 9449, respectively, also 9049, 9649, 9949 are primes &gt; 10, thus the family 9{0,3,4,6,9}9 splits into the two families 9{0,3,6,9}9 and 94{0,3,6,9}9.
* 251, 281, 521, 821, 881 are primes &gt; 10, and they are subsequences of 9251, 9281, 9521, 9821, 9881, respectively, also 9001, 9221, 9551, 9851 are primes &gt; 10, thus the family 9{0,2,5,8}1 splits into the numbers {91, 901, 921, 951, 981, 9021, 9051, 9081, 9201, 9501, 9581, 9801, 90581, 95081, 95801}.

If the methods we have discussed cannot be used to rule out or shrink ''x''{''Y''}''z'' where ''Y'' = {''y''&lt;sub&gt;1&lt;/sub&gt;, ''y''&lt;sub&gt;2&lt;/sub&gt;, ..., ''y''&lt;sub&gt;''n''&lt;/sub&gt;}, then we can replace ''x''{''Y''}''z'' by ''xy''&lt;sub&gt;1&lt;/sub&gt;{''Y''}''z'' ∪ ''xy''&lt;sub&gt;2&lt;/sub&gt;{''Y''}''z'' ∪ ... ∪ ''xy''&lt;sub&gt;''n''&lt;/sub&gt;{''Y''}''z'' and re-run the methods on this new [[:w:Formal language|language]].

If all remain families are linear families (i.e. of the form ''x''{''y''}''z'', where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''), then we search the smallest (probable) primes in these families and add these primes to the list.

e.g. in decimal (base ''b'' = 10):

* The smallest prime in the family 5{0}27 is 5000000000000000000000000000027.
* The smallest prime in the family {5}1 is 555555555551.
* The smallest prime in the family 8{5}1 is 8555555555555555555551, but 8555555555555555555551 is not a minimal element since 555555555551 is a subsequence of 8555555555555555555551.

There is no guarantee that the techniques discussed will ever terminate, but in practice they often do. They are able to determine the set of the minimal elements in base ''b'' for 2 ≤ ''b'' ≤ 16 and ''b'' = 18, 20, 22, 24, 30. The bases ''b'' = 17, 19, 21, 23, 25 ≤ ''b'' ≤ 29, 31 ≤ ''b'' ≤ 36 are solved with the exception of 771 families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'').

The following is a &quot;[[:w:Semi-algorithm|semi-algorithm]]&quot; that is guaranteed to solve the Athena problem for a given base ''b'', but it is not so easy to implement:

# ''M'' = ''[[:w:Empty string|∅]]''
# while (''L'' ≠ ''∅'') do
# choose ''x'', a shortest string in ''L''
# ''M'' := ''M'' ∪ {''x''}
# ''L'' := ''L'' − ''sup''({''x''})

In practice, for arbitrary ''L'', we cannot feasibly carry out step 5. Instead, we work with ''L''&amp;#39;, some regular overapproximation to ''L'', until we can show ''L''&amp;#39; = ''∅'' (which implies ''L'' = ''∅''). In practice, ''L''&amp;#39; is usually chosen to be a finite [[:w:Union (set theory)|union]] of sets of the form ''L''&lt;sub&gt;1&lt;/sub&gt;{''L''&lt;sub&gt;2&lt;/sub&gt;}''L''&lt;sub&gt;3&lt;/sub&gt;, where each of ''L''&lt;sub&gt;1&lt;/sub&gt;, ''L''&lt;sub&gt;2&lt;/sub&gt;, ''L''&lt;sub&gt;3&lt;/sub&gt; is finite. In the case we consider in this project, we then have to determine whether such a family contains a prime or not.

Thus, the [[:w:Time complexity|time complexity]] of the Athena problem in base ''b'' may be ''[[:w:Big O notation|O]]''(''[[:w:E (mathematical_constant)|e]]''&lt;sup&gt;''[[:w:Euler's constant|γ]]''×(''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(*b*)&lt;/sup&gt;), and the [[:w:CPU time|CPU time]] of the Athena problem in base ''b'' may be longer than [[:w:Age of the universe|the age of the universe]] for bases ''b'' = 19, 23, 25, 27, 29, 31, 32, 33, 34, 35, also, Athena problem in bases ''b'' around 500 may be [[:w:NP-complete|NP-complete]] or [[:w:NP-hard|NP-hard]], or an [[:w:Undecidable problem|undecidable problem]], or an example of [[:w:Gödel's incompleteness theorems|Gödel's incompleteness theorems]] (like the [[:w:Continuum hypothesis|continuum hypothesis]] and the [[:w:Halting problem|halting problem]]).

To solve the Athena problem, we need to determine whether a given family contains a prime. In practice, if family ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''Y'' is a set of digits in base ''b'') could not be ruled out as only containing composites and ''Y'' contains two or more digits, then a relatively small prime &gt; ''b'' could always be found in this family. Intuitively, this is because there are a large number of small strings in such a family, and at least one is likely to be prime (e.g. there are 2&lt;sup&gt;''n''−2&lt;/sup&gt; strings of length ''n'' in the family 1{3,7}9, and there are over a thousand strings of length 12 in the family 1{3,7}9, thus it is very impossible that these numbers are all composite). In the case ''Y'' contains only one digit, this family is of the form ''x''{''y''}''z'', and there is only a single string of each length &gt; (the length of ''x'' + the length of ''z''), and it is not known if the following [[:w:Decision problem|decision problem]] is recursively solvable (just like [[:w:Sierpiński number|Sierpiński problem]] and [[:w:Riesel number|Riesel problem]], Sierpiński problem and Riesel problem can be generalized to other bases ''b'' (references: http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm), in fact, Athena problem base ''b'' covers the Sierpiński problem base ''b'' and the Riesel problem base ''b'' with ''k'' &lt; ''b'', i.e. finding the smallest prime of the form ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 and ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (or prove such prime does not exist) with ''k'' &lt; ''b'' (specially, for bases ''b'' such that the conjectured smallest Sierpiński number or the conjectured smallest Riesel number is &lt; ''b'', Athena problem base ''b'' covers the Sierpiński problem base ''b'' or the Riesel problem base ''b'', respectively), since the smallest prime of the form ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 and ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (if exists) must be a minimal element in base ''b'', also, Athena problem base ''b'' covers finding the smallest prime of these forms in base ''b'' (or proving that such prime does not exist): (''b''&lt;sup&gt;''n''&lt;/sup&gt;−1)/(''b''−1) (for this form, ''n'' must be prime, and we want ''n'' ≥ 2) (references of this form: http://www.fermatquotient.com/PrimSerien/GenRepu.txt, https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html, http://www.primenumbers.net/Henri/us/MersFermus.htm, http://www.bitman.name/math/table/379, https://pzktupel.de/Primetables/TableRepunitGen.php, https://oeis.org/A084740, https://oeis.org/A084738, https://oeis.org/A128164, https://oeis.org/A285642; or for prime bases ''b'': https://oeis.org/A065854, https://oeis.org/A279068), ''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 (for this form, ''n'' must be power of 2, and we want ''n'' ≥ 1) (references of this form: http://jeppesn.dk/generalized-fermat.html, http://www.noprimeleftbehind.net/crus/GFN-primes.htm, https://web.archive.org/web/20231002190634/http://yves.gallot.pagesperso-orange.fr/primes/index.html, https://pzktupel.de/Primetables/TableFermatGFBB.php, https://oeis.org/A079706, https://oeis.org/A084712, https://oeis.org/A228101), (''b''&lt;sup&gt;''n''&lt;/sup&gt;+1)/2 (for odd ''b'') (for this form, ''n'' must be power of 2, and we want ''n'' ≥ 2) (reference of this form: http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt), (''sqrt''(''b'')×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1)/(''sqrt''(''b'')+1) (for square ''b'') (for this form, 2×''n''+1 must be prime, and we want ''n'' ≥ 2) (references of this form: http://www.fermatquotient.com/PrimSerien/GenRepuP.txt, http://www.primenumbers.net/Henri/us/MersFermus.htm, http://www.bitman.name/math/table/488, https://pzktupel.de/Primetables/TableWagstaffGen.php, https://oeis.org/A084742, https://oeis.org/A084741; or for bases ''b'' with ''sqrt''(''b'') prime: https://oeis.org/A065507), ((''b''−2)×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1)/(''b''−1) (''n'' ≥ 2) (reference of this form: https://oeis.org/A243404), 2×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 (''n'' ≥ 1) (references of this form: https://www.mersenneforum.org/showthread.php?t=6918, https://www.mersenneforum.org/showthread.php?t=19725, https://oeis.org/A119624), 2×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (''n'' ≥ 1) (references of this form: https://www.mersenneforum.org/showthread.php?t=24576, https://www.mersenneforum.org/attachment.php?attachmentid=20976&amp;d=1567314217, https://oeis.org/A119591), ''b''&lt;sup&gt;''n''&lt;/sup&gt;+2 (''n'' ≥ 1) (references of this form: https://oeis.org/A138066, https://oeis.org/A084713, https://oeis.org/A138067), ''b''&lt;sup&gt;''n''&lt;/sup&gt;−2 (''n'' ≥ 2) (references of this form: https://www.primepuzzles.net/puzzles/puzz_887.htm, https://oeis.org/A250200, https://oeis.org/A255707, https://oeis.org/A084714; or for prime bases ''b'': https://oeis.org/A292201), (''b''−1)×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 (''n'' ≥ 1) (references of this form: http://www.noprimeleftbehind.net/Williams-primes-MP.htm, http://www.bitman.name/math/table/477, https://pzktupel.de/Primetables/TableWilliams2.php, https://oeis.org/A305531; or for prime bases ''b'': https://oeis.org/A087139), (''b''−1)×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (''n'' ≥ 1) (references of this form: https://harvey563.tripod.com/wills.txt, http://www.noprimeleftbehind.net/Williams-primes-MM.htm, http://www.bitman.name/math/table/484, https://pzktupel.de/Primetables/TableWilliams1.php; or for prime bases ''b'': https://oeis.org/A122396), ''b''&lt;sup&gt;''n''&lt;/sup&gt;+(''b''−1) (''n'' ≥ 1) (references of this form: http://www.bitman.name/math/table/795, https://pzktupel.de/Primetables/TableWilliams6.php, https://oeis.org/A076845, https://oeis.org/A076846, https://oeis.org/A078178, https://oeis.org/A078179), ''b''&lt;sup&gt;''n''&lt;/sup&gt;−(''b''−1) (''n'' ≥ 2) (references of this form: http://www.bitman.name/math/table/792, https://pzktupel.de/Primetables/TableWilliams5.php, https://oeis.org/A113516, https://oeis.org/A343589; or for prime bases ''b'': https://cs.uwaterloo.ca/journals/JIS/VOL3/mccranie.html, http://www.bitman.name/math/table/435)):

Problem: Given strings ''x'', ''z'' (may be empty), a digit ''y'', and a base ''b'' (''x'' does not [[:w:Leading zero|start with the digit 0]], ''z'' ends with a digit which [[:w:Coprime integers|coprime]] to ''b'', ''y'' is not 0 if ''x'' is empty, ''y'' is coprime to ''b'' if ''z'' is empty), does there exist a prime number whose base-''b'' expansion is of the form ''xy''&lt;sub&gt;''n''&lt;/sub&gt;''z'' for some ''n'' ≥ 0?

Some families can be ruled out to contain no prime &gt; ''b'' by [[:w:Covering set|covering congruence]], [[:w:Factorization of polynomials|algebraic factorization]] (e.g. [[:w:Difference of two squares|difference of two squares]], [[:w:Sum of two cubes|sum of two cubes]], [[:w:Sophie Germain's identity|Sophie Germain's identity of ''x''&lt;sup&gt;4&lt;/sup&gt;+4×''y''&lt;sup&gt;4&lt;/sup&gt;]]), or combine of them, e.g.

* The base 9 family 2{7}: Always divisible by 2 or 5
* The base 11 family 2{5}: Always divisible by 2 or 3
* The base 14 family B{0}1: Always divisible by 3 or 5
* The base 13 family 95{0}3: Always divisible by 5, 7, or 17
* The base 16 family {4}D: Always divisible by 3, 7, or 13
* The base 16 family {8}F: Always divisible by 3, 7, or 13
* The base 21 family {7}D: Always divisible by 2, 13, or 17
* The base 23 family {D}GA: Always divisible by 2, 5, 7, 37, or 79
* The base 9 family {1}: Can be written as (9&lt;sup&gt;''n''&lt;/sup&gt;−1)/8 and can be factored as (3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (3&lt;sup&gt;''n''&lt;/sup&gt;+1) / 8
* The base 8 family 1{0}1: Can be written as 8&lt;sup&gt;''n''&lt;/sup&gt;+1 and can be factored as (2&lt;sup&gt;''n''&lt;/sup&gt;+1) × (4&lt;sup&gt;''n''&lt;/sup&gt;−2&lt;sup&gt;''n''&lt;/sup&gt;+1)
* The base 9 family 3{8}: Can be written as 4×9&lt;sup&gt;''n''&lt;/sup&gt;−1 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (2×3&lt;sup&gt;''n''&lt;/sup&gt;+1)
* The base 16 family 1{5}: Can be written as (4×16&lt;sup&gt;''n''&lt;/sup&gt;−1)/3 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (2×3&lt;sup&gt;''n''&lt;/sup&gt;+1) / 3
* The base 16 family {4}1: Can be written as (4×16&lt;sup&gt;''n''&lt;/sup&gt;−49)/15 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−7) × (2×3&lt;sup&gt;''n''&lt;/sup&gt;+7) / 15
* The base 27 family 7{Q}: Can be written as 8×27&lt;sup&gt;''n''&lt;/sup&gt;−1 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (4×9&lt;sup&gt;''n''&lt;/sup&gt;+2×3&lt;sup&gt;''n''&lt;/sup&gt;+1)
* The base 27 family 9{G}: Can be written as (125×27&lt;sup&gt;''n''&lt;/sup&gt;−8)/13 and can be factored as (5×3&lt;sup&gt;''n''&lt;/sup&gt;−2) × (25×9&lt;sup&gt;''n''&lt;/sup&gt;+10×3&lt;sup&gt;''n''&lt;/sup&gt;+4)
* The base 16 family {C}D: Can be written as (4×16&lt;sup&gt;''n''&lt;/sup&gt;+1)/5 and can be factored as (2×4&lt;sup&gt;''n''&lt;/sup&gt;−2×2&lt;sup&gt;''n''&lt;/sup&gt;+1) × (2×4&lt;sup&gt;''n''&lt;/sup&gt;+2×2&lt;sup&gt;''n''&lt;/sup&gt;+1) / 5
* The base 14 family 8{D}: Can be written as 9×14&lt;sup&gt;''n''&lt;/sup&gt;−1, it is divisible by 5 if ''n'' is odd and can be factored as (3×14&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (3×14&lt;sup&gt;''n''/2&lt;/sup&gt;+1) if ''n'' is even
* The base 12 family {B}9B: Can be written as 12&lt;sup&gt;''n''&lt;/sup&gt;−25, it is divisible by 13 if ''n'' is odd and can be factored as (12&lt;sup&gt;''n''/2&lt;/sup&gt;−5) × (12&lt;sup&gt;''n''/2&lt;/sup&gt;+5) if ''n'' is even
* The base 14 family {D}5: Can be written as 14&lt;sup&gt;''n''&lt;/sup&gt;−9, it is divisible by 5 if ''n'' is odd and can be factored as (14&lt;sup&gt;''n''/2&lt;/sup&gt;−3) × (14&lt;sup&gt;''n''/2&lt;/sup&gt;+3) if ''n'' is even
* The base 17 family 1{9}: Can be written as (25×17&lt;sup&gt;''n''&lt;/sup&gt;−9)/16, it is divisible by 2 if ''n'' is odd and can be factored as (5×17&lt;sup&gt;''n''/2&lt;/sup&gt;−3) × (5×17&lt;sup&gt;''n''/2&lt;/sup&gt;+3) / 16 if ''n'' is even
* The base 17 family 7{9}: Can be written as (121×17&lt;sup&gt;''n''&lt;/sup&gt;−9)/16, it is divisible by 2 if ''n'' is odd and can be factored as (11×17&lt;sup&gt;''n''/2&lt;/sup&gt;−3) × (11×17&lt;sup&gt;''n''/2&lt;/sup&gt;+3) / 16 if ''n'' is even
* The base 19 family 1{6}: Can be written as (4×19&lt;sup&gt;''n''&lt;/sup&gt;−1)/3, it is divisible by 5 if ''n'' is odd and can be factored as (2×19&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (2×19&lt;sup&gt;''n''/2&lt;/sup&gt;+1) / 3 if ''n'' is even
* The base 19 family 7{2}: Can be written as (64×19&lt;sup&gt;''n''&lt;/sup&gt;−1)/9, it is divisible by 5 if ''n'' is odd and can be factored as (8×19&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (8×19&lt;sup&gt;''n''/2&lt;/sup&gt;+1) / 9 if ''n'' is even
* The base 24 family 3{N}: Can be written as 4×24&lt;sup&gt;''n''&lt;/sup&gt;−1, it is divisible by 5 if ''n'' is odd and can be factored as (2×24&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (2×24&lt;sup&gt;''n''/2&lt;/sup&gt;+1) if ''n'' is even

By the [[:w:Prime number theorem|prime number theorem]], the [[:w:Probability|chance]] that a [[:w:Random number|random]] ''n''-digit base ''b'' number is prime is [[:w:Asymptotic analysis|approximately]] 1/''n'' (more accurately, the chance is approximately 1/(''n''×''ln''(''b'')), where ''ln'' is the [[:w:Natural logarithm|natural logarithm]]). If one conjectures the numbers ''x''{''y''}''z'' behave similarly (i.e. the numbers ''x''{''y''}''z'' is a [[:w:Pseudorandomness|pseudorandom sequence]]) you would expect [[:w:Harmonic_series (mathematics)|1/1 + 1/2 + 1/3 + 1/4 + ... = ∞]] primes of the form ''x''{''y''}''z'' (of course, this does not always happen, since some ''x''{''y''}''z'' families can be ruled out to contain no prime &gt; ''b'' (by covering congruence, algebraic factorization, or combine of them), but it is at least a reasonable conjecture in the absence of evidence to the contrary. Hence, the [[:w:Heuristic argument|heuristic argument]] suggests there are always infinitely many primes in family ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') if it cannot be ruled out to contain no prime or only contain finitely many primes, by covering congruence, algebraic factorization, or combine of them. However, some families ''x''{''y''}''z'' could not be proven to contain no primes &gt; ''b'' (by covering congruence, algebraic factorization, or combine of them) but no primes &gt; ''b'' could be found in the family, even after searching through numbers with over 100000 digits. In such a case, the only way to proceed is to [[:w:Primality test|test the primality]] of larger and larger numbers of such form and hope a prime is eventually discovered. e.g. the smallest (probable) prime in the family A{3}A in base ''b'' = 13 is A3&lt;sub&gt;592197&lt;/sub&gt;A, its algebraic form is (41×13&lt;sup&gt;592198&lt;/sup&gt;+27)/4, when written in decimal contains 659677 digits (it is only probable prime, i.e. not definitely prime, since technically, probable primality tests were used to show this (which have a ''very'' small chance of making an error, see https://t5k.org/notes/prp_prob.html) because all known primality tests run far too slowly to run on numbers of this size unless either [https://t5k.org/prove/prove3_1.html ''N''−1] or [https://t5k.org/prove/prove3_2.html ''N''+1] (or both) can be ≥ 1/3 factored).

The numbers in family ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') are of the form (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1) for some fixed ''a'', ''b'', ''c'' such that ''a'' ≥ 1, ''b'' ≥ 2 (''b'' is the base), ''c'' ≠ 0, ''gcd''(''a'',''c'') = 1, ''gcd''(''b'',''c'') = 1. Except in the [[:w:Special case|special case]] ''c'' = ±1 and ''gcd''(''a''+''c'',''b''−1) = 1, when ''n'' is large the known [[:w:Primality test|primality test]]s for such a number are too inefficient to run. In this case one must resort to a [[:w:Probabilistic algorithm|probable]] primality test such as a [[:w:Miller–Rabin primality test|Miller–Rabin primality test]] or a [[:w:Baillie–PSW primality test|Baillie–PSW primality test]], unless a divisor of the number can be found. Since we are testing many numbers in an [[:w:Exponential growth|exponential sequence]], it is possible to use a sieving process to find divisors rather than using [[:w:Trial division|trial division]].

To do this, we made use of Geoffrey Reynolds' ''srsieve'' software (download: https://pzktupel.de/Software/srsieve_1.1.4.7z). This program uses the [[:w:Baby-step giant-step|baby-step giant-step]] [[:w:Algorithm|algorithm]] to find all primes ''p'' which divide ''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'' where ''p'' and ''n'' lie in a [[:w:Interval_(mathematics)|specified range]]. Since this program cannot handle the general case (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1) when ''gcd''(''a''+''c'',''b''−1) &gt; 1 we only used it to sieve the sequence ''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'' for primes ''p'' not dividing ''gcd''(''a''+''c'',''b''−1), and initialized the list of candidates to not include ''n'' for which there is some prime ''p'' dividing ''gcd''(''a''+''c'',''b''−1) for which ''p'' dividing (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1). The program had to be modified slightly to remove a check which would prevent it from running in the case when ''a'', ''b'', and ''c'' were all odd (since then 2 divides ''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'', but 2 may not divide (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1)).

Once the numbers with small divisors had been removed, it remained to test the remaining numbers using a probable primality test. For this we used the software ''LLR'' by Jean Penné. (download: http://jpenne.free.fr/index2.html). Although undocumented, it is possible to run this program on numbers of the form (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1) when ''gcd''(''a''+''c'',''b''−1) &gt; 1, so this program required no modifications. A script was also written which allowed one to run ''srsieve'' while ''LLR'' was testing the remaining candidates, so that when a divisor was found by srsieve on a number which had not yet been tested by ''LLR'' it would be removed from the list of candidates.

For the primes &lt; 10&lt;sup&gt;25000&lt;/sup&gt; for the &quot;easy&quot; bases (bases ''b'' with ≤ 150 primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; (base ''b'' = 26 has 83 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 3 unsolved families, base ''b'' = 36 has 75 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 4 unsolved families, base ''b'' = 17 has 99 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 18 unsolved families, base ''b'' = 21 has 80 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 12 unsolved families, base ''b'' = 19 has 201 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 23 unsolved families), i.e. bases *b* = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 26, 28, 30, 36), we employed ''CM'' by Andreas Enge (download: https://www.multiprecision.org/cm/download.html), an elliptic curve primality proving implementation.

== Data ==

These are the results of the Athena problem in bases 2 ≤ ''b'' ≤ 36 (we stop at base 36 since this base is the maximum base for which it is possible to write the numbers with the [[:w:Symbol|symbol]]s 0, 1, 2, ..., 9 and A, B, C, ..., Z (i.e. the 10 [[:w:Arabic numerals|Arabic numerals]] and the 26 [[:w:Latin script|Latin letters]]): (some large primes are only probable primes, i.e. not definitely primes, since they are too large to be [[:w:Elliptic curve primality|ECPP proved]] and [[:w:Pocklington primality test#Extensions and variants|neither ''N''−1 nor ''N''+1 can be ≥ 1/3 factored]], all of them pass the [[:w:Baillie–PSW primality test|Baillie–PSW primality test]] and the [[:w:Strong pseudoprime|strong primality test]] (i.e. the [[:w:Miller–Rabin primality test|Miller–Rabin primality test]]) with all prime bases ''p'' ≤ 61, however, all primes &lt; 10&lt;sup&gt;25000&lt;/sup&gt; for bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 36 are definitely primes, most of them &gt; 10&lt;sup&gt;299&lt;/sup&gt; are proven primes with [[:w:Elliptic curve primality|ECPP proving]], others &gt; 10&lt;sup&gt;299&lt;/sup&gt; are proven primes with [[:w:Pocklington primality test#Extensions and variants|''N''−1 or ''N''+1 proving]])

All numbers are written in base ''b'', [[:w:Senary#Base 36 as senary compression|using A to Z to represent digit values 10 to 35]], &quot;{}&quot; means repeating, e.g. family 12{3}45 means the sequence {1245, 12345, 123345, 1233345, 12333345, 123333345, ...} (where the members are expressed as base ''b'' strings), subscripts are used to indicate repetitions of digits, e.g. 123&lt;sub&gt;4&lt;/sub&gt;567 means 123333567 (all subscripts are written in decimal).

Base 2: 1 Athena prime (the largest of which has 2 digits (it is 11, and its value is 3 in decimal)): {11}

Base 3: 3 Athena primes (the largest of which has 3 digits (it is 111, and its value is 13 in decimal)): {12, 21, 111}

Base 4: 5 Athena primes (the largest of which has 3 digits (it is 221, and its value is 41 in decimal)): {11, 13, 23, 31, 221}

Base 5: 22 Athena primes (the largest of which has 96 digits (it is 10&lt;sub&gt;93&lt;/sub&gt;13, and its algebraic form is 5&lt;sup&gt;95&lt;/sup&gt;+8)): {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 Athena primes (the largest of which has 5 digits (it is 40041, and its value is 5209 in decimal)): {11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041}

Base 7: 71 Athena primes (the largest of which has 17 digits (it is 3&lt;sub&gt;16&lt;/sub&gt;1, and its algebraic form is (7&lt;sup&gt;17&lt;/sup&gt;−5)/2)): {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: 75 Athena primes (the largest of which has 221 digits (it is 4&lt;sub&gt;220&lt;/sub&gt;7, and its algebraic form is (4×8&lt;sup&gt;221&lt;/sup&gt;+17)/7)): {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: 151 Athena primes (the largest of which has 1161 digits (it is 30&lt;sub&gt;1158&lt;/sub&gt;11, and its algebraic form is 3×9&lt;sup&gt;1160&lt;/sup&gt;+10)): {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: 77 Athena primes (the largest of which has 31 digits (it is 50&lt;sub&gt;28&lt;/sub&gt;27, and its algebraic form is 5×10&lt;sup&gt;30&lt;/sup&gt;+27)): {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: 1068 Athena primes (including 1 unproven probable prime: 57&lt;sub&gt;62668&lt;/sub&gt;), the largest of which has 62669 digits (it is 57&lt;sub&gt;62668&lt;/sub&gt;, and its algebraic form is (57×11&lt;sup&gt;62668&lt;/sup&gt;−7)/10), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel11 Data of Athena (probable) primes base 11]

Base 12: 106 Athena primes (the largest of which has 42 digits (it is 40&lt;sub&gt;39&lt;/sub&gt;77, and its algebraic form is 4×12&lt;sup&gt;41&lt;/sup&gt;+91)): {11, 15, 17, 1B, 25, 27, 31, 35, 37, 3B, 45, 4B, 51, 57, 5B, 61, 67, 6B, 75, 81, 85, 87, 8B, 91, 95, A7, AB, B5, B7, 221, 241, 2A1, 2B1, 2BB, 401, 421, 447, 471, 497, 565, 655, 665, 701, 70B, 721, 747, 771, 77B, 797, 7A1, 7BB, 907, 90B, 9BB, A41, B21, B2B, 2001, 200B, 202B, 222B, 229B, 292B, 299B, 4441, 4707, 4777, 6A05, 6AA5, 729B, 7441, 7B41, 929B, 9777, 992B, 9947, 997B, 9997, A0A1, A201, A605, A6A5, AA65, B001, B0B1, BB01, BB41, 600A5, 7999B, 9999B, AAAA1, B04A1, B0B9B, BAA01, BAAA1, BB09B, BBBB1, 44AAA1, A00065, BBBAA1, AAA0001, B00099B, AA000001, BBBBBB99B, B0000000000000000000000000009B, 400000000000000000000000000000000000000077}

Base 13: 3197 Athena primes (including 4 unproven probable primes: C5&lt;sub&gt;23755&lt;/sub&gt;C, 80&lt;sub&gt;32017&lt;/sub&gt;111, 95&lt;sub&gt;197420&lt;/sub&gt;, A3&lt;sub&gt;592197&lt;/sub&gt;A), the largest of which has 592199 digits (it is A3&lt;sub&gt;592197&lt;/sub&gt;A, and its algebraic form is (41×13&lt;sup&gt;592198&lt;/sup&gt;+27)/4), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel13 Data of Athena (probable) primes base 13]

Base 14: 650 Athena primes, the largest of which has 19699 digits (it is 4D&lt;sub&gt;19698&lt;/sub&gt;, and its algebraic form is 5×14&lt;sup&gt;19698&lt;/sup&gt;−1), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel14 Data of Athena primes base 14]

Base 15: 1284 Athena primes, the largest of which has 157 digits (it is 7&lt;sub&gt;155&lt;/sub&gt;97, and its algebraic form is (15&lt;sup&gt;157&lt;/sup&gt;+59)/2), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel15 Data of Athena primes base 15]

Base 16: 2347 Athena primes (including 3 unproven probable primes: DB&lt;sub&gt;32234&lt;/sub&gt;, 4&lt;sub&gt;72785&lt;/sub&gt;DD, 3&lt;sub&gt;116137&lt;/sub&gt;AF), the largest of which has 116139 digits (it is 3&lt;sub&gt;116137&lt;/sub&gt;AF, and its algebraic form is (16&lt;sup&gt;116139&lt;/sup&gt;+619)/5), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel16 Data of Athena (probable) primes base 16]

Base 17: 10415 known Athena primes (including many unproven probable primes) and 12 unsolved families (1{7}, 1F{0}7, 4{7}A, 70F{0}D, 8{B}9, 9{5}9, A{D}F, B{0}B3, {B}E9, {B}EE, F1{9}, FD0{D}, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel17 Data of known Athena (probable) primes base 17]

Base 18: 549 Athena primes, the largest of which has 6271 digits (it is C0&lt;sub&gt;6268&lt;/sub&gt;C5, and its algebraic form is 12×18&lt;sup&gt;6270&lt;/sup&gt;+221), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel18 Data of Athena primes base 18]

Base 19: 31417 known Athena primes (including many unproven probable primes) and 17 unsolved families (4B5{0}H, {5}3, 5{H}05, 5{H}0H, 5{H}5, 66{B}, 71{0}177, 7AF{0}H, 97{0}3, C{H}C, EE1{6}, F{7}5, F{B}G, F{D}F, H0F{0}7A, HB{0}5B5, II{D}, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel19 Data of known Athena (probable) primes base 19]

Base 20: 3314 Athena primes, the largest of which has 6271 digits (it is G0&lt;sub&gt;6269&lt;/sub&gt;D, and its algebraic form is 16×20&lt;sup&gt;6270&lt;/sup&gt;+13), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel20 Data of Athena primes base 20]

Base 21: 13386 known Athena primes (including many unproven probable primes) and 8 unsolved families (5{0}DJ, {9}D, B3{0}EB, B{H}6H, C{F}0K, {F}35, G{0}FK, H{0}7771, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel21 Data of known Athena (probable) primes base 21]

Base 22: 8003 Athena primes (including 1 unproven probable prime: BK&lt;sub&gt;22001&lt;/sub&gt;5), the largest of which has 22003 digits (it is BK&lt;sub&gt;22001&lt;/sub&gt;5, and its algebraic form is (251×22&lt;sup&gt;22002&lt;/sup&gt;−335)/21), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel22 Data of Athena (probable) primes base 22]

Base 23: 65178 known Athena primes (including many unproven probable primes) and 87 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel23 Data of known Athena (probable) primes base 23] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left23 Data of unsolved families for Athena problem base 23]

Base 24: 3409 Athena primes, the largest of which has 8134 digits (it is N00N&lt;sub&gt;8129&lt;/sub&gt;LN, and its algebraic form is 13249×24&lt;sup&gt;8131&lt;/sup&gt;−49), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel24 Data of Athena primes base 24]

Base 25: 133639 known Athena primes (including many unproven probable primes) and 85 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel25 Data of known Athena (probable) primes base 25] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left25 Data of unsolved families for Athena problem base 25]

Base 26: 25256 known Athena primes (including 7 unproven probable primes: 5&lt;sub&gt;19391&lt;/sub&gt;6F, 7&lt;sub&gt;20279&lt;/sub&gt;OL, LD0&lt;sub&gt;20975&lt;/sub&gt;7, 6K&lt;sub&gt;23300&lt;/sub&gt;5, J0&lt;sub&gt;44303&lt;/sub&gt;KCB, M0&lt;sub&gt;61186&lt;/sub&gt;2BB, 85M&lt;sub&gt;197060&lt;/sub&gt;B) and 3 unsolved families ({A}6F, {H}MH, {I}GL, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel26 Data of known Athena (probable) primes base 26]

Base 27: 102852 known Athena primes (including many unproven probable primes) and 44 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel27 Data of known Athena (probable) primes base 27] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left27 Data of unsolved families for Athena problem base 27]

Base 28: 25528 known Athena primes (including 3 unproven probable primes: N6&lt;sub&gt;24051&lt;/sub&gt;LR, 5OA&lt;sub&gt;31238&lt;/sub&gt;F, O4O&lt;sub&gt;94535&lt;/sub&gt;9) and 1 unsolved family (O{A}F, no primes or probable primes with length ≤ 900000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel28 Data of known Athena (probable) primes base 28]

Base 29: 355242 known Athena primes (including many unproven probable primes) and 125 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel29 Data of known Athena (probable) primes base 29] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left29 Data of unsolved families for Athena problem base 29]

Base 30: 2619 Athena primes (including 1 unproven probable prime: I0&lt;sub&gt;24608&lt;/sub&gt;D), the largest of which has 34206 digits (it is OT&lt;sub&gt;34205&lt;/sub&gt;, and its algebraic form is 25×30&lt;sup&gt;34205&lt;/sup&gt;−1), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel30 Data of Athena (probable) primes base 30]

Base 31: 569323 known Athena primes (including many unproven probable primes) and 77 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel31 Data of known Athena (probable) primes base 31] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left31 Data of unsolved families for Athena problem base 31]

Base 32: 168882 known Athena primes (including many unproven probable primes) and 120 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel32 Data of known Athena (probable) primes base 32] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left32 Data of unsolved families for Athena problem base 32]

Base 33: 280012 known Athena primes (including many unproven probable primes) and 81 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel33 Data of known Athena (probable) primes base 33] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left33 Data of unsolved families for Athena problem base 33]

Base 34: 184785 known Athena primes (including many unproven probable primes) and 47 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel34 Data of known Athena (probable) primes base 34] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left34 Data of unsolved families for Athena problem base 34]

Base 35: 720002 known Athena primes (including many unproven probable primes) and 60 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel35 Data of known Athena (probable) primes base 35] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left35 Data of unsolved families for Athena problem base 35]

Base 36: 35286 known Athena primes (including 3 unproven probable primes: 7K&lt;sub&gt;26567&lt;/sub&gt;Z, S0&lt;sub&gt;75007&lt;/sub&gt;8H, P&lt;sub&gt;81993&lt;/sub&gt;SZ) and 4 unsolved families (B{0}EUV, HM{0}N, N{0}YYN, O{L}Z, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel36 Data of known Athena (probable) primes base 36]

== Condensed table for bases 2 ≤ ''b'' ≤ 36 ==

{|class=&quot;wikitable&quot;
||''b''||number of primes (or probable prime, which is a minimal prime assuming its primality) in the Athena problem base ''b''||base-''b'' form of the top 10 known primes (or probable prime, which is a minimal prime assuming its primality) in the Athena problem base ''b'' (write &quot;''d''&lt;sub&gt;''n''&lt;/sub&gt;&quot; if there are 5 or more (''n'') consecutive same digits ''d'')||length of the top 10 known primes (or probable prime, which is a minimal prime assuming its primality) in the Athena problem base ''b''||length of the top 10 known primes (or probable prime, which is a minimal prime assuming its primality) in the Athena problem base ''b'' in decimal||algebraic ((''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1)) form of the top 10 known primes (or probable prime, which is a minimal prime assuming its primality) in the Athena problem base ''b''||''factordb'' entry of the top 10 known primes (or probable prime, which is a minimal prime assuming its primality) in the Athena problem base ''b''||the top 10 known primes (or probable prime, which is a minimal prime assuming its primality) in the Athena problem base ''b'' written in base ''b'' (use lower case letters instead of upper case letters)||number of left families in the Athena problem base ''b'' (all of these left families are linear families)||searching limit of length for the left families in the Athena problem base ''b'' (if there are different searching limits for the left families in base ''b'', choose the lowest searching limit)||
|-
||2||1||11||2||1||3||http://factordb.com/index.php?id=3&amp;open=ecm||http://factordb.com/index.php?showid=3&amp;base=2||0||–||
|-
||3||3||111&lt;br&gt;21&lt;br&gt;12||3&lt;br&gt;2&lt;br&gt;2||2&lt;br&gt;1&lt;br&gt;1||13&lt;br&gt;7&lt;br&gt;5||http://factordb.com/index.php?id=13&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=7&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=5&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=13&amp;base=3&lt;br&gt;http://factordb.com/index.php?showid=7&amp;base=3&lt;br&gt;http://factordb.com/index.php?showid=5&amp;base=3&lt;nowiki/&gt;||0||–||
|-
||4||5||221&lt;br&gt;31&lt;br&gt;23&lt;br&gt;13&lt;br&gt;11||3&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2||2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;1&lt;br&gt;1||41&lt;br&gt;13&lt;br&gt;11&lt;br&gt;7&lt;br&gt;5||http://factordb.com/index.php?id=41&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=13&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=11&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=7&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=5&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=41&amp;base=4&lt;br&gt;http://factordb.com/index.php?showid=13&amp;base=4&lt;br&gt;http://factordb.com/index.php?showid=11&amp;base=4&lt;br&gt;http://factordb.com/index.php?showid=7&amp;base=4&lt;br&gt;http://factordb.com/index.php?showid=5&amp;base=4&lt;nowiki/&gt;||0||–||
|-
||5||22||10&lt;sub&gt;93&lt;/sub&gt;13&lt;br&gt;300031&lt;br&gt;44441&lt;br&gt;33331&lt;br&gt;33001&lt;br&gt;30301&lt;br&gt;14444&lt;br&gt;10103&lt;br&gt;3101&lt;br&gt;414||96&lt;br&gt;6&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;4&lt;br&gt;3||67&lt;br&gt;4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;3&lt;br&gt;3&lt;br&gt;3||5&lt;sup&gt;95&lt;/sup&gt;+8&lt;br&gt;9391&lt;br&gt;3121&lt;br&gt;2341&lt;br&gt;2251&lt;br&gt;1951&lt;br&gt;1249&lt;br&gt;653&lt;br&gt;401&lt;br&gt;109||http://factordb.com/index.php?id=1100000000034686071&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=9391&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=3121&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=2341&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=2251&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1951&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1249&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=653&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=401&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=109&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000000034686071&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=9391&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=3121&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=2341&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=2251&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=1951&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=1249&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=653&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=401&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=109&amp;base=5&lt;nowiki/&gt;||0||–||
|-
||6||11||40041&lt;br&gt;4441&lt;br&gt;4401&lt;br&gt;51&lt;br&gt;45&lt;br&gt;35&lt;br&gt;31&lt;br&gt;25&lt;br&gt;21&lt;br&gt;15||5&lt;br&gt;4&lt;br&gt;4&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2||4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2||5209&lt;br&gt;1033&lt;br&gt;1009&lt;br&gt;31&lt;br&gt;29&lt;br&gt;23&lt;br&gt;19&lt;br&gt;17&lt;br&gt;13&lt;br&gt;11||http://factordb.com/index.php?id=5209&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1033&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1009&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=31&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=29&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=23&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=19&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=17&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=13&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=11&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=5209&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=1033&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=1009&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=31&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=29&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=23&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=19&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=17&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=13&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=11&amp;base=6&lt;nowiki/&gt;||0||–||
|-
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||33||280012~280093||DP&lt;sub&gt;95093&lt;/sub&gt;M5&lt;br&gt;HJ0&lt;sub&gt;94295&lt;/sub&gt;J&lt;br&gt;90&lt;sub&gt;93597&lt;/sub&gt;Q&lt;br&gt;9F0&lt;sub&gt;93157&lt;/sub&gt;N&lt;br&gt;7&lt;sub&gt;89449&lt;/sub&gt;333H&lt;br&gt;K3&lt;sub&gt;80751&lt;/sub&gt;6K&lt;br&gt;D&lt;sub&gt;80107&lt;/sub&gt;9UD&lt;br&gt;VFU&lt;sub&gt;72204&lt;/sub&gt;FK&lt;br&gt;J&lt;sub&gt;68715&lt;/sub&gt;2BJ&lt;br&gt;DF0&lt;sub&gt;68367&lt;/sub&gt;J||95096&lt;br&gt;94298&lt;br&gt;93599&lt;br&gt;93160&lt;br&gt;89453&lt;br&gt;80754&lt;br&gt;80110&lt;br&gt;72208&lt;br&gt;68718&lt;br&gt;68370||144405&lt;br&gt;143193&lt;br&gt;142131&lt;br&gt;141465&lt;br&gt;135835&lt;br&gt;122626&lt;br&gt;121648&lt;br&gt;109649&lt;br&gt;104350&lt;br&gt;103821||(441×33&lt;sup&gt;95095&lt;/sup&gt;−3833)/32&lt;br&gt;580×33&lt;sup&gt;94296&lt;/sup&gt;+19&lt;br&gt;9×33&lt;sup&gt;93598&lt;/sup&gt;+26&lt;br&gt;312×33&lt;sup&gt;93158&lt;/sup&gt;+23&lt;br&gt;(7×33&lt;sup&gt;89453&lt;/sup&gt;−4743239)/32&lt;br&gt;(643×33&lt;sup&gt;80753&lt;/sup&gt;+3709)/32&lt;br&gt;(13×33&lt;sup&gt;80110&lt;/sup&gt;−121453)/32&lt;br&gt;(16623×33&lt;sup&gt;72206&lt;/sup&gt;−8095)/16&lt;br&gt;(19×33&lt;sup&gt;68718&lt;/sup&gt;−600883)/32&lt;br&gt;444×33&lt;sup&gt;68368&lt;/sup&gt;+19||http://factordb.com/index.php?id=1100000005652348775&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000005652362811&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000005652375073&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000005652389776&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000005652430746&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000005652446200&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000005652461592&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004614764298&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004614770536&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004614784274&amp;open=prime&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000005652348775&amp;base=33&lt;br&gt;http://factordb.com/index.php?showid=1100000005652362811&amp;base=33&lt;br&gt;http://factordb.com/index.php?showid=1100000005652375073&amp;base=33&lt;br&gt;http://factordb.com/index.php?showid=1100000005652389776&amp;base=33&lt;br&gt;http://factordb.com/index.php?showid=1100000005652430746&amp;base=33&lt;br&gt;http://factordb.com/index.php?showid=1100000005652446200&amp;base=33&lt;br&gt;http://factordb.com/index.php?showid=1100000005652461592&amp;base=33&lt;br&gt;http://factordb.com/index.php?showid=1100000004614764298&amp;base=33&lt;br&gt;http://factordb.com/index.php?showid=1100000004614770536&amp;base=33&lt;br&gt;http://factordb.com/index.php?showid=1100000004614784274&amp;base=33&lt;nowiki/&gt;||81||100000||
|-
||34||184785~184832||GFGC&lt;sub&gt;99996&lt;/sub&gt;5&lt;br&gt;90&lt;sub&gt;97950&lt;/sub&gt;FJ&lt;br&gt;NM0&lt;sub&gt;85218&lt;/sub&gt;KX&lt;br&gt;F&lt;sub&gt;83189&lt;/sub&gt;H2HP&lt;br&gt;P&lt;sub&gt;79441&lt;/sub&gt;444P&lt;br&gt;6&lt;sub&gt;77027&lt;/sub&gt;8X&lt;br&gt;XQIQ&lt;sub&gt;72241&lt;/sub&gt;D&lt;br&gt;T&lt;sub&gt;66530&lt;/sub&gt;IF&lt;br&gt;4&lt;sub&gt;66152&lt;/sub&gt;B&lt;br&gt;2EEC&lt;sub&gt;66039&lt;/sub&gt;7||100000&lt;br&gt;97953&lt;br&gt;85222&lt;br&gt;83193&lt;br&gt;79445&lt;br&gt;77029&lt;br&gt;72245&lt;br&gt;66532&lt;br&gt;66153&lt;br&gt;66043||153148&lt;br&gt;150013&lt;br&gt;130516&lt;br&gt;127408&lt;br&gt;121669&lt;br&gt;117968&lt;br&gt;110642&lt;br&gt;101893&lt;br&gt;101312&lt;br&gt;101143||(209246×34&lt;sup&gt;99997&lt;/sup&gt;−81)/11&lt;br&gt;9×34&lt;sup&gt;97952&lt;/sup&gt;+529&lt;br&gt;804×34&lt;sup&gt;85220&lt;/sup&gt;+713&lt;br&gt;(5×34&lt;sup&gt;83193&lt;/sup&gt;+700233)/11&lt;br&gt;(25×34&lt;sup&gt;79445&lt;/sup&gt;−28062367)/33&lt;br&gt;(2×34&lt;sup&gt;77029&lt;/sup&gt;+1043)/11&lt;br&gt;(1288676×34&lt;sup&gt;72242&lt;/sup&gt;−455)/33&lt;br&gt;(29×34&lt;sup&gt;66532&lt;/sup&gt;−12833)/33&lt;br&gt;(4×34&lt;sup&gt;66153&lt;/sup&gt;+227)/33&lt;br&gt;(30826×34&lt;sup&gt;66040&lt;/sup&gt;−59)/11||http://factordb.com/index.php?id=1100000004702891268&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004702894713&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004702900996&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004702910376&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004702913746&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004702918600&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004399656529&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004399657696&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004399658651&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004399659716&amp;open=prime&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000004702891268&amp;base=34&lt;br&gt;http://factordb.com/index.php?showid=1100000004702894713&amp;base=34&lt;br&gt;http://factordb.com/index.php?showid=1100000004702900996&amp;base=34&lt;br&gt;http://factordb.com/index.php?showid=1100000004702910376&amp;base=34&lt;br&gt;http://factordb.com/index.php?showid=1100000004702913746&amp;base=34&lt;br&gt;http://factordb.com/index.php?showid=1100000004702918600&amp;base=34&lt;br&gt;http://factordb.com/index.php?showid=1100000004399656529&amp;base=34&lt;br&gt;http://factordb.com/index.php?showid=1100000004399657696&amp;base=34&lt;br&gt;http://factordb.com/index.php?showid=1100000004399658651&amp;base=34&lt;br&gt;http://factordb.com/index.php?showid=1100000004399659716&amp;base=34&lt;nowiki/&gt;||47||100000||
|-
||35||720002~720062||N0N&lt;sub&gt;99971&lt;/sub&gt;9&lt;br&gt;V0&lt;sub&gt;83669&lt;/sub&gt;E73&lt;br&gt;N&lt;sub&gt;81563&lt;/sub&gt;K7N&lt;br&gt;BJ0&lt;sub&gt;81279&lt;/sub&gt;N&lt;br&gt;J0&lt;sub&gt;80062&lt;/sub&gt;FUH&lt;br&gt;43V&lt;sub&gt;79754&lt;/sub&gt;&lt;br&gt;9&lt;sub&gt;76600&lt;/sub&gt;K3&lt;br&gt;LB&lt;sub&gt;71366&lt;/sub&gt;PB&lt;br&gt;Q&lt;sub&gt;64150&lt;/sub&gt;H&lt;br&gt;50&lt;sub&gt;63397&lt;/sub&gt;5R||99974&lt;br&gt;83673&lt;br&gt;81566&lt;br&gt;81282&lt;br&gt;80066&lt;br&gt;79756&lt;br&gt;76602&lt;br&gt;71369&lt;br&gt;64151&lt;br&gt;63400||154367&lt;br&gt;129197&lt;br&gt;125944&lt;br&gt;125505&lt;br&gt;123628&lt;br&gt;123148&lt;br&gt;118279&lt;br&gt;110199&lt;br&gt;99054&lt;br&gt;97894||(27393×35&lt;sup&gt;99972&lt;/sup&gt;−499)/34&lt;br&gt;31×35&lt;sup&gt;83672&lt;/sup&gt;+17398&lt;br&gt;(23×35&lt;sup&gt;81566&lt;/sup&gt;−144013)/34&lt;br&gt;404×35&lt;sup&gt;81280&lt;/sup&gt;+23&lt;br&gt;19×35&lt;sup&gt;80065&lt;/sup&gt;+19442&lt;br&gt;(4893×35&lt;sup&gt;79754&lt;/sup&gt;−31)/34&lt;br&gt;(9×35&lt;sup&gt;76602&lt;/sup&gt;+12877)/34&lt;br&gt;(725×35&lt;sup&gt;71368&lt;/sup&gt;+16649)/34&lt;br&gt;(13×35&lt;sup&gt;64151&lt;/sup&gt;−166)/17&lt;br&gt;5×35&lt;sup&gt;63399&lt;/sup&gt;+202||http://factordb.com/index.php?id=1100000008248342445&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008248353306&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008248375642&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008248397018&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008248412468&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008248418540&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008248423670&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008192119974&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008192126630&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008192129294&amp;open=prime&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000008248342445&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008248353306&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008248375642&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008248397018&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008248412468&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008248418540&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008248423670&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008192119974&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008192126630&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008192129294&amp;base=35&lt;nowiki/&gt;||60||100000||
|-
||36||35286~35290||P&lt;sub&gt;81993&lt;/sub&gt;SZ&lt;br&gt;S0&lt;sub&gt;75007&lt;/sub&gt;8H&lt;br&gt;7K&lt;sub&gt;26567&lt;/sub&gt;Z&lt;br&gt;J&lt;sub&gt;10117&lt;/sub&gt;LJ&lt;br&gt;VL0&lt;sub&gt;7258&lt;/sub&gt;J&lt;br&gt;EO0&lt;sub&gt;6177&lt;/sub&gt;V&lt;br&gt;FZ&lt;sub&gt;5777&lt;/sub&gt;3P&lt;br&gt;T09&lt;sub&gt;4618&lt;/sub&gt;1&lt;br&gt;RY&lt;sub&gt;4562&lt;/sub&gt;H&lt;br&gt;OZ&lt;sub&gt;3932&lt;/sub&gt;AZ||81995&lt;br&gt;75010&lt;br&gt;26569&lt;br&gt;10119&lt;br&gt;7261&lt;br&gt;6180&lt;br&gt;5780&lt;br&gt;4621&lt;br&gt;4564&lt;br&gt;3935||127609&lt;br&gt;116739&lt;br&gt;41349&lt;br&gt;15748&lt;br&gt;11301&lt;br&gt;9618&lt;br&gt;8996&lt;br&gt;7192&lt;br&gt;7103&lt;br&gt;6124||(5×36&lt;sup&gt;81995&lt;/sup&gt;+821)/7&lt;br&gt;28×36&lt;sup&gt;75009&lt;/sup&gt;+305&lt;br&gt;(53×36&lt;sup&gt;26568&lt;/sup&gt;+101)/7&lt;br&gt;(19×36&lt;sup&gt;10119&lt;/sup&gt;+2501)/35&lt;br&gt;1137×36&lt;sup&gt;7259&lt;/sup&gt;+19&lt;br&gt;528×36&lt;sup&gt;6178&lt;/sup&gt;+31&lt;br&gt;16×36&lt;sup&gt;5779&lt;/sup&gt;−1163&lt;br&gt;(36549×36&lt;sup&gt;4619&lt;/sup&gt;−289)/35&lt;br&gt;(979×36&lt;sup&gt;4563&lt;/sup&gt;−629)/35&lt;br&gt;25×36&lt;sup&gt;3934&lt;/sup&gt;−901||http://factordb.com/index.php?id=1100000002394962083&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004020085177&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003896952461&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362491&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362489&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362488&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362487&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362486&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362485&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000000840634476&amp;open=prime&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000002394962083&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000004020085177&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003896952461&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362491&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362489&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362488&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362487&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362486&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362485&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000000840634476&amp;base=36&lt;nowiki/&gt;||4||200000||
|}

== The fully proof of Athena problem in decimal (base ''b'' = 10) ==

'''Bold''' for the Athena primes, ''x'' ◁ ''y'' means ''x'' is a subsequence of ''y''.

Assume ''p'' is a prime &gt; 10, and the last digit of ''p'' must lie in {1,3,7,9}.

Case 1: ''p'' ends with 1.

In this case we can write ''p'' = ''x''1. If ''x'' contains 1, 3, 4, 6, or 7, then (respectively) '''11''' ◁ ''p'', '''31''' ◁ ''p'', '''41''' ◁ ''p'', '''61''' ◁ ''p'', or '''71''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 2, 5, 8, or 9.

Case 1.1: ''p'' begins with 2.

In this case we can write ''p'' = 2''y''1. If 5 ◁ ''y'', then '''251''' ◁ ''p''. If 8 ◁ ''y'', then '''281''' ◁ ''p''. If 9 ◁ ''y'', then 29 ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 2.

If 22 ◁ ''y'', then '''2221''' ◁ ''p''. Hence we may assume ''y'' contains zero or one 2's.

If ''y'' contains no 2's, then ''p'' ∈ 2{0}1. But then, since the sum of the digits of ''p'' is 3, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' contains exactly one 2, then we can write ''p'' = 2''z''2''w''1, where ''z'',''w'' ∈ {0}. If 0 ◁ ''z'' and 0 ◁ ''w'', then '''20201''' ◁ ''p''. Hence we may assume either ''z'' or ''w'' is empty.

If ''z'' is empty, then ''p'' ∈ 22{0}1, and the smallest prime ''p'' ∈ 22{0}1 is '''22000001'''.

If ''w'' is empty, then ''p'' ∈ 2{0}21, and the smallest prime ''p'' ∈ 2{0}21 is '''20021'''.

Case 1.2: ''p'' begins with 5.

In this case we can write ''p'' = 5''y''1. If 2 ◁ ''y'', then '''521''' ◁ ''p''. If 9 ◁ ''y'', then 59 ◁ ''p''. Hence we may assume all digits of ''y'' are 0, 5, or 8.

If 05 ◁ ''y'', then '''5051''' ◁ ''p''. If 08 ◁ ''y'', then '''5081''' ◁ ''p''. If 50 ◁ ''y'', then '''5501''' ◁ ''p''. If 58 ◁ ''y'', then '''5581''' ◁ ''p''. If 80 ◁ ''y'', then '''5801''' ◁ ''p''. If 85 ◁ ''y'', then '''5851''' ◁ ''p''. Hence we may assume ''y'' ∈ {0} ∪ {5} ∪ {8}.

If ''y'' ∈ {0}, then ''p'' ∈ 5{0}1. But then, since the sum of the digits of ''p'' is 6, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' ∈ {5}, then ''p'' ∈ 5{5}1, and the smallest prime ''p'' ∈ 5{5}1 is '''555555555551'''.

If ''y'' ∈ {8}, since if 88 ◁ ''y'', then 881 ◁ ''p'', hence we may assume ''y'' ∈ {''𝜆'',8}, and thus ''p'' ∈ {51,581}, but 51 and 581 are both composite.

Case 1.3: ''p'' begins with 8.

In this case we can write p = 8''y''1. If 2 ◁ ''y'', then '''821''' ◁ ''p''. If 8 ◁ ''y'', then '''881''' ◁ ''p''. If 9 ◁ ''y'', then 89 ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 5.

If 50 ◁ ''y'', then '''8501''' ◁ ''p''. Hence we may assume y ∈ {0}{5}.

If 005 ◁ ''y'', then '''80051''' ◁ p. Hence we may assume y ∈ {0} ∪ {5} ∪ 0{5}.

If y ∈ {0}, then ''p'' ∈ 8{0}1. But then, since the sum of the digits of ''p'' is 9, ''p'' is divisible by 3, so ''p'' cannot be prime.

If y ∈ {5}, since if 55555555555 ◁ ''y'', then 555555555551 ◁ ''p'', hence we may assume ''y'' ∈ {''𝜆'', 5, 55, 555, 5555, 55555, 555555, 5555555, 55555555, 555555555, 5555555555}, and thus ''p'' ∈ {81, 851, 8551, 85551, 855551, 8555551, 85555551, 855555551, 8555555551, 85555555551, 855555555551}, but all of these numbers are composite.

If y ∈ 0{5}, since if 55555555555 ◁ ''y'', then 555555555551 ◁ ''p'', hence we may assume ''y'' ∈ {0, 05, 055, 0555, 05555, 055555, 0555555, 05555555, 055555555, 0555555555, 05555555555}, and thus ''p'' ∈ {801, 8051, 80551, 805551, 8055551, 80555551, 805555551, 8055555551, 80555555551, 805555555551, 8055555555551}, and of these numbers only 80555551 and 8055555551 are primes, but 80555551 ◁ 8055555551, thus only '''80555551''' is a minimal element.

Case 1.4: ''p'' begins with 9.

In this case we can write p = 9''y''1. If 9 ◁ ''y'', then '''991''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0, 2, 5, or 8.

If 00 ◁ ''y'', then '''9001''' ◁ ''p''. If 22 ◁ ''y'', then '''9221''' ◁ ''p''. If 55 ◁ ''y'', then '''9551''' ◁ ''p''. If 88 ◁ ''y'', then 881 ◁ ''p''. Hence we may assume ''y'' contains at most one 0, at most one 2, at most one 5, and at most one 8.

If ''y'' only contains at most one 0 and does not contain any of {2,5,8}, then ''y'' ∈ {''𝜆'',0}, and thus ''p'' ∈ {91,901}, but 91 and 901 are both composite. If ''y'' only contains at most one 0 and only one of {2,5,8}, then the sum of the digits of ''p'' is divisible by 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Hence we may assume ''y'' contains at least two of {2,5,8}.

If 25 ◁ ''y'', then 251 ◁ ''p''. If 28 ◁ ''y'', then 281 ◁ ''p''. If 52 ◁ ''y'', then 521 ◁ ''p''. If 82 ◁ ''y'', then 821 ◁ ''p''. Hence we may assume ''y'' contains no 2's (since if ''y'' contains 2, then ''y'' cannot contain either 5's or 8's, which is a contradiction).

If 85 ◁ ''y'', then '''9851''' ◁ ''p''. Hence we may assume ''y'' ∈ {58,580,508,058}, and thus ''p'' ∈ {9581,95801,95081,90581}, and of these numbers only 95801 is prime, but 95801 is not a minimal element since 5801 ◁ 95801.

Case 2: ''p'' ends with 3.

In this case we can write p = ''x''3. If ''x'' contains 1, 2, 4, 5, 7, or 8, then (respectively) '''13''' ◁ ''p'', '''23''' ◁ ''p'', '''43''' ◁ ''p'', '''53''' ◁ ''p'', '''73''' ◁ ''p'', or '''83''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 3, 6, or 9, and thus all digits of ''p'' are 0, 3, 6, or 9. But then, since the digits of ''p'' all have a common factor 3, ''p'' is divisible by 3, so ''p'' cannot be prime.

Case 3: ''p'' ends with 7.

In this case we can write ''p'' = ''x''7. If ''x'' contains 1, 3, 4, 6, or 9, then (respectively) '''17''' ◁ ''p'', '''37''' ◁ ''p'', '''47''' ◁ ''p'', '''67''' ◁ ''p'', or '''97''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 2, 5, 7, or 8.

Case 3.1: ''p'' begins with 2.

In this case we can write ''p'' = 2''y''7. If 2 ◁ ''y'', then '''227''' ◁ ''p''. If 5 ◁ ''y'', then '''257''' ◁ ''p''. If 7 ◁ ''y'', then '''277''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 8.

If 08 ◁ ''y'', then '''2087''' ◁ ''p''. If 88 ◁ ''y'', then 887 ◁ ''p''. Hence we may assume ''y'' ∈ {0} ∪ 8{0}.

If ''y'' ∈ {0}, then ''p'' ∈ 2{0}7. But then, since the sum of the digits of ''p'' is 9, ''p'' is divisible by 3, so ''p'' cannot be prime.

If y ∈ 8{0}, then ''p'' ∈ 28{0}7. But then ''p'' is divisible by 7, since for ''n'' ≥ 0 we have 7 × 40&lt;sub&gt;''n''&lt;/sub&gt;1 = 280&lt;sub&gt;''n''&lt;/sub&gt;7.

Case 3.2: ''p'' begins with 5.

In this case we can write ''p'' = 5''y''7. If 5 ◁ ''y'', then '''557''' ◁ ''p''. If 7 ◁ ''y'', then '''577''' ◁ ''p''. If 8 ◁ ''y'', then '''587''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 2.

If 22 ◁ ''y'', then 227 ◁ ''p''. Hence we may assume ''y'' contains zero or one 2's.

If ''y'' contains no 2's, then ''p'' ∈ 5{0}7. But then, since the sum of the digits of ''p'' is 12, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' contains exactly one 2, then we can write ''p'' = 5''z''2''w''7, where ''z'',''w'' ∈ {0}. If 0 ◁ ''z'' and 0 ◁ ''w'', then '''50207''' ◁ ''p''. Hence we may assume either ''z'' or ''w'' is empty.

If ''z'' is empty, then ''p'' ∈ 52{0}7, and the smallest prime ''p'' ∈ 52{0}7 is '''5200007'''.

If ''w'' is empty, then ''p'' ∈ 5{0}27, and the smallest prime ''p'' ∈ 5{0}27 is '''5000000000000000000000000000027'''.

Case 3.3: ''p'' begins with 7.

In this case we can write ''p'' = 7''y''7. If 2 ◁ ''y'', then '''727''' ◁ ''p''. If 5 ◁ ''y'', then '''757''' ◁ ''p''. If 8 ◁ ''y'', then '''787''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 7, and thus all digits of ''p'' are 0 or 7. But then, since the digits of ''p'' all have a common factor 7, ''p'' is divisible by 7, so ''p'' cannot be prime.

Case 3.4: ''p'' begins with 8.

In this case we can write ''p'' = 8''y''7. If 2 ◁ ''y'', then '''827''' ◁ ''p''. If 5 ◁ ''y'', then '''857''' ◁ ''p''. If 7 ◁ ''y'', then '''877''' ◁ ''p''. If 8 ◁ ''y'', then '''887''' ◁ ''p''. Hence we may assume ''y'' ∈ {0}, and thus ''p'' ∈ 8{0}7. But then, since the sum of the digits of ''p'' is 15, ''p'' is divisible by 3, so ''p'' cannot be prime.

Case 4: ''p'' ends with 9.

In this case we can write ''p'' = ''x''9. If ''x'' contains 1, 2, 5, 7, or 8, then (respectively) '''19''' ◁ ''p'', '''29''' ◁ ''p'', '''59''' ◁ ''p'', '''79''' ◁ ''p'', or '''89''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 3, 4, 6, or 9.

If 44 ◁ ''x'', then '''449''' ◁ ''p''. Hence we may assume ''x'' contains zero or one 4's.

If x contains no 4's, then all digits of ''x'' are 0, 3, 6, or 9, and thus all digits of ''p'' are 0, 3, 6, or 9. But then, since the digits of ''p'' all have a common factor 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Hence we may assume that ''x'' contains exactly one 4.

Case 4.1: ''p'' begins with 3.

In this case we can write ''p'' = 3''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. We must have '''349''' ◁ ''p''.

Case 4.2: ''p'' begins with 4.

In this case we can write ''p'' = 4''y''9, where all digits of ''y'' are 0, 3, 6, or 9. If 0 ◁ ''y'', then '''409''' ◁ ''p''. If 3 ◁ ''y'', then 43 ◁ ''p''. If 9 ◁ ''y'', then '''499''' ◁ ''p''. Hence we may assume ''y'' ∈ {6}, and thus ''p'' ∈ 4{6}9. But then ''p'' is divisible by 7, since for ''n'' ≥ 0 we have 7 × 6&lt;sub&gt;''n''&lt;/sub&gt;7 = 46&lt;sub&gt;''n''&lt;/sub&gt;9.

Case 4.3: ''p'' begins with 6.

In this case we can write p = 6''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. If 0 ◁ ''z'', then 409 ◁ ''p''. If 3 ◁ ''z'', then 43 ◁ ''p''. If 6 ◁ ''z'', then '''6469''' ◁ ''p''. If 9 ◁ ''z'', then 499 ◁ ''p''. Hence we may assume ''z'' is empty.

If 3 ◁ ''y'', then 349 ◁ ''p''. If 9 ◁ ''y'', then '''6949''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 6.

If 06 ◁ ''y'', then '''60649''' ◁ ''p''. Hence we may assume ''y'' ∈ {6}{0}.

If 666 ◁ ''y'', then '''666649''' ◁ ''p''. If 00000 ◁ ''y'', then '''60000049''' ◁ ''p''. Hence we may assume ''y'' ∈ {''𝜆'', 0, 00, 000, 0000, 6, 60, 600, 6000, 60000, 66, 660, 6600, 66000, 660000}, and thus ''p'' ∈ {649, 6049, 60049, 600049, 6000049, 6649, 66049, 660049, 6600049, 66000049, 66649, 666049, 6660049, 66600049, 666000049}, and of these numbers only '''66000049''' and '''66600049''' are primes.

Case 4.4: ''p'' begins with 9.

In this case we can write p = 9''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. If 0 ◁ ''y'', then '''9049''' ◁ ''p''. If 3 ◁ ''y'', then 349 ◁ ''p''. If 6 ◁ ''y'', then '''9649''' ◁ ''p''. If 9 ◁ ''y'', then '''9949''' ◁ ''p''. Hence we may assume ''y'' is empty.

If 0 ◁ ''z'', then 409 ◁ ''p''. If 3 ◁ ''z'', then 43 ◁ ''p''. If 9 ◁ ''z'', then 499 ◁ ''p''. Hence we may assume ''z'' ∈ {6}, and thus ''p'' ∈ 94{6}9, and the smallest prime ''p'' ∈ 94{6}9 is 946669.

[[Category:Number theory]]</text>
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    <revision>
      <id>2819768</id>
      <parentid>2819767</parentid>
      <timestamp>2026-07-29T19:31:07Z</timestamp>
      <contributor>
        <username>Athene241</username>
        <id>3100061</id>
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      <comment>/* Condensed table for bases 2 ≤ b ≤ 36 */</comment>
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      <text bytes="128796" sha1="o3omh8ibpfsvloq6qkec5uakw0sbw46" xml:space="preserve">{{mathematics}}
'''Athena problem''' is an [[:w:List of unsolved problems in mathematics|unsolved problem]] in [[:w:Number theory|number theory]] and [[:w:Formal language theory|formal language theory]] and [[:w:Order theory|order theory]], this problem is named after the ancient Greek goddess [[:w:Athena|Athena]] (which is associated with [[:w:Wisdom|wisdom]]). Athena problem is: Give a [[:w:Natural number|natural number]] ''b'' &gt; 1, find the [[:w:Set (mathematics)|set]] of the [[:w:Minimal element|minimal element]]s of the set of the &quot;[[:w:Prime number|prime number]] [[:w:Greater than|&gt;]] ''b''&quot; [[:w:Numerical digit|digit]] [[:w:String (computer science)|string]]s in the [[:w:Positional numeral system|positional numeral system]] with [[:w:Radix|base]] ''b'' for the [[:w:Subsequence|subsequence]] [[:w:Partially ordered set|ordering]]. (A string ''x'' is a subsequence of another string ''y'', if ''x'' can be obtained from ''y'' by deleting zero or more of the [[:w:Character (computing)|character]]s in ''y''. For example, 514 is a subsequence of 352148, &quot;string&quot; is a subsequence of &quot;meistersinger&quot;. In contrast, 758 is not a subsequence of 378259, &quot;abc&quot; is not a subsequence of &quot;cbacacba&quot;, since the characters must be in the same order) (Unlike [[:w:Substring|substring]], subsequence is not required to occupy consecutive positions within the original sequences, e.g. the [[:w:Longest common subsequence|longest common subsequence problem]] is different from the [[:w:Longest common substring|longest common substring problem]])

Using [[:w:Formal language theory|formal language theory]] terminology, Athena problem is finding the [[:w:Set (mathematics)|set]] of the [[:w:Minimal element|minimal element]]s of the [[:w:Formal language|language]] of base-''b'' [[:w:Representation (mathematics)|representation]]s of the [[:w:Prime number|prime number]]s [[:w:Greater than|&gt;]] ''b'' (which is a set of [[:w:String (computer science)|string]]s of [[:w:Symbol|symbol]]s over the [[:w:Alphabet (formal languages)|alphabet]] ''Σ''&lt;sub&gt;''b''&lt;/sub&gt; := {0, 1, ..., ''b''−1}), under the subsequence ordering (i.e. the [[:w:Binary relation|binary relation]] &quot;is a subsequence of&quot;, which is a [[:w:Partially ordered set|partial ordering]]), for a given natural number ''b'' &gt; 1 (You can draw this partial ordering as a [[:w:Hasse diagram|Hasse diagram]] to find all [[:w:Minimal element|minimal element]]s), this set is called '''Athena set base ''b''''', and the prime numbers in this set is called '''Athena primes base ''b'''''.

By [[:w:Higman's lemma|Higman's lemma]], there are no [[:w:Infinite set|infinite]] [[:w:Antichain|antichain]]s for the subsequence ordering (i.e. the subsequence ordering is always a [[:w:Well-quasi-ordering|well quasi order]]) (i.e. under the subsequence ordering (i.e. the [[:w:Binary relation|binary relation]] &quot;is a subsequence of&quot;, which is a [[:w:Partially ordered set|partial ordering]]), every set of pairwise incomparable (i.e. not [[:w:Comparability|comparable]]) strings is finite), thus there must be only finitely many such minimal elements. In other words, the Athena set in every base ''b'' must be a [[:w:Finite set|finite set]], and every base ''b'' ≥ 2 has only finitely many Athena primes, e.g. in [[:w:Decimal|decimal]] (base ''b'' = 10), the Athena set has exactly 77 [[:w:Element of a set|element]]s (the Athena primes in decimal (base ''b'' = 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}.

For bases 2 ≤ ''b'' ≤ 36, Athena problem is fully solved in bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 24, and also solved in bases ''b'' = 11, 13, 16, 22, 30 if [[:w:Probable prime|probable prime]]s are allowed. For the unsolved bases ''b'' = 17, 19, 21, 23, 25, 26, 27, 28, 29, 31, 32, 34, 35, 36, Athena problem is solved (if probable primes are allowed) except 771 [[:w:Indexed family|families]] of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be [[:w:Empty string|empty]]) of digits in base ''b'', ''y'' is a digit in base ''b'') = sequence {''xz'', ''xyz'', ''xyyz'', ''xyyyz'', ''xyyyyz'', ''xyyyyyz'', ...} (i.e. &quot;''xy''&lt;sup&gt;+&lt;/sup&gt;''z''&quot; in [[:w:Regular expression|regular expression]]), all of these 771 families contain no primes &gt; ''b'' or probable primes &gt; ''b'' with length ≤ 100000.

== Solve the problem ==

To solve the Athena problem for a given base ''b'', we must [[:w:Computing|compute]] the elements up to families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''), and find the smallest prime &gt; ''b'' in all such families.

We call families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') &quot;linear&quot; families, and we reduce these families by removing all trailing digits ''y'' from ''x'', and removing all leading digits ''y'' from ''z'', to make the families be easier, e.g. family 12333{3}33345 in base ''b'' is reduced to family 12{3}45 in base ''b'', since they are in fact the same family. Our [[:w:Algorithm|algorithm]] then proceeds as follows:

* 1. ''M'' := {minimal primes in base ''b'' of length 2 or 3}, ''L'' := union of all ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'') such that ''x'' ≠ 0 and ''gcd''(''z'', ''b'') = 1 and ''Y'' is the set of digits ''y'' in base ''b'' such that ''xyz'' has no subsequence in ''M''.
* 2. While ''L'' contains nonlinear families (families which are not linear families): Explore each family of ''L'', and update ''L''. Examine each family of ''L'' by:
* 2.1. Let ''w'' be the shortest string in the family. If ''w'' has a subsequence in ''M'', then remove the family from ''L''. If ''w'' represents a prime, then add ''w'' to ''M'' and remove the family from ''L''.
* 2.2. If possible, simplify the family.
* 2.3. Using the techniques below (covering congruence, algebraic factorization, or combine of them), check if the family can be proven to only contain composites (only count the numbers &gt; ''b''), and if so then remove the family from ''L''.
* 3. Update ''L'', after each split examine the new families as in step 2.

e.g. in decimal (base ''b'' = 10):

''M'' := {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}

''L'' := {2{0,2}1, 2{0,8}7, 3{0,3,6,9}3, 3{0,3,6,9}9, 4{6}9, 5{0,5,8}1, 5{0,2}7, 6{0,3,6,9}3, 6{0,3,4,6,9}9, 7{0,7}7, 8{0,5}1, 8{0}7, 9{0,2,5,8}1, 9{0,3,6,9}3, 9{0,3,4,6,9}9}

and since 2221 is prime, it follows that the family 2{0,2}1 splits into the families 2{0}1 and 2{0}2{0}1

and since the family 2{0}1 can be proven to contain no primes &gt; base (since all numbers in this family are divisible by 3), it can be removed

and since 20201 is prime, it follows that the family 2{0}2{0}1 splits into the families 2{0}21 and 22{0}1

221 and 2021 are composites, but 20021 is prime, thus add 20021 to ''L''

none of 221, 2201, 22001, 220001, 2200001 are primes, but 22000001 is prime, thus add 22000001 to ''L''

and since the family 3{0,3,6,9}3 can be proven to contain no primes &gt; base (since all numbers in this family are divisible by 3), it can be removed

etc.

Since the number of possible (first digit,last digit) (also called (initial digit,final digit)) combos ([[:w:Ordered pair|ordered pair]]s) of a prime &gt; ''b'' in base ''b'' is (''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(''b'') (all digits except 0 can be the first digit of a prime &gt; ''b'' in base ''b'' (thus ''b''−1 possible digits), but only the digits coprime to ''b'' can be the last digit of a prime &gt; ''b'' in base ''b'' (thus ''eulerphi''(''b'') possible digits), and by the [[:w:Rule of product|rule of product]], there are (''b''−1)×''eulerphi''(''b'') choices of the (first digit,last digit) combo, also, both &quot;numbers of primes in the set of the Athena problem in base ''b''&quot; and &quot;length of the largest prime in the set of the Athena problem in base ''b''&quot; are [[:w:Asymptotic analysis|roughly]] ''[[:w:E (mathematical_constant)|e]]''&lt;sup&gt;''[[:w:Euler's constant|γ]]''×(''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(*b*)&lt;/sup&gt;.

Shrinking the family ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''Y'' is a set of digits in base ''b'')

* If ''y'' ∈ ''Y'' and the string ''xyyz'' represents a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or has a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''}''z'' ∪ ''x''{''Y'' \ ''y''}''y''{''Y'' \ ''y''}''z''.
* If ''y''&lt;sub&gt;1&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;2&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;1&lt;/sub&gt; ≠ ''y''&lt;sub&gt;2&lt;/sub&gt; and the string ''xy''&lt;sub&gt;1&lt;/sub&gt;''y''&lt;sub&gt;2&lt;/sub&gt;''z'' represents a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or has a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''&lt;sub&gt;1&lt;/sub&gt;}{''Y'' \ ''y''&lt;sub&gt;2&lt;/sub&gt;}''z''.
* If ''y''&lt;sub&gt;1&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;2&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;1&lt;/sub&gt; ≠ ''y''&lt;sub&gt;2&lt;/sub&gt; and both the strings ''xy''&lt;sub&gt;1&lt;/sub&gt;''y''&lt;sub&gt;2&lt;/sub&gt;''z'' and ''xy''&lt;sub&gt;2&lt;/sub&gt;''y''&lt;sub&gt;1&lt;/sub&gt;''z'' represent a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or have a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''&lt;sub&gt;1&lt;/sub&gt;}''z'' ∪ ''x''{''Y'' \ ''y''&lt;sub&gt;2&lt;/sub&gt;}''z''.

e.g. in decimal (base ''b'' = 10):

* 2221 is a prime &gt; 10, thus the family 2{0,2}1 splits into the two families 2{0}1 and 2{0}2{0}1.
* 227 is a prime &gt; 10, and it is a subsequence of 5227, thus the family 5{0,2}7 splits into the two families 5{0}7 and 5{0}2{0}7.
* 449 is a prime &gt; 10, and it is a subsequence of 6449, thus the family 6{0,3,4,6,9}9 splits into the two families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9.
* Both 5051 and 5501 are primes &gt; 10, thus the family 5{0,5}1 splits into the two families 5{0}1 and 5{5}1 = {5}1.
* 8501 is a prime &gt; 10, thus the family 8{0,5}1 splits into the family 8{0}{5}1.
* 887 is a prime &gt; 10, and it is a subsequence of 2887, also 2087 is a prime &gt; 10, thus the family 2{0,8}7 splits into the two families 2{0}7 and 28{0}7.
* 349 and 449 are primes &gt; 10, and they are subsequences of 9349 and 9449, respectively, also 9049, 9649, 9949 are primes &gt; 10, thus the family 9{0,3,4,6,9}9 splits into the two families 9{0,3,6,9}9 and 94{0,3,6,9}9.
* 251, 281, 521, 821, 881 are primes &gt; 10, and they are subsequences of 9251, 9281, 9521, 9821, 9881, respectively, also 9001, 9221, 9551, 9851 are primes &gt; 10, thus the family 9{0,2,5,8}1 splits into the numbers {91, 901, 921, 951, 981, 9021, 9051, 9081, 9201, 9501, 9581, 9801, 90581, 95081, 95801}.

If the methods we have discussed cannot be used to rule out or shrink ''x''{''Y''}''z'' where ''Y'' = {''y''&lt;sub&gt;1&lt;/sub&gt;, ''y''&lt;sub&gt;2&lt;/sub&gt;, ..., ''y''&lt;sub&gt;''n''&lt;/sub&gt;}, then we can replace ''x''{''Y''}''z'' by ''xy''&lt;sub&gt;1&lt;/sub&gt;{''Y''}''z'' ∪ ''xy''&lt;sub&gt;2&lt;/sub&gt;{''Y''}''z'' ∪ ... ∪ ''xy''&lt;sub&gt;''n''&lt;/sub&gt;{''Y''}''z'' and re-run the methods on this new [[:w:Formal language|language]].

If all remain families are linear families (i.e. of the form ''x''{''y''}''z'', where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''), then we search the smallest (probable) primes in these families and add these primes to the list.

e.g. in decimal (base ''b'' = 10):

* The smallest prime in the family 5{0}27 is 5000000000000000000000000000027.
* The smallest prime in the family {5}1 is 555555555551.
* The smallest prime in the family 8{5}1 is 8555555555555555555551, but 8555555555555555555551 is not a minimal element since 555555555551 is a subsequence of 8555555555555555555551.

There is no guarantee that the techniques discussed will ever terminate, but in practice they often do. They are able to determine the set of the minimal elements in base ''b'' for 2 ≤ ''b'' ≤ 16 and ''b'' = 18, 20, 22, 24, 30. The bases ''b'' = 17, 19, 21, 23, 25 ≤ ''b'' ≤ 29, 31 ≤ ''b'' ≤ 36 are solved with the exception of 771 families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'').

The following is a &quot;[[:w:Semi-algorithm|semi-algorithm]]&quot; that is guaranteed to solve the Athena problem for a given base ''b'', but it is not so easy to implement:

# ''M'' = ''[[:w:Empty string|∅]]''
# while (''L'' ≠ ''∅'') do
# choose ''x'', a shortest string in ''L''
# ''M'' := ''M'' ∪ {''x''}
# ''L'' := ''L'' − ''sup''({''x''})

In practice, for arbitrary ''L'', we cannot feasibly carry out step 5. Instead, we work with ''L''&amp;#39;, some regular overapproximation to ''L'', until we can show ''L''&amp;#39; = ''∅'' (which implies ''L'' = ''∅''). In practice, ''L''&amp;#39; is usually chosen to be a finite [[:w:Union (set theory)|union]] of sets of the form ''L''&lt;sub&gt;1&lt;/sub&gt;{''L''&lt;sub&gt;2&lt;/sub&gt;}''L''&lt;sub&gt;3&lt;/sub&gt;, where each of ''L''&lt;sub&gt;1&lt;/sub&gt;, ''L''&lt;sub&gt;2&lt;/sub&gt;, ''L''&lt;sub&gt;3&lt;/sub&gt; is finite. In the case we consider in this project, we then have to determine whether such a family contains a prime or not.

Thus, the [[:w:Time complexity|time complexity]] of the Athena problem in base ''b'' may be ''[[:w:Big O notation|O]]''(''[[:w:E (mathematical_constant)|e]]''&lt;sup&gt;''[[:w:Euler's constant|γ]]''×(''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(*b*)&lt;/sup&gt;), and the [[:w:CPU time|CPU time]] of the Athena problem in base ''b'' may be longer than [[:w:Age of the universe|the age of the universe]] for bases ''b'' = 19, 23, 25, 27, 29, 31, 32, 33, 34, 35, also, Athena problem in bases ''b'' around 500 may be [[:w:NP-complete|NP-complete]] or [[:w:NP-hard|NP-hard]], or an [[:w:Undecidable problem|undecidable problem]], or an example of [[:w:Gödel's incompleteness theorems|Gödel's incompleteness theorems]] (like the [[:w:Continuum hypothesis|continuum hypothesis]] and the [[:w:Halting problem|halting problem]]).

To solve the Athena problem, we need to determine whether a given family contains a prime. In practice, if family ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''Y'' is a set of digits in base ''b'') could not be ruled out as only containing composites and ''Y'' contains two or more digits, then a relatively small prime &gt; ''b'' could always be found in this family. Intuitively, this is because there are a large number of small strings in such a family, and at least one is likely to be prime (e.g. there are 2&lt;sup&gt;''n''−2&lt;/sup&gt; strings of length ''n'' in the family 1{3,7}9, and there are over a thousand strings of length 12 in the family 1{3,7}9, thus it is very impossible that these numbers are all composite). In the case ''Y'' contains only one digit, this family is of the form ''x''{''y''}''z'', and there is only a single string of each length &gt; (the length of ''x'' + the length of ''z''), and it is not known if the following [[:w:Decision problem|decision problem]] is recursively solvable (just like [[:w:Sierpiński number|Sierpiński problem]] and [[:w:Riesel number|Riesel problem]], Sierpiński problem and Riesel problem can be generalized to other bases ''b'' (references: http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm), in fact, Athena problem base ''b'' covers the Sierpiński problem base ''b'' and the Riesel problem base ''b'' with ''k'' &lt; ''b'', i.e. finding the smallest prime of the form ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 and ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (or prove such prime does not exist) with ''k'' &lt; ''b'' (specially, for bases ''b'' such that the conjectured smallest Sierpiński number or the conjectured smallest Riesel number is &lt; ''b'', Athena problem base ''b'' covers the Sierpiński problem base ''b'' or the Riesel problem base ''b'', respectively), since the smallest prime of the form ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 and ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (if exists) must be a minimal element in base ''b'', also, Athena problem base ''b'' covers finding the smallest prime of these forms in base ''b'' (or proving that such prime does not exist): (''b''&lt;sup&gt;''n''&lt;/sup&gt;−1)/(''b''−1) (for this form, ''n'' must be prime, and we want ''n'' ≥ 2) (references of this form: http://www.fermatquotient.com/PrimSerien/GenRepu.txt, https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html, http://www.primenumbers.net/Henri/us/MersFermus.htm, http://www.bitman.name/math/table/379, https://pzktupel.de/Primetables/TableRepunitGen.php, https://oeis.org/A084740, https://oeis.org/A084738, https://oeis.org/A128164, https://oeis.org/A285642; or for prime bases ''b'': https://oeis.org/A065854, https://oeis.org/A279068), ''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 (for this form, ''n'' must be power of 2, and we want ''n'' ≥ 1) (references of this form: http://jeppesn.dk/generalized-fermat.html, http://www.noprimeleftbehind.net/crus/GFN-primes.htm, https://web.archive.org/web/20231002190634/http://yves.gallot.pagesperso-orange.fr/primes/index.html, https://pzktupel.de/Primetables/TableFermatGFBB.php, https://oeis.org/A079706, https://oeis.org/A084712, https://oeis.org/A228101), (''b''&lt;sup&gt;''n''&lt;/sup&gt;+1)/2 (for odd ''b'') (for this form, ''n'' must be power of 2, and we want ''n'' ≥ 2) (reference of this form: http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt), (''sqrt''(''b'')×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1)/(''sqrt''(''b'')+1) (for square ''b'') (for this form, 2×''n''+1 must be prime, and we want ''n'' ≥ 2) (references of this form: http://www.fermatquotient.com/PrimSerien/GenRepuP.txt, http://www.primenumbers.net/Henri/us/MersFermus.htm, http://www.bitman.name/math/table/488, https://pzktupel.de/Primetables/TableWagstaffGen.php, https://oeis.org/A084742, https://oeis.org/A084741; or for bases ''b'' with ''sqrt''(''b'') prime: https://oeis.org/A065507), ((''b''−2)×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1)/(''b''−1) (''n'' ≥ 2) (reference of this form: https://oeis.org/A243404), 2×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 (''n'' ≥ 1) (references of this form: https://www.mersenneforum.org/showthread.php?t=6918, https://www.mersenneforum.org/showthread.php?t=19725, https://oeis.org/A119624), 2×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (''n'' ≥ 1) (references of this form: https://www.mersenneforum.org/showthread.php?t=24576, https://www.mersenneforum.org/attachment.php?attachmentid=20976&amp;d=1567314217, https://oeis.org/A119591), ''b''&lt;sup&gt;''n''&lt;/sup&gt;+2 (''n'' ≥ 1) (references of this form: https://oeis.org/A138066, https://oeis.org/A084713, https://oeis.org/A138067), ''b''&lt;sup&gt;''n''&lt;/sup&gt;−2 (''n'' ≥ 2) (references of this form: https://www.primepuzzles.net/puzzles/puzz_887.htm, https://oeis.org/A250200, https://oeis.org/A255707, https://oeis.org/A084714; or for prime bases ''b'': https://oeis.org/A292201), (''b''−1)×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 (''n'' ≥ 1) (references of this form: http://www.noprimeleftbehind.net/Williams-primes-MP.htm, http://www.bitman.name/math/table/477, https://pzktupel.de/Primetables/TableWilliams2.php, https://oeis.org/A305531; or for prime bases ''b'': https://oeis.org/A087139), (''b''−1)×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (''n'' ≥ 1) (references of this form: https://harvey563.tripod.com/wills.txt, http://www.noprimeleftbehind.net/Williams-primes-MM.htm, http://www.bitman.name/math/table/484, https://pzktupel.de/Primetables/TableWilliams1.php; or for prime bases ''b'': https://oeis.org/A122396), ''b''&lt;sup&gt;''n''&lt;/sup&gt;+(''b''−1) (''n'' ≥ 1) (references of this form: http://www.bitman.name/math/table/795, https://pzktupel.de/Primetables/TableWilliams6.php, https://oeis.org/A076845, https://oeis.org/A076846, https://oeis.org/A078178, https://oeis.org/A078179), ''b''&lt;sup&gt;''n''&lt;/sup&gt;−(''b''−1) (''n'' ≥ 2) (references of this form: http://www.bitman.name/math/table/792, https://pzktupel.de/Primetables/TableWilliams5.php, https://oeis.org/A113516, https://oeis.org/A343589; or for prime bases ''b'': https://cs.uwaterloo.ca/journals/JIS/VOL3/mccranie.html, http://www.bitman.name/math/table/435)):

Problem: Given strings ''x'', ''z'' (may be empty), a digit ''y'', and a base ''b'' (''x'' does not [[:w:Leading zero|start with the digit 0]], ''z'' ends with a digit which [[:w:Coprime integers|coprime]] to ''b'', ''y'' is not 0 if ''x'' is empty, ''y'' is coprime to ''b'' if ''z'' is empty), does there exist a prime number whose base-''b'' expansion is of the form ''xy''&lt;sub&gt;''n''&lt;/sub&gt;''z'' for some ''n'' ≥ 0?

Some families can be ruled out to contain no prime &gt; ''b'' by [[:w:Covering set|covering congruence]], [[:w:Factorization of polynomials|algebraic factorization]] (e.g. [[:w:Difference of two squares|difference of two squares]], [[:w:Sum of two cubes|sum of two cubes]], [[:w:Sophie Germain's identity|Sophie Germain's identity of ''x''&lt;sup&gt;4&lt;/sup&gt;+4×''y''&lt;sup&gt;4&lt;/sup&gt;]]), or combine of them, e.g.

* The base 9 family 2{7}: Always divisible by 2 or 5
* The base 11 family 2{5}: Always divisible by 2 or 3
* The base 14 family B{0}1: Always divisible by 3 or 5
* The base 13 family 95{0}3: Always divisible by 5, 7, or 17
* The base 16 family {4}D: Always divisible by 3, 7, or 13
* The base 16 family {8}F: Always divisible by 3, 7, or 13
* The base 21 family {7}D: Always divisible by 2, 13, or 17
* The base 23 family {D}GA: Always divisible by 2, 5, 7, 37, or 79
* The base 9 family {1}: Can be written as (9&lt;sup&gt;''n''&lt;/sup&gt;−1)/8 and can be factored as (3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (3&lt;sup&gt;''n''&lt;/sup&gt;+1) / 8
* The base 8 family 1{0}1: Can be written as 8&lt;sup&gt;''n''&lt;/sup&gt;+1 and can be factored as (2&lt;sup&gt;''n''&lt;/sup&gt;+1) × (4&lt;sup&gt;''n''&lt;/sup&gt;−2&lt;sup&gt;''n''&lt;/sup&gt;+1)
* The base 9 family 3{8}: Can be written as 4×9&lt;sup&gt;''n''&lt;/sup&gt;−1 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (2×3&lt;sup&gt;''n''&lt;/sup&gt;+1)
* The base 16 family 1{5}: Can be written as (4×16&lt;sup&gt;''n''&lt;/sup&gt;−1)/3 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (2×3&lt;sup&gt;''n''&lt;/sup&gt;+1) / 3
* The base 16 family {4}1: Can be written as (4×16&lt;sup&gt;''n''&lt;/sup&gt;−49)/15 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−7) × (2×3&lt;sup&gt;''n''&lt;/sup&gt;+7) / 15
* The base 27 family 7{Q}: Can be written as 8×27&lt;sup&gt;''n''&lt;/sup&gt;−1 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (4×9&lt;sup&gt;''n''&lt;/sup&gt;+2×3&lt;sup&gt;''n''&lt;/sup&gt;+1)
* The base 27 family 9{G}: Can be written as (125×27&lt;sup&gt;''n''&lt;/sup&gt;−8)/13 and can be factored as (5×3&lt;sup&gt;''n''&lt;/sup&gt;−2) × (25×9&lt;sup&gt;''n''&lt;/sup&gt;+10×3&lt;sup&gt;''n''&lt;/sup&gt;+4)
* The base 16 family {C}D: Can be written as (4×16&lt;sup&gt;''n''&lt;/sup&gt;+1)/5 and can be factored as (2×4&lt;sup&gt;''n''&lt;/sup&gt;−2×2&lt;sup&gt;''n''&lt;/sup&gt;+1) × (2×4&lt;sup&gt;''n''&lt;/sup&gt;+2×2&lt;sup&gt;''n''&lt;/sup&gt;+1) / 5
* The base 14 family 8{D}: Can be written as 9×14&lt;sup&gt;''n''&lt;/sup&gt;−1, it is divisible by 5 if ''n'' is odd and can be factored as (3×14&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (3×14&lt;sup&gt;''n''/2&lt;/sup&gt;+1) if ''n'' is even
* The base 12 family {B}9B: Can be written as 12&lt;sup&gt;''n''&lt;/sup&gt;−25, it is divisible by 13 if ''n'' is odd and can be factored as (12&lt;sup&gt;''n''/2&lt;/sup&gt;−5) × (12&lt;sup&gt;''n''/2&lt;/sup&gt;+5) if ''n'' is even
* The base 14 family {D}5: Can be written as 14&lt;sup&gt;''n''&lt;/sup&gt;−9, it is divisible by 5 if ''n'' is odd and can be factored as (14&lt;sup&gt;''n''/2&lt;/sup&gt;−3) × (14&lt;sup&gt;''n''/2&lt;/sup&gt;+3) if ''n'' is even
* The base 17 family 1{9}: Can be written as (25×17&lt;sup&gt;''n''&lt;/sup&gt;−9)/16, it is divisible by 2 if ''n'' is odd and can be factored as (5×17&lt;sup&gt;''n''/2&lt;/sup&gt;−3) × (5×17&lt;sup&gt;''n''/2&lt;/sup&gt;+3) / 16 if ''n'' is even
* The base 17 family 7{9}: Can be written as (121×17&lt;sup&gt;''n''&lt;/sup&gt;−9)/16, it is divisible by 2 if ''n'' is odd and can be factored as (11×17&lt;sup&gt;''n''/2&lt;/sup&gt;−3) × (11×17&lt;sup&gt;''n''/2&lt;/sup&gt;+3) / 16 if ''n'' is even
* The base 19 family 1{6}: Can be written as (4×19&lt;sup&gt;''n''&lt;/sup&gt;−1)/3, it is divisible by 5 if ''n'' is odd and can be factored as (2×19&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (2×19&lt;sup&gt;''n''/2&lt;/sup&gt;+1) / 3 if ''n'' is even
* The base 19 family 7{2}: Can be written as (64×19&lt;sup&gt;''n''&lt;/sup&gt;−1)/9, it is divisible by 5 if ''n'' is odd and can be factored as (8×19&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (8×19&lt;sup&gt;''n''/2&lt;/sup&gt;+1) / 9 if ''n'' is even
* The base 24 family 3{N}: Can be written as 4×24&lt;sup&gt;''n''&lt;/sup&gt;−1, it is divisible by 5 if ''n'' is odd and can be factored as (2×24&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (2×24&lt;sup&gt;''n''/2&lt;/sup&gt;+1) if ''n'' is even

By the [[:w:Prime number theorem|prime number theorem]], the [[:w:Probability|chance]] that a [[:w:Random number|random]] ''n''-digit base ''b'' number is prime is [[:w:Asymptotic analysis|approximately]] 1/''n'' (more accurately, the chance is approximately 1/(''n''×''ln''(''b'')), where ''ln'' is the [[:w:Natural logarithm|natural logarithm]]). If one conjectures the numbers ''x''{''y''}''z'' behave similarly (i.e. the numbers ''x''{''y''}''z'' is a [[:w:Pseudorandomness|pseudorandom sequence]]) you would expect [[:w:Harmonic_series (mathematics)|1/1 + 1/2 + 1/3 + 1/4 + ... = ∞]] primes of the form ''x''{''y''}''z'' (of course, this does not always happen, since some ''x''{''y''}''z'' families can be ruled out to contain no prime &gt; ''b'' (by covering congruence, algebraic factorization, or combine of them), but it is at least a reasonable conjecture in the absence of evidence to the contrary. Hence, the [[:w:Heuristic argument|heuristic argument]] suggests there are always infinitely many primes in family ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') if it cannot be ruled out to contain no prime or only contain finitely many primes, by covering congruence, algebraic factorization, or combine of them. However, some families ''x''{''y''}''z'' could not be proven to contain no primes &gt; ''b'' (by covering congruence, algebraic factorization, or combine of them) but no primes &gt; ''b'' could be found in the family, even after searching through numbers with over 100000 digits. In such a case, the only way to proceed is to [[:w:Primality test|test the primality]] of larger and larger numbers of such form and hope a prime is eventually discovered. e.g. the smallest (probable) prime in the family A{3}A in base ''b'' = 13 is A3&lt;sub&gt;592197&lt;/sub&gt;A, its algebraic form is (41×13&lt;sup&gt;592198&lt;/sup&gt;+27)/4, when written in decimal contains 659677 digits (it is only probable prime, i.e. not definitely prime, since technically, probable primality tests were used to show this (which have a ''very'' small chance of making an error, see https://t5k.org/notes/prp_prob.html) because all known primality tests run far too slowly to run on numbers of this size unless either [https://t5k.org/prove/prove3_1.html ''N''−1] or [https://t5k.org/prove/prove3_2.html ''N''+1] (or both) can be ≥ 1/3 factored).

The numbers in family ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') are of the form (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1) for some fixed ''a'', ''b'', ''c'' such that ''a'' ≥ 1, ''b'' ≥ 2 (''b'' is the base), ''c'' ≠ 0, ''gcd''(''a'',''c'') = 1, ''gcd''(''b'',''c'') = 1. Except in the [[:w:Special case|special case]] ''c'' = ±1 and ''gcd''(''a''+''c'',''b''−1) = 1, when ''n'' is large the known [[:w:Primality test|primality test]]s for such a number are too inefficient to run. In this case one must resort to a [[:w:Probabilistic algorithm|probable]] primality test such as a [[:w:Miller–Rabin primality test|Miller–Rabin primality test]] or a [[:w:Baillie–PSW primality test|Baillie–PSW primality test]], unless a divisor of the number can be found. Since we are testing many numbers in an [[:w:Exponential growth|exponential sequence]], it is possible to use a sieving process to find divisors rather than using [[:w:Trial division|trial division]].

To do this, we made use of Geoffrey Reynolds' ''srsieve'' software (download: https://pzktupel.de/Software/srsieve_1.1.4.7z). This program uses the [[:w:Baby-step giant-step|baby-step giant-step]] [[:w:Algorithm|algorithm]] to find all primes ''p'' which divide ''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'' where ''p'' and ''n'' lie in a [[:w:Interval_(mathematics)|specified range]]. Since this program cannot handle the general case (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1) when ''gcd''(''a''+''c'',''b''−1) &gt; 1 we only used it to sieve the sequence ''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'' for primes ''p'' not dividing ''gcd''(''a''+''c'',''b''−1), and initialized the list of candidates to not include ''n'' for which there is some prime ''p'' dividing ''gcd''(''a''+''c'',''b''−1) for which ''p'' dividing (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1). The program had to be modified slightly to remove a check which would prevent it from running in the case when ''a'', ''b'', and ''c'' were all odd (since then 2 divides ''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'', but 2 may not divide (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1)).

Once the numbers with small divisors had been removed, it remained to test the remaining numbers using a probable primality test. For this we used the software ''LLR'' by Jean Penné. (download: http://jpenne.free.fr/index2.html). Although undocumented, it is possible to run this program on numbers of the form (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1) when ''gcd''(''a''+''c'',''b''−1) &gt; 1, so this program required no modifications. A script was also written which allowed one to run ''srsieve'' while ''LLR'' was testing the remaining candidates, so that when a divisor was found by srsieve on a number which had not yet been tested by ''LLR'' it would be removed from the list of candidates.

For the primes &lt; 10&lt;sup&gt;25000&lt;/sup&gt; for the &quot;easy&quot; bases (bases ''b'' with ≤ 150 primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; (base ''b'' = 26 has 83 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 3 unsolved families, base ''b'' = 36 has 75 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 4 unsolved families, base ''b'' = 17 has 99 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 18 unsolved families, base ''b'' = 21 has 80 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 12 unsolved families, base ''b'' = 19 has 201 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 23 unsolved families), i.e. bases *b* = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 26, 28, 30, 36), we employed ''CM'' by Andreas Enge (download: https://www.multiprecision.org/cm/download.html), an elliptic curve primality proving implementation.

== Data ==

These are the results of the Athena problem in bases 2 ≤ ''b'' ≤ 36 (we stop at base 36 since this base is the maximum base for which it is possible to write the numbers with the [[:w:Symbol|symbol]]s 0, 1, 2, ..., 9 and A, B, C, ..., Z (i.e. the 10 [[:w:Arabic numerals|Arabic numerals]] and the 26 [[:w:Latin script|Latin letters]]): (some large primes are only probable primes, i.e. not definitely primes, since they are too large to be [[:w:Elliptic curve primality|ECPP proved]] and [[:w:Pocklington primality test#Extensions and variants|neither ''N''−1 nor ''N''+1 can be ≥ 1/3 factored]], all of them pass the [[:w:Baillie–PSW primality test|Baillie–PSW primality test]] and the [[:w:Strong pseudoprime|strong primality test]] (i.e. the [[:w:Miller–Rabin primality test|Miller–Rabin primality test]]) with all prime bases ''p'' ≤ 61, however, all primes &lt; 10&lt;sup&gt;25000&lt;/sup&gt; for bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 36 are definitely primes, most of them &gt; 10&lt;sup&gt;299&lt;/sup&gt; are proven primes with [[:w:Elliptic curve primality|ECPP proving]], others &gt; 10&lt;sup&gt;299&lt;/sup&gt; are proven primes with [[:w:Pocklington primality test#Extensions and variants|''N''−1 or ''N''+1 proving]])

All numbers are written in base ''b'', [[:w:Senary#Base 36 as senary compression|using A to Z to represent digit values 10 to 35]], &quot;{}&quot; means repeating, e.g. family 12{3}45 means the sequence {1245, 12345, 123345, 1233345, 12333345, 123333345, ...} (where the members are expressed as base ''b'' strings), subscripts are used to indicate repetitions of digits, e.g. 123&lt;sub&gt;4&lt;/sub&gt;567 means 123333567 (all subscripts are written in decimal).

Base 2: 1 Athena prime (the largest of which has 2 digits (it is 11, and its value is 3 in decimal)): {11}

Base 3: 3 Athena primes (the largest of which has 3 digits (it is 111, and its value is 13 in decimal)): {12, 21, 111}

Base 4: 5 Athena primes (the largest of which has 3 digits (it is 221, and its value is 41 in decimal)): {11, 13, 23, 31, 221}

Base 5: 22 Athena primes (the largest of which has 96 digits (it is 10&lt;sub&gt;93&lt;/sub&gt;13, and its algebraic form is 5&lt;sup&gt;95&lt;/sup&gt;+8)): {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 Athena primes (the largest of which has 5 digits (it is 40041, and its value is 5209 in decimal)): {11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041}

Base 7: 71 Athena primes (the largest of which has 17 digits (it is 3&lt;sub&gt;16&lt;/sub&gt;1, and its algebraic form is (7&lt;sup&gt;17&lt;/sup&gt;−5)/2)): {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: 75 Athena primes (the largest of which has 221 digits (it is 4&lt;sub&gt;220&lt;/sub&gt;7, and its algebraic form is (4×8&lt;sup&gt;221&lt;/sup&gt;+17)/7)): {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: 151 Athena primes (the largest of which has 1161 digits (it is 30&lt;sub&gt;1158&lt;/sub&gt;11, and its algebraic form is 3×9&lt;sup&gt;1160&lt;/sup&gt;+10)): {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: 77 Athena primes (the largest of which has 31 digits (it is 50&lt;sub&gt;28&lt;/sub&gt;27, and its algebraic form is 5×10&lt;sup&gt;30&lt;/sup&gt;+27)): {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: 1068 Athena primes (including 1 unproven probable prime: 57&lt;sub&gt;62668&lt;/sub&gt;), the largest of which has 62669 digits (it is 57&lt;sub&gt;62668&lt;/sub&gt;, and its algebraic form is (57×11&lt;sup&gt;62668&lt;/sup&gt;−7)/10), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel11 Data of Athena (probable) primes base 11]

Base 12: 106 Athena primes (the largest of which has 42 digits (it is 40&lt;sub&gt;39&lt;/sub&gt;77, and its algebraic form is 4×12&lt;sup&gt;41&lt;/sup&gt;+91)): {11, 15, 17, 1B, 25, 27, 31, 35, 37, 3B, 45, 4B, 51, 57, 5B, 61, 67, 6B, 75, 81, 85, 87, 8B, 91, 95, A7, AB, B5, B7, 221, 241, 2A1, 2B1, 2BB, 401, 421, 447, 471, 497, 565, 655, 665, 701, 70B, 721, 747, 771, 77B, 797, 7A1, 7BB, 907, 90B, 9BB, A41, B21, B2B, 2001, 200B, 202B, 222B, 229B, 292B, 299B, 4441, 4707, 4777, 6A05, 6AA5, 729B, 7441, 7B41, 929B, 9777, 992B, 9947, 997B, 9997, A0A1, A201, A605, A6A5, AA65, B001, B0B1, BB01, BB41, 600A5, 7999B, 9999B, AAAA1, B04A1, B0B9B, BAA01, BAAA1, BB09B, BBBB1, 44AAA1, A00065, BBBAA1, AAA0001, B00099B, AA000001, BBBBBB99B, B0000000000000000000000000009B, 400000000000000000000000000000000000000077}

Base 13: 3197 Athena primes (including 4 unproven probable primes: C5&lt;sub&gt;23755&lt;/sub&gt;C, 80&lt;sub&gt;32017&lt;/sub&gt;111, 95&lt;sub&gt;197420&lt;/sub&gt;, A3&lt;sub&gt;592197&lt;/sub&gt;A), the largest of which has 592199 digits (it is A3&lt;sub&gt;592197&lt;/sub&gt;A, and its algebraic form is (41×13&lt;sup&gt;592198&lt;/sup&gt;+27)/4), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel13 Data of Athena (probable) primes base 13]

Base 14: 650 Athena primes, the largest of which has 19699 digits (it is 4D&lt;sub&gt;19698&lt;/sub&gt;, and its algebraic form is 5×14&lt;sup&gt;19698&lt;/sup&gt;−1), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel14 Data of Athena primes base 14]

Base 15: 1284 Athena primes, the largest of which has 157 digits (it is 7&lt;sub&gt;155&lt;/sub&gt;97, and its algebraic form is (15&lt;sup&gt;157&lt;/sup&gt;+59)/2), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel15 Data of Athena primes base 15]

Base 16: 2347 Athena primes (including 3 unproven probable primes: DB&lt;sub&gt;32234&lt;/sub&gt;, 4&lt;sub&gt;72785&lt;/sub&gt;DD, 3&lt;sub&gt;116137&lt;/sub&gt;AF), the largest of which has 116139 digits (it is 3&lt;sub&gt;116137&lt;/sub&gt;AF, and its algebraic form is (16&lt;sup&gt;116139&lt;/sup&gt;+619)/5), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel16 Data of Athena (probable) primes base 16]

Base 17: 10415 known Athena primes (including many unproven probable primes) and 12 unsolved families (1{7}, 1F{0}7, 4{7}A, 70F{0}D, 8{B}9, 9{5}9, A{D}F, B{0}B3, {B}E9, {B}EE, F1{9}, FD0{D}, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel17 Data of known Athena (probable) primes base 17]

Base 18: 549 Athena primes, the largest of which has 6271 digits (it is C0&lt;sub&gt;6268&lt;/sub&gt;C5, and its algebraic form is 12×18&lt;sup&gt;6270&lt;/sup&gt;+221), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel18 Data of Athena primes base 18]

Base 19: 31417 known Athena primes (including many unproven probable primes) and 17 unsolved families (4B5{0}H, {5}3, 5{H}05, 5{H}0H, 5{H}5, 66{B}, 71{0}177, 7AF{0}H, 97{0}3, C{H}C, EE1{6}, F{7}5, F{B}G, F{D}F, H0F{0}7A, HB{0}5B5, II{D}, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel19 Data of known Athena (probable) primes base 19]

Base 20: 3314 Athena primes, the largest of which has 6271 digits (it is G0&lt;sub&gt;6269&lt;/sub&gt;D, and its algebraic form is 16×20&lt;sup&gt;6270&lt;/sup&gt;+13), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel20 Data of Athena primes base 20]

Base 21: 13386 known Athena primes (including many unproven probable primes) and 8 unsolved families (5{0}DJ, {9}D, B3{0}EB, B{H}6H, C{F}0K, {F}35, G{0}FK, H{0}7771, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel21 Data of known Athena (probable) primes base 21]

Base 22: 8003 Athena primes (including 1 unproven probable prime: BK&lt;sub&gt;22001&lt;/sub&gt;5), the largest of which has 22003 digits (it is BK&lt;sub&gt;22001&lt;/sub&gt;5, and its algebraic form is (251×22&lt;sup&gt;22002&lt;/sup&gt;−335)/21), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel22 Data of Athena (probable) primes base 22]

Base 23: 65178 known Athena primes (including many unproven probable primes) and 87 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel23 Data of known Athena (probable) primes base 23] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left23 Data of unsolved families for Athena problem base 23]

Base 24: 3409 Athena primes, the largest of which has 8134 digits (it is N00N&lt;sub&gt;8129&lt;/sub&gt;LN, and its algebraic form is 13249×24&lt;sup&gt;8131&lt;/sup&gt;−49), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel24 Data of Athena primes base 24]

Base 25: 133639 known Athena primes (including many unproven probable primes) and 85 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel25 Data of known Athena (probable) primes base 25] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left25 Data of unsolved families for Athena problem base 25]

Base 26: 25256 known Athena primes (including 7 unproven probable primes: 5&lt;sub&gt;19391&lt;/sub&gt;6F, 7&lt;sub&gt;20279&lt;/sub&gt;OL, LD0&lt;sub&gt;20975&lt;/sub&gt;7, 6K&lt;sub&gt;23300&lt;/sub&gt;5, J0&lt;sub&gt;44303&lt;/sub&gt;KCB, M0&lt;sub&gt;61186&lt;/sub&gt;2BB, 85M&lt;sub&gt;197060&lt;/sub&gt;B) and 3 unsolved families ({A}6F, {H}MH, {I}GL, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel26 Data of known Athena (probable) primes base 26]

Base 27: 102852 known Athena primes (including many unproven probable primes) and 44 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel27 Data of known Athena (probable) primes base 27] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left27 Data of unsolved families for Athena problem base 27]

Base 28: 25528 known Athena primes (including 3 unproven probable primes: N6&lt;sub&gt;24051&lt;/sub&gt;LR, 5OA&lt;sub&gt;31238&lt;/sub&gt;F, O4O&lt;sub&gt;94535&lt;/sub&gt;9) and 1 unsolved family (O{A}F, no primes or probable primes with length ≤ 900000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel28 Data of known Athena (probable) primes base 28]

Base 29: 355242 known Athena primes (including many unproven probable primes) and 125 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel29 Data of known Athena (probable) primes base 29] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left29 Data of unsolved families for Athena problem base 29]

Base 30: 2619 Athena primes (including 1 unproven probable prime: I0&lt;sub&gt;24608&lt;/sub&gt;D), the largest of which has 34206 digits (it is OT&lt;sub&gt;34205&lt;/sub&gt;, and its algebraic form is 25×30&lt;sup&gt;34205&lt;/sup&gt;−1), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel30 Data of Athena (probable) primes base 30]

Base 31: 569323 known Athena primes (including many unproven probable primes) and 77 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel31 Data of known Athena (probable) primes base 31] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left31 Data of unsolved families for Athena problem base 31]

Base 32: 168882 known Athena primes (including many unproven probable primes) and 120 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel32 Data of known Athena (probable) primes base 32] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left32 Data of unsolved families for Athena problem base 32]

Base 33: 280012 known Athena primes (including many unproven probable primes) and 81 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel33 Data of known Athena (probable) primes base 33] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left33 Data of unsolved families for Athena problem base 33]

Base 34: 184785 known Athena primes (including many unproven probable primes) and 47 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel34 Data of known Athena (probable) primes base 34] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left34 Data of unsolved families for Athena problem base 34]

Base 35: 720002 known Athena primes (including many unproven probable primes) and 60 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel35 Data of known Athena (probable) primes base 35] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left35 Data of unsolved families for Athena problem base 35]

Base 36: 35286 known Athena primes (including 3 unproven probable primes: 7K&lt;sub&gt;26567&lt;/sub&gt;Z, S0&lt;sub&gt;75007&lt;/sub&gt;8H, P&lt;sub&gt;81993&lt;/sub&gt;SZ) and 4 unsolved families (B{0}EUV, HM{0}N, N{0}YYN, O{L}Z, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel36 Data of known Athena (probable) primes base 36]

== Condensed table for bases 2 ≤ ''b'' ≤ 36 ==

{|class=&quot;wikitable&quot;
||''b''||number of Athena primes (or probable primes, which are Athena primes assuming their primality) base ''b''||base-''b'' form of the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) base ''b'' (write &quot;''d''&lt;sub&gt;''n''&lt;/sub&gt;&quot; if there are 5 or more (''n'') consecutive same digits ''d'')||length of the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) base ''b''||length of the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) base ''b'' in decimal||algebraic ((''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1)) form of the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) base ''b''||''factordb'' entry of the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) base ''b''||the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) base ''b'' written in base ''b'' (use lower case letters instead of upper case letters)||number of unsolved families in the Athena problem base ''b'' (all of these left families are linear families)||searching limit of length for the unsolved families in the Athena problem base ''b'' (if there are different searching limits for the unsolved families in base ''b'', choose the lowest searching limit)||
|-
||2||1||11||2||1||3||http://factordb.com/index.php?id=3&amp;open=ecm||http://factordb.com/index.php?showid=3&amp;base=2||0||–||
|-
||3||3||111&lt;br&gt;21&lt;br&gt;12||3&lt;br&gt;2&lt;br&gt;2||2&lt;br&gt;1&lt;br&gt;1||13&lt;br&gt;7&lt;br&gt;5||http://factordb.com/index.php?id=13&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=7&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=5&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=13&amp;base=3&lt;br&gt;http://factordb.com/index.php?showid=7&amp;base=3&lt;br&gt;http://factordb.com/index.php?showid=5&amp;base=3&lt;nowiki/&gt;||0||–||
|-
||4||5||221&lt;br&gt;31&lt;br&gt;23&lt;br&gt;13&lt;br&gt;11||3&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2||2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;1&lt;br&gt;1||41&lt;br&gt;13&lt;br&gt;11&lt;br&gt;7&lt;br&gt;5||http://factordb.com/index.php?id=41&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=13&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=11&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=7&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=5&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=41&amp;base=4&lt;br&gt;http://factordb.com/index.php?showid=13&amp;base=4&lt;br&gt;http://factordb.com/index.php?showid=11&amp;base=4&lt;br&gt;http://factordb.com/index.php?showid=7&amp;base=4&lt;br&gt;http://factordb.com/index.php?showid=5&amp;base=4&lt;nowiki/&gt;||0||–||
|-
||5||22||10&lt;sub&gt;93&lt;/sub&gt;13&lt;br&gt;300031&lt;br&gt;44441&lt;br&gt;33331&lt;br&gt;33001&lt;br&gt;30301&lt;br&gt;14444&lt;br&gt;10103&lt;br&gt;3101&lt;br&gt;414||96&lt;br&gt;6&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;4&lt;br&gt;3||67&lt;br&gt;4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;3&lt;br&gt;3&lt;br&gt;3||5&lt;sup&gt;95&lt;/sup&gt;+8&lt;br&gt;9391&lt;br&gt;3121&lt;br&gt;2341&lt;br&gt;2251&lt;br&gt;1951&lt;br&gt;1249&lt;br&gt;653&lt;br&gt;401&lt;br&gt;109||http://factordb.com/index.php?id=1100000000034686071&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=9391&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=3121&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=2341&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=2251&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1951&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1249&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=653&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=401&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=109&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000000034686071&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=9391&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=3121&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=2341&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=2251&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=1951&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=1249&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=653&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=401&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=109&amp;base=5&lt;nowiki/&gt;||0||–||
|-
||6||11||40041&lt;br&gt;4441&lt;br&gt;4401&lt;br&gt;51&lt;br&gt;45&lt;br&gt;35&lt;br&gt;31&lt;br&gt;25&lt;br&gt;21&lt;br&gt;15||5&lt;br&gt;4&lt;br&gt;4&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2||4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2||5209&lt;br&gt;1033&lt;br&gt;1009&lt;br&gt;31&lt;br&gt;29&lt;br&gt;23&lt;br&gt;19&lt;br&gt;17&lt;br&gt;13&lt;br&gt;11||http://factordb.com/index.php?id=5209&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1033&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1009&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=31&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=29&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=23&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=19&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=17&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=13&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=11&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=5209&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=1033&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=1009&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=31&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=29&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=23&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=19&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=17&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=13&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=11&amp;base=6&lt;nowiki/&gt;||0||–||
|-
||7||71||3&lt;sub&gt;16&lt;/sub&gt;1&lt;br&gt;510&lt;sub&gt;7&lt;/sub&gt;1&lt;br&gt;3&lt;sub&gt;6&lt;/sub&gt;01&lt;br&gt;1100021&lt;br&gt;531101&lt;br&gt;351101&lt;br&gt;300053&lt;br&gt;150001&lt;br&gt;100121&lt;br&gt;40054||17&lt;br&gt;10&lt;br&gt;8&lt;br&gt;7&lt;br&gt;6&lt;br&gt;6&lt;br&gt;6&lt;br&gt;6&lt;br&gt;6&lt;br&gt;5||15&lt;br&gt;9&lt;br&gt;7&lt;br&gt;6&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;4||(7&lt;sup&gt;17&lt;/sup&gt;−5)/2&lt;br&gt;36×7&lt;sup&gt;8&lt;/sup&gt;+1&lt;br&gt;(7&lt;sup&gt;8&lt;/sup&gt;−47)/2&lt;br&gt;134471&lt;br&gt;91631&lt;br&gt;62819&lt;br&gt;50459&lt;br&gt;28813&lt;br&gt;16871&lt;br&gt;9643||http://factordb.com/index.php?id=116315256993601&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=207532837&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=2882377&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=134471&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=91631&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=62819&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=50459&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=28813&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=16871&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=9643&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=116315256993601&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=207532837&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=2882377&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=134471&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=91631&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=62819&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=50459&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=28813&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=16871&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=9643&amp;base=7&lt;nowiki/&gt;||0||–||
|-
||8||75||4&lt;sub&gt;220&lt;/sub&gt;7&lt;br&gt;5&lt;sub&gt;13&lt;/sub&gt;25&lt;br&gt;7&lt;sub&gt;12&lt;/sub&gt;1&lt;br&gt;77774&lt;sub&gt;6&lt;/sub&gt;1&lt;br&gt;74&lt;sub&gt;7&lt;/sub&gt;1&lt;br&gt;4&lt;sub&gt;8&lt;/sub&gt;1&lt;br&gt;5&lt;sub&gt;5&lt;/sub&gt;025&lt;br&gt;5550525&lt;br&gt;5500525&lt;br&gt;4&lt;sub&gt;5&lt;/sub&gt;77||221&lt;br&gt;15&lt;br&gt;13&lt;br&gt;11&lt;br&gt;9&lt;br&gt;9&lt;br&gt;8&lt;br&gt;7&lt;br&gt;7&lt;br&gt;7||200&lt;br&gt;14&lt;br&gt;12&lt;br&gt;10&lt;br&gt;9&lt;br&gt;8&lt;br&gt;8&lt;br&gt;7&lt;br&gt;7&lt;br&gt;7||(4×8&lt;sup&gt;221&lt;/sup&gt;+17)/7&lt;br&gt;(5×8&lt;sup&gt;15&lt;/sup&gt;−173)/7&lt;br&gt;8&lt;sup&gt;13&lt;/sup&gt;−7&lt;br&gt;(28669×8&lt;sup&gt;7&lt;/sup&gt;−25)/7&lt;br&gt;(53×8&lt;sup&gt;8&lt;/sup&gt;−25)/7&lt;br&gt;(4×8&lt;sup&gt;9&lt;/sup&gt;−25)/7&lt;br&gt;(5×8&lt;sup&gt;8&lt;/sup&gt;−2413)/7&lt;br&gt;1495381&lt;br&gt;1474901&lt;br&gt;(4×8&lt;sup&gt;7&lt;/sup&gt;+185)/7||http://factordb.com/index.php?id=1100000000416605822&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=25131694349141&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=549755813881&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=8589035809&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=127027489&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=76695841&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=11983381&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1495381&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1474901&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1198399&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000000416605822&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=25131694349141&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=549755813881&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=8589035809&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=127027489&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=76695841&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=11983381&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=1495381&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=1474901&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=1198399&amp;base=8&lt;nowiki/&gt;||0||–||
|-
||9||151||30&lt;sub&gt;1158&lt;/sub&gt;11&lt;br&gt;27&lt;sub&gt;686&lt;/sub&gt;07&lt;br&gt;76&lt;sub&gt;329&lt;/sub&gt;2&lt;br&gt;561&lt;sub&gt;36&lt;/sub&gt;&lt;br&gt;10&lt;sub&gt;25&lt;/sub&gt;57&lt;br&gt;30&lt;sub&gt;20&lt;/sub&gt;51&lt;br&gt;8&lt;sub&gt;19&lt;/sub&gt;335&lt;br&gt;727&lt;sub&gt;15&lt;/sub&gt;07&lt;br&gt;51&lt;sub&gt;13&lt;/sub&gt;61&lt;br&gt;10&lt;sub&gt;11&lt;/sub&gt;507||1161&lt;br&gt;689&lt;br&gt;331&lt;br&gt;38&lt;br&gt;28&lt;br&gt;23&lt;br&gt;22&lt;br&gt;19&lt;br&gt;16&lt;br&gt;15||1108&lt;br&gt;657&lt;br&gt;316&lt;br&gt;37&lt;br&gt;26&lt;br&gt;22&lt;br&gt;21&lt;br&gt;19&lt;br&gt;16&lt;br&gt;14||3×9&lt;sup&gt;1160&lt;/sup&gt;+10&lt;br&gt;(23×9&lt;sup&gt;688&lt;/sup&gt;−511)/8&lt;br&gt;(31×9&lt;sup&gt;330&lt;/sup&gt;−19)/4&lt;br&gt;(409×9&lt;sup&gt;36&lt;/sup&gt;−1)/8&lt;br&gt;9&lt;sup&gt;27&lt;/sup&gt;+52&lt;br&gt;3×9&lt;sup&gt;22&lt;/sup&gt;+46&lt;br&gt;9&lt;sup&gt;22&lt;/sup&gt;−454&lt;br&gt;(527×9&lt;sup&gt;17&lt;/sup&gt;−511)/8&lt;br&gt;(41×9&lt;sup&gt;15&lt;/sup&gt;+359)/8&lt;br&gt;9&lt;sup&gt;14&lt;/sup&gt;+412||http://factordb.com/index.php?id=1100000002376318423&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000002495467486&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000002359003642&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000001554010824&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1100000002512830927&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1100000000032261811&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1100000002495736583&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1100000003446800389&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1055192051985121&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=22876792455373&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000002376318423&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=1100000002495467486&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=1100000002359003642&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=1100000001554010824&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=1100000002512830927&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=1100000000032261811&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=1100000002495736583&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=1100000003446800389&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=1055192051985121&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=22876792455373&amp;base=9&lt;nowiki/&gt;||0||–||
|-
||10||77||50&lt;sub&gt;28&lt;/sub&gt;27&lt;br&gt;5&lt;sub&gt;11&lt;/sub&gt;1&lt;br&gt;805&lt;sub&gt;5&lt;/sub&gt;1&lt;br&gt;66600049&lt;br&gt;66000049&lt;br&gt;60&lt;sub&gt;5&lt;/sub&gt;49&lt;br&gt;220&lt;sub&gt;5&lt;/sub&gt;1&lt;br&gt;5200007&lt;br&gt;946669&lt;br&gt;666649||31&lt;br&gt;12&lt;br&gt;8&lt;br&gt;8&lt;br&gt;8&lt;br&gt;8&lt;br&gt;8&lt;br&gt;7&lt;br&gt;6&lt;br&gt;6||31&lt;br&gt;12&lt;br&gt;8&lt;br&gt;8&lt;br&gt;8&lt;br&gt;8&lt;br&gt;8&lt;br&gt;7&lt;br&gt;6&lt;br&gt;6||5×10&lt;sup&gt;30&lt;/sup&gt;+27&lt;br&gt;(5×10&lt;sup&gt;12&lt;/sup&gt;−41)/9&lt;br&gt;(725×10&lt;sup&gt;6&lt;/sup&gt;−41)/9&lt;br&gt;66600049&lt;br&gt;66000049&lt;br&gt;6×10&lt;sup&gt;7&lt;/sup&gt;+49&lt;br&gt;22×10&lt;sup&gt;6&lt;/sup&gt;+1&lt;br&gt;5200007&lt;br&gt;946669&lt;br&gt;666649||http://factordb.com/index.php?id=1100000000204142046&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=555555555551&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=80555551&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=66600049&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=66000049&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=60000049&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=22000001&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=5200007&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=946669&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=666649&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000000204142046&amp;base=10&lt;br&gt;http://factordb.com/index.php?showid=555555555551&amp;base=10&lt;br&gt;http://factordb.com/index.php?showid=80555551&amp;base=10&lt;br&gt;http://factordb.com/index.php?showid=66600049&amp;base=10&lt;br&gt;http://factordb.com/index.php?showid=66000049&amp;base=10&lt;br&gt;http://factordb.com/index.php?showid=60000049&amp;base=10&lt;br&gt;http://factordb.com/index.php?showid=22000001&amp;base=10&lt;br&gt;http://factordb.com/index.php?showid=5200007&amp;base=10&lt;br&gt;http://factordb.com/index.php?showid=946669&amp;base=10&lt;br&gt;http://factordb.com/index.php?showid=666649&amp;base=10&lt;nowiki/&gt;||0||–||
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||24||3409||N00N&lt;sub&gt;8129&lt;/sub&gt;LN&lt;br&gt;88N&lt;sub&gt;5951&lt;/sub&gt;&lt;br&gt;A0&lt;sub&gt;2951&lt;/sub&gt;8ID&lt;br&gt;D&lt;sub&gt;2698&lt;/sub&gt;LD&lt;br&gt;N&lt;sub&gt;2644&lt;/sub&gt;LLN&lt;br&gt;BC0&lt;sub&gt;331&lt;/sub&gt;B&lt;br&gt;20&lt;sub&gt;313&lt;/sub&gt;7&lt;br&gt;C7&lt;sub&gt;298&lt;/sub&gt;&lt;br&gt;D0&lt;sub&gt;259&lt;/sub&gt;KKD&lt;br&gt;I0&lt;sub&gt;241&lt;/sub&gt;I5||8134&lt;br&gt;5953&lt;br&gt;2955&lt;br&gt;2700&lt;br&gt;2647&lt;br&gt;334&lt;br&gt;315&lt;br&gt;299&lt;br&gt;263&lt;br&gt;244||11227&lt;br&gt;8216&lt;br&gt;4079&lt;br&gt;3727&lt;br&gt;3654&lt;br&gt;461&lt;br&gt;434&lt;br&gt;413&lt;br&gt;363&lt;br&gt;337||13249×24&lt;sup&gt;8131&lt;/sup&gt;−49&lt;br&gt;201×24&lt;sup&gt;5951&lt;/sup&gt;−1&lt;br&gt;10×24&lt;sup&gt;2954&lt;/sup&gt;+5053&lt;br&gt;(13×24&lt;sup&gt;2700&lt;/sup&gt;+4403)/23&lt;br&gt;24&lt;sup&gt;2647&lt;/sup&gt;−1201&lt;br&gt;276×24&lt;sup&gt;332&lt;/sup&gt;+11&lt;br&gt;2×24&lt;sup&gt;314&lt;/sup&gt;+7&lt;br&gt;(283×24&lt;sup&gt;298&lt;/sup&gt;−7)/23&lt;br&gt;13×24&lt;sup&gt;262&lt;/sup&gt;+12013&lt;br&gt;18×24&lt;sup&gt;243&lt;/sup&gt;+437||http://factordb.com/index.php?id=1100000003593391606&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003593275880&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003593269654&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003593269876&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003593270089&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000002633359842&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000002355610241&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000002326181235&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003593270725&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000002633360037&amp;open=prime&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000003593391606&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000003593275880&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000003593269654&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000003593269876&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000003593270089&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000002633359842&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000002355610241&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000002326181235&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000003593270725&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000002633360037&amp;base=24&lt;nowiki/&gt;||0||–||
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|-
||36||35286~35290||P&lt;sub&gt;81993&lt;/sub&gt;SZ&lt;br&gt;S0&lt;sub&gt;75007&lt;/sub&gt;8H&lt;br&gt;7K&lt;sub&gt;26567&lt;/sub&gt;Z&lt;br&gt;J&lt;sub&gt;10117&lt;/sub&gt;LJ&lt;br&gt;VL0&lt;sub&gt;7258&lt;/sub&gt;J&lt;br&gt;EO0&lt;sub&gt;6177&lt;/sub&gt;V&lt;br&gt;FZ&lt;sub&gt;5777&lt;/sub&gt;3P&lt;br&gt;T09&lt;sub&gt;4618&lt;/sub&gt;1&lt;br&gt;RY&lt;sub&gt;4562&lt;/sub&gt;H&lt;br&gt;OZ&lt;sub&gt;3932&lt;/sub&gt;AZ||81995&lt;br&gt;75010&lt;br&gt;26569&lt;br&gt;10119&lt;br&gt;7261&lt;br&gt;6180&lt;br&gt;5780&lt;br&gt;4621&lt;br&gt;4564&lt;br&gt;3935||127609&lt;br&gt;116739&lt;br&gt;41349&lt;br&gt;15748&lt;br&gt;11301&lt;br&gt;9618&lt;br&gt;8996&lt;br&gt;7192&lt;br&gt;7103&lt;br&gt;6124||(5×36&lt;sup&gt;81995&lt;/sup&gt;+821)/7&lt;br&gt;28×36&lt;sup&gt;75009&lt;/sup&gt;+305&lt;br&gt;(53×36&lt;sup&gt;26568&lt;/sup&gt;+101)/7&lt;br&gt;(19×36&lt;sup&gt;10119&lt;/sup&gt;+2501)/35&lt;br&gt;1137×36&lt;sup&gt;7259&lt;/sup&gt;+19&lt;br&gt;528×36&lt;sup&gt;6178&lt;/sup&gt;+31&lt;br&gt;16×36&lt;sup&gt;5779&lt;/sup&gt;−1163&lt;br&gt;(36549×36&lt;sup&gt;4619&lt;/sup&gt;−289)/35&lt;br&gt;(979×36&lt;sup&gt;4563&lt;/sup&gt;−629)/35&lt;br&gt;25×36&lt;sup&gt;3934&lt;/sup&gt;−901||http://factordb.com/index.php?id=1100000002394962083&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004020085177&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003896952461&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362491&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362489&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362488&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362487&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362486&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362485&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000000840634476&amp;open=prime&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000002394962083&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000004020085177&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003896952461&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362491&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362489&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362488&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362487&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362486&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362485&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000000840634476&amp;base=36&lt;nowiki/&gt;||4||200000||
|}

== The fully proof of Athena problem in decimal (base ''b'' = 10) ==

'''Bold''' for the Athena primes, ''x'' ◁ ''y'' means ''x'' is a subsequence of ''y''.

Assume ''p'' is a prime &gt; 10, and the last digit of ''p'' must lie in {1,3,7,9}.

Case 1: ''p'' ends with 1.

In this case we can write ''p'' = ''x''1. If ''x'' contains 1, 3, 4, 6, or 7, then (respectively) '''11''' ◁ ''p'', '''31''' ◁ ''p'', '''41''' ◁ ''p'', '''61''' ◁ ''p'', or '''71''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 2, 5, 8, or 9.

Case 1.1: ''p'' begins with 2.

In this case we can write ''p'' = 2''y''1. If 5 ◁ ''y'', then '''251''' ◁ ''p''. If 8 ◁ ''y'', then '''281''' ◁ ''p''. If 9 ◁ ''y'', then 29 ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 2.

If 22 ◁ ''y'', then '''2221''' ◁ ''p''. Hence we may assume ''y'' contains zero or one 2's.

If ''y'' contains no 2's, then ''p'' ∈ 2{0}1. But then, since the sum of the digits of ''p'' is 3, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' contains exactly one 2, then we can write ''p'' = 2''z''2''w''1, where ''z'',''w'' ∈ {0}. If 0 ◁ ''z'' and 0 ◁ ''w'', then '''20201''' ◁ ''p''. Hence we may assume either ''z'' or ''w'' is empty.

If ''z'' is empty, then ''p'' ∈ 22{0}1, and the smallest prime ''p'' ∈ 22{0}1 is '''22000001'''.

If ''w'' is empty, then ''p'' ∈ 2{0}21, and the smallest prime ''p'' ∈ 2{0}21 is '''20021'''.

Case 1.2: ''p'' begins with 5.

In this case we can write ''p'' = 5''y''1. If 2 ◁ ''y'', then '''521''' ◁ ''p''. If 9 ◁ ''y'', then 59 ◁ ''p''. Hence we may assume all digits of ''y'' are 0, 5, or 8.

If 05 ◁ ''y'', then '''5051''' ◁ ''p''. If 08 ◁ ''y'', then '''5081''' ◁ ''p''. If 50 ◁ ''y'', then '''5501''' ◁ ''p''. If 58 ◁ ''y'', then '''5581''' ◁ ''p''. If 80 ◁ ''y'', then '''5801''' ◁ ''p''. If 85 ◁ ''y'', then '''5851''' ◁ ''p''. Hence we may assume ''y'' ∈ {0} ∪ {5} ∪ {8}.

If ''y'' ∈ {0}, then ''p'' ∈ 5{0}1. But then, since the sum of the digits of ''p'' is 6, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' ∈ {5}, then ''p'' ∈ 5{5}1, and the smallest prime ''p'' ∈ 5{5}1 is '''555555555551'''.

If ''y'' ∈ {8}, since if 88 ◁ ''y'', then 881 ◁ ''p'', hence we may assume ''y'' ∈ {''𝜆'',8}, and thus ''p'' ∈ {51,581}, but 51 and 581 are both composite.

Case 1.3: ''p'' begins with 8.

In this case we can write p = 8''y''1. If 2 ◁ ''y'', then '''821''' ◁ ''p''. If 8 ◁ ''y'', then '''881''' ◁ ''p''. If 9 ◁ ''y'', then 89 ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 5.

If 50 ◁ ''y'', then '''8501''' ◁ ''p''. Hence we may assume y ∈ {0}{5}.

If 005 ◁ ''y'', then '''80051''' ◁ p. Hence we may assume y ∈ {0} ∪ {5} ∪ 0{5}.

If y ∈ {0}, then ''p'' ∈ 8{0}1. But then, since the sum of the digits of ''p'' is 9, ''p'' is divisible by 3, so ''p'' cannot be prime.

If y ∈ {5}, since if 55555555555 ◁ ''y'', then 555555555551 ◁ ''p'', hence we may assume ''y'' ∈ {''𝜆'', 5, 55, 555, 5555, 55555, 555555, 5555555, 55555555, 555555555, 5555555555}, and thus ''p'' ∈ {81, 851, 8551, 85551, 855551, 8555551, 85555551, 855555551, 8555555551, 85555555551, 855555555551}, but all of these numbers are composite.

If y ∈ 0{5}, since if 55555555555 ◁ ''y'', then 555555555551 ◁ ''p'', hence we may assume ''y'' ∈ {0, 05, 055, 0555, 05555, 055555, 0555555, 05555555, 055555555, 0555555555, 05555555555}, and thus ''p'' ∈ {801, 8051, 80551, 805551, 8055551, 80555551, 805555551, 8055555551, 80555555551, 805555555551, 8055555555551}, and of these numbers only 80555551 and 8055555551 are primes, but 80555551 ◁ 8055555551, thus only '''80555551''' is a minimal element.

Case 1.4: ''p'' begins with 9.

In this case we can write p = 9''y''1. If 9 ◁ ''y'', then '''991''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0, 2, 5, or 8.

If 00 ◁ ''y'', then '''9001''' ◁ ''p''. If 22 ◁ ''y'', then '''9221''' ◁ ''p''. If 55 ◁ ''y'', then '''9551''' ◁ ''p''. If 88 ◁ ''y'', then 881 ◁ ''p''. Hence we may assume ''y'' contains at most one 0, at most one 2, at most one 5, and at most one 8.

If ''y'' only contains at most one 0 and does not contain any of {2,5,8}, then ''y'' ∈ {''𝜆'',0}, and thus ''p'' ∈ {91,901}, but 91 and 901 are both composite. If ''y'' only contains at most one 0 and only one of {2,5,8}, then the sum of the digits of ''p'' is divisible by 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Hence we may assume ''y'' contains at least two of {2,5,8}.

If 25 ◁ ''y'', then 251 ◁ ''p''. If 28 ◁ ''y'', then 281 ◁ ''p''. If 52 ◁ ''y'', then 521 ◁ ''p''. If 82 ◁ ''y'', then 821 ◁ ''p''. Hence we may assume ''y'' contains no 2's (since if ''y'' contains 2, then ''y'' cannot contain either 5's or 8's, which is a contradiction).

If 85 ◁ ''y'', then '''9851''' ◁ ''p''. Hence we may assume ''y'' ∈ {58,580,508,058}, and thus ''p'' ∈ {9581,95801,95081,90581}, and of these numbers only 95801 is prime, but 95801 is not a minimal element since 5801 ◁ 95801.

Case 2: ''p'' ends with 3.

In this case we can write p = ''x''3. If ''x'' contains 1, 2, 4, 5, 7, or 8, then (respectively) '''13''' ◁ ''p'', '''23''' ◁ ''p'', '''43''' ◁ ''p'', '''53''' ◁ ''p'', '''73''' ◁ ''p'', or '''83''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 3, 6, or 9, and thus all digits of ''p'' are 0, 3, 6, or 9. But then, since the digits of ''p'' all have a common factor 3, ''p'' is divisible by 3, so ''p'' cannot be prime.

Case 3: ''p'' ends with 7.

In this case we can write ''p'' = ''x''7. If ''x'' contains 1, 3, 4, 6, or 9, then (respectively) '''17''' ◁ ''p'', '''37''' ◁ ''p'', '''47''' ◁ ''p'', '''67''' ◁ ''p'', or '''97''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 2, 5, 7, or 8.

Case 3.1: ''p'' begins with 2.

In this case we can write ''p'' = 2''y''7. If 2 ◁ ''y'', then '''227''' ◁ ''p''. If 5 ◁ ''y'', then '''257''' ◁ ''p''. If 7 ◁ ''y'', then '''277''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 8.

If 08 ◁ ''y'', then '''2087''' ◁ ''p''. If 88 ◁ ''y'', then 887 ◁ ''p''. Hence we may assume ''y'' ∈ {0} ∪ 8{0}.

If ''y'' ∈ {0}, then ''p'' ∈ 2{0}7. But then, since the sum of the digits of ''p'' is 9, ''p'' is divisible by 3, so ''p'' cannot be prime.

If y ∈ 8{0}, then ''p'' ∈ 28{0}7. But then ''p'' is divisible by 7, since for ''n'' ≥ 0 we have 7 × 40&lt;sub&gt;''n''&lt;/sub&gt;1 = 280&lt;sub&gt;''n''&lt;/sub&gt;7.

Case 3.2: ''p'' begins with 5.

In this case we can write ''p'' = 5''y''7. If 5 ◁ ''y'', then '''557''' ◁ ''p''. If 7 ◁ ''y'', then '''577''' ◁ ''p''. If 8 ◁ ''y'', then '''587''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 2.

If 22 ◁ ''y'', then 227 ◁ ''p''. Hence we may assume ''y'' contains zero or one 2's.

If ''y'' contains no 2's, then ''p'' ∈ 5{0}7. But then, since the sum of the digits of ''p'' is 12, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' contains exactly one 2, then we can write ''p'' = 5''z''2''w''7, where ''z'',''w'' ∈ {0}. If 0 ◁ ''z'' and 0 ◁ ''w'', then '''50207''' ◁ ''p''. Hence we may assume either ''z'' or ''w'' is empty.

If ''z'' is empty, then ''p'' ∈ 52{0}7, and the smallest prime ''p'' ∈ 52{0}7 is '''5200007'''.

If ''w'' is empty, then ''p'' ∈ 5{0}27, and the smallest prime ''p'' ∈ 5{0}27 is '''5000000000000000000000000000027'''.

Case 3.3: ''p'' begins with 7.

In this case we can write ''p'' = 7''y''7. If 2 ◁ ''y'', then '''727''' ◁ ''p''. If 5 ◁ ''y'', then '''757''' ◁ ''p''. If 8 ◁ ''y'', then '''787''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 7, and thus all digits of ''p'' are 0 or 7. But then, since the digits of ''p'' all have a common factor 7, ''p'' is divisible by 7, so ''p'' cannot be prime.

Case 3.4: ''p'' begins with 8.

In this case we can write ''p'' = 8''y''7. If 2 ◁ ''y'', then '''827''' ◁ ''p''. If 5 ◁ ''y'', then '''857''' ◁ ''p''. If 7 ◁ ''y'', then '''877''' ◁ ''p''. If 8 ◁ ''y'', then '''887''' ◁ ''p''. Hence we may assume ''y'' ∈ {0}, and thus ''p'' ∈ 8{0}7. But then, since the sum of the digits of ''p'' is 15, ''p'' is divisible by 3, so ''p'' cannot be prime.

Case 4: ''p'' ends with 9.

In this case we can write ''p'' = ''x''9. If ''x'' contains 1, 2, 5, 7, or 8, then (respectively) '''19''' ◁ ''p'', '''29''' ◁ ''p'', '''59''' ◁ ''p'', '''79''' ◁ ''p'', or '''89''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 3, 4, 6, or 9.

If 44 ◁ ''x'', then '''449''' ◁ ''p''. Hence we may assume ''x'' contains zero or one 4's.

If x contains no 4's, then all digits of ''x'' are 0, 3, 6, or 9, and thus all digits of ''p'' are 0, 3, 6, or 9. But then, since the digits of ''p'' all have a common factor 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Hence we may assume that ''x'' contains exactly one 4.

Case 4.1: ''p'' begins with 3.

In this case we can write ''p'' = 3''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. We must have '''349''' ◁ ''p''.

Case 4.2: ''p'' begins with 4.

In this case we can write ''p'' = 4''y''9, where all digits of ''y'' are 0, 3, 6, or 9. If 0 ◁ ''y'', then '''409''' ◁ ''p''. If 3 ◁ ''y'', then 43 ◁ ''p''. If 9 ◁ ''y'', then '''499''' ◁ ''p''. Hence we may assume ''y'' ∈ {6}, and thus ''p'' ∈ 4{6}9. But then ''p'' is divisible by 7, since for ''n'' ≥ 0 we have 7 × 6&lt;sub&gt;''n''&lt;/sub&gt;7 = 46&lt;sub&gt;''n''&lt;/sub&gt;9.

Case 4.3: ''p'' begins with 6.

In this case we can write p = 6''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. If 0 ◁ ''z'', then 409 ◁ ''p''. If 3 ◁ ''z'', then 43 ◁ ''p''. If 6 ◁ ''z'', then '''6469''' ◁ ''p''. If 9 ◁ ''z'', then 499 ◁ ''p''. Hence we may assume ''z'' is empty.

If 3 ◁ ''y'', then 349 ◁ ''p''. If 9 ◁ ''y'', then '''6949''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 6.

If 06 ◁ ''y'', then '''60649''' ◁ ''p''. Hence we may assume ''y'' ∈ {6}{0}.

If 666 ◁ ''y'', then '''666649''' ◁ ''p''. If 00000 ◁ ''y'', then '''60000049''' ◁ ''p''. Hence we may assume ''y'' ∈ {''𝜆'', 0, 00, 000, 0000, 6, 60, 600, 6000, 60000, 66, 660, 6600, 66000, 660000}, and thus ''p'' ∈ {649, 6049, 60049, 600049, 6000049, 6649, 66049, 660049, 6600049, 66000049, 66649, 666049, 6660049, 66600049, 666000049}, and of these numbers only '''66000049''' and '''66600049''' are primes.

Case 4.4: ''p'' begins with 9.

In this case we can write p = 9''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. If 0 ◁ ''y'', then '''9049''' ◁ ''p''. If 3 ◁ ''y'', then 349 ◁ ''p''. If 6 ◁ ''y'', then '''9649''' ◁ ''p''. If 9 ◁ ''y'', then '''9949''' ◁ ''p''. Hence we may assume ''y'' is empty.

If 0 ◁ ''z'', then 409 ◁ ''p''. If 3 ◁ ''z'', then 43 ◁ ''p''. If 9 ◁ ''z'', then 499 ◁ ''p''. Hence we may assume ''z'' ∈ {6}, and thus ''p'' ∈ 94{6}9, and the smallest prime ''p'' ∈ 94{6}9 is 946669.

[[Category:Number theory]]</text>
      <sha1>o3omh8ibpfsvloq6qkec5uakw0sbw46</sha1>
    </revision>
    <revision>
      <id>2819838</id>
      <parentid>2819768</parentid>
      <timestamp>2026-07-30T08:51:43Z</timestamp>
      <contributor>
        <username>Athene241</username>
        <id>3100061</id>
      </contributor>
      <comment>/* Solve the problem */</comment>
      <origin>2819838</origin>
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      <format>text/x-wiki</format>
      <text bytes="128777" sha1="s7fqb14njwrvyfkmy0p784hnm5ql1w2" xml:space="preserve">{{mathematics}}
'''Athena problem''' is an [[:w:List of unsolved problems in mathematics|unsolved problem]] in [[:w:Number theory|number theory]] and [[:w:Formal language theory|formal language theory]] and [[:w:Order theory|order theory]], this problem is named after the ancient Greek goddess [[:w:Athena|Athena]] (which is associated with [[:w:Wisdom|wisdom]]). Athena problem is: Give a [[:w:Natural number|natural number]] ''b'' &gt; 1, find the [[:w:Set (mathematics)|set]] of the [[:w:Minimal element|minimal element]]s of the set of the &quot;[[:w:Prime number|prime number]] [[:w:Greater than|&gt;]] ''b''&quot; [[:w:Numerical digit|digit]] [[:w:String (computer science)|string]]s in the [[:w:Positional numeral system|positional numeral system]] with [[:w:Radix|base]] ''b'' for the [[:w:Subsequence|subsequence]] [[:w:Partially ordered set|ordering]]. (A string ''x'' is a subsequence of another string ''y'', if ''x'' can be obtained from ''y'' by deleting zero or more of the [[:w:Character (computing)|character]]s in ''y''. For example, 514 is a subsequence of 352148, &quot;string&quot; is a subsequence of &quot;meistersinger&quot;. In contrast, 758 is not a subsequence of 378259, &quot;abc&quot; is not a subsequence of &quot;cbacacba&quot;, since the characters must be in the same order) (Unlike [[:w:Substring|substring]], subsequence is not required to occupy consecutive positions within the original sequences, e.g. the [[:w:Longest common subsequence|longest common subsequence problem]] is different from the [[:w:Longest common substring|longest common substring problem]])

Using [[:w:Formal language theory|formal language theory]] terminology, Athena problem is finding the [[:w:Set (mathematics)|set]] of the [[:w:Minimal element|minimal element]]s of the [[:w:Formal language|language]] of base-''b'' [[:w:Representation (mathematics)|representation]]s of the [[:w:Prime number|prime number]]s [[:w:Greater than|&gt;]] ''b'' (which is a set of [[:w:String (computer science)|string]]s of [[:w:Symbol|symbol]]s over the [[:w:Alphabet (formal languages)|alphabet]] ''Σ''&lt;sub&gt;''b''&lt;/sub&gt; := {0, 1, ..., ''b''−1}), under the subsequence ordering (i.e. the [[:w:Binary relation|binary relation]] &quot;is a subsequence of&quot;, which is a [[:w:Partially ordered set|partial ordering]]), for a given natural number ''b'' &gt; 1 (You can draw this partial ordering as a [[:w:Hasse diagram|Hasse diagram]] to find all [[:w:Minimal element|minimal element]]s), this set is called '''Athena set base ''b''''', and the prime numbers in this set is called '''Athena primes base ''b'''''.

By [[:w:Higman's lemma|Higman's lemma]], there are no [[:w:Infinite set|infinite]] [[:w:Antichain|antichain]]s for the subsequence ordering (i.e. the subsequence ordering is always a [[:w:Well-quasi-ordering|well quasi order]]) (i.e. under the subsequence ordering (i.e. the [[:w:Binary relation|binary relation]] &quot;is a subsequence of&quot;, which is a [[:w:Partially ordered set|partial ordering]]), every set of pairwise incomparable (i.e. not [[:w:Comparability|comparable]]) strings is finite), thus there must be only finitely many such minimal elements. In other words, the Athena set in every base ''b'' must be a [[:w:Finite set|finite set]], and every base ''b'' ≥ 2 has only finitely many Athena primes, e.g. in [[:w:Decimal|decimal]] (base ''b'' = 10), the Athena set has exactly 77 [[:w:Element of a set|element]]s (the Athena primes in decimal (base ''b'' = 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}.

For bases 2 ≤ ''b'' ≤ 36, Athena problem is fully solved in bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 24, and also solved in bases ''b'' = 11, 13, 16, 22, 30 if [[:w:Probable prime|probable prime]]s are allowed. For the unsolved bases ''b'' = 17, 19, 21, 23, 25, 26, 27, 28, 29, 31, 32, 34, 35, 36, Athena problem is solved (if probable primes are allowed) except 771 [[:w:Indexed family|families]] of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be [[:w:Empty string|empty]]) of digits in base ''b'', ''y'' is a digit in base ''b'') = sequence {''xz'', ''xyz'', ''xyyz'', ''xyyyz'', ''xyyyyz'', ''xyyyyyz'', ...} (i.e. &quot;''xy''&lt;sup&gt;+&lt;/sup&gt;''z''&quot; in [[:w:Regular expression|regular expression]]), all of these 771 families contain no primes &gt; ''b'' or probable primes &gt; ''b'' with length ≤ 100000.

== Solve the problem ==

To solve the Athena problem for a given base ''b'', we must [[:w:Computing|compute]] the elements up to families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''), and find the smallest prime &gt; ''b'' in all such families.

We call families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') &quot;linear&quot; families, and we reduce these families by removing all trailing digits ''y'' from ''x'', and removing all leading digits ''y'' from ''z'', to make the families be easier, e.g. family 12333{3}33345 in base ''b'' is reduced to family 12{3}45 in base ''b'', since they are in fact the same family. Our [[:w:Algorithm|algorithm]] then proceeds as follows:

* 1. ''M'' := {minimal primes in base ''b'' of length 2 or 3}, ''L'' := union of all ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'') such that ''x'' ≠ 0 and ''gcd''(''z'', ''b'') = 1 and ''Y'' is the set of digits ''y'' in base ''b'' such that ''xyz'' has no subsequence in ''M''.
* 2. While ''L'' contains nonlinear families (families which are not linear families): Explore each family of ''L'', and update ''L''. Examine each family of ''L'' by:
* 2.1. Let ''w'' be the shortest string in the family. If ''w'' has a subsequence in ''M'', then remove the family from ''L''. If ''w'' represents a prime, then add ''w'' to ''M'' and remove the family from ''L''.
* 2.2. If possible, simplify the family.
* 2.3. Using the techniques below (covering congruence, algebraic factorization, or combine of them), check if the family can be proven to only contain composites (only count the numbers &gt; ''b''), and if so then remove the family from ''L''.
* 3. Update ''L'', after each split examine the new families as in step 2.

e.g. in decimal (base ''b'' = 10):

''M'' := {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}

''L'' := {2{0,2}1, 2{0,8}7, 3{0,3,6,9}3, 3{0,3,6,9}9, 4{6}9, 5{0,5,8}1, 5{0,2}7, 6{0,3,6,9}3, 6{0,3,4,6,9}9, 7{0,7}7, 8{0,5}1, 8{0}7, 9{0,2,5,8}1, 9{0,3,6,9}3, 9{0,3,4,6,9}9}

and since 2221 is prime, it follows that the family 2{0,2}1 splits into the families 2{0}1 and 2{0}2{0}1

and since the family 2{0}1 can be proven to contain no primes &gt; base (since all numbers in this family are divisible by 3), it can be removed

and since 20201 is prime, it follows that the family 2{0}2{0}1 splits into the families 2{0}21 and 22{0}1

221 and 2021 are composites, but 20021 is prime, thus add 20021 to ''L''

none of 221, 2201, 22001, 220001, 2200001 are primes, but 22000001 is prime, thus add 22000001 to ''L''

and since the family 3{0,3,6,9}3 can be proven to contain no primes &gt; base (since all numbers in this family are divisible by 3), it can be removed

etc.

Since the number of possible (first digit,last digit) (also called (initial digit,final digit)) combos ([[:w:Ordered pair|ordered pair]]s) of a prime &gt; ''b'' in base ''b'' is (''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(''b'') (all digits except 0 can be the first digit of a prime &gt; ''b'' in base ''b'' (thus ''b''−1 possible digits), but only the digits coprime to ''b'' can be the last digit of a prime &gt; ''b'' in base ''b'' (thus ''eulerphi''(''b'') possible digits), and by the [[:w:Rule of product|rule of product]], there are (''b''−1)×''eulerphi''(''b'') choices of the (first digit,last digit) combo, also, both &quot;numbers of Athena primes in base ''b''&quot; and &quot;length of the largest Athena prime in base ''b''&quot; are [[:w:Asymptotic analysis|roughly]] ''[[:w:E (mathematical_constant)|e]]''&lt;sup&gt;''[[:w:Euler's constant|γ]]''×(''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(*b*)&lt;/sup&gt;.

Shrinking the family ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''Y'' is a set of digits in base ''b'')

* If ''y'' ∈ ''Y'' and the string ''xyyz'' represents a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or has a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''}''z'' ∪ ''x''{''Y'' \ ''y''}''y''{''Y'' \ ''y''}''z''.
* If ''y''&lt;sub&gt;1&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;2&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;1&lt;/sub&gt; ≠ ''y''&lt;sub&gt;2&lt;/sub&gt; and the string ''xy''&lt;sub&gt;1&lt;/sub&gt;''y''&lt;sub&gt;2&lt;/sub&gt;''z'' represents a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or has a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''&lt;sub&gt;1&lt;/sub&gt;}{''Y'' \ ''y''&lt;sub&gt;2&lt;/sub&gt;}''z''.
* If ''y''&lt;sub&gt;1&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;2&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;1&lt;/sub&gt; ≠ ''y''&lt;sub&gt;2&lt;/sub&gt; and both the strings ''xy''&lt;sub&gt;1&lt;/sub&gt;''y''&lt;sub&gt;2&lt;/sub&gt;''z'' and ''xy''&lt;sub&gt;2&lt;/sub&gt;''y''&lt;sub&gt;1&lt;/sub&gt;''z'' represent a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or have a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''&lt;sub&gt;1&lt;/sub&gt;}''z'' ∪ ''x''{''Y'' \ ''y''&lt;sub&gt;2&lt;/sub&gt;}''z''.

e.g. in decimal (base ''b'' = 10):

* 2221 is a prime &gt; 10, thus the family 2{0,2}1 splits into the two families 2{0}1 and 2{0}2{0}1.
* 227 is a prime &gt; 10, and it is a subsequence of 5227, thus the family 5{0,2}7 splits into the two families 5{0}7 and 5{0}2{0}7.
* 449 is a prime &gt; 10, and it is a subsequence of 6449, thus the family 6{0,3,4,6,9}9 splits into the two families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9.
* Both 5051 and 5501 are primes &gt; 10, thus the family 5{0,5}1 splits into the two families 5{0}1 and 5{5}1 = {5}1.
* 8501 is a prime &gt; 10, thus the family 8{0,5}1 splits into the family 8{0}{5}1.
* 887 is a prime &gt; 10, and it is a subsequence of 2887, also 2087 is a prime &gt; 10, thus the family 2{0,8}7 splits into the two families 2{0}7 and 28{0}7.
* 349 and 449 are primes &gt; 10, and they are subsequences of 9349 and 9449, respectively, also 9049, 9649, 9949 are primes &gt; 10, thus the family 9{0,3,4,6,9}9 splits into the two families 9{0,3,6,9}9 and 94{0,3,6,9}9.
* 251, 281, 521, 821, 881 are primes &gt; 10, and they are subsequences of 9251, 9281, 9521, 9821, 9881, respectively, also 9001, 9221, 9551, 9851 are primes &gt; 10, thus the family 9{0,2,5,8}1 splits into the numbers {91, 901, 921, 951, 981, 9021, 9051, 9081, 9201, 9501, 9581, 9801, 90581, 95081, 95801}.

If the methods we have discussed cannot be used to rule out or shrink ''x''{''Y''}''z'' where ''Y'' = {''y''&lt;sub&gt;1&lt;/sub&gt;, ''y''&lt;sub&gt;2&lt;/sub&gt;, ..., ''y''&lt;sub&gt;''n''&lt;/sub&gt;}, then we can replace ''x''{''Y''}''z'' by ''xy''&lt;sub&gt;1&lt;/sub&gt;{''Y''}''z'' ∪ ''xy''&lt;sub&gt;2&lt;/sub&gt;{''Y''}''z'' ∪ ... ∪ ''xy''&lt;sub&gt;''n''&lt;/sub&gt;{''Y''}''z'' and re-run the methods on this new [[:w:Formal language|language]].

If all remain families are linear families (i.e. of the form ''x''{''y''}''z'', where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''), then we search the smallest (probable) primes in these families and add these primes to the list.

e.g. in decimal (base ''b'' = 10):

* The smallest prime in the family 5{0}27 is 5000000000000000000000000000027.
* The smallest prime in the family {5}1 is 555555555551.
* The smallest prime in the family 8{5}1 is 8555555555555555555551, but 8555555555555555555551 is not a minimal element since 555555555551 is a subsequence of 8555555555555555555551.

There is no guarantee that the techniques discussed will ever terminate, but in practice they often do. They are able to determine the Athena set in base ''b'' for 2 ≤ ''b'' ≤ 16 and ''b'' = 18, 20, 22, 24, 30. The bases ''b'' = 17, 19, 21, 23, 25 ≤ ''b'' ≤ 29, 31 ≤ ''b'' ≤ 36 are solved with the exception of 771 families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'').

The following is a &quot;[[:w:Semi-algorithm|semi-algorithm]]&quot; that is guaranteed to solve the Athena problem for a given base ''b'', but it is not so easy to implement:

# ''M'' = ''[[:w:Empty string|∅]]''
# while (''L'' ≠ ''∅'') do
# choose ''x'', a shortest string in ''L''
# ''M'' := ''M'' ∪ {''x''}
# ''L'' := ''L'' − ''sup''({''x''})

In practice, for arbitrary ''L'', we cannot feasibly carry out step 5. Instead, we work with ''L''&amp;#39;, some regular overapproximation to ''L'', until we can show ''L''&amp;#39; = ''∅'' (which implies ''L'' = ''∅''). In practice, ''L''&amp;#39; is usually chosen to be a finite [[:w:Union (set theory)|union]] of sets of the form ''L''&lt;sub&gt;1&lt;/sub&gt;{''L''&lt;sub&gt;2&lt;/sub&gt;}''L''&lt;sub&gt;3&lt;/sub&gt;, where each of ''L''&lt;sub&gt;1&lt;/sub&gt;, ''L''&lt;sub&gt;2&lt;/sub&gt;, ''L''&lt;sub&gt;3&lt;/sub&gt; is finite. In the case we consider in this project, we then have to determine whether such a family contains a prime or not.

Thus, the [[:w:Time complexity|time complexity]] of the Athena problem in base ''b'' may be ''[[:w:Big O notation|O]]''(''[[:w:E (mathematical_constant)|e]]''&lt;sup&gt;''[[:w:Euler's constant|γ]]''×(''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(*b*)&lt;/sup&gt;), and the [[:w:CPU time|CPU time]] of the Athena problem in base ''b'' may be longer than [[:w:Age of the universe|the age of the universe]] for bases ''b'' = 19, 23, 25, 27, 29, 31, 32, 33, 34, 35, also, Athena problem in bases ''b'' around 500 may be [[:w:NP-complete|NP-complete]] or [[:w:NP-hard|NP-hard]], or an [[:w:Undecidable problem|undecidable problem]], or an example of [[:w:Gödel's incompleteness theorems|Gödel's incompleteness theorems]] (like the [[:w:Continuum hypothesis|continuum hypothesis]] and the [[:w:Halting problem|halting problem]]).

To solve the Athena problem (i.e. to [[:w:Computing|compute]] the Athena set), we need to determine whether a given family contains a prime. In practice, if family ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''Y'' is a set of digits in base ''b'') could not be ruled out as only containing composites and ''Y'' contains two or more digits, then a relatively small prime &gt; ''b'' could always be found in this family. Intuitively, this is because there are a large number of small strings in such a family, and at least one is likely to be prime (e.g. there are 2&lt;sup&gt;''n''−2&lt;/sup&gt; strings of length ''n'' in the family 1{3,7}9, and there are over a thousand strings of length 12 in the family 1{3,7}9, thus it is very impossible that these numbers are all composite). In the case ''Y'' contains only one digit, this family is of the form ''x''{''y''}''z'', and there is only a single string of each length &gt; (the length of ''x'' + the length of ''z''), and it is not known if the following [[:w:Decision problem|decision problem]] is recursively solvable (just like [[:w:Sierpiński number|Sierpiński problem]] and [[:w:Riesel number|Riesel problem]], Sierpiński problem and Riesel problem can be generalized to other bases ''b'' (references: http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm), in fact, Athena problem base ''b'' covers the Sierpiński problem base ''b'' and the Riesel problem base ''b'' with ''k'' &lt; ''b'', i.e. finding the smallest prime of the form ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 and ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (or prove such prime does not exist) with ''k'' &lt; ''b'' (specially, for bases ''b'' such that the conjectured smallest Sierpiński number or the conjectured smallest Riesel number is &lt; ''b'', Athena problem base ''b'' covers the Sierpiński problem base ''b'' or the Riesel problem base ''b'', respectively), since the smallest prime of the form ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 and ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (if exists) must be a minimal element in base ''b'', also, Athena problem base ''b'' covers finding the smallest prime of these forms in base ''b'' (or proving that such prime does not exist): (''b''&lt;sup&gt;''n''&lt;/sup&gt;−1)/(''b''−1) (for this form, ''n'' must be prime, and we want ''n'' ≥ 2) (references of this form: http://www.fermatquotient.com/PrimSerien/GenRepu.txt, https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html, http://www.primenumbers.net/Henri/us/MersFermus.htm, http://www.bitman.name/math/table/379, https://pzktupel.de/Primetables/TableRepunitGen.php, https://oeis.org/A084740, https://oeis.org/A084738, https://oeis.org/A128164, https://oeis.org/A285642; or for prime bases ''b'': https://oeis.org/A065854, https://oeis.org/A279068), ''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 (for this form, ''n'' must be power of 2, and we want ''n'' ≥ 1) (references of this form: http://jeppesn.dk/generalized-fermat.html, http://www.noprimeleftbehind.net/crus/GFN-primes.htm, https://web.archive.org/web/20231002190634/http://yves.gallot.pagesperso-orange.fr/primes/index.html, https://pzktupel.de/Primetables/TableFermatGFBB.php, https://oeis.org/A079706, https://oeis.org/A084712, https://oeis.org/A228101), (''b''&lt;sup&gt;''n''&lt;/sup&gt;+1)/2 (for odd ''b'') (for this form, ''n'' must be power of 2, and we want ''n'' ≥ 2) (reference of this form: http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt), (''sqrt''(''b'')×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1)/(''sqrt''(''b'')+1) (for square ''b'') (for this form, 2×''n''+1 must be prime, and we want ''n'' ≥ 2) (references of this form: http://www.fermatquotient.com/PrimSerien/GenRepuP.txt, http://www.primenumbers.net/Henri/us/MersFermus.htm, http://www.bitman.name/math/table/488, https://pzktupel.de/Primetables/TableWagstaffGen.php, https://oeis.org/A084742, https://oeis.org/A084741; or for bases ''b'' with ''sqrt''(''b'') prime: https://oeis.org/A065507), ((''b''−2)×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1)/(''b''−1) (''n'' ≥ 2) (reference of this form: https://oeis.org/A243404), 2×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 (''n'' ≥ 1) (references of this form: https://www.mersenneforum.org/showthread.php?t=6918, https://www.mersenneforum.org/showthread.php?t=19725, https://oeis.org/A119624), 2×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (''n'' ≥ 1) (references of this form: https://www.mersenneforum.org/showthread.php?t=24576, https://www.mersenneforum.org/attachment.php?attachmentid=20976&amp;d=1567314217, https://oeis.org/A119591), ''b''&lt;sup&gt;''n''&lt;/sup&gt;+2 (''n'' ≥ 1) (references of this form: https://oeis.org/A138066, https://oeis.org/A084713, https://oeis.org/A138067), ''b''&lt;sup&gt;''n''&lt;/sup&gt;−2 (''n'' ≥ 2) (references of this form: https://www.primepuzzles.net/puzzles/puzz_887.htm, https://oeis.org/A250200, https://oeis.org/A255707, https://oeis.org/A084714; or for prime bases ''b'': https://oeis.org/A292201), (''b''−1)×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 (''n'' ≥ 1) (references of this form: http://www.noprimeleftbehind.net/Williams-primes-MP.htm, http://www.bitman.name/math/table/477, https://pzktupel.de/Primetables/TableWilliams2.php, https://oeis.org/A305531; or for prime bases ''b'': https://oeis.org/A087139), (''b''−1)×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (''n'' ≥ 1) (references of this form: https://harvey563.tripod.com/wills.txt, http://www.noprimeleftbehind.net/Williams-primes-MM.htm, http://www.bitman.name/math/table/484, https://pzktupel.de/Primetables/TableWilliams1.php; or for prime bases ''b'': https://oeis.org/A122396), ''b''&lt;sup&gt;''n''&lt;/sup&gt;+(''b''−1) (''n'' ≥ 1) (references of this form: http://www.bitman.name/math/table/795, https://pzktupel.de/Primetables/TableWilliams6.php, https://oeis.org/A076845, https://oeis.org/A076846, https://oeis.org/A078178, https://oeis.org/A078179), ''b''&lt;sup&gt;''n''&lt;/sup&gt;−(''b''−1) (''n'' ≥ 2) (references of this form: http://www.bitman.name/math/table/792, https://pzktupel.de/Primetables/TableWilliams5.php, https://oeis.org/A113516, https://oeis.org/A343589; or for prime bases ''b'': https://cs.uwaterloo.ca/journals/JIS/VOL3/mccranie.html, http://www.bitman.name/math/table/435)):

Problem: Given strings ''x'', ''z'' (may be empty), a digit ''y'', and a base ''b'' (''x'' does not [[:w:Leading zero|start with the digit 0]], ''z'' ends with a digit which [[:w:Coprime integers|coprime]] to ''b'', ''y'' is not 0 if ''x'' is empty, ''y'' is coprime to ''b'' if ''z'' is empty), does there exist a prime number whose base-''b'' expansion is of the form ''xy''&lt;sub&gt;''n''&lt;/sub&gt;''z'' for some ''n'' ≥ 0?

Some families can be ruled out to contain no prime &gt; ''b'' by [[:w:Covering set|covering congruence]], [[:w:Factorization of polynomials|algebraic factorization]] (e.g. [[:w:Difference of two squares|difference of two squares]], [[:w:Sum of two cubes|sum of two cubes]], [[:w:Sophie Germain's identity|Sophie Germain's identity of ''x''&lt;sup&gt;4&lt;/sup&gt;+4×''y''&lt;sup&gt;4&lt;/sup&gt;]]), or combine of them, e.g.

* The base 9 family 2{7}: Always divisible by 2 or 5
* The base 11 family 2{5}: Always divisible by 2 or 3
* The base 14 family B{0}1: Always divisible by 3 or 5
* The base 13 family 95{0}3: Always divisible by 5, 7, or 17
* The base 16 family {4}D: Always divisible by 3, 7, or 13
* The base 16 family {8}F: Always divisible by 3, 7, or 13
* The base 21 family {7}D: Always divisible by 2, 13, or 17
* The base 23 family {D}GA: Always divisible by 2, 5, 7, 37, or 79
* The base 9 family {1}: Can be written as (9&lt;sup&gt;''n''&lt;/sup&gt;−1)/8 and can be factored as (3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (3&lt;sup&gt;''n''&lt;/sup&gt;+1) / 8
* The base 8 family 1{0}1: Can be written as 8&lt;sup&gt;''n''&lt;/sup&gt;+1 and can be factored as (2&lt;sup&gt;''n''&lt;/sup&gt;+1) × (4&lt;sup&gt;''n''&lt;/sup&gt;−2&lt;sup&gt;''n''&lt;/sup&gt;+1)
* The base 9 family 3{8}: Can be written as 4×9&lt;sup&gt;''n''&lt;/sup&gt;−1 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (2×3&lt;sup&gt;''n''&lt;/sup&gt;+1)
* The base 16 family 1{5}: Can be written as (4×16&lt;sup&gt;''n''&lt;/sup&gt;−1)/3 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (2×3&lt;sup&gt;''n''&lt;/sup&gt;+1) / 3
* The base 16 family {4}1: Can be written as (4×16&lt;sup&gt;''n''&lt;/sup&gt;−49)/15 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−7) × (2×3&lt;sup&gt;''n''&lt;/sup&gt;+7) / 15
* The base 27 family 7{Q}: Can be written as 8×27&lt;sup&gt;''n''&lt;/sup&gt;−1 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (4×9&lt;sup&gt;''n''&lt;/sup&gt;+2×3&lt;sup&gt;''n''&lt;/sup&gt;+1)
* The base 27 family 9{G}: Can be written as (125×27&lt;sup&gt;''n''&lt;/sup&gt;−8)/13 and can be factored as (5×3&lt;sup&gt;''n''&lt;/sup&gt;−2) × (25×9&lt;sup&gt;''n''&lt;/sup&gt;+10×3&lt;sup&gt;''n''&lt;/sup&gt;+4)
* The base 16 family {C}D: Can be written as (4×16&lt;sup&gt;''n''&lt;/sup&gt;+1)/5 and can be factored as (2×4&lt;sup&gt;''n''&lt;/sup&gt;−2×2&lt;sup&gt;''n''&lt;/sup&gt;+1) × (2×4&lt;sup&gt;''n''&lt;/sup&gt;+2×2&lt;sup&gt;''n''&lt;/sup&gt;+1) / 5
* The base 14 family 8{D}: Can be written as 9×14&lt;sup&gt;''n''&lt;/sup&gt;−1, it is divisible by 5 if ''n'' is odd and can be factored as (3×14&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (3×14&lt;sup&gt;''n''/2&lt;/sup&gt;+1) if ''n'' is even
* The base 12 family {B}9B: Can be written as 12&lt;sup&gt;''n''&lt;/sup&gt;−25, it is divisible by 13 if ''n'' is odd and can be factored as (12&lt;sup&gt;''n''/2&lt;/sup&gt;−5) × (12&lt;sup&gt;''n''/2&lt;/sup&gt;+5) if ''n'' is even
* The base 14 family {D}5: Can be written as 14&lt;sup&gt;''n''&lt;/sup&gt;−9, it is divisible by 5 if ''n'' is odd and can be factored as (14&lt;sup&gt;''n''/2&lt;/sup&gt;−3) × (14&lt;sup&gt;''n''/2&lt;/sup&gt;+3) if ''n'' is even
* The base 17 family 1{9}: Can be written as (25×17&lt;sup&gt;''n''&lt;/sup&gt;−9)/16, it is divisible by 2 if ''n'' is odd and can be factored as (5×17&lt;sup&gt;''n''/2&lt;/sup&gt;−3) × (5×17&lt;sup&gt;''n''/2&lt;/sup&gt;+3) / 16 if ''n'' is even
* The base 17 family 7{9}: Can be written as (121×17&lt;sup&gt;''n''&lt;/sup&gt;−9)/16, it is divisible by 2 if ''n'' is odd and can be factored as (11×17&lt;sup&gt;''n''/2&lt;/sup&gt;−3) × (11×17&lt;sup&gt;''n''/2&lt;/sup&gt;+3) / 16 if ''n'' is even
* The base 19 family 1{6}: Can be written as (4×19&lt;sup&gt;''n''&lt;/sup&gt;−1)/3, it is divisible by 5 if ''n'' is odd and can be factored as (2×19&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (2×19&lt;sup&gt;''n''/2&lt;/sup&gt;+1) / 3 if ''n'' is even
* The base 19 family 7{2}: Can be written as (64×19&lt;sup&gt;''n''&lt;/sup&gt;−1)/9, it is divisible by 5 if ''n'' is odd and can be factored as (8×19&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (8×19&lt;sup&gt;''n''/2&lt;/sup&gt;+1) / 9 if ''n'' is even
* The base 24 family 3{N}: Can be written as 4×24&lt;sup&gt;''n''&lt;/sup&gt;−1, it is divisible by 5 if ''n'' is odd and can be factored as (2×24&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (2×24&lt;sup&gt;''n''/2&lt;/sup&gt;+1) if ''n'' is even

By the [[:w:Prime number theorem|prime number theorem]], the [[:w:Probability|chance]] that a [[:w:Random number|random]] ''n''-digit base ''b'' number is prime is [[:w:Asymptotic analysis|approximately]] 1/''n'' (more accurately, the chance is approximately 1/(''n''×''ln''(''b'')), where ''ln'' is the [[:w:Natural logarithm|natural logarithm]]). If one conjectures the numbers ''x''{''y''}''z'' behave similarly (i.e. the numbers ''x''{''y''}''z'' is a [[:w:Pseudorandomness|pseudorandom sequence]]) you would expect [[:w:Harmonic_series (mathematics)|1/1 + 1/2 + 1/3 + 1/4 + ... = ∞]] primes of the form ''x''{''y''}''z'' (of course, this does not always happen, since some ''x''{''y''}''z'' families can be ruled out to contain no prime &gt; ''b'' (by covering congruence, algebraic factorization, or combine of them), but it is at least a reasonable conjecture in the absence of evidence to the contrary. Hence, the [[:w:Heuristic argument|heuristic argument]] suggests there are always infinitely many primes in family ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') if it cannot be ruled out to contain no prime or only contain finitely many primes, by covering congruence, algebraic factorization, or combine of them. However, some families ''x''{''y''}''z'' could not be proven to contain no primes &gt; ''b'' (by covering congruence, algebraic factorization, or combine of them) but no primes &gt; ''b'' could be found in the family, even after searching through numbers with over 100000 digits. In such a case, the only way to proceed is to [[:w:Primality test|test the primality]] of larger and larger numbers of such form and hope a prime is eventually discovered. e.g. the smallest (probable) prime in the family A{3}A in base ''b'' = 13 is A3&lt;sub&gt;592197&lt;/sub&gt;A, its algebraic form is (41×13&lt;sup&gt;592198&lt;/sup&gt;+27)/4, when written in decimal contains 659677 digits (it is only probable prime, i.e. not definitely prime, since technically, probable primality tests were used to show this (which have a ''very'' small chance of making an error, see https://t5k.org/notes/prp_prob.html) because all known primality tests run far too slowly to run on numbers of this size unless either [https://t5k.org/prove/prove3_1.html ''N''−1] or [https://t5k.org/prove/prove3_2.html ''N''+1] (or both) can be ≥ 1/3 factored).

The numbers in family ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') are of the form (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1) for some fixed ''a'', ''b'', ''c'' such that ''a'' ≥ 1, ''b'' ≥ 2 (''b'' is the base), ''c'' ≠ 0, ''gcd''(''a'',''c'') = 1, ''gcd''(''b'',''c'') = 1. Except in the [[:w:Special case|special case]] ''c'' = ±1 and ''gcd''(''a''+''c'',''b''−1) = 1, when ''n'' is large the known [[:w:Primality test|primality test]]s for such a number are too inefficient to run. In this case one must resort to a [[:w:Probabilistic algorithm|probable]] primality test such as a [[:w:Miller–Rabin primality test|Miller–Rabin primality test]] or a [[:w:Baillie–PSW primality test|Baillie–PSW primality test]], unless a divisor of the number can be found. Since we are testing many numbers in an [[:w:Exponential growth|exponential sequence]], it is possible to use a sieving process to find divisors rather than using [[:w:Trial division|trial division]].

To do this, we made use of Geoffrey Reynolds' ''srsieve'' software (download: https://pzktupel.de/Software/srsieve_1.1.4.7z). This program uses the [[:w:Baby-step giant-step|baby-step giant-step]] [[:w:Algorithm|algorithm]] to find all primes ''p'' which divide ''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'' where ''p'' and ''n'' lie in a [[:w:Interval_(mathematics)|specified range]]. Since this program cannot handle the general case (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1) when ''gcd''(''a''+''c'',''b''−1) &gt; 1 we only used it to sieve the sequence ''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'' for primes ''p'' not dividing ''gcd''(''a''+''c'',''b''−1), and initialized the list of candidates to not include ''n'' for which there is some prime ''p'' dividing ''gcd''(''a''+''c'',''b''−1) for which ''p'' dividing (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1). The program had to be modified slightly to remove a check which would prevent it from running in the case when ''a'', ''b'', and ''c'' were all odd (since then 2 divides ''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'', but 2 may not divide (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1)).

Once the numbers with small divisors had been removed, it remained to test the remaining numbers using a probable primality test. For this we used the software ''LLR'' by Jean Penné. (download: http://jpenne.free.fr/index2.html). Although undocumented, it is possible to run this program on numbers of the form (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1) when ''gcd''(''a''+''c'',''b''−1) &gt; 1, so this program required no modifications. A script was also written which allowed one to run ''srsieve'' while ''LLR'' was testing the remaining candidates, so that when a divisor was found by srsieve on a number which had not yet been tested by ''LLR'' it would be removed from the list of candidates.

For the primes &lt; 10&lt;sup&gt;25000&lt;/sup&gt; for the &quot;easy&quot; bases (bases ''b'' with ≤ 150 primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; (base ''b'' = 26 has 83 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 3 unsolved families, base ''b'' = 36 has 75 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 4 unsolved families, base ''b'' = 17 has 99 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 18 unsolved families, base ''b'' = 21 has 80 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 12 unsolved families, base ''b'' = 19 has 201 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 23 unsolved families), i.e. bases *b* = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 26, 28, 30, 36), we employed ''CM'' by Andreas Enge (download: https://www.multiprecision.org/cm/download.html), an elliptic curve primality proving implementation.

== Data ==

These are the results of the Athena problem in bases 2 ≤ ''b'' ≤ 36 (we stop at base 36 since this base is the maximum base for which it is possible to write the numbers with the [[:w:Symbol|symbol]]s 0, 1, 2, ..., 9 and A, B, C, ..., Z (i.e. the 10 [[:w:Arabic numerals|Arabic numerals]] and the 26 [[:w:Latin script|Latin letters]]): (some large primes are only probable primes, i.e. not definitely primes, since they are too large to be [[:w:Elliptic curve primality|ECPP proved]] and [[:w:Pocklington primality test#Extensions and variants|neither ''N''−1 nor ''N''+1 can be ≥ 1/3 factored]], all of them pass the [[:w:Baillie–PSW primality test|Baillie–PSW primality test]] and the [[:w:Strong pseudoprime|strong primality test]] (i.e. the [[:w:Miller–Rabin primality test|Miller–Rabin primality test]]) with all prime bases ''p'' ≤ 61, however, all primes &lt; 10&lt;sup&gt;25000&lt;/sup&gt; for bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 36 are definitely primes, most of them &gt; 10&lt;sup&gt;299&lt;/sup&gt; are proven primes with [[:w:Elliptic curve primality|ECPP proving]], others &gt; 10&lt;sup&gt;299&lt;/sup&gt; are proven primes with [[:w:Pocklington primality test#Extensions and variants|''N''−1 or ''N''+1 proving]])

All numbers are written in base ''b'', [[:w:Senary#Base 36 as senary compression|using A to Z to represent digit values 10 to 35]], &quot;{}&quot; means repeating, e.g. family 12{3}45 means the sequence {1245, 12345, 123345, 1233345, 12333345, 123333345, ...} (where the members are expressed as base ''b'' strings), subscripts are used to indicate repetitions of digits, e.g. 123&lt;sub&gt;4&lt;/sub&gt;567 means 123333567 (all subscripts are written in decimal).

Base 2: 1 Athena prime (the largest of which has 2 digits (it is 11, and its value is 3 in decimal)): {11}

Base 3: 3 Athena primes (the largest of which has 3 digits (it is 111, and its value is 13 in decimal)): {12, 21, 111}

Base 4: 5 Athena primes (the largest of which has 3 digits (it is 221, and its value is 41 in decimal)): {11, 13, 23, 31, 221}

Base 5: 22 Athena primes (the largest of which has 96 digits (it is 10&lt;sub&gt;93&lt;/sub&gt;13, and its algebraic form is 5&lt;sup&gt;95&lt;/sup&gt;+8)): {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 Athena primes (the largest of which has 5 digits (it is 40041, and its value is 5209 in decimal)): {11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041}

Base 7: 71 Athena primes (the largest of which has 17 digits (it is 3&lt;sub&gt;16&lt;/sub&gt;1, and its algebraic form is (7&lt;sup&gt;17&lt;/sup&gt;−5)/2)): {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: 75 Athena primes (the largest of which has 221 digits (it is 4&lt;sub&gt;220&lt;/sub&gt;7, and its algebraic form is (4×8&lt;sup&gt;221&lt;/sup&gt;+17)/7)): {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: 151 Athena primes (the largest of which has 1161 digits (it is 30&lt;sub&gt;1158&lt;/sub&gt;11, and its algebraic form is 3×9&lt;sup&gt;1160&lt;/sup&gt;+10)): {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: 77 Athena primes (the largest of which has 31 digits (it is 50&lt;sub&gt;28&lt;/sub&gt;27, and its algebraic form is 5×10&lt;sup&gt;30&lt;/sup&gt;+27)): {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: 1068 Athena primes (including 1 unproven probable prime: 57&lt;sub&gt;62668&lt;/sub&gt;), the largest of which has 62669 digits (it is 57&lt;sub&gt;62668&lt;/sub&gt;, and its algebraic form is (57×11&lt;sup&gt;62668&lt;/sup&gt;−7)/10), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel11 Data of Athena (probable) primes base 11]

Base 12: 106 Athena primes (the largest of which has 42 digits (it is 40&lt;sub&gt;39&lt;/sub&gt;77, and its algebraic form is 4×12&lt;sup&gt;41&lt;/sup&gt;+91)): {11, 15, 17, 1B, 25, 27, 31, 35, 37, 3B, 45, 4B, 51, 57, 5B, 61, 67, 6B, 75, 81, 85, 87, 8B, 91, 95, A7, AB, B5, B7, 221, 241, 2A1, 2B1, 2BB, 401, 421, 447, 471, 497, 565, 655, 665, 701, 70B, 721, 747, 771, 77B, 797, 7A1, 7BB, 907, 90B, 9BB, A41, B21, B2B, 2001, 200B, 202B, 222B, 229B, 292B, 299B, 4441, 4707, 4777, 6A05, 6AA5, 729B, 7441, 7B41, 929B, 9777, 992B, 9947, 997B, 9997, A0A1, A201, A605, A6A5, AA65, B001, B0B1, BB01, BB41, 600A5, 7999B, 9999B, AAAA1, B04A1, B0B9B, BAA01, BAAA1, BB09B, BBBB1, 44AAA1, A00065, BBBAA1, AAA0001, B00099B, AA000001, BBBBBB99B, B0000000000000000000000000009B, 400000000000000000000000000000000000000077}

Base 13: 3197 Athena primes (including 4 unproven probable primes: C5&lt;sub&gt;23755&lt;/sub&gt;C, 80&lt;sub&gt;32017&lt;/sub&gt;111, 95&lt;sub&gt;197420&lt;/sub&gt;, A3&lt;sub&gt;592197&lt;/sub&gt;A), the largest of which has 592199 digits (it is A3&lt;sub&gt;592197&lt;/sub&gt;A, and its algebraic form is (41×13&lt;sup&gt;592198&lt;/sup&gt;+27)/4), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel13 Data of Athena (probable) primes base 13]

Base 14: 650 Athena primes, the largest of which has 19699 digits (it is 4D&lt;sub&gt;19698&lt;/sub&gt;, and its algebraic form is 5×14&lt;sup&gt;19698&lt;/sup&gt;−1), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel14 Data of Athena primes base 14]

Base 15: 1284 Athena primes, the largest of which has 157 digits (it is 7&lt;sub&gt;155&lt;/sub&gt;97, and its algebraic form is (15&lt;sup&gt;157&lt;/sup&gt;+59)/2), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel15 Data of Athena primes base 15]

Base 16: 2347 Athena primes (including 3 unproven probable primes: DB&lt;sub&gt;32234&lt;/sub&gt;, 4&lt;sub&gt;72785&lt;/sub&gt;DD, 3&lt;sub&gt;116137&lt;/sub&gt;AF), the largest of which has 116139 digits (it is 3&lt;sub&gt;116137&lt;/sub&gt;AF, and its algebraic form is (16&lt;sup&gt;116139&lt;/sup&gt;+619)/5), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel16 Data of Athena (probable) primes base 16]

Base 17: 10415 known Athena primes (including many unproven probable primes) and 12 unsolved families (1{7}, 1F{0}7, 4{7}A, 70F{0}D, 8{B}9, 9{5}9, A{D}F, B{0}B3, {B}E9, {B}EE, F1{9}, FD0{D}, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel17 Data of known Athena (probable) primes base 17]

Base 18: 549 Athena primes, the largest of which has 6271 digits (it is C0&lt;sub&gt;6268&lt;/sub&gt;C5, and its algebraic form is 12×18&lt;sup&gt;6270&lt;/sup&gt;+221), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel18 Data of Athena primes base 18]

Base 19: 31417 known Athena primes (including many unproven probable primes) and 17 unsolved families (4B5{0}H, {5}3, 5{H}05, 5{H}0H, 5{H}5, 66{B}, 71{0}177, 7AF{0}H, 97{0}3, C{H}C, EE1{6}, F{7}5, F{B}G, F{D}F, H0F{0}7A, HB{0}5B5, II{D}, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel19 Data of known Athena (probable) primes base 19]

Base 20: 3314 Athena primes, the largest of which has 6271 digits (it is G0&lt;sub&gt;6269&lt;/sub&gt;D, and its algebraic form is 16×20&lt;sup&gt;6270&lt;/sup&gt;+13), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel20 Data of Athena primes base 20]

Base 21: 13386 known Athena primes (including many unproven probable primes) and 8 unsolved families (5{0}DJ, {9}D, B3{0}EB, B{H}6H, C{F}0K, {F}35, G{0}FK, H{0}7771, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel21 Data of known Athena (probable) primes base 21]

Base 22: 8003 Athena primes (including 1 unproven probable prime: BK&lt;sub&gt;22001&lt;/sub&gt;5), the largest of which has 22003 digits (it is BK&lt;sub&gt;22001&lt;/sub&gt;5, and its algebraic form is (251×22&lt;sup&gt;22002&lt;/sup&gt;−335)/21), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel22 Data of Athena (probable) primes base 22]

Base 23: 65178 known Athena primes (including many unproven probable primes) and 87 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel23 Data of known Athena (probable) primes base 23] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left23 Data of unsolved families for Athena problem base 23]

Base 24: 3409 Athena primes, the largest of which has 8134 digits (it is N00N&lt;sub&gt;8129&lt;/sub&gt;LN, and its algebraic form is 13249×24&lt;sup&gt;8131&lt;/sup&gt;−49), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel24 Data of Athena primes base 24]

Base 25: 133639 known Athena primes (including many unproven probable primes) and 85 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel25 Data of known Athena (probable) primes base 25] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left25 Data of unsolved families for Athena problem base 25]

Base 26: 25256 known Athena primes (including 7 unproven probable primes: 5&lt;sub&gt;19391&lt;/sub&gt;6F, 7&lt;sub&gt;20279&lt;/sub&gt;OL, LD0&lt;sub&gt;20975&lt;/sub&gt;7, 6K&lt;sub&gt;23300&lt;/sub&gt;5, J0&lt;sub&gt;44303&lt;/sub&gt;KCB, M0&lt;sub&gt;61186&lt;/sub&gt;2BB, 85M&lt;sub&gt;197060&lt;/sub&gt;B) and 3 unsolved families ({A}6F, {H}MH, {I}GL, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel26 Data of known Athena (probable) primes base 26]

Base 27: 102852 known Athena primes (including many unproven probable primes) and 44 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel27 Data of known Athena (probable) primes base 27] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left27 Data of unsolved families for Athena problem base 27]

Base 28: 25528 known Athena primes (including 3 unproven probable primes: N6&lt;sub&gt;24051&lt;/sub&gt;LR, 5OA&lt;sub&gt;31238&lt;/sub&gt;F, O4O&lt;sub&gt;94535&lt;/sub&gt;9) and 1 unsolved family (O{A}F, no primes or probable primes with length ≤ 900000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel28 Data of known Athena (probable) primes base 28]

Base 29: 355242 known Athena primes (including many unproven probable primes) and 125 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel29 Data of known Athena (probable) primes base 29] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left29 Data of unsolved families for Athena problem base 29]

Base 30: 2619 Athena primes (including 1 unproven probable prime: I0&lt;sub&gt;24608&lt;/sub&gt;D), the largest of which has 34206 digits (it is OT&lt;sub&gt;34205&lt;/sub&gt;, and its algebraic form is 25×30&lt;sup&gt;34205&lt;/sup&gt;−1), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel30 Data of Athena (probable) primes base 30]

Base 31: 569323 known Athena primes (including many unproven probable primes) and 77 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel31 Data of known Athena (probable) primes base 31] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left31 Data of unsolved families for Athena problem base 31]

Base 32: 168882 known Athena primes (including many unproven probable primes) and 120 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel32 Data of known Athena (probable) primes base 32] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left32 Data of unsolved families for Athena problem base 32]

Base 33: 280012 known Athena primes (including many unproven probable primes) and 81 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel33 Data of known Athena (probable) primes base 33] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left33 Data of unsolved families for Athena problem base 33]

Base 34: 184785 known Athena primes (including many unproven probable primes) and 47 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel34 Data of known Athena (probable) primes base 34] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left34 Data of unsolved families for Athena problem base 34]

Base 35: 720002 known Athena primes (including many unproven probable primes) and 60 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel35 Data of known Athena (probable) primes base 35] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left35 Data of unsolved families for Athena problem base 35]

Base 36: 35286 known Athena primes (including 3 unproven probable primes: 7K&lt;sub&gt;26567&lt;/sub&gt;Z, S0&lt;sub&gt;75007&lt;/sub&gt;8H, P&lt;sub&gt;81993&lt;/sub&gt;SZ) and 4 unsolved families (B{0}EUV, HM{0}N, N{0}YYN, O{L}Z, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel36 Data of known Athena (probable) primes base 36]

== Condensed table for bases 2 ≤ ''b'' ≤ 36 ==

{|class=&quot;wikitable&quot;
||''b''||number of Athena primes (or probable primes, which are Athena primes assuming their primality) base ''b''||base-''b'' form of the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) base ''b'' (write &quot;''d''&lt;sub&gt;''n''&lt;/sub&gt;&quot; if there are 5 or more (''n'') consecutive same digits ''d'')||length of the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) base ''b''||length of the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) base ''b'' in decimal||algebraic ((''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1)) form of the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) base ''b''||''factordb'' entry of the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) base ''b''||the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) base ''b'' written in base ''b'' (use lower case letters instead of upper case letters)||number of unsolved families in the Athena problem base ''b'' (all of these left families are linear families)||searching limit of length for the unsolved families in the Athena problem base ''b'' (if there are different searching limits for the unsolved families in base ''b'', choose the lowest searching limit)||
|-
||2||1||11||2||1||3||http://factordb.com/index.php?id=3&amp;open=ecm||http://factordb.com/index.php?showid=3&amp;base=2||0||–||
|-
||3||3||111&lt;br&gt;21&lt;br&gt;12||3&lt;br&gt;2&lt;br&gt;2||2&lt;br&gt;1&lt;br&gt;1||13&lt;br&gt;7&lt;br&gt;5||http://factordb.com/index.php?id=13&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=7&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=5&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=13&amp;base=3&lt;br&gt;http://factordb.com/index.php?showid=7&amp;base=3&lt;br&gt;http://factordb.com/index.php?showid=5&amp;base=3&lt;nowiki/&gt;||0||–||
|-
||4||5||221&lt;br&gt;31&lt;br&gt;23&lt;br&gt;13&lt;br&gt;11||3&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2||2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;1&lt;br&gt;1||41&lt;br&gt;13&lt;br&gt;11&lt;br&gt;7&lt;br&gt;5||http://factordb.com/index.php?id=41&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=13&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=11&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=7&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=5&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=41&amp;base=4&lt;br&gt;http://factordb.com/index.php?showid=13&amp;base=4&lt;br&gt;http://factordb.com/index.php?showid=11&amp;base=4&lt;br&gt;http://factordb.com/index.php?showid=7&amp;base=4&lt;br&gt;http://factordb.com/index.php?showid=5&amp;base=4&lt;nowiki/&gt;||0||–||
|-
||5||22||10&lt;sub&gt;93&lt;/sub&gt;13&lt;br&gt;300031&lt;br&gt;44441&lt;br&gt;33331&lt;br&gt;33001&lt;br&gt;30301&lt;br&gt;14444&lt;br&gt;10103&lt;br&gt;3101&lt;br&gt;414||96&lt;br&gt;6&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;4&lt;br&gt;3||67&lt;br&gt;4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;3&lt;br&gt;3&lt;br&gt;3||5&lt;sup&gt;95&lt;/sup&gt;+8&lt;br&gt;9391&lt;br&gt;3121&lt;br&gt;2341&lt;br&gt;2251&lt;br&gt;1951&lt;br&gt;1249&lt;br&gt;653&lt;br&gt;401&lt;br&gt;109||http://factordb.com/index.php?id=1100000000034686071&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=9391&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=3121&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=2341&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=2251&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1951&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1249&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=653&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=401&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=109&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000000034686071&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=9391&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=3121&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=2341&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=2251&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=1951&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=1249&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=653&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=401&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=109&amp;base=5&lt;nowiki/&gt;||0||–||
|-
||6||11||40041&lt;br&gt;4441&lt;br&gt;4401&lt;br&gt;51&lt;br&gt;45&lt;br&gt;35&lt;br&gt;31&lt;br&gt;25&lt;br&gt;21&lt;br&gt;15||5&lt;br&gt;4&lt;br&gt;4&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2||4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2||5209&lt;br&gt;1033&lt;br&gt;1009&lt;br&gt;31&lt;br&gt;29&lt;br&gt;23&lt;br&gt;19&lt;br&gt;17&lt;br&gt;13&lt;br&gt;11||http://factordb.com/index.php?id=5209&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1033&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1009&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=31&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=29&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=23&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=19&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=17&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=13&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=11&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=5209&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=1033&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=1009&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=31&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=29&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=23&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=19&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=17&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=13&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=11&amp;base=6&lt;nowiki/&gt;||0||–||
|-
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||33||280012~280093||DP&lt;sub&gt;95093&lt;/sub&gt;M5&lt;br&gt;HJ0&lt;sub&gt;94295&lt;/sub&gt;J&lt;br&gt;90&lt;sub&gt;93597&lt;/sub&gt;Q&lt;br&gt;9F0&lt;sub&gt;93157&lt;/sub&gt;N&lt;br&gt;7&lt;sub&gt;89449&lt;/sub&gt;333H&lt;br&gt;K3&lt;sub&gt;80751&lt;/sub&gt;6K&lt;br&gt;D&lt;sub&gt;80107&lt;/sub&gt;9UD&lt;br&gt;VFU&lt;sub&gt;72204&lt;/sub&gt;FK&lt;br&gt;J&lt;sub&gt;68715&lt;/sub&gt;2BJ&lt;br&gt;DF0&lt;sub&gt;68367&lt;/sub&gt;J||95096&lt;br&gt;94298&lt;br&gt;93599&lt;br&gt;93160&lt;br&gt;89453&lt;br&gt;80754&lt;br&gt;80110&lt;br&gt;72208&lt;br&gt;68718&lt;br&gt;68370||144405&lt;br&gt;143193&lt;br&gt;142131&lt;br&gt;141465&lt;br&gt;135835&lt;br&gt;122626&lt;br&gt;121648&lt;br&gt;109649&lt;br&gt;104350&lt;br&gt;103821||(441×33&lt;sup&gt;95095&lt;/sup&gt;−3833)/32&lt;br&gt;580×33&lt;sup&gt;94296&lt;/sup&gt;+19&lt;br&gt;9×33&lt;sup&gt;93598&lt;/sup&gt;+26&lt;br&gt;312×33&lt;sup&gt;93158&lt;/sup&gt;+23&lt;br&gt;(7×33&lt;sup&gt;89453&lt;/sup&gt;−4743239)/32&lt;br&gt;(643×33&lt;sup&gt;80753&lt;/sup&gt;+3709)/32&lt;br&gt;(13×33&lt;sup&gt;80110&lt;/sup&gt;−121453)/32&lt;br&gt;(16623×33&lt;sup&gt;72206&lt;/sup&gt;−8095)/16&lt;br&gt;(19×33&lt;sup&gt;68718&lt;/sup&gt;−600883)/32&lt;br&gt;444×33&lt;sup&gt;68368&lt;/sup&gt;+19||http://factordb.com/index.php?id=1100000005652348775&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000005652362811&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000005652375073&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000005652389776&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000005652430746&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000005652446200&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000005652461592&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004614764298&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004614770536&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004614784274&amp;open=prime&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000005652348775&amp;base=33&lt;br&gt;http://factordb.com/index.php?showid=1100000005652362811&amp;base=33&lt;br&gt;http://factordb.com/index.php?showid=1100000005652375073&amp;base=33&lt;br&gt;http://factordb.com/index.php?showid=1100000005652389776&amp;base=33&lt;br&gt;http://factordb.com/index.php?showid=1100000005652430746&amp;base=33&lt;br&gt;http://factordb.com/index.php?showid=1100000005652446200&amp;base=33&lt;br&gt;http://factordb.com/index.php?showid=1100000005652461592&amp;base=33&lt;br&gt;http://factordb.com/index.php?showid=1100000004614764298&amp;base=33&lt;br&gt;http://factordb.com/index.php?showid=1100000004614770536&amp;base=33&lt;br&gt;http://factordb.com/index.php?showid=1100000004614784274&amp;base=33&lt;nowiki/&gt;||81||100000||
|-
||34||184785~184832||GFGC&lt;sub&gt;99996&lt;/sub&gt;5&lt;br&gt;90&lt;sub&gt;97950&lt;/sub&gt;FJ&lt;br&gt;NM0&lt;sub&gt;85218&lt;/sub&gt;KX&lt;br&gt;F&lt;sub&gt;83189&lt;/sub&gt;H2HP&lt;br&gt;P&lt;sub&gt;79441&lt;/sub&gt;444P&lt;br&gt;6&lt;sub&gt;77027&lt;/sub&gt;8X&lt;br&gt;XQIQ&lt;sub&gt;72241&lt;/sub&gt;D&lt;br&gt;T&lt;sub&gt;66530&lt;/sub&gt;IF&lt;br&gt;4&lt;sub&gt;66152&lt;/sub&gt;B&lt;br&gt;2EEC&lt;sub&gt;66039&lt;/sub&gt;7||100000&lt;br&gt;97953&lt;br&gt;85222&lt;br&gt;83193&lt;br&gt;79445&lt;br&gt;77029&lt;br&gt;72245&lt;br&gt;66532&lt;br&gt;66153&lt;br&gt;66043||153148&lt;br&gt;150013&lt;br&gt;130516&lt;br&gt;127408&lt;br&gt;121669&lt;br&gt;117968&lt;br&gt;110642&lt;br&gt;101893&lt;br&gt;101312&lt;br&gt;101143||(209246×34&lt;sup&gt;99997&lt;/sup&gt;−81)/11&lt;br&gt;9×34&lt;sup&gt;97952&lt;/sup&gt;+529&lt;br&gt;804×34&lt;sup&gt;85220&lt;/sup&gt;+713&lt;br&gt;(5×34&lt;sup&gt;83193&lt;/sup&gt;+700233)/11&lt;br&gt;(25×34&lt;sup&gt;79445&lt;/sup&gt;−28062367)/33&lt;br&gt;(2×34&lt;sup&gt;77029&lt;/sup&gt;+1043)/11&lt;br&gt;(1288676×34&lt;sup&gt;72242&lt;/sup&gt;−455)/33&lt;br&gt;(29×34&lt;sup&gt;66532&lt;/sup&gt;−12833)/33&lt;br&gt;(4×34&lt;sup&gt;66153&lt;/sup&gt;+227)/33&lt;br&gt;(30826×34&lt;sup&gt;66040&lt;/sup&gt;−59)/11||http://factordb.com/index.php?id=1100000004702891268&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004702894713&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004702900996&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004702910376&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004702913746&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004702918600&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004399656529&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004399657696&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004399658651&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004399659716&amp;open=prime&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000004702891268&amp;base=34&lt;br&gt;http://factordb.com/index.php?showid=1100000004702894713&amp;base=34&lt;br&gt;http://factordb.com/index.php?showid=1100000004702900996&amp;base=34&lt;br&gt;http://factordb.com/index.php?showid=1100000004702910376&amp;base=34&lt;br&gt;http://factordb.com/index.php?showid=1100000004702913746&amp;base=34&lt;br&gt;http://factordb.com/index.php?showid=1100000004702918600&amp;base=34&lt;br&gt;http://factordb.com/index.php?showid=1100000004399656529&amp;base=34&lt;br&gt;http://factordb.com/index.php?showid=1100000004399657696&amp;base=34&lt;br&gt;http://factordb.com/index.php?showid=1100000004399658651&amp;base=34&lt;br&gt;http://factordb.com/index.php?showid=1100000004399659716&amp;base=34&lt;nowiki/&gt;||47||100000||
|-
||35||720002~720062||N0N&lt;sub&gt;99971&lt;/sub&gt;9&lt;br&gt;V0&lt;sub&gt;83669&lt;/sub&gt;E73&lt;br&gt;N&lt;sub&gt;81563&lt;/sub&gt;K7N&lt;br&gt;BJ0&lt;sub&gt;81279&lt;/sub&gt;N&lt;br&gt;J0&lt;sub&gt;80062&lt;/sub&gt;FUH&lt;br&gt;43V&lt;sub&gt;79754&lt;/sub&gt;&lt;br&gt;9&lt;sub&gt;76600&lt;/sub&gt;K3&lt;br&gt;LB&lt;sub&gt;71366&lt;/sub&gt;PB&lt;br&gt;Q&lt;sub&gt;64150&lt;/sub&gt;H&lt;br&gt;50&lt;sub&gt;63397&lt;/sub&gt;5R||99974&lt;br&gt;83673&lt;br&gt;81566&lt;br&gt;81282&lt;br&gt;80066&lt;br&gt;79756&lt;br&gt;76602&lt;br&gt;71369&lt;br&gt;64151&lt;br&gt;63400||154367&lt;br&gt;129197&lt;br&gt;125944&lt;br&gt;125505&lt;br&gt;123628&lt;br&gt;123148&lt;br&gt;118279&lt;br&gt;110199&lt;br&gt;99054&lt;br&gt;97894||(27393×35&lt;sup&gt;99972&lt;/sup&gt;−499)/34&lt;br&gt;31×35&lt;sup&gt;83672&lt;/sup&gt;+17398&lt;br&gt;(23×35&lt;sup&gt;81566&lt;/sup&gt;−144013)/34&lt;br&gt;404×35&lt;sup&gt;81280&lt;/sup&gt;+23&lt;br&gt;19×35&lt;sup&gt;80065&lt;/sup&gt;+19442&lt;br&gt;(4893×35&lt;sup&gt;79754&lt;/sup&gt;−31)/34&lt;br&gt;(9×35&lt;sup&gt;76602&lt;/sup&gt;+12877)/34&lt;br&gt;(725×35&lt;sup&gt;71368&lt;/sup&gt;+16649)/34&lt;br&gt;(13×35&lt;sup&gt;64151&lt;/sup&gt;−166)/17&lt;br&gt;5×35&lt;sup&gt;63399&lt;/sup&gt;+202||http://factordb.com/index.php?id=1100000008248342445&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008248353306&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008248375642&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008248397018&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008248412468&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008248418540&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008248423670&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008192119974&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008192126630&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008192129294&amp;open=prime&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000008248342445&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008248353306&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008248375642&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008248397018&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008248412468&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008248418540&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008248423670&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008192119974&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008192126630&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008192129294&amp;base=35&lt;nowiki/&gt;||60||100000||
|-
||36||35286~35290||P&lt;sub&gt;81993&lt;/sub&gt;SZ&lt;br&gt;S0&lt;sub&gt;75007&lt;/sub&gt;8H&lt;br&gt;7K&lt;sub&gt;26567&lt;/sub&gt;Z&lt;br&gt;J&lt;sub&gt;10117&lt;/sub&gt;LJ&lt;br&gt;VL0&lt;sub&gt;7258&lt;/sub&gt;J&lt;br&gt;EO0&lt;sub&gt;6177&lt;/sub&gt;V&lt;br&gt;FZ&lt;sub&gt;5777&lt;/sub&gt;3P&lt;br&gt;T09&lt;sub&gt;4618&lt;/sub&gt;1&lt;br&gt;RY&lt;sub&gt;4562&lt;/sub&gt;H&lt;br&gt;OZ&lt;sub&gt;3932&lt;/sub&gt;AZ||81995&lt;br&gt;75010&lt;br&gt;26569&lt;br&gt;10119&lt;br&gt;7261&lt;br&gt;6180&lt;br&gt;5780&lt;br&gt;4621&lt;br&gt;4564&lt;br&gt;3935||127609&lt;br&gt;116739&lt;br&gt;41349&lt;br&gt;15748&lt;br&gt;11301&lt;br&gt;9618&lt;br&gt;8996&lt;br&gt;7192&lt;br&gt;7103&lt;br&gt;6124||(5×36&lt;sup&gt;81995&lt;/sup&gt;+821)/7&lt;br&gt;28×36&lt;sup&gt;75009&lt;/sup&gt;+305&lt;br&gt;(53×36&lt;sup&gt;26568&lt;/sup&gt;+101)/7&lt;br&gt;(19×36&lt;sup&gt;10119&lt;/sup&gt;+2501)/35&lt;br&gt;1137×36&lt;sup&gt;7259&lt;/sup&gt;+19&lt;br&gt;528×36&lt;sup&gt;6178&lt;/sup&gt;+31&lt;br&gt;16×36&lt;sup&gt;5779&lt;/sup&gt;−1163&lt;br&gt;(36549×36&lt;sup&gt;4619&lt;/sup&gt;−289)/35&lt;br&gt;(979×36&lt;sup&gt;4563&lt;/sup&gt;−629)/35&lt;br&gt;25×36&lt;sup&gt;3934&lt;/sup&gt;−901||http://factordb.com/index.php?id=1100000002394962083&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004020085177&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003896952461&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362491&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362489&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362488&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362487&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362486&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362485&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000000840634476&amp;open=prime&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000002394962083&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000004020085177&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003896952461&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362491&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362489&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362488&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362487&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362486&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362485&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000000840634476&amp;base=36&lt;nowiki/&gt;||4||200000||
|}

== The fully proof of Athena problem in decimal (base ''b'' = 10) ==

'''Bold''' for the Athena primes, ''x'' ◁ ''y'' means ''x'' is a subsequence of ''y''.

Assume ''p'' is a prime &gt; 10, and the last digit of ''p'' must lie in {1,3,7,9}.

Case 1: ''p'' ends with 1.

In this case we can write ''p'' = ''x''1. If ''x'' contains 1, 3, 4, 6, or 7, then (respectively) '''11''' ◁ ''p'', '''31''' ◁ ''p'', '''41''' ◁ ''p'', '''61''' ◁ ''p'', or '''71''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 2, 5, 8, or 9.

Case 1.1: ''p'' begins with 2.

In this case we can write ''p'' = 2''y''1. If 5 ◁ ''y'', then '''251''' ◁ ''p''. If 8 ◁ ''y'', then '''281''' ◁ ''p''. If 9 ◁ ''y'', then 29 ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 2.

If 22 ◁ ''y'', then '''2221''' ◁ ''p''. Hence we may assume ''y'' contains zero or one 2's.

If ''y'' contains no 2's, then ''p'' ∈ 2{0}1. But then, since the sum of the digits of ''p'' is 3, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' contains exactly one 2, then we can write ''p'' = 2''z''2''w''1, where ''z'',''w'' ∈ {0}. If 0 ◁ ''z'' and 0 ◁ ''w'', then '''20201''' ◁ ''p''. Hence we may assume either ''z'' or ''w'' is empty.

If ''z'' is empty, then ''p'' ∈ 22{0}1, and the smallest prime ''p'' ∈ 22{0}1 is '''22000001'''.

If ''w'' is empty, then ''p'' ∈ 2{0}21, and the smallest prime ''p'' ∈ 2{0}21 is '''20021'''.

Case 1.2: ''p'' begins with 5.

In this case we can write ''p'' = 5''y''1. If 2 ◁ ''y'', then '''521''' ◁ ''p''. If 9 ◁ ''y'', then 59 ◁ ''p''. Hence we may assume all digits of ''y'' are 0, 5, or 8.

If 05 ◁ ''y'', then '''5051''' ◁ ''p''. If 08 ◁ ''y'', then '''5081''' ◁ ''p''. If 50 ◁ ''y'', then '''5501''' ◁ ''p''. If 58 ◁ ''y'', then '''5581''' ◁ ''p''. If 80 ◁ ''y'', then '''5801''' ◁ ''p''. If 85 ◁ ''y'', then '''5851''' ◁ ''p''. Hence we may assume ''y'' ∈ {0} ∪ {5} ∪ {8}.

If ''y'' ∈ {0}, then ''p'' ∈ 5{0}1. But then, since the sum of the digits of ''p'' is 6, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' ∈ {5}, then ''p'' ∈ 5{5}1, and the smallest prime ''p'' ∈ 5{5}1 is '''555555555551'''.

If ''y'' ∈ {8}, since if 88 ◁ ''y'', then 881 ◁ ''p'', hence we may assume ''y'' ∈ {''𝜆'',8}, and thus ''p'' ∈ {51,581}, but 51 and 581 are both composite.

Case 1.3: ''p'' begins with 8.

In this case we can write p = 8''y''1. If 2 ◁ ''y'', then '''821''' ◁ ''p''. If 8 ◁ ''y'', then '''881''' ◁ ''p''. If 9 ◁ ''y'', then 89 ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 5.

If 50 ◁ ''y'', then '''8501''' ◁ ''p''. Hence we may assume y ∈ {0}{5}.

If 005 ◁ ''y'', then '''80051''' ◁ p. Hence we may assume y ∈ {0} ∪ {5} ∪ 0{5}.

If y ∈ {0}, then ''p'' ∈ 8{0}1. But then, since the sum of the digits of ''p'' is 9, ''p'' is divisible by 3, so ''p'' cannot be prime.

If y ∈ {5}, since if 55555555555 ◁ ''y'', then 555555555551 ◁ ''p'', hence we may assume ''y'' ∈ {''𝜆'', 5, 55, 555, 5555, 55555, 555555, 5555555, 55555555, 555555555, 5555555555}, and thus ''p'' ∈ {81, 851, 8551, 85551, 855551, 8555551, 85555551, 855555551, 8555555551, 85555555551, 855555555551}, but all of these numbers are composite.

If y ∈ 0{5}, since if 55555555555 ◁ ''y'', then 555555555551 ◁ ''p'', hence we may assume ''y'' ∈ {0, 05, 055, 0555, 05555, 055555, 0555555, 05555555, 055555555, 0555555555, 05555555555}, and thus ''p'' ∈ {801, 8051, 80551, 805551, 8055551, 80555551, 805555551, 8055555551, 80555555551, 805555555551, 8055555555551}, and of these numbers only 80555551 and 8055555551 are primes, but 80555551 ◁ 8055555551, thus only '''80555551''' is a minimal element.

Case 1.4: ''p'' begins with 9.

In this case we can write p = 9''y''1. If 9 ◁ ''y'', then '''991''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0, 2, 5, or 8.

If 00 ◁ ''y'', then '''9001''' ◁ ''p''. If 22 ◁ ''y'', then '''9221''' ◁ ''p''. If 55 ◁ ''y'', then '''9551''' ◁ ''p''. If 88 ◁ ''y'', then 881 ◁ ''p''. Hence we may assume ''y'' contains at most one 0, at most one 2, at most one 5, and at most one 8.

If ''y'' only contains at most one 0 and does not contain any of {2,5,8}, then ''y'' ∈ {''𝜆'',0}, and thus ''p'' ∈ {91,901}, but 91 and 901 are both composite. If ''y'' only contains at most one 0 and only one of {2,5,8}, then the sum of the digits of ''p'' is divisible by 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Hence we may assume ''y'' contains at least two of {2,5,8}.

If 25 ◁ ''y'', then 251 ◁ ''p''. If 28 ◁ ''y'', then 281 ◁ ''p''. If 52 ◁ ''y'', then 521 ◁ ''p''. If 82 ◁ ''y'', then 821 ◁ ''p''. Hence we may assume ''y'' contains no 2's (since if ''y'' contains 2, then ''y'' cannot contain either 5's or 8's, which is a contradiction).

If 85 ◁ ''y'', then '''9851''' ◁ ''p''. Hence we may assume ''y'' ∈ {58,580,508,058}, and thus ''p'' ∈ {9581,95801,95081,90581}, and of these numbers only 95801 is prime, but 95801 is not a minimal element since 5801 ◁ 95801.

Case 2: ''p'' ends with 3.

In this case we can write p = ''x''3. If ''x'' contains 1, 2, 4, 5, 7, or 8, then (respectively) '''13''' ◁ ''p'', '''23''' ◁ ''p'', '''43''' ◁ ''p'', '''53''' ◁ ''p'', '''73''' ◁ ''p'', or '''83''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 3, 6, or 9, and thus all digits of ''p'' are 0, 3, 6, or 9. But then, since the digits of ''p'' all have a common factor 3, ''p'' is divisible by 3, so ''p'' cannot be prime.

Case 3: ''p'' ends with 7.

In this case we can write ''p'' = ''x''7. If ''x'' contains 1, 3, 4, 6, or 9, then (respectively) '''17''' ◁ ''p'', '''37''' ◁ ''p'', '''47''' ◁ ''p'', '''67''' ◁ ''p'', or '''97''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 2, 5, 7, or 8.

Case 3.1: ''p'' begins with 2.

In this case we can write ''p'' = 2''y''7. If 2 ◁ ''y'', then '''227''' ◁ ''p''. If 5 ◁ ''y'', then '''257''' ◁ ''p''. If 7 ◁ ''y'', then '''277''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 8.

If 08 ◁ ''y'', then '''2087''' ◁ ''p''. If 88 ◁ ''y'', then 887 ◁ ''p''. Hence we may assume ''y'' ∈ {0} ∪ 8{0}.

If ''y'' ∈ {0}, then ''p'' ∈ 2{0}7. But then, since the sum of the digits of ''p'' is 9, ''p'' is divisible by 3, so ''p'' cannot be prime.

If y ∈ 8{0}, then ''p'' ∈ 28{0}7. But then ''p'' is divisible by 7, since for ''n'' ≥ 0 we have 7 × 40&lt;sub&gt;''n''&lt;/sub&gt;1 = 280&lt;sub&gt;''n''&lt;/sub&gt;7.

Case 3.2: ''p'' begins with 5.

In this case we can write ''p'' = 5''y''7. If 5 ◁ ''y'', then '''557''' ◁ ''p''. If 7 ◁ ''y'', then '''577''' ◁ ''p''. If 8 ◁ ''y'', then '''587''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 2.

If 22 ◁ ''y'', then 227 ◁ ''p''. Hence we may assume ''y'' contains zero or one 2's.

If ''y'' contains no 2's, then ''p'' ∈ 5{0}7. But then, since the sum of the digits of ''p'' is 12, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' contains exactly one 2, then we can write ''p'' = 5''z''2''w''7, where ''z'',''w'' ∈ {0}. If 0 ◁ ''z'' and 0 ◁ ''w'', then '''50207''' ◁ ''p''. Hence we may assume either ''z'' or ''w'' is empty.

If ''z'' is empty, then ''p'' ∈ 52{0}7, and the smallest prime ''p'' ∈ 52{0}7 is '''5200007'''.

If ''w'' is empty, then ''p'' ∈ 5{0}27, and the smallest prime ''p'' ∈ 5{0}27 is '''5000000000000000000000000000027'''.

Case 3.3: ''p'' begins with 7.

In this case we can write ''p'' = 7''y''7. If 2 ◁ ''y'', then '''727''' ◁ ''p''. If 5 ◁ ''y'', then '''757''' ◁ ''p''. If 8 ◁ ''y'', then '''787''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 7, and thus all digits of ''p'' are 0 or 7. But then, since the digits of ''p'' all have a common factor 7, ''p'' is divisible by 7, so ''p'' cannot be prime.

Case 3.4: ''p'' begins with 8.

In this case we can write ''p'' = 8''y''7. If 2 ◁ ''y'', then '''827''' ◁ ''p''. If 5 ◁ ''y'', then '''857''' ◁ ''p''. If 7 ◁ ''y'', then '''877''' ◁ ''p''. If 8 ◁ ''y'', then '''887''' ◁ ''p''. Hence we may assume ''y'' ∈ {0}, and thus ''p'' ∈ 8{0}7. But then, since the sum of the digits of ''p'' is 15, ''p'' is divisible by 3, so ''p'' cannot be prime.

Case 4: ''p'' ends with 9.

In this case we can write ''p'' = ''x''9. If ''x'' contains 1, 2, 5, 7, or 8, then (respectively) '''19''' ◁ ''p'', '''29''' ◁ ''p'', '''59''' ◁ ''p'', '''79''' ◁ ''p'', or '''89''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 3, 4, 6, or 9.

If 44 ◁ ''x'', then '''449''' ◁ ''p''. Hence we may assume ''x'' contains zero or one 4's.

If x contains no 4's, then all digits of ''x'' are 0, 3, 6, or 9, and thus all digits of ''p'' are 0, 3, 6, or 9. But then, since the digits of ''p'' all have a common factor 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Hence we may assume that ''x'' contains exactly one 4.

Case 4.1: ''p'' begins with 3.

In this case we can write ''p'' = 3''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. We must have '''349''' ◁ ''p''.

Case 4.2: ''p'' begins with 4.

In this case we can write ''p'' = 4''y''9, where all digits of ''y'' are 0, 3, 6, or 9. If 0 ◁ ''y'', then '''409''' ◁ ''p''. If 3 ◁ ''y'', then 43 ◁ ''p''. If 9 ◁ ''y'', then '''499''' ◁ ''p''. Hence we may assume ''y'' ∈ {6}, and thus ''p'' ∈ 4{6}9. But then ''p'' is divisible by 7, since for ''n'' ≥ 0 we have 7 × 6&lt;sub&gt;''n''&lt;/sub&gt;7 = 46&lt;sub&gt;''n''&lt;/sub&gt;9.

Case 4.3: ''p'' begins with 6.

In this case we can write p = 6''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. If 0 ◁ ''z'', then 409 ◁ ''p''. If 3 ◁ ''z'', then 43 ◁ ''p''. If 6 ◁ ''z'', then '''6469''' ◁ ''p''. If 9 ◁ ''z'', then 499 ◁ ''p''. Hence we may assume ''z'' is empty.

If 3 ◁ ''y'', then 349 ◁ ''p''. If 9 ◁ ''y'', then '''6949''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 6.

If 06 ◁ ''y'', then '''60649''' ◁ ''p''. Hence we may assume ''y'' ∈ {6}{0}.

If 666 ◁ ''y'', then '''666649''' ◁ ''p''. If 00000 ◁ ''y'', then '''60000049''' ◁ ''p''. Hence we may assume ''y'' ∈ {''𝜆'', 0, 00, 000, 0000, 6, 60, 600, 6000, 60000, 66, 660, 6600, 66000, 660000}, and thus ''p'' ∈ {649, 6049, 60049, 600049, 6000049, 6649, 66049, 660049, 6600049, 66000049, 66649, 666049, 6660049, 66600049, 666000049}, and of these numbers only '''66000049''' and '''66600049''' are primes.

Case 4.4: ''p'' begins with 9.

In this case we can write p = 9''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. If 0 ◁ ''y'', then '''9049''' ◁ ''p''. If 3 ◁ ''y'', then 349 ◁ ''p''. If 6 ◁ ''y'', then '''9649''' ◁ ''p''. If 9 ◁ ''y'', then '''9949''' ◁ ''p''. Hence we may assume ''y'' is empty.

If 0 ◁ ''z'', then 409 ◁ ''p''. If 3 ◁ ''z'', then 43 ◁ ''p''. If 9 ◁ ''z'', then 499 ◁ ''p''. Hence we may assume ''z'' ∈ {6}, and thus ''p'' ∈ 94{6}9, and the smallest prime ''p'' ∈ 94{6}9 is 946669.

[[Category:Number theory]]</text>
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    </revision>
    <revision>
      <id>2819839</id>
      <parentid>2819838</parentid>
      <timestamp>2026-07-30T08:52:45Z</timestamp>
      <contributor>
        <username>Athene241</username>
        <id>3100061</id>
      </contributor>
      <comment>/* Condensed table for bases 2 ≤ b ≤ 36 */</comment>
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      <text bytes="128826" sha1="byh3kxbgdm9isv28ufpii4bh0nz6uqe" xml:space="preserve">{{mathematics}}
'''Athena problem''' is an [[:w:List of unsolved problems in mathematics|unsolved problem]] in [[:w:Number theory|number theory]] and [[:w:Formal language theory|formal language theory]] and [[:w:Order theory|order theory]], this problem is named after the ancient Greek goddess [[:w:Athena|Athena]] (which is associated with [[:w:Wisdom|wisdom]]). Athena problem is: Give a [[:w:Natural number|natural number]] ''b'' &gt; 1, find the [[:w:Set (mathematics)|set]] of the [[:w:Minimal element|minimal element]]s of the set of the &quot;[[:w:Prime number|prime number]] [[:w:Greater than|&gt;]] ''b''&quot; [[:w:Numerical digit|digit]] [[:w:String (computer science)|string]]s in the [[:w:Positional numeral system|positional numeral system]] with [[:w:Radix|base]] ''b'' for the [[:w:Subsequence|subsequence]] [[:w:Partially ordered set|ordering]]. (A string ''x'' is a subsequence of another string ''y'', if ''x'' can be obtained from ''y'' by deleting zero or more of the [[:w:Character (computing)|character]]s in ''y''. For example, 514 is a subsequence of 352148, &quot;string&quot; is a subsequence of &quot;meistersinger&quot;. In contrast, 758 is not a subsequence of 378259, &quot;abc&quot; is not a subsequence of &quot;cbacacba&quot;, since the characters must be in the same order) (Unlike [[:w:Substring|substring]], subsequence is not required to occupy consecutive positions within the original sequences, e.g. the [[:w:Longest common subsequence|longest common subsequence problem]] is different from the [[:w:Longest common substring|longest common substring problem]])

Using [[:w:Formal language theory|formal language theory]] terminology, Athena problem is finding the [[:w:Set (mathematics)|set]] of the [[:w:Minimal element|minimal element]]s of the [[:w:Formal language|language]] of base-''b'' [[:w:Representation (mathematics)|representation]]s of the [[:w:Prime number|prime number]]s [[:w:Greater than|&gt;]] ''b'' (which is a set of [[:w:String (computer science)|string]]s of [[:w:Symbol|symbol]]s over the [[:w:Alphabet (formal languages)|alphabet]] ''Σ''&lt;sub&gt;''b''&lt;/sub&gt; := {0, 1, ..., ''b''−1}), under the subsequence ordering (i.e. the [[:w:Binary relation|binary relation]] &quot;is a subsequence of&quot;, which is a [[:w:Partially ordered set|partial ordering]]), for a given natural number ''b'' &gt; 1 (You can draw this partial ordering as a [[:w:Hasse diagram|Hasse diagram]] to find all [[:w:Minimal element|minimal element]]s), this set is called '''Athena set base ''b''''', and the prime numbers in this set is called '''Athena primes base ''b'''''.

By [[:w:Higman's lemma|Higman's lemma]], there are no [[:w:Infinite set|infinite]] [[:w:Antichain|antichain]]s for the subsequence ordering (i.e. the subsequence ordering is always a [[:w:Well-quasi-ordering|well quasi order]]) (i.e. under the subsequence ordering (i.e. the [[:w:Binary relation|binary relation]] &quot;is a subsequence of&quot;, which is a [[:w:Partially ordered set|partial ordering]]), every set of pairwise incomparable (i.e. not [[:w:Comparability|comparable]]) strings is finite), thus there must be only finitely many such minimal elements. In other words, the Athena set in every base ''b'' must be a [[:w:Finite set|finite set]], and every base ''b'' ≥ 2 has only finitely many Athena primes, e.g. in [[:w:Decimal|decimal]] (base ''b'' = 10), the Athena set has exactly 77 [[:w:Element of a set|element]]s (the Athena primes in decimal (base ''b'' = 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}.

For bases 2 ≤ ''b'' ≤ 36, Athena problem is fully solved in bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 24, and also solved in bases ''b'' = 11, 13, 16, 22, 30 if [[:w:Probable prime|probable prime]]s are allowed. For the unsolved bases ''b'' = 17, 19, 21, 23, 25, 26, 27, 28, 29, 31, 32, 34, 35, 36, Athena problem is solved (if probable primes are allowed) except 771 [[:w:Indexed family|families]] of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be [[:w:Empty string|empty]]) of digits in base ''b'', ''y'' is a digit in base ''b'') = sequence {''xz'', ''xyz'', ''xyyz'', ''xyyyz'', ''xyyyyz'', ''xyyyyyz'', ...} (i.e. &quot;''xy''&lt;sup&gt;+&lt;/sup&gt;''z''&quot; in [[:w:Regular expression|regular expression]]), all of these 771 families contain no primes &gt; ''b'' or probable primes &gt; ''b'' with length ≤ 100000.

== Solve the problem ==

To solve the Athena problem for a given base ''b'', we must [[:w:Computing|compute]] the elements up to families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''), and find the smallest prime &gt; ''b'' in all such families.

We call families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') &quot;linear&quot; families, and we reduce these families by removing all trailing digits ''y'' from ''x'', and removing all leading digits ''y'' from ''z'', to make the families be easier, e.g. family 12333{3}33345 in base ''b'' is reduced to family 12{3}45 in base ''b'', since they are in fact the same family. Our [[:w:Algorithm|algorithm]] then proceeds as follows:

* 1. ''M'' := {minimal primes in base ''b'' of length 2 or 3}, ''L'' := union of all ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'') such that ''x'' ≠ 0 and ''gcd''(''z'', ''b'') = 1 and ''Y'' is the set of digits ''y'' in base ''b'' such that ''xyz'' has no subsequence in ''M''.
* 2. While ''L'' contains nonlinear families (families which are not linear families): Explore each family of ''L'', and update ''L''. Examine each family of ''L'' by:
* 2.1. Let ''w'' be the shortest string in the family. If ''w'' has a subsequence in ''M'', then remove the family from ''L''. If ''w'' represents a prime, then add ''w'' to ''M'' and remove the family from ''L''.
* 2.2. If possible, simplify the family.
* 2.3. Using the techniques below (covering congruence, algebraic factorization, or combine of them), check if the family can be proven to only contain composites (only count the numbers &gt; ''b''), and if so then remove the family from ''L''.
* 3. Update ''L'', after each split examine the new families as in step 2.

e.g. in decimal (base ''b'' = 10):

''M'' := {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}

''L'' := {2{0,2}1, 2{0,8}7, 3{0,3,6,9}3, 3{0,3,6,9}9, 4{6}9, 5{0,5,8}1, 5{0,2}7, 6{0,3,6,9}3, 6{0,3,4,6,9}9, 7{0,7}7, 8{0,5}1, 8{0}7, 9{0,2,5,8}1, 9{0,3,6,9}3, 9{0,3,4,6,9}9}

and since 2221 is prime, it follows that the family 2{0,2}1 splits into the families 2{0}1 and 2{0}2{0}1

and since the family 2{0}1 can be proven to contain no primes &gt; base (since all numbers in this family are divisible by 3), it can be removed

and since 20201 is prime, it follows that the family 2{0}2{0}1 splits into the families 2{0}21 and 22{0}1

221 and 2021 are composites, but 20021 is prime, thus add 20021 to ''L''

none of 221, 2201, 22001, 220001, 2200001 are primes, but 22000001 is prime, thus add 22000001 to ''L''

and since the family 3{0,3,6,9}3 can be proven to contain no primes &gt; base (since all numbers in this family are divisible by 3), it can be removed

etc.

Since the number of possible (first digit,last digit) (also called (initial digit,final digit)) combos ([[:w:Ordered pair|ordered pair]]s) of a prime &gt; ''b'' in base ''b'' is (''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(''b'') (all digits except 0 can be the first digit of a prime &gt; ''b'' in base ''b'' (thus ''b''−1 possible digits), but only the digits coprime to ''b'' can be the last digit of a prime &gt; ''b'' in base ''b'' (thus ''eulerphi''(''b'') possible digits), and by the [[:w:Rule of product|rule of product]], there are (''b''−1)×''eulerphi''(''b'') choices of the (first digit,last digit) combo, also, both &quot;numbers of Athena primes in base ''b''&quot; and &quot;length of the largest Athena prime in base ''b''&quot; are [[:w:Asymptotic analysis|roughly]] ''[[:w:E (mathematical_constant)|e]]''&lt;sup&gt;''[[:w:Euler's constant|γ]]''×(''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(*b*)&lt;/sup&gt;.

Shrinking the family ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''Y'' is a set of digits in base ''b'')

* If ''y'' ∈ ''Y'' and the string ''xyyz'' represents a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or has a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''}''z'' ∪ ''x''{''Y'' \ ''y''}''y''{''Y'' \ ''y''}''z''.
* If ''y''&lt;sub&gt;1&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;2&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;1&lt;/sub&gt; ≠ ''y''&lt;sub&gt;2&lt;/sub&gt; and the string ''xy''&lt;sub&gt;1&lt;/sub&gt;''y''&lt;sub&gt;2&lt;/sub&gt;''z'' represents a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or has a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''&lt;sub&gt;1&lt;/sub&gt;}{''Y'' \ ''y''&lt;sub&gt;2&lt;/sub&gt;}''z''.
* If ''y''&lt;sub&gt;1&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;2&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;1&lt;/sub&gt; ≠ ''y''&lt;sub&gt;2&lt;/sub&gt; and both the strings ''xy''&lt;sub&gt;1&lt;/sub&gt;''y''&lt;sub&gt;2&lt;/sub&gt;''z'' and ''xy''&lt;sub&gt;2&lt;/sub&gt;''y''&lt;sub&gt;1&lt;/sub&gt;''z'' represent a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or have a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''&lt;sub&gt;1&lt;/sub&gt;}''z'' ∪ ''x''{''Y'' \ ''y''&lt;sub&gt;2&lt;/sub&gt;}''z''.

e.g. in decimal (base ''b'' = 10):

* 2221 is a prime &gt; 10, thus the family 2{0,2}1 splits into the two families 2{0}1 and 2{0}2{0}1.
* 227 is a prime &gt; 10, and it is a subsequence of 5227, thus the family 5{0,2}7 splits into the two families 5{0}7 and 5{0}2{0}7.
* 449 is a prime &gt; 10, and it is a subsequence of 6449, thus the family 6{0,3,4,6,9}9 splits into the two families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9.
* Both 5051 and 5501 are primes &gt; 10, thus the family 5{0,5}1 splits into the two families 5{0}1 and 5{5}1 = {5}1.
* 8501 is a prime &gt; 10, thus the family 8{0,5}1 splits into the family 8{0}{5}1.
* 887 is a prime &gt; 10, and it is a subsequence of 2887, also 2087 is a prime &gt; 10, thus the family 2{0,8}7 splits into the two families 2{0}7 and 28{0}7.
* 349 and 449 are primes &gt; 10, and they are subsequences of 9349 and 9449, respectively, also 9049, 9649, 9949 are primes &gt; 10, thus the family 9{0,3,4,6,9}9 splits into the two families 9{0,3,6,9}9 and 94{0,3,6,9}9.
* 251, 281, 521, 821, 881 are primes &gt; 10, and they are subsequences of 9251, 9281, 9521, 9821, 9881, respectively, also 9001, 9221, 9551, 9851 are primes &gt; 10, thus the family 9{0,2,5,8}1 splits into the numbers {91, 901, 921, 951, 981, 9021, 9051, 9081, 9201, 9501, 9581, 9801, 90581, 95081, 95801}.

If the methods we have discussed cannot be used to rule out or shrink ''x''{''Y''}''z'' where ''Y'' = {''y''&lt;sub&gt;1&lt;/sub&gt;, ''y''&lt;sub&gt;2&lt;/sub&gt;, ..., ''y''&lt;sub&gt;''n''&lt;/sub&gt;}, then we can replace ''x''{''Y''}''z'' by ''xy''&lt;sub&gt;1&lt;/sub&gt;{''Y''}''z'' ∪ ''xy''&lt;sub&gt;2&lt;/sub&gt;{''Y''}''z'' ∪ ... ∪ ''xy''&lt;sub&gt;''n''&lt;/sub&gt;{''Y''}''z'' and re-run the methods on this new [[:w:Formal language|language]].

If all remain families are linear families (i.e. of the form ''x''{''y''}''z'', where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''), then we search the smallest (probable) primes in these families and add these primes to the list.

e.g. in decimal (base ''b'' = 10):

* The smallest prime in the family 5{0}27 is 5000000000000000000000000000027.
* The smallest prime in the family {5}1 is 555555555551.
* The smallest prime in the family 8{5}1 is 8555555555555555555551, but 8555555555555555555551 is not a minimal element since 555555555551 is a subsequence of 8555555555555555555551.

There is no guarantee that the techniques discussed will ever terminate, but in practice they often do. They are able to determine the Athena set in base ''b'' for 2 ≤ ''b'' ≤ 16 and ''b'' = 18, 20, 22, 24, 30. The bases ''b'' = 17, 19, 21, 23, 25 ≤ ''b'' ≤ 29, 31 ≤ ''b'' ≤ 36 are solved with the exception of 771 families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'').

The following is a &quot;[[:w:Semi-algorithm|semi-algorithm]]&quot; that is guaranteed to solve the Athena problem for a given base ''b'', but it is not so easy to implement:

# ''M'' = ''[[:w:Empty string|∅]]''
# while (''L'' ≠ ''∅'') do
# choose ''x'', a shortest string in ''L''
# ''M'' := ''M'' ∪ {''x''}
# ''L'' := ''L'' − ''sup''({''x''})

In practice, for arbitrary ''L'', we cannot feasibly carry out step 5. Instead, we work with ''L''&amp;#39;, some regular overapproximation to ''L'', until we can show ''L''&amp;#39; = ''∅'' (which implies ''L'' = ''∅''). In practice, ''L''&amp;#39; is usually chosen to be a finite [[:w:Union (set theory)|union]] of sets of the form ''L''&lt;sub&gt;1&lt;/sub&gt;{''L''&lt;sub&gt;2&lt;/sub&gt;}''L''&lt;sub&gt;3&lt;/sub&gt;, where each of ''L''&lt;sub&gt;1&lt;/sub&gt;, ''L''&lt;sub&gt;2&lt;/sub&gt;, ''L''&lt;sub&gt;3&lt;/sub&gt; is finite. In the case we consider in this project, we then have to determine whether such a family contains a prime or not.

Thus, the [[:w:Time complexity|time complexity]] of the Athena problem in base ''b'' may be ''[[:w:Big O notation|O]]''(''[[:w:E (mathematical_constant)|e]]''&lt;sup&gt;''[[:w:Euler's constant|γ]]''×(''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(*b*)&lt;/sup&gt;), and the [[:w:CPU time|CPU time]] of the Athena problem in base ''b'' may be longer than [[:w:Age of the universe|the age of the universe]] for bases ''b'' = 19, 23, 25, 27, 29, 31, 32, 33, 34, 35, also, Athena problem in bases ''b'' around 500 may be [[:w:NP-complete|NP-complete]] or [[:w:NP-hard|NP-hard]], or an [[:w:Undecidable problem|undecidable problem]], or an example of [[:w:Gödel's incompleteness theorems|Gödel's incompleteness theorems]] (like the [[:w:Continuum hypothesis|continuum hypothesis]] and the [[:w:Halting problem|halting problem]]).

To solve the Athena problem (i.e. to [[:w:Computing|compute]] the Athena set), we need to determine whether a given family contains a prime. In practice, if family ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''Y'' is a set of digits in base ''b'') could not be ruled out as only containing composites and ''Y'' contains two or more digits, then a relatively small prime &gt; ''b'' could always be found in this family. Intuitively, this is because there are a large number of small strings in such a family, and at least one is likely to be prime (e.g. there are 2&lt;sup&gt;''n''−2&lt;/sup&gt; strings of length ''n'' in the family 1{3,7}9, and there are over a thousand strings of length 12 in the family 1{3,7}9, thus it is very impossible that these numbers are all composite). In the case ''Y'' contains only one digit, this family is of the form ''x''{''y''}''z'', and there is only a single string of each length &gt; (the length of ''x'' + the length of ''z''), and it is not known if the following [[:w:Decision problem|decision problem]] is recursively solvable (just like [[:w:Sierpiński number|Sierpiński problem]] and [[:w:Riesel number|Riesel problem]], Sierpiński problem and Riesel problem can be generalized to other bases ''b'' (references: http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm), in fact, Athena problem base ''b'' covers the Sierpiński problem base ''b'' and the Riesel problem base ''b'' with ''k'' &lt; ''b'', i.e. finding the smallest prime of the form ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 and ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (or prove such prime does not exist) with ''k'' &lt; ''b'' (specially, for bases ''b'' such that the conjectured smallest Sierpiński number or the conjectured smallest Riesel number is &lt; ''b'', Athena problem base ''b'' covers the Sierpiński problem base ''b'' or the Riesel problem base ''b'', respectively), since the smallest prime of the form ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 and ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (if exists) must be a minimal element in base ''b'', also, Athena problem base ''b'' covers finding the smallest prime of these forms in base ''b'' (or proving that such prime does not exist): (''b''&lt;sup&gt;''n''&lt;/sup&gt;−1)/(''b''−1) (for this form, ''n'' must be prime, and we want ''n'' ≥ 2) (references of this form: http://www.fermatquotient.com/PrimSerien/GenRepu.txt, https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html, http://www.primenumbers.net/Henri/us/MersFermus.htm, http://www.bitman.name/math/table/379, https://pzktupel.de/Primetables/TableRepunitGen.php, https://oeis.org/A084740, https://oeis.org/A084738, https://oeis.org/A128164, https://oeis.org/A285642; or for prime bases ''b'': https://oeis.org/A065854, https://oeis.org/A279068), ''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 (for this form, ''n'' must be power of 2, and we want ''n'' ≥ 1) (references of this form: http://jeppesn.dk/generalized-fermat.html, http://www.noprimeleftbehind.net/crus/GFN-primes.htm, https://web.archive.org/web/20231002190634/http://yves.gallot.pagesperso-orange.fr/primes/index.html, https://pzktupel.de/Primetables/TableFermatGFBB.php, https://oeis.org/A079706, https://oeis.org/A084712, https://oeis.org/A228101), (''b''&lt;sup&gt;''n''&lt;/sup&gt;+1)/2 (for odd ''b'') (for this form, ''n'' must be power of 2, and we want ''n'' ≥ 2) (reference of this form: http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt), (''sqrt''(''b'')×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1)/(''sqrt''(''b'')+1) (for square ''b'') (for this form, 2×''n''+1 must be prime, and we want ''n'' ≥ 2) (references of this form: http://www.fermatquotient.com/PrimSerien/GenRepuP.txt, http://www.primenumbers.net/Henri/us/MersFermus.htm, http://www.bitman.name/math/table/488, https://pzktupel.de/Primetables/TableWagstaffGen.php, https://oeis.org/A084742, https://oeis.org/A084741; or for bases ''b'' with ''sqrt''(''b'') prime: https://oeis.org/A065507), ((''b''−2)×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1)/(''b''−1) (''n'' ≥ 2) (reference of this form: https://oeis.org/A243404), 2×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 (''n'' ≥ 1) (references of this form: https://www.mersenneforum.org/showthread.php?t=6918, https://www.mersenneforum.org/showthread.php?t=19725, https://oeis.org/A119624), 2×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (''n'' ≥ 1) (references of this form: https://www.mersenneforum.org/showthread.php?t=24576, https://www.mersenneforum.org/attachment.php?attachmentid=20976&amp;d=1567314217, https://oeis.org/A119591), ''b''&lt;sup&gt;''n''&lt;/sup&gt;+2 (''n'' ≥ 1) (references of this form: https://oeis.org/A138066, https://oeis.org/A084713, https://oeis.org/A138067), ''b''&lt;sup&gt;''n''&lt;/sup&gt;−2 (''n'' ≥ 2) (references of this form: https://www.primepuzzles.net/puzzles/puzz_887.htm, https://oeis.org/A250200, https://oeis.org/A255707, https://oeis.org/A084714; or for prime bases ''b'': https://oeis.org/A292201), (''b''−1)×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 (''n'' ≥ 1) (references of this form: http://www.noprimeleftbehind.net/Williams-primes-MP.htm, http://www.bitman.name/math/table/477, https://pzktupel.de/Primetables/TableWilliams2.php, https://oeis.org/A305531; or for prime bases ''b'': https://oeis.org/A087139), (''b''−1)×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (''n'' ≥ 1) (references of this form: https://harvey563.tripod.com/wills.txt, http://www.noprimeleftbehind.net/Williams-primes-MM.htm, http://www.bitman.name/math/table/484, https://pzktupel.de/Primetables/TableWilliams1.php; or for prime bases ''b'': https://oeis.org/A122396), ''b''&lt;sup&gt;''n''&lt;/sup&gt;+(''b''−1) (''n'' ≥ 1) (references of this form: http://www.bitman.name/math/table/795, https://pzktupel.de/Primetables/TableWilliams6.php, https://oeis.org/A076845, https://oeis.org/A076846, https://oeis.org/A078178, https://oeis.org/A078179), ''b''&lt;sup&gt;''n''&lt;/sup&gt;−(''b''−1) (''n'' ≥ 2) (references of this form: http://www.bitman.name/math/table/792, https://pzktupel.de/Primetables/TableWilliams5.php, https://oeis.org/A113516, https://oeis.org/A343589; or for prime bases ''b'': https://cs.uwaterloo.ca/journals/JIS/VOL3/mccranie.html, http://www.bitman.name/math/table/435)):

Problem: Given strings ''x'', ''z'' (may be empty), a digit ''y'', and a base ''b'' (''x'' does not [[:w:Leading zero|start with the digit 0]], ''z'' ends with a digit which [[:w:Coprime integers|coprime]] to ''b'', ''y'' is not 0 if ''x'' is empty, ''y'' is coprime to ''b'' if ''z'' is empty), does there exist a prime number whose base-''b'' expansion is of the form ''xy''&lt;sub&gt;''n''&lt;/sub&gt;''z'' for some ''n'' ≥ 0?

Some families can be ruled out to contain no prime &gt; ''b'' by [[:w:Covering set|covering congruence]], [[:w:Factorization of polynomials|algebraic factorization]] (e.g. [[:w:Difference of two squares|difference of two squares]], [[:w:Sum of two cubes|sum of two cubes]], [[:w:Sophie Germain's identity|Sophie Germain's identity of ''x''&lt;sup&gt;4&lt;/sup&gt;+4×''y''&lt;sup&gt;4&lt;/sup&gt;]]), or combine of them, e.g.

* The base 9 family 2{7}: Always divisible by 2 or 5
* The base 11 family 2{5}: Always divisible by 2 or 3
* The base 14 family B{0}1: Always divisible by 3 or 5
* The base 13 family 95{0}3: Always divisible by 5, 7, or 17
* The base 16 family {4}D: Always divisible by 3, 7, or 13
* The base 16 family {8}F: Always divisible by 3, 7, or 13
* The base 21 family {7}D: Always divisible by 2, 13, or 17
* The base 23 family {D}GA: Always divisible by 2, 5, 7, 37, or 79
* The base 9 family {1}: Can be written as (9&lt;sup&gt;''n''&lt;/sup&gt;−1)/8 and can be factored as (3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (3&lt;sup&gt;''n''&lt;/sup&gt;+1) / 8
* The base 8 family 1{0}1: Can be written as 8&lt;sup&gt;''n''&lt;/sup&gt;+1 and can be factored as (2&lt;sup&gt;''n''&lt;/sup&gt;+1) × (4&lt;sup&gt;''n''&lt;/sup&gt;−2&lt;sup&gt;''n''&lt;/sup&gt;+1)
* The base 9 family 3{8}: Can be written as 4×9&lt;sup&gt;''n''&lt;/sup&gt;−1 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (2×3&lt;sup&gt;''n''&lt;/sup&gt;+1)
* The base 16 family 1{5}: Can be written as (4×16&lt;sup&gt;''n''&lt;/sup&gt;−1)/3 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (2×3&lt;sup&gt;''n''&lt;/sup&gt;+1) / 3
* The base 16 family {4}1: Can be written as (4×16&lt;sup&gt;''n''&lt;/sup&gt;−49)/15 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−7) × (2×3&lt;sup&gt;''n''&lt;/sup&gt;+7) / 15
* The base 27 family 7{Q}: Can be written as 8×27&lt;sup&gt;''n''&lt;/sup&gt;−1 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (4×9&lt;sup&gt;''n''&lt;/sup&gt;+2×3&lt;sup&gt;''n''&lt;/sup&gt;+1)
* The base 27 family 9{G}: Can be written as (125×27&lt;sup&gt;''n''&lt;/sup&gt;−8)/13 and can be factored as (5×3&lt;sup&gt;''n''&lt;/sup&gt;−2) × (25×9&lt;sup&gt;''n''&lt;/sup&gt;+10×3&lt;sup&gt;''n''&lt;/sup&gt;+4)
* The base 16 family {C}D: Can be written as (4×16&lt;sup&gt;''n''&lt;/sup&gt;+1)/5 and can be factored as (2×4&lt;sup&gt;''n''&lt;/sup&gt;−2×2&lt;sup&gt;''n''&lt;/sup&gt;+1) × (2×4&lt;sup&gt;''n''&lt;/sup&gt;+2×2&lt;sup&gt;''n''&lt;/sup&gt;+1) / 5
* The base 14 family 8{D}: Can be written as 9×14&lt;sup&gt;''n''&lt;/sup&gt;−1, it is divisible by 5 if ''n'' is odd and can be factored as (3×14&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (3×14&lt;sup&gt;''n''/2&lt;/sup&gt;+1) if ''n'' is even
* The base 12 family {B}9B: Can be written as 12&lt;sup&gt;''n''&lt;/sup&gt;−25, it is divisible by 13 if ''n'' is odd and can be factored as (12&lt;sup&gt;''n''/2&lt;/sup&gt;−5) × (12&lt;sup&gt;''n''/2&lt;/sup&gt;+5) if ''n'' is even
* The base 14 family {D}5: Can be written as 14&lt;sup&gt;''n''&lt;/sup&gt;−9, it is divisible by 5 if ''n'' is odd and can be factored as (14&lt;sup&gt;''n''/2&lt;/sup&gt;−3) × (14&lt;sup&gt;''n''/2&lt;/sup&gt;+3) if ''n'' is even
* The base 17 family 1{9}: Can be written as (25×17&lt;sup&gt;''n''&lt;/sup&gt;−9)/16, it is divisible by 2 if ''n'' is odd and can be factored as (5×17&lt;sup&gt;''n''/2&lt;/sup&gt;−3) × (5×17&lt;sup&gt;''n''/2&lt;/sup&gt;+3) / 16 if ''n'' is even
* The base 17 family 7{9}: Can be written as (121×17&lt;sup&gt;''n''&lt;/sup&gt;−9)/16, it is divisible by 2 if ''n'' is odd and can be factored as (11×17&lt;sup&gt;''n''/2&lt;/sup&gt;−3) × (11×17&lt;sup&gt;''n''/2&lt;/sup&gt;+3) / 16 if ''n'' is even
* The base 19 family 1{6}: Can be written as (4×19&lt;sup&gt;''n''&lt;/sup&gt;−1)/3, it is divisible by 5 if ''n'' is odd and can be factored as (2×19&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (2×19&lt;sup&gt;''n''/2&lt;/sup&gt;+1) / 3 if ''n'' is even
* The base 19 family 7{2}: Can be written as (64×19&lt;sup&gt;''n''&lt;/sup&gt;−1)/9, it is divisible by 5 if ''n'' is odd and can be factored as (8×19&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (8×19&lt;sup&gt;''n''/2&lt;/sup&gt;+1) / 9 if ''n'' is even
* The base 24 family 3{N}: Can be written as 4×24&lt;sup&gt;''n''&lt;/sup&gt;−1, it is divisible by 5 if ''n'' is odd and can be factored as (2×24&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (2×24&lt;sup&gt;''n''/2&lt;/sup&gt;+1) if ''n'' is even

By the [[:w:Prime number theorem|prime number theorem]], the [[:w:Probability|chance]] that a [[:w:Random number|random]] ''n''-digit base ''b'' number is prime is [[:w:Asymptotic analysis|approximately]] 1/''n'' (more accurately, the chance is approximately 1/(''n''×''ln''(''b'')), where ''ln'' is the [[:w:Natural logarithm|natural logarithm]]). If one conjectures the numbers ''x''{''y''}''z'' behave similarly (i.e. the numbers ''x''{''y''}''z'' is a [[:w:Pseudorandomness|pseudorandom sequence]]) you would expect [[:w:Harmonic_series (mathematics)|1/1 + 1/2 + 1/3 + 1/4 + ... = ∞]] primes of the form ''x''{''y''}''z'' (of course, this does not always happen, since some ''x''{''y''}''z'' families can be ruled out to contain no prime &gt; ''b'' (by covering congruence, algebraic factorization, or combine of them), but it is at least a reasonable conjecture in the absence of evidence to the contrary. Hence, the [[:w:Heuristic argument|heuristic argument]] suggests there are always infinitely many primes in family ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') if it cannot be ruled out to contain no prime or only contain finitely many primes, by covering congruence, algebraic factorization, or combine of them. However, some families ''x''{''y''}''z'' could not be proven to contain no primes &gt; ''b'' (by covering congruence, algebraic factorization, or combine of them) but no primes &gt; ''b'' could be found in the family, even after searching through numbers with over 100000 digits. In such a case, the only way to proceed is to [[:w:Primality test|test the primality]] of larger and larger numbers of such form and hope a prime is eventually discovered. e.g. the smallest (probable) prime in the family A{3}A in base ''b'' = 13 is A3&lt;sub&gt;592197&lt;/sub&gt;A, its algebraic form is (41×13&lt;sup&gt;592198&lt;/sup&gt;+27)/4, when written in decimal contains 659677 digits (it is only probable prime, i.e. not definitely prime, since technically, probable primality tests were used to show this (which have a ''very'' small chance of making an error, see https://t5k.org/notes/prp_prob.html) because all known primality tests run far too slowly to run on numbers of this size unless either [https://t5k.org/prove/prove3_1.html ''N''−1] or [https://t5k.org/prove/prove3_2.html ''N''+1] (or both) can be ≥ 1/3 factored).

The numbers in family ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') are of the form (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1) for some fixed ''a'', ''b'', ''c'' such that ''a'' ≥ 1, ''b'' ≥ 2 (''b'' is the base), ''c'' ≠ 0, ''gcd''(''a'',''c'') = 1, ''gcd''(''b'',''c'') = 1. Except in the [[:w:Special case|special case]] ''c'' = ±1 and ''gcd''(''a''+''c'',''b''−1) = 1, when ''n'' is large the known [[:w:Primality test|primality test]]s for such a number are too inefficient to run. In this case one must resort to a [[:w:Probabilistic algorithm|probable]] primality test such as a [[:w:Miller–Rabin primality test|Miller–Rabin primality test]] or a [[:w:Baillie–PSW primality test|Baillie–PSW primality test]], unless a divisor of the number can be found. Since we are testing many numbers in an [[:w:Exponential growth|exponential sequence]], it is possible to use a sieving process to find divisors rather than using [[:w:Trial division|trial division]].

To do this, we made use of Geoffrey Reynolds' ''srsieve'' software (download: https://pzktupel.de/Software/srsieve_1.1.4.7z). This program uses the [[:w:Baby-step giant-step|baby-step giant-step]] [[:w:Algorithm|algorithm]] to find all primes ''p'' which divide ''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'' where ''p'' and ''n'' lie in a [[:w:Interval_(mathematics)|specified range]]. Since this program cannot handle the general case (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1) when ''gcd''(''a''+''c'',''b''−1) &gt; 1 we only used it to sieve the sequence ''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'' for primes ''p'' not dividing ''gcd''(''a''+''c'',''b''−1), and initialized the list of candidates to not include ''n'' for which there is some prime ''p'' dividing ''gcd''(''a''+''c'',''b''−1) for which ''p'' dividing (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1). The program had to be modified slightly to remove a check which would prevent it from running in the case when ''a'', ''b'', and ''c'' were all odd (since then 2 divides ''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'', but 2 may not divide (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1)).

Once the numbers with small divisors had been removed, it remained to test the remaining numbers using a probable primality test. For this we used the software ''LLR'' by Jean Penné. (download: http://jpenne.free.fr/index2.html). Although undocumented, it is possible to run this program on numbers of the form (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1) when ''gcd''(''a''+''c'',''b''−1) &gt; 1, so this program required no modifications. A script was also written which allowed one to run ''srsieve'' while ''LLR'' was testing the remaining candidates, so that when a divisor was found by srsieve on a number which had not yet been tested by ''LLR'' it would be removed from the list of candidates.

For the primes &lt; 10&lt;sup&gt;25000&lt;/sup&gt; for the &quot;easy&quot; bases (bases ''b'' with ≤ 150 primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; (base ''b'' = 26 has 83 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 3 unsolved families, base ''b'' = 36 has 75 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 4 unsolved families, base ''b'' = 17 has 99 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 18 unsolved families, base ''b'' = 21 has 80 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 12 unsolved families, base ''b'' = 19 has 201 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 23 unsolved families), i.e. bases *b* = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 26, 28, 30, 36), we employed ''CM'' by Andreas Enge (download: https://www.multiprecision.org/cm/download.html), an elliptic curve primality proving implementation.

== Data ==

These are the results of the Athena problem in bases 2 ≤ ''b'' ≤ 36 (we stop at base 36 since this base is the maximum base for which it is possible to write the numbers with the [[:w:Symbol|symbol]]s 0, 1, 2, ..., 9 and A, B, C, ..., Z (i.e. the 10 [[:w:Arabic numerals|Arabic numerals]] and the 26 [[:w:Latin script|Latin letters]]): (some large primes are only probable primes, i.e. not definitely primes, since they are too large to be [[:w:Elliptic curve primality|ECPP proved]] and [[:w:Pocklington primality test#Extensions and variants|neither ''N''−1 nor ''N''+1 can be ≥ 1/3 factored]], all of them pass the [[:w:Baillie–PSW primality test|Baillie–PSW primality test]] and the [[:w:Strong pseudoprime|strong primality test]] (i.e. the [[:w:Miller–Rabin primality test|Miller–Rabin primality test]]) with all prime bases ''p'' ≤ 61, however, all primes &lt; 10&lt;sup&gt;25000&lt;/sup&gt; for bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 36 are definitely primes, most of them &gt; 10&lt;sup&gt;299&lt;/sup&gt; are proven primes with [[:w:Elliptic curve primality|ECPP proving]], others &gt; 10&lt;sup&gt;299&lt;/sup&gt; are proven primes with [[:w:Pocklington primality test#Extensions and variants|''N''−1 or ''N''+1 proving]])

All numbers are written in base ''b'', [[:w:Senary#Base 36 as senary compression|using A to Z to represent digit values 10 to 35]], &quot;{}&quot; means repeating, e.g. family 12{3}45 means the sequence {1245, 12345, 123345, 1233345, 12333345, 123333345, ...} (where the members are expressed as base ''b'' strings), subscripts are used to indicate repetitions of digits, e.g. 123&lt;sub&gt;4&lt;/sub&gt;567 means 123333567 (all subscripts are written in decimal).

Base 2: 1 Athena prime (the largest of which has 2 digits (it is 11, and its value is 3 in decimal)): {11}

Base 3: 3 Athena primes (the largest of which has 3 digits (it is 111, and its value is 13 in decimal)): {12, 21, 111}

Base 4: 5 Athena primes (the largest of which has 3 digits (it is 221, and its value is 41 in decimal)): {11, 13, 23, 31, 221}

Base 5: 22 Athena primes (the largest of which has 96 digits (it is 10&lt;sub&gt;93&lt;/sub&gt;13, and its algebraic form is 5&lt;sup&gt;95&lt;/sup&gt;+8)): {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 Athena primes (the largest of which has 5 digits (it is 40041, and its value is 5209 in decimal)): {11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041}

Base 7: 71 Athena primes (the largest of which has 17 digits (it is 3&lt;sub&gt;16&lt;/sub&gt;1, and its algebraic form is (7&lt;sup&gt;17&lt;/sup&gt;−5)/2)): {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: 75 Athena primes (the largest of which has 221 digits (it is 4&lt;sub&gt;220&lt;/sub&gt;7, and its algebraic form is (4×8&lt;sup&gt;221&lt;/sup&gt;+17)/7)): {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: 151 Athena primes (the largest of which has 1161 digits (it is 30&lt;sub&gt;1158&lt;/sub&gt;11, and its algebraic form is 3×9&lt;sup&gt;1160&lt;/sup&gt;+10)): {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: 77 Athena primes (the largest of which has 31 digits (it is 50&lt;sub&gt;28&lt;/sub&gt;27, and its algebraic form is 5×10&lt;sup&gt;30&lt;/sup&gt;+27)): {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: 1068 Athena primes (including 1 unproven probable prime: 57&lt;sub&gt;62668&lt;/sub&gt;), the largest of which has 62669 digits (it is 57&lt;sub&gt;62668&lt;/sub&gt;, and its algebraic form is (57×11&lt;sup&gt;62668&lt;/sup&gt;−7)/10), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel11 Data of Athena (probable) primes base 11]

Base 12: 106 Athena primes (the largest of which has 42 digits (it is 40&lt;sub&gt;39&lt;/sub&gt;77, and its algebraic form is 4×12&lt;sup&gt;41&lt;/sup&gt;+91)): {11, 15, 17, 1B, 25, 27, 31, 35, 37, 3B, 45, 4B, 51, 57, 5B, 61, 67, 6B, 75, 81, 85, 87, 8B, 91, 95, A7, AB, B5, B7, 221, 241, 2A1, 2B1, 2BB, 401, 421, 447, 471, 497, 565, 655, 665, 701, 70B, 721, 747, 771, 77B, 797, 7A1, 7BB, 907, 90B, 9BB, A41, B21, B2B, 2001, 200B, 202B, 222B, 229B, 292B, 299B, 4441, 4707, 4777, 6A05, 6AA5, 729B, 7441, 7B41, 929B, 9777, 992B, 9947, 997B, 9997, A0A1, A201, A605, A6A5, AA65, B001, B0B1, BB01, BB41, 600A5, 7999B, 9999B, AAAA1, B04A1, B0B9B, BAA01, BAAA1, BB09B, BBBB1, 44AAA1, A00065, BBBAA1, AAA0001, B00099B, AA000001, BBBBBB99B, B0000000000000000000000000009B, 400000000000000000000000000000000000000077}

Base 13: 3197 Athena primes (including 4 unproven probable primes: C5&lt;sub&gt;23755&lt;/sub&gt;C, 80&lt;sub&gt;32017&lt;/sub&gt;111, 95&lt;sub&gt;197420&lt;/sub&gt;, A3&lt;sub&gt;592197&lt;/sub&gt;A), the largest of which has 592199 digits (it is A3&lt;sub&gt;592197&lt;/sub&gt;A, and its algebraic form is (41×13&lt;sup&gt;592198&lt;/sup&gt;+27)/4), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel13 Data of Athena (probable) primes base 13]

Base 14: 650 Athena primes, the largest of which has 19699 digits (it is 4D&lt;sub&gt;19698&lt;/sub&gt;, and its algebraic form is 5×14&lt;sup&gt;19698&lt;/sup&gt;−1), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel14 Data of Athena primes base 14]

Base 15: 1284 Athena primes, the largest of which has 157 digits (it is 7&lt;sub&gt;155&lt;/sub&gt;97, and its algebraic form is (15&lt;sup&gt;157&lt;/sup&gt;+59)/2), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel15 Data of Athena primes base 15]

Base 16: 2347 Athena primes (including 3 unproven probable primes: DB&lt;sub&gt;32234&lt;/sub&gt;, 4&lt;sub&gt;72785&lt;/sub&gt;DD, 3&lt;sub&gt;116137&lt;/sub&gt;AF), the largest of which has 116139 digits (it is 3&lt;sub&gt;116137&lt;/sub&gt;AF, and its algebraic form is (16&lt;sup&gt;116139&lt;/sup&gt;+619)/5), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel16 Data of Athena (probable) primes base 16]

Base 17: 10415 known Athena primes (including many unproven probable primes) and 12 unsolved families (1{7}, 1F{0}7, 4{7}A, 70F{0}D, 8{B}9, 9{5}9, A{D}F, B{0}B3, {B}E9, {B}EE, F1{9}, FD0{D}, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel17 Data of known Athena (probable) primes base 17]

Base 18: 549 Athena primes, the largest of which has 6271 digits (it is C0&lt;sub&gt;6268&lt;/sub&gt;C5, and its algebraic form is 12×18&lt;sup&gt;6270&lt;/sup&gt;+221), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel18 Data of Athena primes base 18]

Base 19: 31417 known Athena primes (including many unproven probable primes) and 17 unsolved families (4B5{0}H, {5}3, 5{H}05, 5{H}0H, 5{H}5, 66{B}, 71{0}177, 7AF{0}H, 97{0}3, C{H}C, EE1{6}, F{7}5, F{B}G, F{D}F, H0F{0}7A, HB{0}5B5, II{D}, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel19 Data of known Athena (probable) primes base 19]

Base 20: 3314 Athena primes, the largest of which has 6271 digits (it is G0&lt;sub&gt;6269&lt;/sub&gt;D, and its algebraic form is 16×20&lt;sup&gt;6270&lt;/sup&gt;+13), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel20 Data of Athena primes base 20]

Base 21: 13386 known Athena primes (including many unproven probable primes) and 8 unsolved families (5{0}DJ, {9}D, B3{0}EB, B{H}6H, C{F}0K, {F}35, G{0}FK, H{0}7771, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel21 Data of known Athena (probable) primes base 21]

Base 22: 8003 Athena primes (including 1 unproven probable prime: BK&lt;sub&gt;22001&lt;/sub&gt;5), the largest of which has 22003 digits (it is BK&lt;sub&gt;22001&lt;/sub&gt;5, and its algebraic form is (251×22&lt;sup&gt;22002&lt;/sup&gt;−335)/21), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel22 Data of Athena (probable) primes base 22]

Base 23: 65178 known Athena primes (including many unproven probable primes) and 87 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel23 Data of known Athena (probable) primes base 23] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left23 Data of unsolved families for Athena problem base 23]

Base 24: 3409 Athena primes, the largest of which has 8134 digits (it is N00N&lt;sub&gt;8129&lt;/sub&gt;LN, and its algebraic form is 13249×24&lt;sup&gt;8131&lt;/sup&gt;−49), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel24 Data of Athena primes base 24]

Base 25: 133639 known Athena primes (including many unproven probable primes) and 85 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel25 Data of known Athena (probable) primes base 25] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left25 Data of unsolved families for Athena problem base 25]

Base 26: 25256 known Athena primes (including 7 unproven probable primes: 5&lt;sub&gt;19391&lt;/sub&gt;6F, 7&lt;sub&gt;20279&lt;/sub&gt;OL, LD0&lt;sub&gt;20975&lt;/sub&gt;7, 6K&lt;sub&gt;23300&lt;/sub&gt;5, J0&lt;sub&gt;44303&lt;/sub&gt;KCB, M0&lt;sub&gt;61186&lt;/sub&gt;2BB, 85M&lt;sub&gt;197060&lt;/sub&gt;B) and 3 unsolved families ({A}6F, {H}MH, {I}GL, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel26 Data of known Athena (probable) primes base 26]

Base 27: 102852 known Athena primes (including many unproven probable primes) and 44 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel27 Data of known Athena (probable) primes base 27] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left27 Data of unsolved families for Athena problem base 27]

Base 28: 25528 known Athena primes (including 3 unproven probable primes: N6&lt;sub&gt;24051&lt;/sub&gt;LR, 5OA&lt;sub&gt;31238&lt;/sub&gt;F, O4O&lt;sub&gt;94535&lt;/sub&gt;9) and 1 unsolved family (O{A}F, no primes or probable primes with length ≤ 900000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel28 Data of known Athena (probable) primes base 28]

Base 29: 355242 known Athena primes (including many unproven probable primes) and 125 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel29 Data of known Athena (probable) primes base 29] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left29 Data of unsolved families for Athena problem base 29]

Base 30: 2619 Athena primes (including 1 unproven probable prime: I0&lt;sub&gt;24608&lt;/sub&gt;D), the largest of which has 34206 digits (it is OT&lt;sub&gt;34205&lt;/sub&gt;, and its algebraic form is 25×30&lt;sup&gt;34205&lt;/sup&gt;−1), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel30 Data of Athena (probable) primes base 30]

Base 31: 569323 known Athena primes (including many unproven probable primes) and 77 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel31 Data of known Athena (probable) primes base 31] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left31 Data of unsolved families for Athena problem base 31]

Base 32: 168882 known Athena primes (including many unproven probable primes) and 120 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel32 Data of known Athena (probable) primes base 32] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left32 Data of unsolved families for Athena problem base 32]

Base 33: 280012 known Athena primes (including many unproven probable primes) and 81 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel33 Data of known Athena (probable) primes base 33] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left33 Data of unsolved families for Athena problem base 33]

Base 34: 184785 known Athena primes (including many unproven probable primes) and 47 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel34 Data of known Athena (probable) primes base 34] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left34 Data of unsolved families for Athena problem base 34]

Base 35: 720002 known Athena primes (including many unproven probable primes) and 60 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel35 Data of known Athena (probable) primes base 35] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left35 Data of unsolved families for Athena problem base 35]

Base 36: 35286 known Athena primes (including 3 unproven probable primes: 7K&lt;sub&gt;26567&lt;/sub&gt;Z, S0&lt;sub&gt;75007&lt;/sub&gt;8H, P&lt;sub&gt;81993&lt;/sub&gt;SZ) and 4 unsolved families (B{0}EUV, HM{0}N, N{0}YYN, O{L}Z, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel36 Data of known Athena (probable) primes base 36]

== Condensed table for bases 2 ≤ ''b'' ≤ 36 ==

{|class=&quot;wikitable&quot;
||''b''||number of Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b''||base-''b'' form of the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b'' (write &quot;''d''&lt;sub&gt;''n''&lt;/sub&gt;&quot; if there are 5 or more (''n'') consecutive same digits ''d'')||length of the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b''||length of the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b'' in decimal||algebraic ((''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1)) form of the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b''||''factordb'' entry of the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b''||the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b'' written in base ''b'' (use lower case letters instead of upper case letters)||number of unsolved families in the Athena problem in base ''b'' (all of these left families are linear families)||searching limit of length for the unsolved families in the Athena problem in base ''b'' (if there are different searching limits for the unsolved families in the Athena problem in base ''b'', choose the lowest searching limit)||
|-
||2||1||11||2||1||3||http://factordb.com/index.php?id=3&amp;open=ecm||http://factordb.com/index.php?showid=3&amp;base=2||0||–||
|-
||3||3||111&lt;br&gt;21&lt;br&gt;12||3&lt;br&gt;2&lt;br&gt;2||2&lt;br&gt;1&lt;br&gt;1||13&lt;br&gt;7&lt;br&gt;5||http://factordb.com/index.php?id=13&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=7&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=5&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=13&amp;base=3&lt;br&gt;http://factordb.com/index.php?showid=7&amp;base=3&lt;br&gt;http://factordb.com/index.php?showid=5&amp;base=3&lt;nowiki/&gt;||0||–||
|-
||4||5||221&lt;br&gt;31&lt;br&gt;23&lt;br&gt;13&lt;br&gt;11||3&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2||2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;1&lt;br&gt;1||41&lt;br&gt;13&lt;br&gt;11&lt;br&gt;7&lt;br&gt;5||http://factordb.com/index.php?id=41&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=13&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=11&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=7&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=5&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=41&amp;base=4&lt;br&gt;http://factordb.com/index.php?showid=13&amp;base=4&lt;br&gt;http://factordb.com/index.php?showid=11&amp;base=4&lt;br&gt;http://factordb.com/index.php?showid=7&amp;base=4&lt;br&gt;http://factordb.com/index.php?showid=5&amp;base=4&lt;nowiki/&gt;||0||–||
|-
||5||22||10&lt;sub&gt;93&lt;/sub&gt;13&lt;br&gt;300031&lt;br&gt;44441&lt;br&gt;33331&lt;br&gt;33001&lt;br&gt;30301&lt;br&gt;14444&lt;br&gt;10103&lt;br&gt;3101&lt;br&gt;414||96&lt;br&gt;6&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;4&lt;br&gt;3||67&lt;br&gt;4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;3&lt;br&gt;3&lt;br&gt;3||5&lt;sup&gt;95&lt;/sup&gt;+8&lt;br&gt;9391&lt;br&gt;3121&lt;br&gt;2341&lt;br&gt;2251&lt;br&gt;1951&lt;br&gt;1249&lt;br&gt;653&lt;br&gt;401&lt;br&gt;109||http://factordb.com/index.php?id=1100000000034686071&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=9391&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=3121&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=2341&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=2251&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1951&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1249&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=653&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=401&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=109&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000000034686071&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=9391&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=3121&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=2341&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=2251&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=1951&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=1249&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=653&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=401&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=109&amp;base=5&lt;nowiki/&gt;||0||–||
|-
||6||11||40041&lt;br&gt;4441&lt;br&gt;4401&lt;br&gt;51&lt;br&gt;45&lt;br&gt;35&lt;br&gt;31&lt;br&gt;25&lt;br&gt;21&lt;br&gt;15||5&lt;br&gt;4&lt;br&gt;4&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2||4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2||5209&lt;br&gt;1033&lt;br&gt;1009&lt;br&gt;31&lt;br&gt;29&lt;br&gt;23&lt;br&gt;19&lt;br&gt;17&lt;br&gt;13&lt;br&gt;11||http://factordb.com/index.php?id=5209&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1033&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1009&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=31&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=29&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=23&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=19&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=17&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=13&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=11&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=5209&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=1033&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=1009&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=31&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=29&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=23&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=19&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=17&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=13&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=11&amp;base=6&lt;nowiki/&gt;||0||–||
|-
||7||71||3&lt;sub&gt;16&lt;/sub&gt;1&lt;br&gt;510&lt;sub&gt;7&lt;/sub&gt;1&lt;br&gt;3&lt;sub&gt;6&lt;/sub&gt;01&lt;br&gt;1100021&lt;br&gt;531101&lt;br&gt;351101&lt;br&gt;300053&lt;br&gt;150001&lt;br&gt;100121&lt;br&gt;40054||17&lt;br&gt;10&lt;br&gt;8&lt;br&gt;7&lt;br&gt;6&lt;br&gt;6&lt;br&gt;6&lt;br&gt;6&lt;br&gt;6&lt;br&gt;5||15&lt;br&gt;9&lt;br&gt;7&lt;br&gt;6&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;4||(7&lt;sup&gt;17&lt;/sup&gt;−5)/2&lt;br&gt;36×7&lt;sup&gt;8&lt;/sup&gt;+1&lt;br&gt;(7&lt;sup&gt;8&lt;/sup&gt;−47)/2&lt;br&gt;134471&lt;br&gt;91631&lt;br&gt;62819&lt;br&gt;50459&lt;br&gt;28813&lt;br&gt;16871&lt;br&gt;9643||http://factordb.com/index.php?id=116315256993601&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=207532837&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=2882377&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=134471&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=91631&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=62819&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=50459&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=28813&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=16871&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=9643&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=116315256993601&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=207532837&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=2882377&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=134471&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=91631&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=62819&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=50459&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=28813&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=16871&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=9643&amp;base=7&lt;nowiki/&gt;||0||–||
|-
||8||75||4&lt;sub&gt;220&lt;/sub&gt;7&lt;br&gt;5&lt;sub&gt;13&lt;/sub&gt;25&lt;br&gt;7&lt;sub&gt;12&lt;/sub&gt;1&lt;br&gt;77774&lt;sub&gt;6&lt;/sub&gt;1&lt;br&gt;74&lt;sub&gt;7&lt;/sub&gt;1&lt;br&gt;4&lt;sub&gt;8&lt;/sub&gt;1&lt;br&gt;5&lt;sub&gt;5&lt;/sub&gt;025&lt;br&gt;5550525&lt;br&gt;5500525&lt;br&gt;4&lt;sub&gt;5&lt;/sub&gt;77||221&lt;br&gt;15&lt;br&gt;13&lt;br&gt;11&lt;br&gt;9&lt;br&gt;9&lt;br&gt;8&lt;br&gt;7&lt;br&gt;7&lt;br&gt;7||200&lt;br&gt;14&lt;br&gt;12&lt;br&gt;10&lt;br&gt;9&lt;br&gt;8&lt;br&gt;8&lt;br&gt;7&lt;br&gt;7&lt;br&gt;7||(4×8&lt;sup&gt;221&lt;/sup&gt;+17)/7&lt;br&gt;(5×8&lt;sup&gt;15&lt;/sup&gt;−173)/7&lt;br&gt;8&lt;sup&gt;13&lt;/sup&gt;−7&lt;br&gt;(28669×8&lt;sup&gt;7&lt;/sup&gt;−25)/7&lt;br&gt;(53×8&lt;sup&gt;8&lt;/sup&gt;−25)/7&lt;br&gt;(4×8&lt;sup&gt;9&lt;/sup&gt;−25)/7&lt;br&gt;(5×8&lt;sup&gt;8&lt;/sup&gt;−2413)/7&lt;br&gt;1495381&lt;br&gt;1474901&lt;br&gt;(4×8&lt;sup&gt;7&lt;/sup&gt;+185)/7||http://factordb.com/index.php?id=1100000000416605822&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=25131694349141&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=549755813881&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=8589035809&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=127027489&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=76695841&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=11983381&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1495381&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1474901&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1198399&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000000416605822&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=25131694349141&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=549755813881&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=8589035809&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=127027489&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=76695841&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=11983381&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=1495381&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=1474901&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=1198399&amp;base=8&lt;nowiki/&gt;||0||–||
|-
||9||151||30&lt;sub&gt;1158&lt;/sub&gt;11&lt;br&gt;27&lt;sub&gt;686&lt;/sub&gt;07&lt;br&gt;76&lt;sub&gt;329&lt;/sub&gt;2&lt;br&gt;561&lt;sub&gt;36&lt;/sub&gt;&lt;br&gt;10&lt;sub&gt;25&lt;/sub&gt;57&lt;br&gt;30&lt;sub&gt;20&lt;/sub&gt;51&lt;br&gt;8&lt;sub&gt;19&lt;/sub&gt;335&lt;br&gt;727&lt;sub&gt;15&lt;/sub&gt;07&lt;br&gt;51&lt;sub&gt;13&lt;/sub&gt;61&lt;br&gt;10&lt;sub&gt;11&lt;/sub&gt;507||1161&lt;br&gt;689&lt;br&gt;331&lt;br&gt;38&lt;br&gt;28&lt;br&gt;23&lt;br&gt;22&lt;br&gt;19&lt;br&gt;16&lt;br&gt;15||1108&lt;br&gt;657&lt;br&gt;316&lt;br&gt;37&lt;br&gt;26&lt;br&gt;22&lt;br&gt;21&lt;br&gt;19&lt;br&gt;16&lt;br&gt;14||3×9&lt;sup&gt;1160&lt;/sup&gt;+10&lt;br&gt;(23×9&lt;sup&gt;688&lt;/sup&gt;−511)/8&lt;br&gt;(31×9&lt;sup&gt;330&lt;/sup&gt;−19)/4&lt;br&gt;(409×9&lt;sup&gt;36&lt;/sup&gt;−1)/8&lt;br&gt;9&lt;sup&gt;27&lt;/sup&gt;+52&lt;br&gt;3×9&lt;sup&gt;22&lt;/sup&gt;+46&lt;br&gt;9&lt;sup&gt;22&lt;/sup&gt;−454&lt;br&gt;(527×9&lt;sup&gt;17&lt;/sup&gt;−511)/8&lt;br&gt;(41×9&lt;sup&gt;15&lt;/sup&gt;+359)/8&lt;br&gt;9&lt;sup&gt;14&lt;/sup&gt;+412||http://factordb.com/index.php?id=1100000002376318423&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000002495467486&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000002359003642&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000001554010824&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1100000002512830927&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1100000000032261811&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1100000002495736583&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1100000003446800389&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1055192051985121&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=22876792455373&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000002376318423&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=1100000002495467486&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=1100000002359003642&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=1100000001554010824&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=1100000002512830927&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=1100000000032261811&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=1100000002495736583&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=1100000003446800389&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=1055192051985121&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=22876792455373&amp;base=9&lt;nowiki/&gt;||0||–||
|-
||10||77||50&lt;sub&gt;28&lt;/sub&gt;27&lt;br&gt;5&lt;sub&gt;11&lt;/sub&gt;1&lt;br&gt;805&lt;sub&gt;5&lt;/sub&gt;1&lt;br&gt;66600049&lt;br&gt;66000049&lt;br&gt;60&lt;sub&gt;5&lt;/sub&gt;49&lt;br&gt;220&lt;sub&gt;5&lt;/sub&gt;1&lt;br&gt;5200007&lt;br&gt;946669&lt;br&gt;666649||31&lt;br&gt;12&lt;br&gt;8&lt;br&gt;8&lt;br&gt;8&lt;br&gt;8&lt;br&gt;8&lt;br&gt;7&lt;br&gt;6&lt;br&gt;6||31&lt;br&gt;12&lt;br&gt;8&lt;br&gt;8&lt;br&gt;8&lt;br&gt;8&lt;br&gt;8&lt;br&gt;7&lt;br&gt;6&lt;br&gt;6||5×10&lt;sup&gt;30&lt;/sup&gt;+27&lt;br&gt;(5×10&lt;sup&gt;12&lt;/sup&gt;−41)/9&lt;br&gt;(725×10&lt;sup&gt;6&lt;/sup&gt;−41)/9&lt;br&gt;66600049&lt;br&gt;66000049&lt;br&gt;6×10&lt;sup&gt;7&lt;/sup&gt;+49&lt;br&gt;22×10&lt;sup&gt;6&lt;/sup&gt;+1&lt;br&gt;5200007&lt;br&gt;946669&lt;br&gt;666649||http://factordb.com/index.php?id=1100000000204142046&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=555555555551&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=80555551&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=66600049&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=66000049&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=60000049&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=22000001&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=5200007&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=946669&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=666649&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000000204142046&amp;base=10&lt;br&gt;http://factordb.com/index.php?showid=555555555551&amp;base=10&lt;br&gt;http://factordb.com/index.php?showid=80555551&amp;base=10&lt;br&gt;http://factordb.com/index.php?showid=66600049&amp;base=10&lt;br&gt;http://factordb.com/index.php?showid=66000049&amp;base=10&lt;br&gt;http://factordb.com/index.php?showid=60000049&amp;base=10&lt;br&gt;http://factordb.com/index.php?showid=22000001&amp;base=10&lt;br&gt;http://factordb.com/index.php?showid=5200007&amp;base=10&lt;br&gt;http://factordb.com/index.php?showid=946669&amp;base=10&lt;br&gt;http://factordb.com/index.php?showid=666649&amp;base=10&lt;nowiki/&gt;||0||–||
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||24||3409||N00N&lt;sub&gt;8129&lt;/sub&gt;LN&lt;br&gt;88N&lt;sub&gt;5951&lt;/sub&gt;&lt;br&gt;A0&lt;sub&gt;2951&lt;/sub&gt;8ID&lt;br&gt;D&lt;sub&gt;2698&lt;/sub&gt;LD&lt;br&gt;N&lt;sub&gt;2644&lt;/sub&gt;LLN&lt;br&gt;BC0&lt;sub&gt;331&lt;/sub&gt;B&lt;br&gt;20&lt;sub&gt;313&lt;/sub&gt;7&lt;br&gt;C7&lt;sub&gt;298&lt;/sub&gt;&lt;br&gt;D0&lt;sub&gt;259&lt;/sub&gt;KKD&lt;br&gt;I0&lt;sub&gt;241&lt;/sub&gt;I5||8134&lt;br&gt;5953&lt;br&gt;2955&lt;br&gt;2700&lt;br&gt;2647&lt;br&gt;334&lt;br&gt;315&lt;br&gt;299&lt;br&gt;263&lt;br&gt;244||11227&lt;br&gt;8216&lt;br&gt;4079&lt;br&gt;3727&lt;br&gt;3654&lt;br&gt;461&lt;br&gt;434&lt;br&gt;413&lt;br&gt;363&lt;br&gt;337||13249×24&lt;sup&gt;8131&lt;/sup&gt;−49&lt;br&gt;201×24&lt;sup&gt;5951&lt;/sup&gt;−1&lt;br&gt;10×24&lt;sup&gt;2954&lt;/sup&gt;+5053&lt;br&gt;(13×24&lt;sup&gt;2700&lt;/sup&gt;+4403)/23&lt;br&gt;24&lt;sup&gt;2647&lt;/sup&gt;−1201&lt;br&gt;276×24&lt;sup&gt;332&lt;/sup&gt;+11&lt;br&gt;2×24&lt;sup&gt;314&lt;/sup&gt;+7&lt;br&gt;(283×24&lt;sup&gt;298&lt;/sup&gt;−7)/23&lt;br&gt;13×24&lt;sup&gt;262&lt;/sup&gt;+12013&lt;br&gt;18×24&lt;sup&gt;243&lt;/sup&gt;+437||http://factordb.com/index.php?id=1100000003593391606&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003593275880&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003593269654&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003593269876&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003593270089&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000002633359842&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000002355610241&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000002326181235&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003593270725&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000002633360037&amp;open=prime&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000003593391606&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000003593275880&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000003593269654&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000003593269876&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000003593270089&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000002633359842&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000002355610241&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000002326181235&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000003593270725&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000002633360037&amp;base=24&lt;nowiki/&gt;||0||–||
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||36||35286~35290||P&lt;sub&gt;81993&lt;/sub&gt;SZ&lt;br&gt;S0&lt;sub&gt;75007&lt;/sub&gt;8H&lt;br&gt;7K&lt;sub&gt;26567&lt;/sub&gt;Z&lt;br&gt;J&lt;sub&gt;10117&lt;/sub&gt;LJ&lt;br&gt;VL0&lt;sub&gt;7258&lt;/sub&gt;J&lt;br&gt;EO0&lt;sub&gt;6177&lt;/sub&gt;V&lt;br&gt;FZ&lt;sub&gt;5777&lt;/sub&gt;3P&lt;br&gt;T09&lt;sub&gt;4618&lt;/sub&gt;1&lt;br&gt;RY&lt;sub&gt;4562&lt;/sub&gt;H&lt;br&gt;OZ&lt;sub&gt;3932&lt;/sub&gt;AZ||81995&lt;br&gt;75010&lt;br&gt;26569&lt;br&gt;10119&lt;br&gt;7261&lt;br&gt;6180&lt;br&gt;5780&lt;br&gt;4621&lt;br&gt;4564&lt;br&gt;3935||127609&lt;br&gt;116739&lt;br&gt;41349&lt;br&gt;15748&lt;br&gt;11301&lt;br&gt;9618&lt;br&gt;8996&lt;br&gt;7192&lt;br&gt;7103&lt;br&gt;6124||(5×36&lt;sup&gt;81995&lt;/sup&gt;+821)/7&lt;br&gt;28×36&lt;sup&gt;75009&lt;/sup&gt;+305&lt;br&gt;(53×36&lt;sup&gt;26568&lt;/sup&gt;+101)/7&lt;br&gt;(19×36&lt;sup&gt;10119&lt;/sup&gt;+2501)/35&lt;br&gt;1137×36&lt;sup&gt;7259&lt;/sup&gt;+19&lt;br&gt;528×36&lt;sup&gt;6178&lt;/sup&gt;+31&lt;br&gt;16×36&lt;sup&gt;5779&lt;/sup&gt;−1163&lt;br&gt;(36549×36&lt;sup&gt;4619&lt;/sup&gt;−289)/35&lt;br&gt;(979×36&lt;sup&gt;4563&lt;/sup&gt;−629)/35&lt;br&gt;25×36&lt;sup&gt;3934&lt;/sup&gt;−901||http://factordb.com/index.php?id=1100000002394962083&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004020085177&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003896952461&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362491&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362489&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362488&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362487&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362486&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362485&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000000840634476&amp;open=prime&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000002394962083&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000004020085177&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003896952461&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362491&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362489&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362488&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362487&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362486&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362485&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000000840634476&amp;base=36&lt;nowiki/&gt;||4||200000||
|}

== The fully proof of Athena problem in decimal (base ''b'' = 10) ==

'''Bold''' for the Athena primes, ''x'' ◁ ''y'' means ''x'' is a subsequence of ''y''.

Assume ''p'' is a prime &gt; 10, and the last digit of ''p'' must lie in {1,3,7,9}.

Case 1: ''p'' ends with 1.

In this case we can write ''p'' = ''x''1. If ''x'' contains 1, 3, 4, 6, or 7, then (respectively) '''11''' ◁ ''p'', '''31''' ◁ ''p'', '''41''' ◁ ''p'', '''61''' ◁ ''p'', or '''71''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 2, 5, 8, or 9.

Case 1.1: ''p'' begins with 2.

In this case we can write ''p'' = 2''y''1. If 5 ◁ ''y'', then '''251''' ◁ ''p''. If 8 ◁ ''y'', then '''281''' ◁ ''p''. If 9 ◁ ''y'', then 29 ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 2.

If 22 ◁ ''y'', then '''2221''' ◁ ''p''. Hence we may assume ''y'' contains zero or one 2's.

If ''y'' contains no 2's, then ''p'' ∈ 2{0}1. But then, since the sum of the digits of ''p'' is 3, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' contains exactly one 2, then we can write ''p'' = 2''z''2''w''1, where ''z'',''w'' ∈ {0}. If 0 ◁ ''z'' and 0 ◁ ''w'', then '''20201''' ◁ ''p''. Hence we may assume either ''z'' or ''w'' is empty.

If ''z'' is empty, then ''p'' ∈ 22{0}1, and the smallest prime ''p'' ∈ 22{0}1 is '''22000001'''.

If ''w'' is empty, then ''p'' ∈ 2{0}21, and the smallest prime ''p'' ∈ 2{0}21 is '''20021'''.

Case 1.2: ''p'' begins with 5.

In this case we can write ''p'' = 5''y''1. If 2 ◁ ''y'', then '''521''' ◁ ''p''. If 9 ◁ ''y'', then 59 ◁ ''p''. Hence we may assume all digits of ''y'' are 0, 5, or 8.

If 05 ◁ ''y'', then '''5051''' ◁ ''p''. If 08 ◁ ''y'', then '''5081''' ◁ ''p''. If 50 ◁ ''y'', then '''5501''' ◁ ''p''. If 58 ◁ ''y'', then '''5581''' ◁ ''p''. If 80 ◁ ''y'', then '''5801''' ◁ ''p''. If 85 ◁ ''y'', then '''5851''' ◁ ''p''. Hence we may assume ''y'' ∈ {0} ∪ {5} ∪ {8}.

If ''y'' ∈ {0}, then ''p'' ∈ 5{0}1. But then, since the sum of the digits of ''p'' is 6, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' ∈ {5}, then ''p'' ∈ 5{5}1, and the smallest prime ''p'' ∈ 5{5}1 is '''555555555551'''.

If ''y'' ∈ {8}, since if 88 ◁ ''y'', then 881 ◁ ''p'', hence we may assume ''y'' ∈ {''𝜆'',8}, and thus ''p'' ∈ {51,581}, but 51 and 581 are both composite.

Case 1.3: ''p'' begins with 8.

In this case we can write p = 8''y''1. If 2 ◁ ''y'', then '''821''' ◁ ''p''. If 8 ◁ ''y'', then '''881''' ◁ ''p''. If 9 ◁ ''y'', then 89 ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 5.

If 50 ◁ ''y'', then '''8501''' ◁ ''p''. Hence we may assume y ∈ {0}{5}.

If 005 ◁ ''y'', then '''80051''' ◁ p. Hence we may assume y ∈ {0} ∪ {5} ∪ 0{5}.

If y ∈ {0}, then ''p'' ∈ 8{0}1. But then, since the sum of the digits of ''p'' is 9, ''p'' is divisible by 3, so ''p'' cannot be prime.

If y ∈ {5}, since if 55555555555 ◁ ''y'', then 555555555551 ◁ ''p'', hence we may assume ''y'' ∈ {''𝜆'', 5, 55, 555, 5555, 55555, 555555, 5555555, 55555555, 555555555, 5555555555}, and thus ''p'' ∈ {81, 851, 8551, 85551, 855551, 8555551, 85555551, 855555551, 8555555551, 85555555551, 855555555551}, but all of these numbers are composite.

If y ∈ 0{5}, since if 55555555555 ◁ ''y'', then 555555555551 ◁ ''p'', hence we may assume ''y'' ∈ {0, 05, 055, 0555, 05555, 055555, 0555555, 05555555, 055555555, 0555555555, 05555555555}, and thus ''p'' ∈ {801, 8051, 80551, 805551, 8055551, 80555551, 805555551, 8055555551, 80555555551, 805555555551, 8055555555551}, and of these numbers only 80555551 and 8055555551 are primes, but 80555551 ◁ 8055555551, thus only '''80555551''' is a minimal element.

Case 1.4: ''p'' begins with 9.

In this case we can write p = 9''y''1. If 9 ◁ ''y'', then '''991''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0, 2, 5, or 8.

If 00 ◁ ''y'', then '''9001''' ◁ ''p''. If 22 ◁ ''y'', then '''9221''' ◁ ''p''. If 55 ◁ ''y'', then '''9551''' ◁ ''p''. If 88 ◁ ''y'', then 881 ◁ ''p''. Hence we may assume ''y'' contains at most one 0, at most one 2, at most one 5, and at most one 8.

If ''y'' only contains at most one 0 and does not contain any of {2,5,8}, then ''y'' ∈ {''𝜆'',0}, and thus ''p'' ∈ {91,901}, but 91 and 901 are both composite. If ''y'' only contains at most one 0 and only one of {2,5,8}, then the sum of the digits of ''p'' is divisible by 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Hence we may assume ''y'' contains at least two of {2,5,8}.

If 25 ◁ ''y'', then 251 ◁ ''p''. If 28 ◁ ''y'', then 281 ◁ ''p''. If 52 ◁ ''y'', then 521 ◁ ''p''. If 82 ◁ ''y'', then 821 ◁ ''p''. Hence we may assume ''y'' contains no 2's (since if ''y'' contains 2, then ''y'' cannot contain either 5's or 8's, which is a contradiction).

If 85 ◁ ''y'', then '''9851''' ◁ ''p''. Hence we may assume ''y'' ∈ {58,580,508,058}, and thus ''p'' ∈ {9581,95801,95081,90581}, and of these numbers only 95801 is prime, but 95801 is not a minimal element since 5801 ◁ 95801.

Case 2: ''p'' ends with 3.

In this case we can write p = ''x''3. If ''x'' contains 1, 2, 4, 5, 7, or 8, then (respectively) '''13''' ◁ ''p'', '''23''' ◁ ''p'', '''43''' ◁ ''p'', '''53''' ◁ ''p'', '''73''' ◁ ''p'', or '''83''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 3, 6, or 9, and thus all digits of ''p'' are 0, 3, 6, or 9. But then, since the digits of ''p'' all have a common factor 3, ''p'' is divisible by 3, so ''p'' cannot be prime.

Case 3: ''p'' ends with 7.

In this case we can write ''p'' = ''x''7. If ''x'' contains 1, 3, 4, 6, or 9, then (respectively) '''17''' ◁ ''p'', '''37''' ◁ ''p'', '''47''' ◁ ''p'', '''67''' ◁ ''p'', or '''97''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 2, 5, 7, or 8.

Case 3.1: ''p'' begins with 2.

In this case we can write ''p'' = 2''y''7. If 2 ◁ ''y'', then '''227''' ◁ ''p''. If 5 ◁ ''y'', then '''257''' ◁ ''p''. If 7 ◁ ''y'', then '''277''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 8.

If 08 ◁ ''y'', then '''2087''' ◁ ''p''. If 88 ◁ ''y'', then 887 ◁ ''p''. Hence we may assume ''y'' ∈ {0} ∪ 8{0}.

If ''y'' ∈ {0}, then ''p'' ∈ 2{0}7. But then, since the sum of the digits of ''p'' is 9, ''p'' is divisible by 3, so ''p'' cannot be prime.

If y ∈ 8{0}, then ''p'' ∈ 28{0}7. But then ''p'' is divisible by 7, since for ''n'' ≥ 0 we have 7 × 40&lt;sub&gt;''n''&lt;/sub&gt;1 = 280&lt;sub&gt;''n''&lt;/sub&gt;7.

Case 3.2: ''p'' begins with 5.

In this case we can write ''p'' = 5''y''7. If 5 ◁ ''y'', then '''557''' ◁ ''p''. If 7 ◁ ''y'', then '''577''' ◁ ''p''. If 8 ◁ ''y'', then '''587''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 2.

If 22 ◁ ''y'', then 227 ◁ ''p''. Hence we may assume ''y'' contains zero or one 2's.

If ''y'' contains no 2's, then ''p'' ∈ 5{0}7. But then, since the sum of the digits of ''p'' is 12, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' contains exactly one 2, then we can write ''p'' = 5''z''2''w''7, where ''z'',''w'' ∈ {0}. If 0 ◁ ''z'' and 0 ◁ ''w'', then '''50207''' ◁ ''p''. Hence we may assume either ''z'' or ''w'' is empty.

If ''z'' is empty, then ''p'' ∈ 52{0}7, and the smallest prime ''p'' ∈ 52{0}7 is '''5200007'''.

If ''w'' is empty, then ''p'' ∈ 5{0}27, and the smallest prime ''p'' ∈ 5{0}27 is '''5000000000000000000000000000027'''.

Case 3.3: ''p'' begins with 7.

In this case we can write ''p'' = 7''y''7. If 2 ◁ ''y'', then '''727''' ◁ ''p''. If 5 ◁ ''y'', then '''757''' ◁ ''p''. If 8 ◁ ''y'', then '''787''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 7, and thus all digits of ''p'' are 0 or 7. But then, since the digits of ''p'' all have a common factor 7, ''p'' is divisible by 7, so ''p'' cannot be prime.

Case 3.4: ''p'' begins with 8.

In this case we can write ''p'' = 8''y''7. If 2 ◁ ''y'', then '''827''' ◁ ''p''. If 5 ◁ ''y'', then '''857''' ◁ ''p''. If 7 ◁ ''y'', then '''877''' ◁ ''p''. If 8 ◁ ''y'', then '''887''' ◁ ''p''. Hence we may assume ''y'' ∈ {0}, and thus ''p'' ∈ 8{0}7. But then, since the sum of the digits of ''p'' is 15, ''p'' is divisible by 3, so ''p'' cannot be prime.

Case 4: ''p'' ends with 9.

In this case we can write ''p'' = ''x''9. If ''x'' contains 1, 2, 5, 7, or 8, then (respectively) '''19''' ◁ ''p'', '''29''' ◁ ''p'', '''59''' ◁ ''p'', '''79''' ◁ ''p'', or '''89''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 3, 4, 6, or 9.

If 44 ◁ ''x'', then '''449''' ◁ ''p''. Hence we may assume ''x'' contains zero or one 4's.

If x contains no 4's, then all digits of ''x'' are 0, 3, 6, or 9, and thus all digits of ''p'' are 0, 3, 6, or 9. But then, since the digits of ''p'' all have a common factor 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Hence we may assume that ''x'' contains exactly one 4.

Case 4.1: ''p'' begins with 3.

In this case we can write ''p'' = 3''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. We must have '''349''' ◁ ''p''.

Case 4.2: ''p'' begins with 4.

In this case we can write ''p'' = 4''y''9, where all digits of ''y'' are 0, 3, 6, or 9. If 0 ◁ ''y'', then '''409''' ◁ ''p''. If 3 ◁ ''y'', then 43 ◁ ''p''. If 9 ◁ ''y'', then '''499''' ◁ ''p''. Hence we may assume ''y'' ∈ {6}, and thus ''p'' ∈ 4{6}9. But then ''p'' is divisible by 7, since for ''n'' ≥ 0 we have 7 × 6&lt;sub&gt;''n''&lt;/sub&gt;7 = 46&lt;sub&gt;''n''&lt;/sub&gt;9.

Case 4.3: ''p'' begins with 6.

In this case we can write p = 6''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. If 0 ◁ ''z'', then 409 ◁ ''p''. If 3 ◁ ''z'', then 43 ◁ ''p''. If 6 ◁ ''z'', then '''6469''' ◁ ''p''. If 9 ◁ ''z'', then 499 ◁ ''p''. Hence we may assume ''z'' is empty.

If 3 ◁ ''y'', then 349 ◁ ''p''. If 9 ◁ ''y'', then '''6949''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 6.

If 06 ◁ ''y'', then '''60649''' ◁ ''p''. Hence we may assume ''y'' ∈ {6}{0}.

If 666 ◁ ''y'', then '''666649''' ◁ ''p''. If 00000 ◁ ''y'', then '''60000049''' ◁ ''p''. Hence we may assume ''y'' ∈ {''𝜆'', 0, 00, 000, 0000, 6, 60, 600, 6000, 60000, 66, 660, 6600, 66000, 660000}, and thus ''p'' ∈ {649, 6049, 60049, 600049, 6000049, 6649, 66049, 660049, 6600049, 66000049, 66649, 666049, 6660049, 66600049, 666000049}, and of these numbers only '''66000049''' and '''66600049''' are primes.

Case 4.4: ''p'' begins with 9.

In this case we can write p = 9''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. If 0 ◁ ''y'', then '''9049''' ◁ ''p''. If 3 ◁ ''y'', then 349 ◁ ''p''. If 6 ◁ ''y'', then '''9649''' ◁ ''p''. If 9 ◁ ''y'', then '''9949''' ◁ ''p''. Hence we may assume ''y'' is empty.

If 0 ◁ ''z'', then 409 ◁ ''p''. If 3 ◁ ''z'', then 43 ◁ ''p''. If 9 ◁ ''z'', then 499 ◁ ''p''. Hence we may assume ''z'' ∈ {6}, and thus ''p'' ∈ 94{6}9, and the smallest prime ''p'' ∈ 94{6}9 is 946669.

[[Category:Number theory]]</text>
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      <text bytes="128840" sha1="8rz29c0namfidl5zcdmshhq95yv26wr" xml:space="preserve">{{mathematics}}
'''Athena problem''' is an [[:w:List of unsolved problems in mathematics|unsolved problem]] in [[:w:Number theory|number theory]] and [[:w:Formal language theory|formal language theory]] and [[:w:Order theory|order theory]], this problem is named after the ancient Greek goddess [[:w:Athena|Athena]] (which is associated with [[:w:Wisdom|wisdom]]). Athena problem is: Give a [[:w:Natural number|natural number]] ''b'' &gt; 1, find the [[:w:Set (mathematics)|set]] of the [[:w:Minimal element|minimal element]]s of the set of the &quot;[[:w:Prime number|prime number]] [[:w:Greater than|&gt;]] ''b''&quot; [[:w:Numerical digit|digit]] [[:w:String (computer science)|string]]s in the [[:w:Positional numeral system|positional numeral system]] with [[:w:Radix|base]] ''b'' for the [[:w:Subsequence|subsequence]] [[:w:Partially ordered set|ordering]]. (A string ''x'' is a subsequence of another string ''y'', if ''x'' can be obtained from ''y'' by deleting zero or more of the [[:w:Character (computing)|character]]s in ''y''. For example, 514 is a subsequence of 352148, &quot;string&quot; is a subsequence of &quot;meistersinger&quot;. In contrast, 758 is not a subsequence of 378259, &quot;abc&quot; is not a subsequence of &quot;cbacacba&quot;, since the characters must be in the same order) (Unlike [[:w:Substring|substring]], subsequence is not required to occupy consecutive positions within the original sequences, e.g. the [[:w:Longest common subsequence|longest common subsequence problem]] is different from the [[:w:Longest common substring|longest common substring problem]])

Using [[:w:Formal language theory|formal language theory]] terminology, Athena problem is finding the [[:w:Set (mathematics)|set]] of the [[:w:Minimal element|minimal element]]s of the [[:w:Formal language|language]] of base-''b'' [[:w:Representation (mathematics)|representation]]s of the [[:w:Prime number|prime number]]s [[:w:Greater than|&gt;]] ''b'' (which is a set of [[:w:String (computer science)|string]]s of [[:w:Symbol|symbol]]s over the [[:w:Alphabet (formal languages)|alphabet]] ''Σ''&lt;sub&gt;''b''&lt;/sub&gt; := {0, 1, ..., ''b''−1}), under the subsequence ordering (i.e. the [[:w:Binary relation|binary relation]] &quot;is a subsequence of&quot;, which is a [[:w:Partially ordered set|partial ordering]]), for a given natural number ''b'' &gt; 1 (You can draw this partial ordering as a [[:w:Hasse diagram|Hasse diagram]] to find all [[:w:Minimal element|minimal element]]s), this set is called '''Athena set base ''b''''', and the prime numbers in this set is called '''Athena primes base ''b'''''.

By [[:w:Higman's lemma|Higman's lemma]], there are no [[:w:Infinite set|infinite]] [[:w:Antichain|antichain]]s for the subsequence ordering (i.e. the subsequence ordering is always a [[:w:Well-quasi-ordering|well quasi order]]) (i.e. under the subsequence ordering (i.e. the [[:w:Binary relation|binary relation]] &quot;is a subsequence of&quot;, which is a [[:w:Partially ordered set|partial ordering]]), every set of pairwise incomparable (i.e. not [[:w:Comparability|comparable]]) strings is finite), thus there must be only finitely many such minimal elements. In other words, the Athena set in every base ''b'' must be a [[:w:Finite set|finite set]], and every base ''b'' ≥ 2 has only finitely many Athena primes, e.g. in [[:w:Decimal|decimal]] (base ''b'' = 10), the Athena set has exactly 77 [[:w:Element of a set|element]]s (the Athena primes in decimal (base ''b'' = 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}.

For bases 2 ≤ ''b'' ≤ 36, Athena problem is fully solved in bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 24, and also solved in bases ''b'' = 11, 13, 16, 22, 30 if [[:w:Probable prime|probable prime]]s are allowed. For the unsolved bases ''b'' = 17, 19, 21, 23, 25, 26, 27, 28, 29, 31, 32, 34, 35, 36, Athena problem is solved (if probable primes are allowed) except 771 [[:w:Indexed family|families]] of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be [[:w:Empty string|empty]]) of digits in base ''b'', ''y'' is a digit in base ''b'') = sequence {''xz'', ''xyz'', ''xyyz'', ''xyyyz'', ''xyyyyz'', ''xyyyyyz'', ...} (i.e. &quot;''xy''&lt;sup&gt;+&lt;/sup&gt;''z''&quot; in [[:w:Regular expression|regular expression]]), all of these 771 families contain no primes &gt; ''b'' or probable primes &gt; ''b'' with length ≤ 100000.

== Solve the problem ==

To solve the Athena problem for a given base ''b'', we must [[:w:Computing|compute]] the elements up to families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''), and find the smallest prime &gt; ''b'' in all such families.

We call families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') &quot;linear&quot; families, and we reduce these families by removing all trailing digits ''y'' from ''x'', and removing all leading digits ''y'' from ''z'', to make the families be easier, e.g. family 12333{3}33345 in base ''b'' is reduced to family 12{3}45 in base ''b'', since they are in fact the same family. Our [[:w:Algorithm|algorithm]] then proceeds as follows:

* 1. ''M'' := {minimal primes in base ''b'' of length 2 or 3}, ''L'' := union of all ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'') such that ''x'' ≠ 0 and ''gcd''(''z'', ''b'') = 1 and ''Y'' is the set of digits ''y'' in base ''b'' such that ''xyz'' has no subsequence in ''M''.
* 2. While ''L'' contains nonlinear families (families which are not linear families): Explore each family of ''L'', and update ''L''. Examine each family of ''L'' by:
* 2.1. Let ''w'' be the shortest string in the family. If ''w'' has a subsequence in ''M'', then remove the family from ''L''. If ''w'' represents a prime, then add ''w'' to ''M'' and remove the family from ''L''.
* 2.2. If possible, simplify the family.
* 2.3. Using the techniques below (covering congruence, algebraic factorization, or combine of them), check if the family can be proven to only contain composites (only count the numbers &gt; ''b''), and if so then remove the family from ''L''.
* 3. Update ''L'', after each split examine the new families as in step 2.

e.g. in decimal (base ''b'' = 10):

''M'' := {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}

''L'' := {2{0,2}1, 2{0,8}7, 3{0,3,6,9}3, 3{0,3,6,9}9, 4{6}9, 5{0,5,8}1, 5{0,2}7, 6{0,3,6,9}3, 6{0,3,4,6,9}9, 7{0,7}7, 8{0,5}1, 8{0}7, 9{0,2,5,8}1, 9{0,3,6,9}3, 9{0,3,4,6,9}9}

and since 2221 is prime, it follows that the family 2{0,2}1 splits into the families 2{0}1 and 2{0}2{0}1

and since the family 2{0}1 can be proven to contain no primes &gt; base (since all numbers in this family are divisible by 3), it can be removed

and since 20201 is prime, it follows that the family 2{0}2{0}1 splits into the families 2{0}21 and 22{0}1

221 and 2021 are composites, but 20021 is prime, thus add 20021 to ''L''

none of 221, 2201, 22001, 220001, 2200001 are primes, but 22000001 is prime, thus add 22000001 to ''L''

and since the family 3{0,3,6,9}3 can be proven to contain no primes &gt; base (since all numbers in this family are divisible by 3), it can be removed

etc.

Since the number of possible (first digit,last digit) (also called (initial digit,final digit)) combos ([[:w:Ordered pair|ordered pair]]s) of a prime &gt; ''b'' in base ''b'' is (''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(''b'') (all digits except 0 can be the first digit of a prime &gt; ''b'' in base ''b'' (thus ''b''−1 possible digits), but only the digits coprime to ''b'' can be the last digit of a prime &gt; ''b'' in base ''b'' (thus ''eulerphi''(''b'') possible digits), and by the [[:w:Rule of product|rule of product]], there are (''b''−1)×''eulerphi''(''b'') choices of the (first digit,last digit) combo, also, both &quot;numbers of Athena primes in base ''b''&quot; and &quot;length of the largest Athena prime in base ''b''&quot; are [[:w:Asymptotic analysis|roughly]] ''[[:w:E (mathematical_constant)|e]]''&lt;sup&gt;''[[:w:Euler's constant|γ]]''×(''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(*b*)&lt;/sup&gt;.

Shrinking the family ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''Y'' is a set of digits in base ''b'')

* If ''y'' ∈ ''Y'' and the string ''xyyz'' represents a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or has a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''}''z'' ∪ ''x''{''Y'' \ ''y''}''y''{''Y'' \ ''y''}''z''.
* If ''y''&lt;sub&gt;1&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;2&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;1&lt;/sub&gt; ≠ ''y''&lt;sub&gt;2&lt;/sub&gt; and the string ''xy''&lt;sub&gt;1&lt;/sub&gt;''y''&lt;sub&gt;2&lt;/sub&gt;''z'' represents a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or has a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''&lt;sub&gt;1&lt;/sub&gt;}{''Y'' \ ''y''&lt;sub&gt;2&lt;/sub&gt;}''z''.
* If ''y''&lt;sub&gt;1&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;2&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;1&lt;/sub&gt; ≠ ''y''&lt;sub&gt;2&lt;/sub&gt; and both the strings ''xy''&lt;sub&gt;1&lt;/sub&gt;''y''&lt;sub&gt;2&lt;/sub&gt;''z'' and ''xy''&lt;sub&gt;2&lt;/sub&gt;''y''&lt;sub&gt;1&lt;/sub&gt;''z'' represent a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or have a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''&lt;sub&gt;1&lt;/sub&gt;}''z'' ∪ ''x''{''Y'' \ ''y''&lt;sub&gt;2&lt;/sub&gt;}''z''.

e.g. in decimal (base ''b'' = 10):

* 2221 is a prime &gt; 10, thus the family 2{0,2}1 splits into the two families 2{0}1 and 2{0}2{0}1.
* 227 is a prime &gt; 10, and it is a subsequence of 5227, thus the family 5{0,2}7 splits into the two families 5{0}7 and 5{0}2{0}7.
* 449 is a prime &gt; 10, and it is a subsequence of 6449, thus the family 6{0,3,4,6,9}9 splits into the two families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9.
* Both 5051 and 5501 are primes &gt; 10, thus the family 5{0,5}1 splits into the two families 5{0}1 and 5{5}1 = {5}1.
* 8501 is a prime &gt; 10, thus the family 8{0,5}1 splits into the family 8{0}{5}1.
* 887 is a prime &gt; 10, and it is a subsequence of 2887, also 2087 is a prime &gt; 10, thus the family 2{0,8}7 splits into the two families 2{0}7 and 28{0}7.
* 349 and 449 are primes &gt; 10, and they are subsequences of 9349 and 9449, respectively, also 9049, 9649, 9949 are primes &gt; 10, thus the family 9{0,3,4,6,9}9 splits into the two families 9{0,3,6,9}9 and 94{0,3,6,9}9.
* 251, 281, 521, 821, 881 are primes &gt; 10, and they are subsequences of 9251, 9281, 9521, 9821, 9881, respectively, also 9001, 9221, 9551, 9851 are primes &gt; 10, thus the family 9{0,2,5,8}1 splits into the numbers {91, 901, 921, 951, 981, 9021, 9051, 9081, 9201, 9501, 9581, 9801, 90581, 95081, 95801}.

If the methods we have discussed cannot be used to rule out or shrink ''x''{''Y''}''z'' where ''Y'' = {''y''&lt;sub&gt;1&lt;/sub&gt;, ''y''&lt;sub&gt;2&lt;/sub&gt;, ..., ''y''&lt;sub&gt;''n''&lt;/sub&gt;}, then we can replace ''x''{''Y''}''z'' by ''xy''&lt;sub&gt;1&lt;/sub&gt;{''Y''}''z'' ∪ ''xy''&lt;sub&gt;2&lt;/sub&gt;{''Y''}''z'' ∪ ... ∪ ''xy''&lt;sub&gt;''n''&lt;/sub&gt;{''Y''}''z'' and re-run the methods on this new [[:w:Formal language|language]].

If all remain families are linear families (i.e. of the form ''x''{''y''}''z'', where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''), then we search the smallest (probable) primes in these families and add these primes to the list.

e.g. in decimal (base ''b'' = 10):

* The smallest prime in the family 5{0}27 is 5000000000000000000000000000027.
* The smallest prime in the family {5}1 is 555555555551.
* The smallest prime in the family 8{5}1 is 8555555555555555555551, but 8555555555555555555551 is not a minimal element since 555555555551 is a subsequence of 8555555555555555555551.

There is no guarantee that the techniques discussed will ever terminate, but in practice they often do. They are able to determine the Athena set in base ''b'' for 2 ≤ ''b'' ≤ 16 and ''b'' = 18, 20, 22, 24, 30. The bases ''b'' = 17, 19, 21, 23, 25 ≤ ''b'' ≤ 29, 31 ≤ ''b'' ≤ 36 are solved with the exception of 771 families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'').

The following is a &quot;[[:w:Semi-algorithm|semi-algorithm]]&quot; that is guaranteed to solve the Athena problem for a given base ''b'', but it is not so easy to implement:

# ''M'' = ''[[:w:Empty string|∅]]''
# while (''L'' ≠ ''∅'') do
# choose ''x'', a shortest string in ''L''
# ''M'' := ''M'' ∪ {''x''}
# ''L'' := ''L'' − ''sup''({''x''})

In practice, for arbitrary ''L'', we cannot feasibly carry out step 5. Instead, we work with ''L''&amp;#39;, some regular overapproximation to ''L'', until we can show ''L''&amp;#39; = ''∅'' (which implies ''L'' = ''∅''). In practice, ''L''&amp;#39; is usually chosen to be a finite [[:w:Union (set theory)|union]] of sets of the form ''L''&lt;sub&gt;1&lt;/sub&gt;{''L''&lt;sub&gt;2&lt;/sub&gt;}''L''&lt;sub&gt;3&lt;/sub&gt;, where each of ''L''&lt;sub&gt;1&lt;/sub&gt;, ''L''&lt;sub&gt;2&lt;/sub&gt;, ''L''&lt;sub&gt;3&lt;/sub&gt; is finite. In the case we consider in this project, we then have to determine whether such a family contains a prime or not.

Thus, the [[:w:Time complexity|time complexity]] of the Athena problem in base ''b'' may be ''[[:w:Big O notation|O]]''(''[[:w:E (mathematical_constant)|e]]''&lt;sup&gt;''[[:w:Euler's constant|γ]]''×(''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(*b*)&lt;/sup&gt;), and the [[:w:CPU time|CPU time]] of the Athena problem in base ''b'' may be longer than [[:w:Age of the universe|the age of the universe]] for bases ''b'' = 19, 23, 25, 27, 29, 31, 32, 33, 34, 35, also, Athena problem in bases ''b'' around 500 may be [[:w:NP-complete|NP-complete]] or [[:w:NP-hard|NP-hard]], or an [[:w:Undecidable problem|undecidable problem]], or an example of [[:w:Gödel's incompleteness theorems|Gödel's incompleteness theorems]] (like the [[:w:Continuum hypothesis|continuum hypothesis]] and the [[:w:Halting problem|halting problem]]).

To solve the Athena problem (i.e. to [[:w:Computing|compute]] the Athena set), we need to determine whether a given family contains a prime. In practice, if family ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''Y'' is a set of digits in base ''b'') could not be ruled out as only containing composites and ''Y'' contains two or more digits, then a relatively small prime &gt; ''b'' could always be found in this family. Intuitively, this is because there are a large number of small strings in such a family, and at least one is likely to be prime (e.g. there are 2&lt;sup&gt;''n''−2&lt;/sup&gt; strings of length ''n'' in the family 1{3,7}9, and there are over a thousand strings of length 12 in the family 1{3,7}9, thus it is very impossible that these numbers are all composite). In the case ''Y'' contains only one digit, this family is of the form ''x''{''y''}''z'', and there is only a single string of each length &gt; (the length of ''x'' + the length of ''z''), and it is not known if the following [[:w:Decision problem|decision problem]] is recursively solvable (just like [[:w:Sierpiński number|Sierpiński problem]] and [[:w:Riesel number|Riesel problem]], Sierpiński problem and Riesel problem can be generalized to other bases ''b'' (references: http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm), in fact, Athena problem base ''b'' covers the Sierpiński problem base ''b'' and the Riesel problem base ''b'' with ''k'' &lt; ''b'', i.e. finding the smallest prime of the form ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 and ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (or prove such prime does not exist) with ''k'' &lt; ''b'' (specially, for bases ''b'' such that the conjectured smallest Sierpiński number or the conjectured smallest Riesel number is &lt; ''b'', Athena problem base ''b'' covers the Sierpiński problem base ''b'' or the Riesel problem base ''b'', respectively), since the smallest prime of the form ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 and ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (if exists) must be a minimal element in base ''b'', also, Athena problem base ''b'' covers finding the smallest prime of these forms in base ''b'' (or proving that such prime does not exist): (''b''&lt;sup&gt;''n''&lt;/sup&gt;−1)/(''b''−1) (for this form, ''n'' must be prime, and we want ''n'' ≥ 2) (references of this form: http://www.fermatquotient.com/PrimSerien/GenRepu.txt, https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html, http://www.primenumbers.net/Henri/us/MersFermus.htm, http://www.bitman.name/math/table/379, https://pzktupel.de/Primetables/TableRepunitGen.php, https://oeis.org/A084740, https://oeis.org/A084738, https://oeis.org/A128164, https://oeis.org/A285642; or for prime bases ''b'': https://oeis.org/A065854, https://oeis.org/A279068), ''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 (for this form, ''n'' must be power of 2, and we want ''n'' ≥ 1) (references of this form: http://jeppesn.dk/generalized-fermat.html, http://www.noprimeleftbehind.net/crus/GFN-primes.htm, https://web.archive.org/web/20231002190634/http://yves.gallot.pagesperso-orange.fr/primes/index.html, https://pzktupel.de/Primetables/TableFermatGFBB.php, https://oeis.org/A079706, https://oeis.org/A084712, https://oeis.org/A228101), (''b''&lt;sup&gt;''n''&lt;/sup&gt;+1)/2 (for odd ''b'') (for this form, ''n'' must be power of 2, and we want ''n'' ≥ 2) (reference of this form: http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt), (''sqrt''(''b'')×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1)/(''sqrt''(''b'')+1) (for square ''b'') (for this form, 2×''n''+1 must be prime, and we want ''n'' ≥ 2) (references of this form: http://www.fermatquotient.com/PrimSerien/GenRepuP.txt, http://www.primenumbers.net/Henri/us/MersFermus.htm, http://www.bitman.name/math/table/488, https://pzktupel.de/Primetables/TableWagstaffGen.php, https://oeis.org/A084742, https://oeis.org/A084741; or for bases ''b'' with ''sqrt''(''b'') prime: https://oeis.org/A065507), ((''b''−2)×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1)/(''b''−1) (''n'' ≥ 2) (reference of this form: https://oeis.org/A243404), 2×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 (''n'' ≥ 1) (references of this form: https://www.mersenneforum.org/showthread.php?t=6918, https://www.mersenneforum.org/showthread.php?t=19725, https://oeis.org/A119624), 2×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (''n'' ≥ 1) (references of this form: https://www.mersenneforum.org/showthread.php?t=24576, https://www.mersenneforum.org/attachment.php?attachmentid=20976&amp;d=1567314217, https://oeis.org/A119591), ''b''&lt;sup&gt;''n''&lt;/sup&gt;+2 (''n'' ≥ 1) (references of this form: https://oeis.org/A138066, https://oeis.org/A084713, https://oeis.org/A138067), ''b''&lt;sup&gt;''n''&lt;/sup&gt;−2 (''n'' ≥ 2) (references of this form: https://www.primepuzzles.net/puzzles/puzz_887.htm, https://oeis.org/A250200, https://oeis.org/A255707, https://oeis.org/A084714; or for prime bases ''b'': https://oeis.org/A292201), (''b''−1)×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 (''n'' ≥ 1) (references of this form: http://www.noprimeleftbehind.net/Williams-primes-MP.htm, http://www.bitman.name/math/table/477, https://pzktupel.de/Primetables/TableWilliams2.php, https://oeis.org/A305531; or for prime bases ''b'': https://oeis.org/A087139), (''b''−1)×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (''n'' ≥ 1) (references of this form: https://harvey563.tripod.com/wills.txt, http://www.noprimeleftbehind.net/Williams-primes-MM.htm, http://www.bitman.name/math/table/484, https://pzktupel.de/Primetables/TableWilliams1.php; or for prime bases ''b'': https://oeis.org/A122396), ''b''&lt;sup&gt;''n''&lt;/sup&gt;+(''b''−1) (''n'' ≥ 1) (references of this form: http://www.bitman.name/math/table/795, https://pzktupel.de/Primetables/TableWilliams6.php, https://oeis.org/A076845, https://oeis.org/A076846, https://oeis.org/A078178, https://oeis.org/A078179), ''b''&lt;sup&gt;''n''&lt;/sup&gt;−(''b''−1) (''n'' ≥ 2) (references of this form: http://www.bitman.name/math/table/792, https://pzktupel.de/Primetables/TableWilliams5.php, https://oeis.org/A113516, https://oeis.org/A343589; or for prime bases ''b'': https://cs.uwaterloo.ca/journals/JIS/VOL3/mccranie.html, http://www.bitman.name/math/table/435)):

Problem: Given strings ''x'', ''z'' (may be empty), a digit ''y'', and a base ''b'' (''x'' does not [[:w:Leading zero|start with the digit 0]], ''z'' ends with a digit which [[:w:Coprime integers|coprime]] to ''b'', ''y'' is not 0 if ''x'' is empty, ''y'' is coprime to ''b'' if ''z'' is empty), does there exist a prime number whose base-''b'' expansion is of the form ''xy''&lt;sub&gt;''n''&lt;/sub&gt;''z'' for some ''n'' ≥ 0?

Some families can be ruled out to contain no prime &gt; ''b'' by [[:w:Covering set|covering congruence]], [[:w:Factorization of polynomials|algebraic factorization]] (e.g. [[:w:Difference of two squares|difference of two squares]], [[:w:Sum of two cubes|sum of two cubes]], [[:w:Sophie Germain's identity|Sophie Germain's identity of ''x''&lt;sup&gt;4&lt;/sup&gt;+4×''y''&lt;sup&gt;4&lt;/sup&gt;]]), or combine of them, e.g.

* The base 9 family 2{7}: Always divisible by 2 or 5
* The base 11 family 2{5}: Always divisible by 2 or 3
* The base 14 family B{0}1: Always divisible by 3 or 5
* The base 13 family 95{0}3: Always divisible by 5, 7, or 17
* The base 16 family {4}D: Always divisible by 3, 7, or 13
* The base 16 family {8}F: Always divisible by 3, 7, or 13
* The base 21 family {7}D: Always divisible by 2, 13, or 17
* The base 23 family {D}GA: Always divisible by 2, 5, 7, 37, or 79
* The base 9 family {1}: Can be written as (9&lt;sup&gt;''n''&lt;/sup&gt;−1)/8 and can be factored as (3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (3&lt;sup&gt;''n''&lt;/sup&gt;+1) / 8
* The base 8 family 1{0}1: Can be written as 8&lt;sup&gt;''n''&lt;/sup&gt;+1 and can be factored as (2&lt;sup&gt;''n''&lt;/sup&gt;+1) × (4&lt;sup&gt;''n''&lt;/sup&gt;−2&lt;sup&gt;''n''&lt;/sup&gt;+1)
* The base 9 family 3{8}: Can be written as 4×9&lt;sup&gt;''n''&lt;/sup&gt;−1 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (2×3&lt;sup&gt;''n''&lt;/sup&gt;+1)
* The base 16 family 1{5}: Can be written as (4×16&lt;sup&gt;''n''&lt;/sup&gt;−1)/3 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (2×3&lt;sup&gt;''n''&lt;/sup&gt;+1) / 3
* The base 16 family {4}1: Can be written as (4×16&lt;sup&gt;''n''&lt;/sup&gt;−49)/15 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−7) × (2×3&lt;sup&gt;''n''&lt;/sup&gt;+7) / 15
* The base 27 family 7{Q}: Can be written as 8×27&lt;sup&gt;''n''&lt;/sup&gt;−1 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (4×9&lt;sup&gt;''n''&lt;/sup&gt;+2×3&lt;sup&gt;''n''&lt;/sup&gt;+1)
* The base 27 family 9{G}: Can be written as (125×27&lt;sup&gt;''n''&lt;/sup&gt;−8)/13 and can be factored as (5×3&lt;sup&gt;''n''&lt;/sup&gt;−2) × (25×9&lt;sup&gt;''n''&lt;/sup&gt;+10×3&lt;sup&gt;''n''&lt;/sup&gt;+4)
* The base 16 family {C}D: Can be written as (4×16&lt;sup&gt;''n''&lt;/sup&gt;+1)/5 and can be factored as (2×4&lt;sup&gt;''n''&lt;/sup&gt;−2×2&lt;sup&gt;''n''&lt;/sup&gt;+1) × (2×4&lt;sup&gt;''n''&lt;/sup&gt;+2×2&lt;sup&gt;''n''&lt;/sup&gt;+1) / 5
* The base 14 family 8{D}: Can be written as 9×14&lt;sup&gt;''n''&lt;/sup&gt;−1, it is divisible by 5 if ''n'' is odd and can be factored as (3×14&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (3×14&lt;sup&gt;''n''/2&lt;/sup&gt;+1) if ''n'' is even
* The base 12 family {B}9B: Can be written as 12&lt;sup&gt;''n''&lt;/sup&gt;−25, it is divisible by 13 if ''n'' is odd and can be factored as (12&lt;sup&gt;''n''/2&lt;/sup&gt;−5) × (12&lt;sup&gt;''n''/2&lt;/sup&gt;+5) if ''n'' is even
* The base 14 family {D}5: Can be written as 14&lt;sup&gt;''n''&lt;/sup&gt;−9, it is divisible by 5 if ''n'' is odd and can be factored as (14&lt;sup&gt;''n''/2&lt;/sup&gt;−3) × (14&lt;sup&gt;''n''/2&lt;/sup&gt;+3) if ''n'' is even
* The base 17 family 1{9}: Can be written as (25×17&lt;sup&gt;''n''&lt;/sup&gt;−9)/16, it is divisible by 2 if ''n'' is odd and can be factored as (5×17&lt;sup&gt;''n''/2&lt;/sup&gt;−3) × (5×17&lt;sup&gt;''n''/2&lt;/sup&gt;+3) / 16 if ''n'' is even
* The base 17 family 7{9}: Can be written as (121×17&lt;sup&gt;''n''&lt;/sup&gt;−9)/16, it is divisible by 2 if ''n'' is odd and can be factored as (11×17&lt;sup&gt;''n''/2&lt;/sup&gt;−3) × (11×17&lt;sup&gt;''n''/2&lt;/sup&gt;+3) / 16 if ''n'' is even
* The base 19 family 1{6}: Can be written as (4×19&lt;sup&gt;''n''&lt;/sup&gt;−1)/3, it is divisible by 5 if ''n'' is odd and can be factored as (2×19&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (2×19&lt;sup&gt;''n''/2&lt;/sup&gt;+1) / 3 if ''n'' is even
* The base 19 family 7{2}: Can be written as (64×19&lt;sup&gt;''n''&lt;/sup&gt;−1)/9, it is divisible by 5 if ''n'' is odd and can be factored as (8×19&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (8×19&lt;sup&gt;''n''/2&lt;/sup&gt;+1) / 9 if ''n'' is even
* The base 24 family 3{N}: Can be written as 4×24&lt;sup&gt;''n''&lt;/sup&gt;−1, it is divisible by 5 if ''n'' is odd and can be factored as (2×24&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (2×24&lt;sup&gt;''n''/2&lt;/sup&gt;+1) if ''n'' is even

By the [[:w:Prime number theorem|prime number theorem]], the [[:w:Probability|chance]] that a [[:w:Random number|random]] ''n''-digit base ''b'' number is prime is [[:w:Asymptotic analysis|approximately]] 1/''n'' (more accurately, the chance is approximately 1/(''n''×''ln''(''b'')), where ''ln'' is the [[:w:Natural logarithm|natural logarithm]]). If one conjectures the numbers ''x''{''y''}''z'' behave similarly (i.e. the numbers ''x''{''y''}''z'' is a [[:w:Pseudorandomness|pseudorandom sequence]]) you would expect [[:w:Harmonic_series (mathematics)|1/1 + 1/2 + 1/3 + 1/4 + ... = ∞]] primes of the form ''x''{''y''}''z'' (of course, this does not always happen, since some ''x''{''y''}''z'' families can be ruled out to contain no prime &gt; ''b'' (by covering congruence, algebraic factorization, or combine of them), but it is at least a reasonable conjecture in the absence of evidence to the contrary. Hence, the [[:w:Heuristic argument|heuristic argument]] suggests there are always infinitely many primes in family ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') if it cannot be ruled out to contain no prime or only contain finitely many primes, by covering congruence, algebraic factorization, or combine of them. However, some families ''x''{''y''}''z'' could not be proven to contain no primes &gt; ''b'' (by covering congruence, algebraic factorization, or combine of them) but no primes &gt; ''b'' could be found in the family, even after searching through numbers with over 100000 digits. In such a case, the only way to proceed is to [[:w:Primality test|test the primality]] of larger and larger numbers of such form and hope a prime is eventually discovered. e.g. the smallest (probable) prime in the family A{3}A in base ''b'' = 13 is A3&lt;sub&gt;592197&lt;/sub&gt;A, its algebraic form is (41×13&lt;sup&gt;592198&lt;/sup&gt;+27)/4, when written in decimal contains 659677 digits (it is only probable prime, i.e. not definitely prime, since technically, probable primality tests were used to show this (which have a ''very'' small chance of making an error, see https://t5k.org/notes/prp_prob.html) because all known primality tests run far too slowly to run on numbers of this size unless either [https://t5k.org/prove/prove3_1.html ''N''−1] or [https://t5k.org/prove/prove3_2.html ''N''+1] (or both) can be ≥ 1/3 factored).

The numbers in family ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') are of the form (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1) for some fixed ''a'', ''b'', ''c'' such that ''a'' ≥ 1, ''b'' ≥ 2 (''b'' is the base), ''c'' ≠ 0, ''gcd''(''a'',''c'') = 1, ''gcd''(''b'',''c'') = 1. Except in the [[:w:Special case|special case]] ''c'' = ±1 and ''gcd''(''a''+''c'',''b''−1) = 1, when ''n'' is large the known [[:w:Primality test|primality test]]s for such a number are too inefficient to run. In this case one must resort to a [[:w:Probabilistic algorithm|probable]] primality test such as a [[:w:Miller–Rabin primality test|Miller–Rabin primality test]] or a [[:w:Baillie–PSW primality test|Baillie–PSW primality test]], unless a divisor of the number can be found. Since we are testing many numbers in an [[:w:Exponential growth|exponential sequence]], it is possible to use a sieving process to find divisors rather than using [[:w:Trial division|trial division]].

To do this, we made use of Geoffrey Reynolds' ''srsieve'' software (download: https://pzktupel.de/Software/srsieve_1.1.4.7z). This program uses the [[:w:Baby-step giant-step|baby-step giant-step]] [[:w:Algorithm|algorithm]] to find all primes ''p'' which divide ''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'' where ''p'' and ''n'' lie in a [[:w:Interval_(mathematics)|specified range]]. Since this program cannot handle the general case (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1) when ''gcd''(''a''+''c'',''b''−1) &gt; 1 we only used it to sieve the sequence ''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'' for primes ''p'' not dividing ''gcd''(''a''+''c'',''b''−1), and initialized the list of candidates to not include ''n'' for which there is some prime ''p'' dividing ''gcd''(''a''+''c'',''b''−1) for which ''p'' dividing (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1). The program had to be modified slightly to remove a check which would prevent it from running in the case when ''a'', ''b'', and ''c'' were all odd (since then 2 divides ''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'', but 2 may not divide (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1)).

Once the numbers with small divisors had been removed, it remained to test the remaining numbers using a probable primality test. For this we used the software ''LLR'' by Jean Penné. (download: http://jpenne.free.fr/index2.html). Although undocumented, it is possible to run this program on numbers of the form (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1) when ''gcd''(''a''+''c'',''b''−1) &gt; 1, so this program required no modifications. A script was also written which allowed one to run ''srsieve'' while ''LLR'' was testing the remaining candidates, so that when a divisor was found by srsieve on a number which had not yet been tested by ''LLR'' it would be removed from the list of candidates.

For the primes &lt; 10&lt;sup&gt;25000&lt;/sup&gt; for the &quot;easy&quot; bases (bases ''b'' with ≤ 150 primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; (base ''b'' = 26 has 83 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 3 unsolved families, base ''b'' = 36 has 75 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 4 unsolved families, base ''b'' = 17 has 99 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 18 unsolved families, base ''b'' = 21 has 80 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 12 unsolved families, base ''b'' = 19 has 201 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 23 unsolved families), i.e. bases *b* = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 26, 28, 30, 36), we employed ''CM'' by Andreas Enge (download: https://www.multiprecision.org/cm/download.html), an elliptic curve primality proving implementation.

== Data ==

These are the results of the Athena problem in bases 2 ≤ ''b'' ≤ 36 (we stop at base 36 since this base is the maximum base for which it is possible to write the numbers with the [[:w:Symbol|symbol]]s 0, 1, 2, ..., 9 and A, B, C, ..., Z (i.e. the 10 [[:w:Arabic numerals|Arabic numerals]] and the 26 [[:w:Latin script|Latin letters]]): (some large Athena primes are only probable primes, i.e. not definitely primes, since they are too large to be [[:w:Elliptic curve primality|ECPP proved]] and [[:w:Pocklington primality test#Extensions and variants|neither ''N''−1 nor ''N''+1 can be ≥ 1/3 factored]], all of them pass the [[:w:Baillie–PSW primality test|Baillie–PSW primality test]] and the [[:w:Strong pseudoprime|strong primality test]] (i.e. the [[:w:Miller–Rabin primality test|Miller–Rabin primality test]]) with all prime bases ''p'' ≤ 61, however, all Athena primes &lt; 10&lt;sup&gt;25000&lt;/sup&gt; for bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 36 are definitely primes, most of them &gt; 10&lt;sup&gt;299&lt;/sup&gt; are proven primes with [[:w:Elliptic curve primality|ECPP proving]], others &gt; 10&lt;sup&gt;299&lt;/sup&gt; are proven primes with [[:w:Pocklington primality test#Extensions and variants|''N''−1 or ''N''+1 proving]])

All numbers are written in base ''b'', [[:w:Senary#Base 36 as senary compression|using A to Z to represent digit values 10 to 35]], &quot;{}&quot; means repeating, e.g. family 12{3}45 means the sequence {1245, 12345, 123345, 1233345, 12333345, 123333345, ...} (where the members are expressed as base ''b'' strings), subscripts are used to indicate repetitions of digits, e.g. 123&lt;sub&gt;4&lt;/sub&gt;567 means 123333567 (all subscripts are written in decimal).

Base 2: 1 Athena prime (the largest of which has 2 digits (it is 11, and its value is 3 in decimal)): {11}

Base 3: 3 Athena primes (the largest of which has 3 digits (it is 111, and its value is 13 in decimal)): {12, 21, 111}

Base 4: 5 Athena primes (the largest of which has 3 digits (it is 221, and its value is 41 in decimal)): {11, 13, 23, 31, 221}

Base 5: 22 Athena primes (the largest of which has 96 digits (it is 10&lt;sub&gt;93&lt;/sub&gt;13, and its algebraic form is 5&lt;sup&gt;95&lt;/sup&gt;+8)): {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 Athena primes (the largest of which has 5 digits (it is 40041, and its value is 5209 in decimal)): {11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041}

Base 7: 71 Athena primes (the largest of which has 17 digits (it is 3&lt;sub&gt;16&lt;/sub&gt;1, and its algebraic form is (7&lt;sup&gt;17&lt;/sup&gt;−5)/2)): {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: 75 Athena primes (the largest of which has 221 digits (it is 4&lt;sub&gt;220&lt;/sub&gt;7, and its algebraic form is (4×8&lt;sup&gt;221&lt;/sup&gt;+17)/7)): {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: 151 Athena primes (the largest of which has 1161 digits (it is 30&lt;sub&gt;1158&lt;/sub&gt;11, and its algebraic form is 3×9&lt;sup&gt;1160&lt;/sup&gt;+10)): {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: 77 Athena primes (the largest of which has 31 digits (it is 50&lt;sub&gt;28&lt;/sub&gt;27, and its algebraic form is 5×10&lt;sup&gt;30&lt;/sup&gt;+27)): {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: 1068 Athena primes (including 1 unproven probable prime: 57&lt;sub&gt;62668&lt;/sub&gt;), the largest of which has 62669 digits (it is 57&lt;sub&gt;62668&lt;/sub&gt;, and its algebraic form is (57×11&lt;sup&gt;62668&lt;/sup&gt;−7)/10), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel11 Data of Athena (probable) primes base 11]

Base 12: 106 Athena primes (the largest of which has 42 digits (it is 40&lt;sub&gt;39&lt;/sub&gt;77, and its algebraic form is 4×12&lt;sup&gt;41&lt;/sup&gt;+91)): {11, 15, 17, 1B, 25, 27, 31, 35, 37, 3B, 45, 4B, 51, 57, 5B, 61, 67, 6B, 75, 81, 85, 87, 8B, 91, 95, A7, AB, B5, B7, 221, 241, 2A1, 2B1, 2BB, 401, 421, 447, 471, 497, 565, 655, 665, 701, 70B, 721, 747, 771, 77B, 797, 7A1, 7BB, 907, 90B, 9BB, A41, B21, B2B, 2001, 200B, 202B, 222B, 229B, 292B, 299B, 4441, 4707, 4777, 6A05, 6AA5, 729B, 7441, 7B41, 929B, 9777, 992B, 9947, 997B, 9997, A0A1, A201, A605, A6A5, AA65, B001, B0B1, BB01, BB41, 600A5, 7999B, 9999B, AAAA1, B04A1, B0B9B, BAA01, BAAA1, BB09B, BBBB1, 44AAA1, A00065, BBBAA1, AAA0001, B00099B, AA000001, BBBBBB99B, B0000000000000000000000000009B, 400000000000000000000000000000000000000077}

Base 13: 3197 Athena primes (including 4 unproven probable primes: C5&lt;sub&gt;23755&lt;/sub&gt;C, 80&lt;sub&gt;32017&lt;/sub&gt;111, 95&lt;sub&gt;197420&lt;/sub&gt;, A3&lt;sub&gt;592197&lt;/sub&gt;A), the largest of which has 592199 digits (it is A3&lt;sub&gt;592197&lt;/sub&gt;A, and its algebraic form is (41×13&lt;sup&gt;592198&lt;/sup&gt;+27)/4), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel13 Data of Athena (probable) primes base 13]

Base 14: 650 Athena primes, the largest of which has 19699 digits (it is 4D&lt;sub&gt;19698&lt;/sub&gt;, and its algebraic form is 5×14&lt;sup&gt;19698&lt;/sup&gt;−1), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel14 Data of Athena primes base 14]

Base 15: 1284 Athena primes, the largest of which has 157 digits (it is 7&lt;sub&gt;155&lt;/sub&gt;97, and its algebraic form is (15&lt;sup&gt;157&lt;/sup&gt;+59)/2), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel15 Data of Athena primes base 15]

Base 16: 2347 Athena primes (including 3 unproven probable primes: DB&lt;sub&gt;32234&lt;/sub&gt;, 4&lt;sub&gt;72785&lt;/sub&gt;DD, 3&lt;sub&gt;116137&lt;/sub&gt;AF), the largest of which has 116139 digits (it is 3&lt;sub&gt;116137&lt;/sub&gt;AF, and its algebraic form is (16&lt;sup&gt;116139&lt;/sup&gt;+619)/5), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel16 Data of Athena (probable) primes base 16]

Base 17: 10415 known Athena primes (including many unproven probable primes) and 12 unsolved families (1{7}, 1F{0}7, 4{7}A, 70F{0}D, 8{B}9, 9{5}9, A{D}F, B{0}B3, {B}E9, {B}EE, F1{9}, FD0{D}, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel17 Data of known Athena (probable) primes base 17]

Base 18: 549 Athena primes, the largest of which has 6271 digits (it is C0&lt;sub&gt;6268&lt;/sub&gt;C5, and its algebraic form is 12×18&lt;sup&gt;6270&lt;/sup&gt;+221), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel18 Data of Athena primes base 18]

Base 19: 31417 known Athena primes (including many unproven probable primes) and 17 unsolved families (4B5{0}H, {5}3, 5{H}05, 5{H}0H, 5{H}5, 66{B}, 71{0}177, 7AF{0}H, 97{0}3, C{H}C, EE1{6}, F{7}5, F{B}G, F{D}F, H0F{0}7A, HB{0}5B5, II{D}, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel19 Data of known Athena (probable) primes base 19]

Base 20: 3314 Athena primes, the largest of which has 6271 digits (it is G0&lt;sub&gt;6269&lt;/sub&gt;D, and its algebraic form is 16×20&lt;sup&gt;6270&lt;/sup&gt;+13), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel20 Data of Athena primes base 20]

Base 21: 13386 known Athena primes (including many unproven probable primes) and 8 unsolved families (5{0}DJ, {9}D, B3{0}EB, B{H}6H, C{F}0K, {F}35, G{0}FK, H{0}7771, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel21 Data of known Athena (probable) primes base 21]

Base 22: 8003 Athena primes (including 1 unproven probable prime: BK&lt;sub&gt;22001&lt;/sub&gt;5), the largest of which has 22003 digits (it is BK&lt;sub&gt;22001&lt;/sub&gt;5, and its algebraic form is (251×22&lt;sup&gt;22002&lt;/sup&gt;−335)/21), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel22 Data of Athena (probable) primes base 22]

Base 23: 65178 known Athena primes (including many unproven probable primes) and 87 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel23 Data of known Athena (probable) primes base 23] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left23 Data of unsolved families for Athena problem base 23]

Base 24: 3409 Athena primes, the largest of which has 8134 digits (it is N00N&lt;sub&gt;8129&lt;/sub&gt;LN, and its algebraic form is 13249×24&lt;sup&gt;8131&lt;/sup&gt;−49), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel24 Data of Athena primes base 24]

Base 25: 133639 known Athena primes (including many unproven probable primes) and 85 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel25 Data of known Athena (probable) primes base 25] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left25 Data of unsolved families for Athena problem base 25]

Base 26: 25256 known Athena primes (including 7 unproven probable primes: 5&lt;sub&gt;19391&lt;/sub&gt;6F, 7&lt;sub&gt;20279&lt;/sub&gt;OL, LD0&lt;sub&gt;20975&lt;/sub&gt;7, 6K&lt;sub&gt;23300&lt;/sub&gt;5, J0&lt;sub&gt;44303&lt;/sub&gt;KCB, M0&lt;sub&gt;61186&lt;/sub&gt;2BB, 85M&lt;sub&gt;197060&lt;/sub&gt;B) and 3 unsolved families ({A}6F, {H}MH, {I}GL, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel26 Data of known Athena (probable) primes base 26]

Base 27: 102852 known Athena primes (including many unproven probable primes) and 44 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel27 Data of known Athena (probable) primes base 27] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left27 Data of unsolved families for Athena problem base 27]

Base 28: 25528 known Athena primes (including 3 unproven probable primes: N6&lt;sub&gt;24051&lt;/sub&gt;LR, 5OA&lt;sub&gt;31238&lt;/sub&gt;F, O4O&lt;sub&gt;94535&lt;/sub&gt;9) and 1 unsolved family (O{A}F, no primes or probable primes with length ≤ 900000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel28 Data of known Athena (probable) primes base 28]

Base 29: 355242 known Athena primes (including many unproven probable primes) and 125 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel29 Data of known Athena (probable) primes base 29] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left29 Data of unsolved families for Athena problem base 29]

Base 30: 2619 Athena primes (including 1 unproven probable prime: I0&lt;sub&gt;24608&lt;/sub&gt;D), the largest of which has 34206 digits (it is OT&lt;sub&gt;34205&lt;/sub&gt;, and its algebraic form is 25×30&lt;sup&gt;34205&lt;/sup&gt;−1), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel30 Data of Athena (probable) primes base 30]

Base 31: 569323 known Athena primes (including many unproven probable primes) and 77 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel31 Data of known Athena (probable) primes base 31] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left31 Data of unsolved families for Athena problem base 31]

Base 32: 168882 known Athena primes (including many unproven probable primes) and 120 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel32 Data of known Athena (probable) primes base 32] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left32 Data of unsolved families for Athena problem base 32]

Base 33: 280012 known Athena primes (including many unproven probable primes) and 81 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel33 Data of known Athena (probable) primes base 33] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left33 Data of unsolved families for Athena problem base 33]

Base 34: 184785 known Athena primes (including many unproven probable primes) and 47 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel34 Data of known Athena (probable) primes base 34] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left34 Data of unsolved families for Athena problem base 34]

Base 35: 720002 known Athena primes (including many unproven probable primes) and 60 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel35 Data of known Athena (probable) primes base 35] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left35 Data of unsolved families for Athena problem base 35]

Base 36: 35286 known Athena primes (including 3 unproven probable primes: 7K&lt;sub&gt;26567&lt;/sub&gt;Z, S0&lt;sub&gt;75007&lt;/sub&gt;8H, P&lt;sub&gt;81993&lt;/sub&gt;SZ) and 4 unsolved families (B{0}EUV, HM{0}N, N{0}YYN, O{L}Z, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel36 Data of known Athena (probable) primes base 36]

== Condensed table for bases 2 ≤ ''b'' ≤ 36 ==

{|class=&quot;wikitable&quot;
||''b''||number of Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b''||base-''b'' form of the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b'' (write &quot;''d''&lt;sub&gt;''n''&lt;/sub&gt;&quot; if there are 5 or more (''n'') consecutive same digits ''d'')||length of the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b''||length of the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b'' in decimal||algebraic ((''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1)) form of the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b''||''factordb'' entry of the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b''||the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b'' written in base ''b'' (use lower case letters instead of upper case letters)||number of unsolved families in the Athena problem in base ''b'' (all of these left families are linear families)||searching limit of length for the unsolved families in the Athena problem in base ''b'' (if there are different searching limits for the unsolved families in the Athena problem in base ''b'', choose the lowest searching limit)||
|-
||2||1||11||2||1||3||http://factordb.com/index.php?id=3&amp;open=ecm||http://factordb.com/index.php?showid=3&amp;base=2||0||–||
|-
||3||3||111&lt;br&gt;21&lt;br&gt;12||3&lt;br&gt;2&lt;br&gt;2||2&lt;br&gt;1&lt;br&gt;1||13&lt;br&gt;7&lt;br&gt;5||http://factordb.com/index.php?id=13&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=7&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=5&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=13&amp;base=3&lt;br&gt;http://factordb.com/index.php?showid=7&amp;base=3&lt;br&gt;http://factordb.com/index.php?showid=5&amp;base=3&lt;nowiki/&gt;||0||–||
|-
||4||5||221&lt;br&gt;31&lt;br&gt;23&lt;br&gt;13&lt;br&gt;11||3&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2||2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;1&lt;br&gt;1||41&lt;br&gt;13&lt;br&gt;11&lt;br&gt;7&lt;br&gt;5||http://factordb.com/index.php?id=41&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=13&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=11&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=7&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=5&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=41&amp;base=4&lt;br&gt;http://factordb.com/index.php?showid=13&amp;base=4&lt;br&gt;http://factordb.com/index.php?showid=11&amp;base=4&lt;br&gt;http://factordb.com/index.php?showid=7&amp;base=4&lt;br&gt;http://factordb.com/index.php?showid=5&amp;base=4&lt;nowiki/&gt;||0||–||
|-
||5||22||10&lt;sub&gt;93&lt;/sub&gt;13&lt;br&gt;300031&lt;br&gt;44441&lt;br&gt;33331&lt;br&gt;33001&lt;br&gt;30301&lt;br&gt;14444&lt;br&gt;10103&lt;br&gt;3101&lt;br&gt;414||96&lt;br&gt;6&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;4&lt;br&gt;3||67&lt;br&gt;4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;3&lt;br&gt;3&lt;br&gt;3||5&lt;sup&gt;95&lt;/sup&gt;+8&lt;br&gt;9391&lt;br&gt;3121&lt;br&gt;2341&lt;br&gt;2251&lt;br&gt;1951&lt;br&gt;1249&lt;br&gt;653&lt;br&gt;401&lt;br&gt;109||http://factordb.com/index.php?id=1100000000034686071&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=9391&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=3121&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=2341&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=2251&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1951&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1249&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=653&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=401&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=109&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000000034686071&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=9391&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=3121&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=2341&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=2251&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=1951&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=1249&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=653&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=401&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=109&amp;base=5&lt;nowiki/&gt;||0||–||
|-
||6||11||40041&lt;br&gt;4441&lt;br&gt;4401&lt;br&gt;51&lt;br&gt;45&lt;br&gt;35&lt;br&gt;31&lt;br&gt;25&lt;br&gt;21&lt;br&gt;15||5&lt;br&gt;4&lt;br&gt;4&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2||4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2||5209&lt;br&gt;1033&lt;br&gt;1009&lt;br&gt;31&lt;br&gt;29&lt;br&gt;23&lt;br&gt;19&lt;br&gt;17&lt;br&gt;13&lt;br&gt;11||http://factordb.com/index.php?id=5209&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1033&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1009&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=31&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=29&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=23&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=19&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=17&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=13&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=11&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=5209&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=1033&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=1009&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=31&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=29&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=23&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=19&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=17&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=13&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=11&amp;base=6&lt;nowiki/&gt;||0||–||
|-
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||33||280012~280093||DP&lt;sub&gt;95093&lt;/sub&gt;M5&lt;br&gt;HJ0&lt;sub&gt;94295&lt;/sub&gt;J&lt;br&gt;90&lt;sub&gt;93597&lt;/sub&gt;Q&lt;br&gt;9F0&lt;sub&gt;93157&lt;/sub&gt;N&lt;br&gt;7&lt;sub&gt;89449&lt;/sub&gt;333H&lt;br&gt;K3&lt;sub&gt;80751&lt;/sub&gt;6K&lt;br&gt;D&lt;sub&gt;80107&lt;/sub&gt;9UD&lt;br&gt;VFU&lt;sub&gt;72204&lt;/sub&gt;FK&lt;br&gt;J&lt;sub&gt;68715&lt;/sub&gt;2BJ&lt;br&gt;DF0&lt;sub&gt;68367&lt;/sub&gt;J||95096&lt;br&gt;94298&lt;br&gt;93599&lt;br&gt;93160&lt;br&gt;89453&lt;br&gt;80754&lt;br&gt;80110&lt;br&gt;72208&lt;br&gt;68718&lt;br&gt;68370||144405&lt;br&gt;143193&lt;br&gt;142131&lt;br&gt;141465&lt;br&gt;135835&lt;br&gt;122626&lt;br&gt;121648&lt;br&gt;109649&lt;br&gt;104350&lt;br&gt;103821||(441×33&lt;sup&gt;95095&lt;/sup&gt;−3833)/32&lt;br&gt;580×33&lt;sup&gt;94296&lt;/sup&gt;+19&lt;br&gt;9×33&lt;sup&gt;93598&lt;/sup&gt;+26&lt;br&gt;312×33&lt;sup&gt;93158&lt;/sup&gt;+23&lt;br&gt;(7×33&lt;sup&gt;89453&lt;/sup&gt;−4743239)/32&lt;br&gt;(643×33&lt;sup&gt;80753&lt;/sup&gt;+3709)/32&lt;br&gt;(13×33&lt;sup&gt;80110&lt;/sup&gt;−121453)/32&lt;br&gt;(16623×33&lt;sup&gt;72206&lt;/sup&gt;−8095)/16&lt;br&gt;(19×33&lt;sup&gt;68718&lt;/sup&gt;−600883)/32&lt;br&gt;444×33&lt;sup&gt;68368&lt;/sup&gt;+19||http://factordb.com/index.php?id=1100000005652348775&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000005652362811&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000005652375073&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000005652389776&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000005652430746&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000005652446200&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000005652461592&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004614764298&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004614770536&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004614784274&amp;open=prime&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000005652348775&amp;base=33&lt;br&gt;http://factordb.com/index.php?showid=1100000005652362811&amp;base=33&lt;br&gt;http://factordb.com/index.php?showid=1100000005652375073&amp;base=33&lt;br&gt;http://factordb.com/index.php?showid=1100000005652389776&amp;base=33&lt;br&gt;http://factordb.com/index.php?showid=1100000005652430746&amp;base=33&lt;br&gt;http://factordb.com/index.php?showid=1100000005652446200&amp;base=33&lt;br&gt;http://factordb.com/index.php?showid=1100000005652461592&amp;base=33&lt;br&gt;http://factordb.com/index.php?showid=1100000004614764298&amp;base=33&lt;br&gt;http://factordb.com/index.php?showid=1100000004614770536&amp;base=33&lt;br&gt;http://factordb.com/index.php?showid=1100000004614784274&amp;base=33&lt;nowiki/&gt;||81||100000||
|-
||34||184785~184832||GFGC&lt;sub&gt;99996&lt;/sub&gt;5&lt;br&gt;90&lt;sub&gt;97950&lt;/sub&gt;FJ&lt;br&gt;NM0&lt;sub&gt;85218&lt;/sub&gt;KX&lt;br&gt;F&lt;sub&gt;83189&lt;/sub&gt;H2HP&lt;br&gt;P&lt;sub&gt;79441&lt;/sub&gt;444P&lt;br&gt;6&lt;sub&gt;77027&lt;/sub&gt;8X&lt;br&gt;XQIQ&lt;sub&gt;72241&lt;/sub&gt;D&lt;br&gt;T&lt;sub&gt;66530&lt;/sub&gt;IF&lt;br&gt;4&lt;sub&gt;66152&lt;/sub&gt;B&lt;br&gt;2EEC&lt;sub&gt;66039&lt;/sub&gt;7||100000&lt;br&gt;97953&lt;br&gt;85222&lt;br&gt;83193&lt;br&gt;79445&lt;br&gt;77029&lt;br&gt;72245&lt;br&gt;66532&lt;br&gt;66153&lt;br&gt;66043||153148&lt;br&gt;150013&lt;br&gt;130516&lt;br&gt;127408&lt;br&gt;121669&lt;br&gt;117968&lt;br&gt;110642&lt;br&gt;101893&lt;br&gt;101312&lt;br&gt;101143||(209246×34&lt;sup&gt;99997&lt;/sup&gt;−81)/11&lt;br&gt;9×34&lt;sup&gt;97952&lt;/sup&gt;+529&lt;br&gt;804×34&lt;sup&gt;85220&lt;/sup&gt;+713&lt;br&gt;(5×34&lt;sup&gt;83193&lt;/sup&gt;+700233)/11&lt;br&gt;(25×34&lt;sup&gt;79445&lt;/sup&gt;−28062367)/33&lt;br&gt;(2×34&lt;sup&gt;77029&lt;/sup&gt;+1043)/11&lt;br&gt;(1288676×34&lt;sup&gt;72242&lt;/sup&gt;−455)/33&lt;br&gt;(29×34&lt;sup&gt;66532&lt;/sup&gt;−12833)/33&lt;br&gt;(4×34&lt;sup&gt;66153&lt;/sup&gt;+227)/33&lt;br&gt;(30826×34&lt;sup&gt;66040&lt;/sup&gt;−59)/11||http://factordb.com/index.php?id=1100000004702891268&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004702894713&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004702900996&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004702910376&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004702913746&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004702918600&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004399656529&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004399657696&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004399658651&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004399659716&amp;open=prime&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000004702891268&amp;base=34&lt;br&gt;http://factordb.com/index.php?showid=1100000004702894713&amp;base=34&lt;br&gt;http://factordb.com/index.php?showid=1100000004702900996&amp;base=34&lt;br&gt;http://factordb.com/index.php?showid=1100000004702910376&amp;base=34&lt;br&gt;http://factordb.com/index.php?showid=1100000004702913746&amp;base=34&lt;br&gt;http://factordb.com/index.php?showid=1100000004702918600&amp;base=34&lt;br&gt;http://factordb.com/index.php?showid=1100000004399656529&amp;base=34&lt;br&gt;http://factordb.com/index.php?showid=1100000004399657696&amp;base=34&lt;br&gt;http://factordb.com/index.php?showid=1100000004399658651&amp;base=34&lt;br&gt;http://factordb.com/index.php?showid=1100000004399659716&amp;base=34&lt;nowiki/&gt;||47||100000||
|-
||35||720002~720062||N0N&lt;sub&gt;99971&lt;/sub&gt;9&lt;br&gt;V0&lt;sub&gt;83669&lt;/sub&gt;E73&lt;br&gt;N&lt;sub&gt;81563&lt;/sub&gt;K7N&lt;br&gt;BJ0&lt;sub&gt;81279&lt;/sub&gt;N&lt;br&gt;J0&lt;sub&gt;80062&lt;/sub&gt;FUH&lt;br&gt;43V&lt;sub&gt;79754&lt;/sub&gt;&lt;br&gt;9&lt;sub&gt;76600&lt;/sub&gt;K3&lt;br&gt;LB&lt;sub&gt;71366&lt;/sub&gt;PB&lt;br&gt;Q&lt;sub&gt;64150&lt;/sub&gt;H&lt;br&gt;50&lt;sub&gt;63397&lt;/sub&gt;5R||99974&lt;br&gt;83673&lt;br&gt;81566&lt;br&gt;81282&lt;br&gt;80066&lt;br&gt;79756&lt;br&gt;76602&lt;br&gt;71369&lt;br&gt;64151&lt;br&gt;63400||154367&lt;br&gt;129197&lt;br&gt;125944&lt;br&gt;125505&lt;br&gt;123628&lt;br&gt;123148&lt;br&gt;118279&lt;br&gt;110199&lt;br&gt;99054&lt;br&gt;97894||(27393×35&lt;sup&gt;99972&lt;/sup&gt;−499)/34&lt;br&gt;31×35&lt;sup&gt;83672&lt;/sup&gt;+17398&lt;br&gt;(23×35&lt;sup&gt;81566&lt;/sup&gt;−144013)/34&lt;br&gt;404×35&lt;sup&gt;81280&lt;/sup&gt;+23&lt;br&gt;19×35&lt;sup&gt;80065&lt;/sup&gt;+19442&lt;br&gt;(4893×35&lt;sup&gt;79754&lt;/sup&gt;−31)/34&lt;br&gt;(9×35&lt;sup&gt;76602&lt;/sup&gt;+12877)/34&lt;br&gt;(725×35&lt;sup&gt;71368&lt;/sup&gt;+16649)/34&lt;br&gt;(13×35&lt;sup&gt;64151&lt;/sup&gt;−166)/17&lt;br&gt;5×35&lt;sup&gt;63399&lt;/sup&gt;+202||http://factordb.com/index.php?id=1100000008248342445&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008248353306&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008248375642&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008248397018&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008248412468&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008248418540&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008248423670&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008192119974&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008192126630&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008192129294&amp;open=prime&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000008248342445&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008248353306&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008248375642&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008248397018&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008248412468&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008248418540&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008248423670&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008192119974&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008192126630&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008192129294&amp;base=35&lt;nowiki/&gt;||60||100000||
|-
||36||35286~35290||P&lt;sub&gt;81993&lt;/sub&gt;SZ&lt;br&gt;S0&lt;sub&gt;75007&lt;/sub&gt;8H&lt;br&gt;7K&lt;sub&gt;26567&lt;/sub&gt;Z&lt;br&gt;J&lt;sub&gt;10117&lt;/sub&gt;LJ&lt;br&gt;VL0&lt;sub&gt;7258&lt;/sub&gt;J&lt;br&gt;EO0&lt;sub&gt;6177&lt;/sub&gt;V&lt;br&gt;FZ&lt;sub&gt;5777&lt;/sub&gt;3P&lt;br&gt;T09&lt;sub&gt;4618&lt;/sub&gt;1&lt;br&gt;RY&lt;sub&gt;4562&lt;/sub&gt;H&lt;br&gt;OZ&lt;sub&gt;3932&lt;/sub&gt;AZ||81995&lt;br&gt;75010&lt;br&gt;26569&lt;br&gt;10119&lt;br&gt;7261&lt;br&gt;6180&lt;br&gt;5780&lt;br&gt;4621&lt;br&gt;4564&lt;br&gt;3935||127609&lt;br&gt;116739&lt;br&gt;41349&lt;br&gt;15748&lt;br&gt;11301&lt;br&gt;9618&lt;br&gt;8996&lt;br&gt;7192&lt;br&gt;7103&lt;br&gt;6124||(5×36&lt;sup&gt;81995&lt;/sup&gt;+821)/7&lt;br&gt;28×36&lt;sup&gt;75009&lt;/sup&gt;+305&lt;br&gt;(53×36&lt;sup&gt;26568&lt;/sup&gt;+101)/7&lt;br&gt;(19×36&lt;sup&gt;10119&lt;/sup&gt;+2501)/35&lt;br&gt;1137×36&lt;sup&gt;7259&lt;/sup&gt;+19&lt;br&gt;528×36&lt;sup&gt;6178&lt;/sup&gt;+31&lt;br&gt;16×36&lt;sup&gt;5779&lt;/sup&gt;−1163&lt;br&gt;(36549×36&lt;sup&gt;4619&lt;/sup&gt;−289)/35&lt;br&gt;(979×36&lt;sup&gt;4563&lt;/sup&gt;−629)/35&lt;br&gt;25×36&lt;sup&gt;3934&lt;/sup&gt;−901||http://factordb.com/index.php?id=1100000002394962083&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004020085177&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003896952461&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362491&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362489&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362488&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362487&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362486&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362485&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000000840634476&amp;open=prime&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000002394962083&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000004020085177&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003896952461&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362491&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362489&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362488&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362487&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362486&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362485&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000000840634476&amp;base=36&lt;nowiki/&gt;||4||200000||
|}

== The fully proof of Athena problem in decimal (base ''b'' = 10) ==

'''Bold''' for the Athena primes, ''x'' ◁ ''y'' means ''x'' is a subsequence of ''y''.

Assume ''p'' is a prime &gt; 10, and the last digit of ''p'' must lie in {1,3,7,9}.

Case 1: ''p'' ends with 1.

In this case we can write ''p'' = ''x''1. If ''x'' contains 1, 3, 4, 6, or 7, then (respectively) '''11''' ◁ ''p'', '''31''' ◁ ''p'', '''41''' ◁ ''p'', '''61''' ◁ ''p'', or '''71''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 2, 5, 8, or 9.

Case 1.1: ''p'' begins with 2.

In this case we can write ''p'' = 2''y''1. If 5 ◁ ''y'', then '''251''' ◁ ''p''. If 8 ◁ ''y'', then '''281''' ◁ ''p''. If 9 ◁ ''y'', then 29 ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 2.

If 22 ◁ ''y'', then '''2221''' ◁ ''p''. Hence we may assume ''y'' contains zero or one 2's.

If ''y'' contains no 2's, then ''p'' ∈ 2{0}1. But then, since the sum of the digits of ''p'' is 3, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' contains exactly one 2, then we can write ''p'' = 2''z''2''w''1, where ''z'',''w'' ∈ {0}. If 0 ◁ ''z'' and 0 ◁ ''w'', then '''20201''' ◁ ''p''. Hence we may assume either ''z'' or ''w'' is empty.

If ''z'' is empty, then ''p'' ∈ 22{0}1, and the smallest prime ''p'' ∈ 22{0}1 is '''22000001'''.

If ''w'' is empty, then ''p'' ∈ 2{0}21, and the smallest prime ''p'' ∈ 2{0}21 is '''20021'''.

Case 1.2: ''p'' begins with 5.

In this case we can write ''p'' = 5''y''1. If 2 ◁ ''y'', then '''521''' ◁ ''p''. If 9 ◁ ''y'', then 59 ◁ ''p''. Hence we may assume all digits of ''y'' are 0, 5, or 8.

If 05 ◁ ''y'', then '''5051''' ◁ ''p''. If 08 ◁ ''y'', then '''5081''' ◁ ''p''. If 50 ◁ ''y'', then '''5501''' ◁ ''p''. If 58 ◁ ''y'', then '''5581''' ◁ ''p''. If 80 ◁ ''y'', then '''5801''' ◁ ''p''. If 85 ◁ ''y'', then '''5851''' ◁ ''p''. Hence we may assume ''y'' ∈ {0} ∪ {5} ∪ {8}.

If ''y'' ∈ {0}, then ''p'' ∈ 5{0}1. But then, since the sum of the digits of ''p'' is 6, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' ∈ {5}, then ''p'' ∈ 5{5}1, and the smallest prime ''p'' ∈ 5{5}1 is '''555555555551'''.

If ''y'' ∈ {8}, since if 88 ◁ ''y'', then 881 ◁ ''p'', hence we may assume ''y'' ∈ {''𝜆'',8}, and thus ''p'' ∈ {51,581}, but 51 and 581 are both composite.

Case 1.3: ''p'' begins with 8.

In this case we can write p = 8''y''1. If 2 ◁ ''y'', then '''821''' ◁ ''p''. If 8 ◁ ''y'', then '''881''' ◁ ''p''. If 9 ◁ ''y'', then 89 ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 5.

If 50 ◁ ''y'', then '''8501''' ◁ ''p''. Hence we may assume y ∈ {0}{5}.

If 005 ◁ ''y'', then '''80051''' ◁ p. Hence we may assume y ∈ {0} ∪ {5} ∪ 0{5}.

If y ∈ {0}, then ''p'' ∈ 8{0}1. But then, since the sum of the digits of ''p'' is 9, ''p'' is divisible by 3, so ''p'' cannot be prime.

If y ∈ {5}, since if 55555555555 ◁ ''y'', then 555555555551 ◁ ''p'', hence we may assume ''y'' ∈ {''𝜆'', 5, 55, 555, 5555, 55555, 555555, 5555555, 55555555, 555555555, 5555555555}, and thus ''p'' ∈ {81, 851, 8551, 85551, 855551, 8555551, 85555551, 855555551, 8555555551, 85555555551, 855555555551}, but all of these numbers are composite.

If y ∈ 0{5}, since if 55555555555 ◁ ''y'', then 555555555551 ◁ ''p'', hence we may assume ''y'' ∈ {0, 05, 055, 0555, 05555, 055555, 0555555, 05555555, 055555555, 0555555555, 05555555555}, and thus ''p'' ∈ {801, 8051, 80551, 805551, 8055551, 80555551, 805555551, 8055555551, 80555555551, 805555555551, 8055555555551}, and of these numbers only 80555551 and 8055555551 are primes, but 80555551 ◁ 8055555551, thus only '''80555551''' is a minimal element.

Case 1.4: ''p'' begins with 9.

In this case we can write p = 9''y''1. If 9 ◁ ''y'', then '''991''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0, 2, 5, or 8.

If 00 ◁ ''y'', then '''9001''' ◁ ''p''. If 22 ◁ ''y'', then '''9221''' ◁ ''p''. If 55 ◁ ''y'', then '''9551''' ◁ ''p''. If 88 ◁ ''y'', then 881 ◁ ''p''. Hence we may assume ''y'' contains at most one 0, at most one 2, at most one 5, and at most one 8.

If ''y'' only contains at most one 0 and does not contain any of {2,5,8}, then ''y'' ∈ {''𝜆'',0}, and thus ''p'' ∈ {91,901}, but 91 and 901 are both composite. If ''y'' only contains at most one 0 and only one of {2,5,8}, then the sum of the digits of ''p'' is divisible by 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Hence we may assume ''y'' contains at least two of {2,5,8}.

If 25 ◁ ''y'', then 251 ◁ ''p''. If 28 ◁ ''y'', then 281 ◁ ''p''. If 52 ◁ ''y'', then 521 ◁ ''p''. If 82 ◁ ''y'', then 821 ◁ ''p''. Hence we may assume ''y'' contains no 2's (since if ''y'' contains 2, then ''y'' cannot contain either 5's or 8's, which is a contradiction).

If 85 ◁ ''y'', then '''9851''' ◁ ''p''. Hence we may assume ''y'' ∈ {58,580,508,058}, and thus ''p'' ∈ {9581,95801,95081,90581}, and of these numbers only 95801 is prime, but 95801 is not a minimal element since 5801 ◁ 95801.

Case 2: ''p'' ends with 3.

In this case we can write p = ''x''3. If ''x'' contains 1, 2, 4, 5, 7, or 8, then (respectively) '''13''' ◁ ''p'', '''23''' ◁ ''p'', '''43''' ◁ ''p'', '''53''' ◁ ''p'', '''73''' ◁ ''p'', or '''83''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 3, 6, or 9, and thus all digits of ''p'' are 0, 3, 6, or 9. But then, since the digits of ''p'' all have a common factor 3, ''p'' is divisible by 3, so ''p'' cannot be prime.

Case 3: ''p'' ends with 7.

In this case we can write ''p'' = ''x''7. If ''x'' contains 1, 3, 4, 6, or 9, then (respectively) '''17''' ◁ ''p'', '''37''' ◁ ''p'', '''47''' ◁ ''p'', '''67''' ◁ ''p'', or '''97''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 2, 5, 7, or 8.

Case 3.1: ''p'' begins with 2.

In this case we can write ''p'' = 2''y''7. If 2 ◁ ''y'', then '''227''' ◁ ''p''. If 5 ◁ ''y'', then '''257''' ◁ ''p''. If 7 ◁ ''y'', then '''277''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 8.

If 08 ◁ ''y'', then '''2087''' ◁ ''p''. If 88 ◁ ''y'', then 887 ◁ ''p''. Hence we may assume ''y'' ∈ {0} ∪ 8{0}.

If ''y'' ∈ {0}, then ''p'' ∈ 2{0}7. But then, since the sum of the digits of ''p'' is 9, ''p'' is divisible by 3, so ''p'' cannot be prime.

If y ∈ 8{0}, then ''p'' ∈ 28{0}7. But then ''p'' is divisible by 7, since for ''n'' ≥ 0 we have 7 × 40&lt;sub&gt;''n''&lt;/sub&gt;1 = 280&lt;sub&gt;''n''&lt;/sub&gt;7.

Case 3.2: ''p'' begins with 5.

In this case we can write ''p'' = 5''y''7. If 5 ◁ ''y'', then '''557''' ◁ ''p''. If 7 ◁ ''y'', then '''577''' ◁ ''p''. If 8 ◁ ''y'', then '''587''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 2.

If 22 ◁ ''y'', then 227 ◁ ''p''. Hence we may assume ''y'' contains zero or one 2's.

If ''y'' contains no 2's, then ''p'' ∈ 5{0}7. But then, since the sum of the digits of ''p'' is 12, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' contains exactly one 2, then we can write ''p'' = 5''z''2''w''7, where ''z'',''w'' ∈ {0}. If 0 ◁ ''z'' and 0 ◁ ''w'', then '''50207''' ◁ ''p''. Hence we may assume either ''z'' or ''w'' is empty.

If ''z'' is empty, then ''p'' ∈ 52{0}7, and the smallest prime ''p'' ∈ 52{0}7 is '''5200007'''.

If ''w'' is empty, then ''p'' ∈ 5{0}27, and the smallest prime ''p'' ∈ 5{0}27 is '''5000000000000000000000000000027'''.

Case 3.3: ''p'' begins with 7.

In this case we can write ''p'' = 7''y''7. If 2 ◁ ''y'', then '''727''' ◁ ''p''. If 5 ◁ ''y'', then '''757''' ◁ ''p''. If 8 ◁ ''y'', then '''787''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 7, and thus all digits of ''p'' are 0 or 7. But then, since the digits of ''p'' all have a common factor 7, ''p'' is divisible by 7, so ''p'' cannot be prime.

Case 3.4: ''p'' begins with 8.

In this case we can write ''p'' = 8''y''7. If 2 ◁ ''y'', then '''827''' ◁ ''p''. If 5 ◁ ''y'', then '''857''' ◁ ''p''. If 7 ◁ ''y'', then '''877''' ◁ ''p''. If 8 ◁ ''y'', then '''887''' ◁ ''p''. Hence we may assume ''y'' ∈ {0}, and thus ''p'' ∈ 8{0}7. But then, since the sum of the digits of ''p'' is 15, ''p'' is divisible by 3, so ''p'' cannot be prime.

Case 4: ''p'' ends with 9.

In this case we can write ''p'' = ''x''9. If ''x'' contains 1, 2, 5, 7, or 8, then (respectively) '''19''' ◁ ''p'', '''29''' ◁ ''p'', '''59''' ◁ ''p'', '''79''' ◁ ''p'', or '''89''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 3, 4, 6, or 9.

If 44 ◁ ''x'', then '''449''' ◁ ''p''. Hence we may assume ''x'' contains zero or one 4's.

If x contains no 4's, then all digits of ''x'' are 0, 3, 6, or 9, and thus all digits of ''p'' are 0, 3, 6, or 9. But then, since the digits of ''p'' all have a common factor 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Hence we may assume that ''x'' contains exactly one 4.

Case 4.1: ''p'' begins with 3.

In this case we can write ''p'' = 3''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. We must have '''349''' ◁ ''p''.

Case 4.2: ''p'' begins with 4.

In this case we can write ''p'' = 4''y''9, where all digits of ''y'' are 0, 3, 6, or 9. If 0 ◁ ''y'', then '''409''' ◁ ''p''. If 3 ◁ ''y'', then 43 ◁ ''p''. If 9 ◁ ''y'', then '''499''' ◁ ''p''. Hence we may assume ''y'' ∈ {6}, and thus ''p'' ∈ 4{6}9. But then ''p'' is divisible by 7, since for ''n'' ≥ 0 we have 7 × 6&lt;sub&gt;''n''&lt;/sub&gt;7 = 46&lt;sub&gt;''n''&lt;/sub&gt;9.

Case 4.3: ''p'' begins with 6.

In this case we can write p = 6''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. If 0 ◁ ''z'', then 409 ◁ ''p''. If 3 ◁ ''z'', then 43 ◁ ''p''. If 6 ◁ ''z'', then '''6469''' ◁ ''p''. If 9 ◁ ''z'', then 499 ◁ ''p''. Hence we may assume ''z'' is empty.

If 3 ◁ ''y'', then 349 ◁ ''p''. If 9 ◁ ''y'', then '''6949''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 6.

If 06 ◁ ''y'', then '''60649''' ◁ ''p''. Hence we may assume ''y'' ∈ {6}{0}.

If 666 ◁ ''y'', then '''666649''' ◁ ''p''. If 00000 ◁ ''y'', then '''60000049''' ◁ ''p''. Hence we may assume ''y'' ∈ {''𝜆'', 0, 00, 000, 0000, 6, 60, 600, 6000, 60000, 66, 660, 6600, 66000, 660000}, and thus ''p'' ∈ {649, 6049, 60049, 600049, 6000049, 6649, 66049, 660049, 6600049, 66000049, 66649, 666049, 6660049, 66600049, 666000049}, and of these numbers only '''66000049''' and '''66600049''' are primes.

Case 4.4: ''p'' begins with 9.

In this case we can write p = 9''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. If 0 ◁ ''y'', then '''9049''' ◁ ''p''. If 3 ◁ ''y'', then 349 ◁ ''p''. If 6 ◁ ''y'', then '''9649''' ◁ ''p''. If 9 ◁ ''y'', then '''9949''' ◁ ''p''. Hence we may assume ''y'' is empty.

If 0 ◁ ''z'', then 409 ◁ ''p''. If 3 ◁ ''z'', then 43 ◁ ''p''. If 9 ◁ ''z'', then 499 ◁ ''p''. Hence we may assume ''z'' ∈ {6}, and thus ''p'' ∈ 94{6}9, and the smallest prime ''p'' ∈ 94{6}9 is 946669.

[[Category:Number theory]]</text>
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'''Athena problem''' is an [[:w:List of unsolved problems in mathematics|unsolved problem]] in [[:w:Number theory|number theory]] and [[:w:Formal language theory|formal language theory]] and [[:w:Order theory|order theory]], this problem is named after the ancient Greek goddess [[:w:Athena|Athena]] (which is associated with [[:w:Wisdom|wisdom]]). Athena problem is: Give a [[:w:Natural number|natural number]] ''b'' &gt; 1, find the [[:w:Set (mathematics)|set]] of the [[:w:Minimal element|minimal element]]s of the set of the &quot;[[:w:Prime number|prime number]] [[:w:Greater than|&gt;]] ''b''&quot; [[:w:Numerical digit|digit]] [[:w:String (computer science)|string]]s in the [[:w:Positional numeral system|positional numeral system]] with [[:w:Radix|base]] ''b'' for the [[:w:Subsequence|subsequence]] [[:w:Partially ordered set|ordering]]. (A string ''x'' is a subsequence of another string ''y'', if ''x'' can be obtained from ''y'' by deleting zero or more of the [[:w:Character (computing)|character]]s in ''y''. For example, 514 is a subsequence of 352148, &quot;string&quot; is a subsequence of &quot;meistersinger&quot;. In contrast, 758 is not a subsequence of 378259, &quot;abc&quot; is not a subsequence of &quot;cbacacba&quot;, since the characters must be in the same order) (Unlike [[:w:Substring|substring]], subsequence is not required to occupy consecutive positions within the original sequences, e.g. the [[:w:Longest common subsequence|longest common subsequence problem]] is different from the [[:w:Longest common substring|longest common substring problem]])

Using [[:w:Formal language theory|formal language theory]] terminology, Athena problem is finding the [[:w:Set (mathematics)|set]] of the [[:w:Minimal element|minimal element]]s of the [[:w:Formal language|language]] of base-''b'' [[:w:Representation (mathematics)|representation]]s of the [[:w:Prime number|prime number]]s [[:w:Greater than|&gt;]] ''b'' (which is a set of [[:w:String (computer science)|string]]s of [[:w:Symbol|symbol]]s over the [[:w:Alphabet (formal languages)|alphabet]] ''Σ''&lt;sub&gt;''b''&lt;/sub&gt; := {0, 1, ..., ''b''−1}), under the subsequence ordering (i.e. the [[:w:Binary relation|binary relation]] &quot;is a subsequence of&quot;, which is a [[:w:Partially ordered set|partial ordering]]), for a given natural number ''b'' &gt; 1 (You can draw this partial ordering as a [[:w:Hasse diagram|Hasse diagram]] to find all [[:w:Minimal element|minimal element]]s), this set is called '''Athena set''', and the prime numbers in this set is called '''Athena primes'''.

By [[:w:Higman's lemma|Higman's lemma]], there are no [[:w:Infinite set|infinite]] [[:w:Antichain|antichain]]s for the subsequence ordering (i.e. the subsequence ordering is always a [[:w:Well-quasi-ordering|well quasi order]]) (i.e. under the subsequence ordering (i.e. the [[:w:Binary relation|binary relation]] &quot;is a subsequence of&quot;, which is a [[:w:Partially ordered set|partial ordering]]), every set of pairwise incomparable (i.e. not [[:w:Comparability|comparable]]) strings is finite), thus there must be only finitely many such minimal elements. In other words, the Athena set in every base ''b'' must be a [[:w:Finite set|finite set]], and every base ''b'' ≥ 2 has only finitely many Athena primes, e.g. in [[:w:Decimal|decimal]] (base ''b'' = 10), the Athena set has exactly 77 [[:w:Element of a set|element]]s (the Athena primes in decimal (base ''b'' = 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}.

For bases 2 ≤ ''b'' ≤ 36, Athena problem is fully solved in bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 24, and also solved in bases ''b'' = 11, 13, 16, 22, 30 if [[:w:Probable prime|probable prime]]s are allowed. For the unsolved bases ''b'' = 17, 19, 21, 23, 25, 26, 27, 28, 29, 31, 32, 34, 35, 36, Athena problem is solved (if probable primes are allowed) except 771 [[:w:Indexed family|families]] of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be [[:w:Empty string|empty]]) of digits in base ''b'', ''y'' is a digit in base ''b'') = sequence {''xz'', ''xyz'', ''xyyz'', ''xyyyz'', ''xyyyyz'', ''xyyyyyz'', ...} (i.e. &quot;''xy''&lt;sup&gt;+&lt;/sup&gt;''z''&quot; in [[:w:Regular expression|regular expression]]), all of these 771 families contain no primes &gt; ''b'' or probable primes &gt; ''b'' with length ≤ 100000.

== Solve the problem ==

To solve the Athena problem for a given base ''b'', we must [[:w:Computing|compute]] the elements up to families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''), and find the smallest prime &gt; ''b'' in all such families.

We call families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') &quot;linear&quot; families, and we reduce these families by removing all trailing digits ''y'' from ''x'', and removing all leading digits ''y'' from ''z'', to make the families be easier, e.g. family 12333{3}33345 in base ''b'' is reduced to family 12{3}45 in base ''b'', since they are in fact the same family. Our [[:w:Algorithm|algorithm]] then proceeds as follows:

* 1. ''M'' := {minimal primes in base ''b'' of length 2 or 3}, ''L'' := union of all ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'') such that ''x'' ≠ 0 and ''gcd''(''z'', ''b'') = 1 and ''Y'' is the set of digits ''y'' in base ''b'' such that ''xyz'' has no subsequence in ''M''.
* 2. While ''L'' contains nonlinear families (families which are not linear families): Explore each family of ''L'', and update ''L''. Examine each family of ''L'' by:
* 2.1. Let ''w'' be the shortest string in the family. If ''w'' has a subsequence in ''M'', then remove the family from ''L''. If ''w'' represents a prime, then add ''w'' to ''M'' and remove the family from ''L''.
* 2.2. If possible, simplify the family.
* 2.3. Using the techniques below (covering congruence, algebraic factorization, or combine of them), check if the family can be proven to only contain composites (only count the numbers &gt; ''b''), and if so then remove the family from ''L''.
* 3. Update ''L'', after each split examine the new families as in step 2.

e.g. in decimal (base ''b'' = 10):

''M'' := {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}

''L'' := {2{0,2}1, 2{0,8}7, 3{0,3,6,9}3, 3{0,3,6,9}9, 4{6}9, 5{0,5,8}1, 5{0,2}7, 6{0,3,6,9}3, 6{0,3,4,6,9}9, 7{0,7}7, 8{0,5}1, 8{0}7, 9{0,2,5,8}1, 9{0,3,6,9}3, 9{0,3,4,6,9}9}

and since 2221 is prime, it follows that the family 2{0,2}1 splits into the families 2{0}1 and 2{0}2{0}1

and since the family 2{0}1 can be proven to contain no primes &gt; base (since all numbers in this family are divisible by 3), it can be removed

and since 20201 is prime, it follows that the family 2{0}2{0}1 splits into the families 2{0}21 and 22{0}1

221 and 2021 are composites, but 20021 is prime, thus add 20021 to ''L''

none of 221, 2201, 22001, 220001, 2200001 are primes, but 22000001 is prime, thus add 22000001 to ''L''

and since the family 3{0,3,6,9}3 can be proven to contain no primes &gt; base (since all numbers in this family are divisible by 3), it can be removed

etc.

Since the number of possible (first digit,last digit) (also called (initial digit,final digit)) combos ([[:w:Ordered pair|ordered pair]]s) of a prime &gt; ''b'' in base ''b'' is (''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(''b'') (all digits except 0 can be the first digit of a prime &gt; ''b'' in base ''b'' (thus ''b''−1 possible digits), but only the digits coprime to ''b'' can be the last digit of a prime &gt; ''b'' in base ''b'' (thus ''eulerphi''(''b'') possible digits), and by the [[:w:Rule of product|rule of product]], there are (''b''−1)×''eulerphi''(''b'') choices of the (first digit,last digit) combo, also, both &quot;numbers of Athena primes in base ''b''&quot; and &quot;length of the largest Athena prime in base ''b''&quot; are [[:w:Asymptotic analysis|roughly]] ''[[:w:E (mathematical_constant)|e]]''&lt;sup&gt;''[[:w:Euler's constant|γ]]''×(''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(*b*)&lt;/sup&gt;.

Shrinking the family ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''Y'' is a set of digits in base ''b'')

* If ''y'' ∈ ''Y'' and the string ''xyyz'' represents a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or has a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''}''z'' ∪ ''x''{''Y'' \ ''y''}''y''{''Y'' \ ''y''}''z''.
* If ''y''&lt;sub&gt;1&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;2&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;1&lt;/sub&gt; ≠ ''y''&lt;sub&gt;2&lt;/sub&gt; and the string ''xy''&lt;sub&gt;1&lt;/sub&gt;''y''&lt;sub&gt;2&lt;/sub&gt;''z'' represents a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or has a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''&lt;sub&gt;1&lt;/sub&gt;}{''Y'' \ ''y''&lt;sub&gt;2&lt;/sub&gt;}''z''.
* If ''y''&lt;sub&gt;1&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;2&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;1&lt;/sub&gt; ≠ ''y''&lt;sub&gt;2&lt;/sub&gt; and both the strings ''xy''&lt;sub&gt;1&lt;/sub&gt;''y''&lt;sub&gt;2&lt;/sub&gt;''z'' and ''xy''&lt;sub&gt;2&lt;/sub&gt;''y''&lt;sub&gt;1&lt;/sub&gt;''z'' represent a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or have a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''&lt;sub&gt;1&lt;/sub&gt;}''z'' ∪ ''x''{''Y'' \ ''y''&lt;sub&gt;2&lt;/sub&gt;}''z''.

e.g. in decimal (base ''b'' = 10):

* 2221 is a prime &gt; 10, thus the family 2{0,2}1 splits into the two families 2{0}1 and 2{0}2{0}1.
* 227 is a prime &gt; 10, and it is a subsequence of 5227, thus the family 5{0,2}7 splits into the two families 5{0}7 and 5{0}2{0}7.
* 449 is a prime &gt; 10, and it is a subsequence of 6449, thus the family 6{0,3,4,6,9}9 splits into the two families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9.
* Both 5051 and 5501 are primes &gt; 10, thus the family 5{0,5}1 splits into the two families 5{0}1 and 5{5}1 = {5}1.
* 8501 is a prime &gt; 10, thus the family 8{0,5}1 splits into the family 8{0}{5}1.
* 887 is a prime &gt; 10, and it is a subsequence of 2887, also 2087 is a prime &gt; 10, thus the family 2{0,8}7 splits into the two families 2{0}7 and 28{0}7.
* 349 and 449 are primes &gt; 10, and they are subsequences of 9349 and 9449, respectively, also 9049, 9649, 9949 are primes &gt; 10, thus the family 9{0,3,4,6,9}9 splits into the two families 9{0,3,6,9}9 and 94{0,3,6,9}9.
* 251, 281, 521, 821, 881 are primes &gt; 10, and they are subsequences of 9251, 9281, 9521, 9821, 9881, respectively, also 9001, 9221, 9551, 9851 are primes &gt; 10, thus the family 9{0,2,5,8}1 splits into the numbers {91, 901, 921, 951, 981, 9021, 9051, 9081, 9201, 9501, 9581, 9801, 90581, 95081, 95801}.

If the methods we have discussed cannot be used to rule out or shrink ''x''{''Y''}''z'' where ''Y'' = {''y''&lt;sub&gt;1&lt;/sub&gt;, ''y''&lt;sub&gt;2&lt;/sub&gt;, ..., ''y''&lt;sub&gt;''n''&lt;/sub&gt;}, then we can replace ''x''{''Y''}''z'' by ''xy''&lt;sub&gt;1&lt;/sub&gt;{''Y''}''z'' ∪ ''xy''&lt;sub&gt;2&lt;/sub&gt;{''Y''}''z'' ∪ ... ∪ ''xy''&lt;sub&gt;''n''&lt;/sub&gt;{''Y''}''z'' and re-run the methods on this new [[:w:Formal language|language]].

If all remain families are linear families (i.e. of the form ''x''{''y''}''z'', where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''), then we search the smallest (probable) primes in these families and add these primes to the list.

e.g. in decimal (base ''b'' = 10):

* The smallest prime in the family 5{0}27 is 5000000000000000000000000000027.
* The smallest prime in the family {5}1 is 555555555551.
* The smallest prime in the family 8{5}1 is 8555555555555555555551, but 8555555555555555555551 is not a minimal element since 555555555551 is a subsequence of 8555555555555555555551.

There is no guarantee that the techniques discussed will ever terminate, but in practice they often do. They are able to determine the Athena set in base ''b'' for 2 ≤ ''b'' ≤ 16 and ''b'' = 18, 20, 22, 24, 30. The bases ''b'' = 17, 19, 21, 23, 25 ≤ ''b'' ≤ 29, 31 ≤ ''b'' ≤ 36 are solved with the exception of 771 families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'').

The following is a &quot;[[:w:Semi-algorithm|semi-algorithm]]&quot; that is guaranteed to solve the Athena problem for a given base ''b'', but it is not so easy to implement:

# ''M'' = ''[[:w:Empty string|∅]]''
# while (''L'' ≠ ''∅'') do
# choose ''x'', a shortest string in ''L''
# ''M'' := ''M'' ∪ {''x''}
# ''L'' := ''L'' − ''sup''({''x''})

In practice, for arbitrary ''L'', we cannot feasibly carry out step 5. Instead, we work with ''L''&amp;#39;, some regular overapproximation to ''L'', until we can show ''L''&amp;#39; = ''∅'' (which implies ''L'' = ''∅''). In practice, ''L''&amp;#39; is usually chosen to be a finite [[:w:Union (set theory)|union]] of sets of the form ''L''&lt;sub&gt;1&lt;/sub&gt;{''L''&lt;sub&gt;2&lt;/sub&gt;}''L''&lt;sub&gt;3&lt;/sub&gt;, where each of ''L''&lt;sub&gt;1&lt;/sub&gt;, ''L''&lt;sub&gt;2&lt;/sub&gt;, ''L''&lt;sub&gt;3&lt;/sub&gt; is finite. In the case we consider in this project, we then have to determine whether such a family contains a prime or not.

Thus, the [[:w:Time complexity|time complexity]] of the Athena problem in base ''b'' may be ''[[:w:Big O notation|O]]''(''[[:w:E (mathematical_constant)|e]]''&lt;sup&gt;''[[:w:Euler's constant|γ]]''×(''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(*b*)&lt;/sup&gt;), and the [[:w:CPU time|CPU time]] of the Athena problem in base ''b'' may be longer than [[:w:Age of the universe|the age of the universe]] for bases ''b'' = 19, 23, 25, 27, 29, 31, 32, 33, 34, 35, also, Athena problem in bases ''b'' around 500 may be [[:w:NP-complete|NP-complete]] or [[:w:NP-hard|NP-hard]], or an [[:w:Undecidable problem|undecidable problem]], or an example of [[:w:Gödel's incompleteness theorems|Gödel's incompleteness theorems]] (like the [[:w:Continuum hypothesis|continuum hypothesis]] and the [[:w:Halting problem|halting problem]]).

To solve the Athena problem (i.e. to [[:w:Computing|compute]] the Athena set), we need to determine whether a given family contains a prime. In practice, if family ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''Y'' is a set of digits in base ''b'') could not be ruled out as only containing composites and ''Y'' contains two or more digits, then a relatively small prime &gt; ''b'' could always be found in this family. Intuitively, this is because there are a large number of small strings in such a family, and at least one is likely to be prime (e.g. there are 2&lt;sup&gt;''n''−2&lt;/sup&gt; strings of length ''n'' in the family 1{3,7}9, and there are over a thousand strings of length 12 in the family 1{3,7}9, thus it is very impossible that these numbers are all composite). In the case ''Y'' contains only one digit, this family is of the form ''x''{''y''}''z'', and there is only a single string of each length &gt; (the length of ''x'' + the length of ''z''), and it is not known if the following [[:w:Decision problem|decision problem]] is recursively solvable (just like [[:w:Sierpiński number|Sierpiński problem]] and [[:w:Riesel number|Riesel problem]], Sierpiński problem and Riesel problem can be generalized to other bases ''b'' (references: http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm), in fact, Athena problem base ''b'' covers the Sierpiński problem base ''b'' and the Riesel problem base ''b'' with ''k'' &lt; ''b'', i.e. finding the smallest prime of the form ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 and ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (or prove such prime does not exist) with ''k'' &lt; ''b'' (specially, for bases ''b'' such that the conjectured smallest Sierpiński number or the conjectured smallest Riesel number is &lt; ''b'', Athena problem base ''b'' covers the Sierpiński problem base ''b'' or the Riesel problem base ''b'', respectively), since the smallest prime of the form ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 and ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (if exists) must be a minimal element in base ''b'', also, Athena problem base ''b'' covers finding the smallest prime of these forms in base ''b'' (or proving that such prime does not exist): (''b''&lt;sup&gt;''n''&lt;/sup&gt;−1)/(''b''−1) (for this form, ''n'' must be prime, and we want ''n'' ≥ 2) (references of this form: http://www.fermatquotient.com/PrimSerien/GenRepu.txt, https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html, http://www.primenumbers.net/Henri/us/MersFermus.htm, http://www.bitman.name/math/table/379, https://pzktupel.de/Primetables/TableRepunitGen.php, https://oeis.org/A084740, https://oeis.org/A084738, https://oeis.org/A128164, https://oeis.org/A285642; or for prime bases ''b'': https://oeis.org/A065854, https://oeis.org/A279068), ''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 (for this form, ''n'' must be power of 2, and we want ''n'' ≥ 1) (references of this form: http://jeppesn.dk/generalized-fermat.html, http://www.noprimeleftbehind.net/crus/GFN-primes.htm, https://web.archive.org/web/20231002190634/http://yves.gallot.pagesperso-orange.fr/primes/index.html, https://pzktupel.de/Primetables/TableFermatGFBB.php, https://oeis.org/A079706, https://oeis.org/A084712, https://oeis.org/A228101), (''b''&lt;sup&gt;''n''&lt;/sup&gt;+1)/2 (for odd ''b'') (for this form, ''n'' must be power of 2, and we want ''n'' ≥ 2) (reference of this form: http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt), (''sqrt''(''b'')×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1)/(''sqrt''(''b'')+1) (for square ''b'') (for this form, 2×''n''+1 must be prime, and we want ''n'' ≥ 2) (references of this form: http://www.fermatquotient.com/PrimSerien/GenRepuP.txt, http://www.primenumbers.net/Henri/us/MersFermus.htm, http://www.bitman.name/math/table/488, https://pzktupel.de/Primetables/TableWagstaffGen.php, https://oeis.org/A084742, https://oeis.org/A084741; or for bases ''b'' with ''sqrt''(''b'') prime: https://oeis.org/A065507), ((''b''−2)×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1)/(''b''−1) (''n'' ≥ 2) (reference of this form: https://oeis.org/A243404), 2×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 (''n'' ≥ 1) (references of this form: https://www.mersenneforum.org/showthread.php?t=6918, https://www.mersenneforum.org/showthread.php?t=19725, https://oeis.org/A119624), 2×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (''n'' ≥ 1) (references of this form: https://www.mersenneforum.org/showthread.php?t=24576, https://www.mersenneforum.org/attachment.php?attachmentid=20976&amp;d=1567314217, https://oeis.org/A119591), ''b''&lt;sup&gt;''n''&lt;/sup&gt;+2 (''n'' ≥ 1) (references of this form: https://oeis.org/A138066, https://oeis.org/A084713, https://oeis.org/A138067), ''b''&lt;sup&gt;''n''&lt;/sup&gt;−2 (''n'' ≥ 2) (references of this form: https://www.primepuzzles.net/puzzles/puzz_887.htm, https://oeis.org/A250200, https://oeis.org/A255707, https://oeis.org/A084714; or for prime bases ''b'': https://oeis.org/A292201), (''b''−1)×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 (''n'' ≥ 1) (references of this form: http://www.noprimeleftbehind.net/Williams-primes-MP.htm, http://www.bitman.name/math/table/477, https://pzktupel.de/Primetables/TableWilliams2.php, https://oeis.org/A305531; or for prime bases ''b'': https://oeis.org/A087139), (''b''−1)×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (''n'' ≥ 1) (references of this form: https://harvey563.tripod.com/wills.txt, http://www.noprimeleftbehind.net/Williams-primes-MM.htm, http://www.bitman.name/math/table/484, https://pzktupel.de/Primetables/TableWilliams1.php; or for prime bases ''b'': https://oeis.org/A122396), ''b''&lt;sup&gt;''n''&lt;/sup&gt;+(''b''−1) (''n'' ≥ 1) (references of this form: http://www.bitman.name/math/table/795, https://pzktupel.de/Primetables/TableWilliams6.php, https://oeis.org/A076845, https://oeis.org/A076846, https://oeis.org/A078178, https://oeis.org/A078179), ''b''&lt;sup&gt;''n''&lt;/sup&gt;−(''b''−1) (''n'' ≥ 2) (references of this form: http://www.bitman.name/math/table/792, https://pzktupel.de/Primetables/TableWilliams5.php, https://oeis.org/A113516, https://oeis.org/A343589; or for prime bases ''b'': https://cs.uwaterloo.ca/journals/JIS/VOL3/mccranie.html, http://www.bitman.name/math/table/435)):

Problem: Given strings ''x'', ''z'' (may be empty), a digit ''y'', and a base ''b'' (''x'' does not [[:w:Leading zero|start with the digit 0]], ''z'' ends with a digit which [[:w:Coprime integers|coprime]] to ''b'', ''y'' is not 0 if ''x'' is empty, ''y'' is coprime to ''b'' if ''z'' is empty), does there exist a prime number whose base-''b'' expansion is of the form ''xy''&lt;sub&gt;''n''&lt;/sub&gt;''z'' for some ''n'' ≥ 0?

Some families can be ruled out to contain no prime &gt; ''b'' by [[:w:Covering set|covering congruence]], [[:w:Factorization of polynomials|algebraic factorization]] (e.g. [[:w:Difference of two squares|difference of two squares]], [[:w:Sum of two cubes|sum of two cubes]], [[:w:Sophie Germain's identity|Sophie Germain's identity of ''x''&lt;sup&gt;4&lt;/sup&gt;+4×''y''&lt;sup&gt;4&lt;/sup&gt;]]), or combine of them, e.g.

* The base 9 family 2{7}: Always divisible by 2 or 5
* The base 11 family 2{5}: Always divisible by 2 or 3
* The base 14 family B{0}1: Always divisible by 3 or 5
* The base 13 family 95{0}3: Always divisible by 5, 7, or 17
* The base 16 family {4}D: Always divisible by 3, 7, or 13
* The base 16 family {8}F: Always divisible by 3, 7, or 13
* The base 21 family {7}D: Always divisible by 2, 13, or 17
* The base 23 family {D}GA: Always divisible by 2, 5, 7, 37, or 79
* The base 9 family {1}: Can be written as (9&lt;sup&gt;''n''&lt;/sup&gt;−1)/8 and can be factored as (3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (3&lt;sup&gt;''n''&lt;/sup&gt;+1) / 8
* The base 8 family 1{0}1: Can be written as 8&lt;sup&gt;''n''&lt;/sup&gt;+1 and can be factored as (2&lt;sup&gt;''n''&lt;/sup&gt;+1) × (4&lt;sup&gt;''n''&lt;/sup&gt;−2&lt;sup&gt;''n''&lt;/sup&gt;+1)
* The base 9 family 3{8}: Can be written as 4×9&lt;sup&gt;''n''&lt;/sup&gt;−1 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (2×3&lt;sup&gt;''n''&lt;/sup&gt;+1)
* The base 16 family 1{5}: Can be written as (4×16&lt;sup&gt;''n''&lt;/sup&gt;−1)/3 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (2×3&lt;sup&gt;''n''&lt;/sup&gt;+1) / 3
* The base 16 family {4}1: Can be written as (4×16&lt;sup&gt;''n''&lt;/sup&gt;−49)/15 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−7) × (2×3&lt;sup&gt;''n''&lt;/sup&gt;+7) / 15
* The base 27 family 7{Q}: Can be written as 8×27&lt;sup&gt;''n''&lt;/sup&gt;−1 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (4×9&lt;sup&gt;''n''&lt;/sup&gt;+2×3&lt;sup&gt;''n''&lt;/sup&gt;+1)
* The base 27 family 9{G}: Can be written as (125×27&lt;sup&gt;''n''&lt;/sup&gt;−8)/13 and can be factored as (5×3&lt;sup&gt;''n''&lt;/sup&gt;−2) × (25×9&lt;sup&gt;''n''&lt;/sup&gt;+10×3&lt;sup&gt;''n''&lt;/sup&gt;+4)
* The base 16 family {C}D: Can be written as (4×16&lt;sup&gt;''n''&lt;/sup&gt;+1)/5 and can be factored as (2×4&lt;sup&gt;''n''&lt;/sup&gt;−2×2&lt;sup&gt;''n''&lt;/sup&gt;+1) × (2×4&lt;sup&gt;''n''&lt;/sup&gt;+2×2&lt;sup&gt;''n''&lt;/sup&gt;+1) / 5
* The base 14 family 8{D}: Can be written as 9×14&lt;sup&gt;''n''&lt;/sup&gt;−1, it is divisible by 5 if ''n'' is odd and can be factored as (3×14&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (3×14&lt;sup&gt;''n''/2&lt;/sup&gt;+1) if ''n'' is even
* The base 12 family {B}9B: Can be written as 12&lt;sup&gt;''n''&lt;/sup&gt;−25, it is divisible by 13 if ''n'' is odd and can be factored as (12&lt;sup&gt;''n''/2&lt;/sup&gt;−5) × (12&lt;sup&gt;''n''/2&lt;/sup&gt;+5) if ''n'' is even
* The base 14 family {D}5: Can be written as 14&lt;sup&gt;''n''&lt;/sup&gt;−9, it is divisible by 5 if ''n'' is odd and can be factored as (14&lt;sup&gt;''n''/2&lt;/sup&gt;−3) × (14&lt;sup&gt;''n''/2&lt;/sup&gt;+3) if ''n'' is even
* The base 17 family 1{9}: Can be written as (25×17&lt;sup&gt;''n''&lt;/sup&gt;−9)/16, it is divisible by 2 if ''n'' is odd and can be factored as (5×17&lt;sup&gt;''n''/2&lt;/sup&gt;−3) × (5×17&lt;sup&gt;''n''/2&lt;/sup&gt;+3) / 16 if ''n'' is even
* The base 17 family 7{9}: Can be written as (121×17&lt;sup&gt;''n''&lt;/sup&gt;−9)/16, it is divisible by 2 if ''n'' is odd and can be factored as (11×17&lt;sup&gt;''n''/2&lt;/sup&gt;−3) × (11×17&lt;sup&gt;''n''/2&lt;/sup&gt;+3) / 16 if ''n'' is even
* The base 19 family 1{6}: Can be written as (4×19&lt;sup&gt;''n''&lt;/sup&gt;−1)/3, it is divisible by 5 if ''n'' is odd and can be factored as (2×19&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (2×19&lt;sup&gt;''n''/2&lt;/sup&gt;+1) / 3 if ''n'' is even
* The base 19 family 7{2}: Can be written as (64×19&lt;sup&gt;''n''&lt;/sup&gt;−1)/9, it is divisible by 5 if ''n'' is odd and can be factored as (8×19&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (8×19&lt;sup&gt;''n''/2&lt;/sup&gt;+1) / 9 if ''n'' is even
* The base 24 family 3{N}: Can be written as 4×24&lt;sup&gt;''n''&lt;/sup&gt;−1, it is divisible by 5 if ''n'' is odd and can be factored as (2×24&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (2×24&lt;sup&gt;''n''/2&lt;/sup&gt;+1) if ''n'' is even

By the [[:w:Prime number theorem|prime number theorem]], the [[:w:Probability|chance]] that a [[:w:Random number|random]] ''n''-digit base ''b'' number is prime is [[:w:Asymptotic analysis|approximately]] 1/''n'' (more accurately, the chance is approximately 1/(''n''×''ln''(''b'')), where ''ln'' is the [[:w:Natural logarithm|natural logarithm]]). If one conjectures the numbers ''x''{''y''}''z'' behave similarly (i.e. the numbers ''x''{''y''}''z'' is a [[:w:Pseudorandomness|pseudorandom sequence]]) you would expect [[:w:Harmonic_series (mathematics)|1/1 + 1/2 + 1/3 + 1/4 + ... = ∞]] primes of the form ''x''{''y''}''z'' (of course, this does not always happen, since some ''x''{''y''}''z'' families can be ruled out to contain no prime &gt; ''b'' (by covering congruence, algebraic factorization, or combine of them), but it is at least a reasonable conjecture in the absence of evidence to the contrary. Hence, the [[:w:Heuristic argument|heuristic argument]] suggests there are always infinitely many primes in family ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') if it cannot be ruled out to contain no prime or only contain finitely many primes, by covering congruence, algebraic factorization, or combine of them. However, some families ''x''{''y''}''z'' could not be proven to contain no primes &gt; ''b'' (by covering congruence, algebraic factorization, or combine of them) but no primes &gt; ''b'' could be found in the family, even after searching through numbers with over 100000 digits. In such a case, the only way to proceed is to [[:w:Primality test|test the primality]] of larger and larger numbers of such form and hope a prime is eventually discovered. e.g. the smallest (probable) prime in the family A{3}A in base ''b'' = 13 is A3&lt;sub&gt;592197&lt;/sub&gt;A, its algebraic form is (41×13&lt;sup&gt;592198&lt;/sup&gt;+27)/4, when written in decimal contains 659677 digits (it is only probable prime, i.e. not definitely prime, since technically, probable primality tests were used to show this (which have a ''very'' small chance of making an error, see https://t5k.org/notes/prp_prob.html) because all known primality tests run far too slowly to run on numbers of this size unless either [https://t5k.org/prove/prove3_1.html ''N''−1] or [https://t5k.org/prove/prove3_2.html ''N''+1] (or both) can be ≥ 1/3 factored).

The numbers in family ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') are of the form (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1) for some fixed ''a'', ''b'', ''c'' such that ''a'' ≥ 1, ''b'' ≥ 2 (''b'' is the base), ''c'' ≠ 0, ''gcd''(''a'',''c'') = 1, ''gcd''(''b'',''c'') = 1. Except in the [[:w:Special case|special case]] ''c'' = ±1 and ''gcd''(''a''+''c'',''b''−1) = 1, when ''n'' is large the known [[:w:Primality test|primality test]]s for such a number are too inefficient to run. In this case one must resort to a [[:w:Probabilistic algorithm|probable]] primality test such as a [[:w:Miller–Rabin primality test|Miller–Rabin primality test]] or a [[:w:Baillie–PSW primality test|Baillie–PSW primality test]], unless a divisor of the number can be found. Since we are testing many numbers in an [[:w:Exponential growth|exponential sequence]], it is possible to use a sieving process to find divisors rather than using [[:w:Trial division|trial division]].

To do this, we made use of Geoffrey Reynolds' ''srsieve'' software (download: https://pzktupel.de/Software/srsieve_1.1.4.7z). This program uses the [[:w:Baby-step giant-step|baby-step giant-step]] [[:w:Algorithm|algorithm]] to find all primes ''p'' which divide ''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'' where ''p'' and ''n'' lie in a [[:w:Interval_(mathematics)|specified range]]. Since this program cannot handle the general case (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1) when ''gcd''(''a''+''c'',''b''−1) &gt; 1 we only used it to sieve the sequence ''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'' for primes ''p'' not dividing ''gcd''(''a''+''c'',''b''−1), and initialized the list of candidates to not include ''n'' for which there is some prime ''p'' dividing ''gcd''(''a''+''c'',''b''−1) for which ''p'' dividing (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1). The program had to be modified slightly to remove a check which would prevent it from running in the case when ''a'', ''b'', and ''c'' were all odd (since then 2 divides ''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'', but 2 may not divide (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1)).

Once the numbers with small divisors had been removed, it remained to test the remaining numbers using a probable primality test. For this we used the software ''LLR'' by Jean Penné. (download: http://jpenne.free.fr/index2.html). Although undocumented, it is possible to run this program on numbers of the form (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1) when ''gcd''(''a''+''c'',''b''−1) &gt; 1, so this program required no modifications. A script was also written which allowed one to run ''srsieve'' while ''LLR'' was testing the remaining candidates, so that when a divisor was found by srsieve on a number which had not yet been tested by ''LLR'' it would be removed from the list of candidates.

For the primes &lt; 10&lt;sup&gt;25000&lt;/sup&gt; for the &quot;easy&quot; bases (bases ''b'' with ≤ 150 primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; (base ''b'' = 26 has 83 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 3 unsolved families, base ''b'' = 36 has 75 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 4 unsolved families, base ''b'' = 17 has 99 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 18 unsolved families, base ''b'' = 21 has 80 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 12 unsolved families, base ''b'' = 19 has 201 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 23 unsolved families), i.e. bases *b* = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 26, 28, 30, 36), we employed ''CM'' by Andreas Enge (download: https://www.multiprecision.org/cm/download.html), an elliptic curve primality proving implementation.

== Data ==

These are the results of the Athena problem in bases 2 ≤ ''b'' ≤ 36 (we stop at base 36 since this base is the maximum base for which it is possible to write the numbers with the [[:w:Symbol|symbol]]s 0, 1, 2, ..., 9 and A, B, C, ..., Z (i.e. the 10 [[:w:Arabic numerals|Arabic numerals]] and the 26 [[:w:Latin script|Latin letters]]): (some large Athena primes are only probable primes, i.e. not definitely primes, since they are too large to be [[:w:Elliptic curve primality|ECPP proved]] and [[:w:Pocklington primality test#Extensions and variants|neither ''N''−1 nor ''N''+1 can be ≥ 1/3 factored]], all of them pass the [[:w:Baillie–PSW primality test|Baillie–PSW primality test]] and the [[:w:Strong pseudoprime|strong primality test]] (i.e. the [[:w:Miller–Rabin primality test|Miller–Rabin primality test]]) with all prime bases ''p'' ≤ 61, however, all Athena primes &lt; 10&lt;sup&gt;25000&lt;/sup&gt; for bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 36 are definitely primes, most of them &gt; 10&lt;sup&gt;299&lt;/sup&gt; are proven primes with [[:w:Elliptic curve primality|ECPP proving]], others &gt; 10&lt;sup&gt;299&lt;/sup&gt; are proven primes with [[:w:Pocklington primality test#Extensions and variants|''N''−1 or ''N''+1 proving]])

All numbers are written in base ''b'', [[:w:Senary#Base 36 as senary compression|using A to Z to represent digit values 10 to 35]], &quot;{}&quot; means repeating, e.g. family 12{3}45 means the sequence {1245, 12345, 123345, 1233345, 12333345, 123333345, ...} (where the members are expressed as base ''b'' strings), subscripts are used to indicate repetitions of digits, e.g. 123&lt;sub&gt;4&lt;/sub&gt;567 means 123333567 (all subscripts are written in decimal).

Base 2: 1 Athena prime (the largest of which has 2 digits (it is 11, and its value is 3 in decimal)): {11}

Base 3: 3 Athena primes (the largest of which has 3 digits (it is 111, and its value is 13 in decimal)): {12, 21, 111}

Base 4: 5 Athena primes (the largest of which has 3 digits (it is 221, and its value is 41 in decimal)): {11, 13, 23, 31, 221}

Base 5: 22 Athena primes (the largest of which has 96 digits (it is 10&lt;sub&gt;93&lt;/sub&gt;13, and its algebraic form is 5&lt;sup&gt;95&lt;/sup&gt;+8)): {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 Athena primes (the largest of which has 5 digits (it is 40041, and its value is 5209 in decimal)): {11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041}

Base 7: 71 Athena primes (the largest of which has 17 digits (it is 3&lt;sub&gt;16&lt;/sub&gt;1, and its algebraic form is (7&lt;sup&gt;17&lt;/sup&gt;−5)/2)): {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: 75 Athena primes (the largest of which has 221 digits (it is 4&lt;sub&gt;220&lt;/sub&gt;7, and its algebraic form is (4×8&lt;sup&gt;221&lt;/sup&gt;+17)/7)): {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: 151 Athena primes (the largest of which has 1161 digits (it is 30&lt;sub&gt;1158&lt;/sub&gt;11, and its algebraic form is 3×9&lt;sup&gt;1160&lt;/sup&gt;+10)): {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: 77 Athena primes (the largest of which has 31 digits (it is 50&lt;sub&gt;28&lt;/sub&gt;27, and its algebraic form is 5×10&lt;sup&gt;30&lt;/sup&gt;+27)): {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: 1068 Athena primes (including 1 unproven probable prime: 57&lt;sub&gt;62668&lt;/sub&gt;), the largest of which has 62669 digits (it is 57&lt;sub&gt;62668&lt;/sub&gt;, and its algebraic form is (57×11&lt;sup&gt;62668&lt;/sup&gt;−7)/10), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel11 Data of Athena (probable) primes base 11]

Base 12: 106 Athena primes (the largest of which has 42 digits (it is 40&lt;sub&gt;39&lt;/sub&gt;77, and its algebraic form is 4×12&lt;sup&gt;41&lt;/sup&gt;+91)): {11, 15, 17, 1B, 25, 27, 31, 35, 37, 3B, 45, 4B, 51, 57, 5B, 61, 67, 6B, 75, 81, 85, 87, 8B, 91, 95, A7, AB, B5, B7, 221, 241, 2A1, 2B1, 2BB, 401, 421, 447, 471, 497, 565, 655, 665, 701, 70B, 721, 747, 771, 77B, 797, 7A1, 7BB, 907, 90B, 9BB, A41, B21, B2B, 2001, 200B, 202B, 222B, 229B, 292B, 299B, 4441, 4707, 4777, 6A05, 6AA5, 729B, 7441, 7B41, 929B, 9777, 992B, 9947, 997B, 9997, A0A1, A201, A605, A6A5, AA65, B001, B0B1, BB01, BB41, 600A5, 7999B, 9999B, AAAA1, B04A1, B0B9B, BAA01, BAAA1, BB09B, BBBB1, 44AAA1, A00065, BBBAA1, AAA0001, B00099B, AA000001, BBBBBB99B, B0000000000000000000000000009B, 400000000000000000000000000000000000000077}

Base 13: 3197 Athena primes (including 4 unproven probable primes: C5&lt;sub&gt;23755&lt;/sub&gt;C, 80&lt;sub&gt;32017&lt;/sub&gt;111, 95&lt;sub&gt;197420&lt;/sub&gt;, A3&lt;sub&gt;592197&lt;/sub&gt;A), the largest of which has 592199 digits (it is A3&lt;sub&gt;592197&lt;/sub&gt;A, and its algebraic form is (41×13&lt;sup&gt;592198&lt;/sup&gt;+27)/4), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel13 Data of Athena (probable) primes base 13]

Base 14: 650 Athena primes, the largest of which has 19699 digits (it is 4D&lt;sub&gt;19698&lt;/sub&gt;, and its algebraic form is 5×14&lt;sup&gt;19698&lt;/sup&gt;−1), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel14 Data of Athena primes base 14]

Base 15: 1284 Athena primes, the largest of which has 157 digits (it is 7&lt;sub&gt;155&lt;/sub&gt;97, and its algebraic form is (15&lt;sup&gt;157&lt;/sup&gt;+59)/2), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel15 Data of Athena primes base 15]

Base 16: 2347 Athena primes (including 3 unproven probable primes: DB&lt;sub&gt;32234&lt;/sub&gt;, 4&lt;sub&gt;72785&lt;/sub&gt;DD, 3&lt;sub&gt;116137&lt;/sub&gt;AF), the largest of which has 116139 digits (it is 3&lt;sub&gt;116137&lt;/sub&gt;AF, and its algebraic form is (16&lt;sup&gt;116139&lt;/sup&gt;+619)/5), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel16 Data of Athena (probable) primes base 16]

Base 17: 10415 known Athena primes (including many unproven probable primes) and 12 unsolved families (1{7}, 1F{0}7, 4{7}A, 70F{0}D, 8{B}9, 9{5}9, A{D}F, B{0}B3, {B}E9, {B}EE, F1{9}, FD0{D}, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel17 Data of known Athena (probable) primes base 17]

Base 18: 549 Athena primes, the largest of which has 6271 digits (it is C0&lt;sub&gt;6268&lt;/sub&gt;C5, and its algebraic form is 12×18&lt;sup&gt;6270&lt;/sup&gt;+221), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel18 Data of Athena primes base 18]

Base 19: 31417 known Athena primes (including many unproven probable primes) and 17 unsolved families (4B5{0}H, {5}3, 5{H}05, 5{H}0H, 5{H}5, 66{B}, 71{0}177, 7AF{0}H, 97{0}3, C{H}C, EE1{6}, F{7}5, F{B}G, F{D}F, H0F{0}7A, HB{0}5B5, II{D}, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel19 Data of known Athena (probable) primes base 19]

Base 20: 3314 Athena primes, the largest of which has 6271 digits (it is G0&lt;sub&gt;6269&lt;/sub&gt;D, and its algebraic form is 16×20&lt;sup&gt;6270&lt;/sup&gt;+13), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel20 Data of Athena primes base 20]

Base 21: 13386 known Athena primes (including many unproven probable primes) and 8 unsolved families (5{0}DJ, {9}D, B3{0}EB, B{H}6H, C{F}0K, {F}35, G{0}FK, H{0}7771, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel21 Data of known Athena (probable) primes base 21]

Base 22: 8003 Athena primes (including 1 unproven probable prime: BK&lt;sub&gt;22001&lt;/sub&gt;5), the largest of which has 22003 digits (it is BK&lt;sub&gt;22001&lt;/sub&gt;5, and its algebraic form is (251×22&lt;sup&gt;22002&lt;/sup&gt;−335)/21), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel22 Data of Athena (probable) primes base 22]

Base 23: 65178 known Athena primes (including many unproven probable primes) and 87 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel23 Data of known Athena (probable) primes base 23] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left23 Data of unsolved families for Athena problem base 23]

Base 24: 3409 Athena primes, the largest of which has 8134 digits (it is N00N&lt;sub&gt;8129&lt;/sub&gt;LN, and its algebraic form is 13249×24&lt;sup&gt;8131&lt;/sup&gt;−49), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel24 Data of Athena primes base 24]

Base 25: 133639 known Athena primes (including many unproven probable primes) and 85 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel25 Data of known Athena (probable) primes base 25] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left25 Data of unsolved families for Athena problem base 25]

Base 26: 25256 known Athena primes (including 7 unproven probable primes: 5&lt;sub&gt;19391&lt;/sub&gt;6F, 7&lt;sub&gt;20279&lt;/sub&gt;OL, LD0&lt;sub&gt;20975&lt;/sub&gt;7, 6K&lt;sub&gt;23300&lt;/sub&gt;5, J0&lt;sub&gt;44303&lt;/sub&gt;KCB, M0&lt;sub&gt;61186&lt;/sub&gt;2BB, 85M&lt;sub&gt;197060&lt;/sub&gt;B) and 3 unsolved families ({A}6F, {H}MH, {I}GL, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel26 Data of known Athena (probable) primes base 26]

Base 27: 102852 known Athena primes (including many unproven probable primes) and 44 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel27 Data of known Athena (probable) primes base 27] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left27 Data of unsolved families for Athena problem base 27]

Base 28: 25528 known Athena primes (including 3 unproven probable primes: N6&lt;sub&gt;24051&lt;/sub&gt;LR, 5OA&lt;sub&gt;31238&lt;/sub&gt;F, O4O&lt;sub&gt;94535&lt;/sub&gt;9) and 1 unsolved family (O{A}F, no primes or probable primes with length ≤ 900000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel28 Data of known Athena (probable) primes base 28]

Base 29: 355242 known Athena primes (including many unproven probable primes) and 125 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel29 Data of known Athena (probable) primes base 29] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left29 Data of unsolved families for Athena problem base 29]

Base 30: 2619 Athena primes (including 1 unproven probable prime: I0&lt;sub&gt;24608&lt;/sub&gt;D), the largest of which has 34206 digits (it is OT&lt;sub&gt;34205&lt;/sub&gt;, and its algebraic form is 25×30&lt;sup&gt;34205&lt;/sup&gt;−1), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel30 Data of Athena (probable) primes base 30]

Base 31: 569323 known Athena primes (including many unproven probable primes) and 77 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel31 Data of known Athena (probable) primes base 31] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left31 Data of unsolved families for Athena problem base 31]

Base 32: 168882 known Athena primes (including many unproven probable primes) and 120 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel32 Data of known Athena (probable) primes base 32] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left32 Data of unsolved families for Athena problem base 32]

Base 33: 280012 known Athena primes (including many unproven probable primes) and 81 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel33 Data of known Athena (probable) primes base 33] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left33 Data of unsolved families for Athena problem base 33]

Base 34: 184785 known Athena primes (including many unproven probable primes) and 47 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel34 Data of known Athena (probable) primes base 34] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left34 Data of unsolved families for Athena problem base 34]

Base 35: 720002 known Athena primes (including many unproven probable primes) and 60 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel35 Data of known Athena (probable) primes base 35] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left35 Data of unsolved families for Athena problem base 35]

Base 36: 35286 known Athena primes (including 3 unproven probable primes: 7K&lt;sub&gt;26567&lt;/sub&gt;Z, S0&lt;sub&gt;75007&lt;/sub&gt;8H, P&lt;sub&gt;81993&lt;/sub&gt;SZ) and 4 unsolved families (B{0}EUV, HM{0}N, N{0}YYN, O{L}Z, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel36 Data of known Athena (probable) primes base 36]

== Condensed table for bases 2 ≤ ''b'' ≤ 36 ==

{|class=&quot;wikitable&quot;
||''b''||number of Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b''||base-''b'' form of the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b'' (write &quot;''d''&lt;sub&gt;''n''&lt;/sub&gt;&quot; if there are 5 or more (''n'') consecutive same digits ''d'')||length of the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b''||length of the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b'' in decimal||algebraic ((''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1)) form of the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b''||''factordb'' entry of the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b''||the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b'' written in base ''b'' (use lower case letters instead of upper case letters)||number of unsolved families in the Athena problem in base ''b'' (all of these left families are linear families)||searching limit of length for the unsolved families in the Athena problem in base ''b'' (if there are different searching limits for the unsolved families in the Athena problem in base ''b'', choose the lowest searching limit)||
|-
||2||1||11||2||1||3||http://factordb.com/index.php?id=3&amp;open=ecm||http://factordb.com/index.php?showid=3&amp;base=2||0||–||
|-
||3||3||111&lt;br&gt;21&lt;br&gt;12||3&lt;br&gt;2&lt;br&gt;2||2&lt;br&gt;1&lt;br&gt;1||13&lt;br&gt;7&lt;br&gt;5||http://factordb.com/index.php?id=13&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=7&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=5&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=13&amp;base=3&lt;br&gt;http://factordb.com/index.php?showid=7&amp;base=3&lt;br&gt;http://factordb.com/index.php?showid=5&amp;base=3&lt;nowiki/&gt;||0||–||
|-
||4||5||221&lt;br&gt;31&lt;br&gt;23&lt;br&gt;13&lt;br&gt;11||3&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2||2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;1&lt;br&gt;1||41&lt;br&gt;13&lt;br&gt;11&lt;br&gt;7&lt;br&gt;5||http://factordb.com/index.php?id=41&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=13&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=11&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=7&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=5&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=41&amp;base=4&lt;br&gt;http://factordb.com/index.php?showid=13&amp;base=4&lt;br&gt;http://factordb.com/index.php?showid=11&amp;base=4&lt;br&gt;http://factordb.com/index.php?showid=7&amp;base=4&lt;br&gt;http://factordb.com/index.php?showid=5&amp;base=4&lt;nowiki/&gt;||0||–||
|-
||5||22||10&lt;sub&gt;93&lt;/sub&gt;13&lt;br&gt;300031&lt;br&gt;44441&lt;br&gt;33331&lt;br&gt;33001&lt;br&gt;30301&lt;br&gt;14444&lt;br&gt;10103&lt;br&gt;3101&lt;br&gt;414||96&lt;br&gt;6&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;4&lt;br&gt;3||67&lt;br&gt;4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;3&lt;br&gt;3&lt;br&gt;3||5&lt;sup&gt;95&lt;/sup&gt;+8&lt;br&gt;9391&lt;br&gt;3121&lt;br&gt;2341&lt;br&gt;2251&lt;br&gt;1951&lt;br&gt;1249&lt;br&gt;653&lt;br&gt;401&lt;br&gt;109||http://factordb.com/index.php?id=1100000000034686071&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=9391&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=3121&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=2341&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=2251&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1951&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1249&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=653&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=401&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=109&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000000034686071&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=9391&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=3121&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=2341&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=2251&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=1951&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=1249&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=653&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=401&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=109&amp;base=5&lt;nowiki/&gt;||0||–||
|-
||6||11||40041&lt;br&gt;4441&lt;br&gt;4401&lt;br&gt;51&lt;br&gt;45&lt;br&gt;35&lt;br&gt;31&lt;br&gt;25&lt;br&gt;21&lt;br&gt;15||5&lt;br&gt;4&lt;br&gt;4&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2||4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2||5209&lt;br&gt;1033&lt;br&gt;1009&lt;br&gt;31&lt;br&gt;29&lt;br&gt;23&lt;br&gt;19&lt;br&gt;17&lt;br&gt;13&lt;br&gt;11||http://factordb.com/index.php?id=5209&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1033&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1009&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=31&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=29&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=23&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=19&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=17&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=13&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=11&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=5209&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=1033&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=1009&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=31&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=29&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=23&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=19&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=17&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=13&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=11&amp;base=6&lt;nowiki/&gt;||0||–||
|-
||7||71||3&lt;sub&gt;16&lt;/sub&gt;1&lt;br&gt;510&lt;sub&gt;7&lt;/sub&gt;1&lt;br&gt;3&lt;sub&gt;6&lt;/sub&gt;01&lt;br&gt;1100021&lt;br&gt;531101&lt;br&gt;351101&lt;br&gt;300053&lt;br&gt;150001&lt;br&gt;100121&lt;br&gt;40054||17&lt;br&gt;10&lt;br&gt;8&lt;br&gt;7&lt;br&gt;6&lt;br&gt;6&lt;br&gt;6&lt;br&gt;6&lt;br&gt;6&lt;br&gt;5||15&lt;br&gt;9&lt;br&gt;7&lt;br&gt;6&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;4||(7&lt;sup&gt;17&lt;/sup&gt;−5)/2&lt;br&gt;36×7&lt;sup&gt;8&lt;/sup&gt;+1&lt;br&gt;(7&lt;sup&gt;8&lt;/sup&gt;−47)/2&lt;br&gt;134471&lt;br&gt;91631&lt;br&gt;62819&lt;br&gt;50459&lt;br&gt;28813&lt;br&gt;16871&lt;br&gt;9643||http://factordb.com/index.php?id=116315256993601&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=207532837&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=2882377&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=134471&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=91631&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=62819&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=50459&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=28813&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=16871&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=9643&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=116315256993601&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=207532837&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=2882377&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=134471&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=91631&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=62819&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=50459&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=28813&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=16871&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=9643&amp;base=7&lt;nowiki/&gt;||0||–||
|-
||8||75||4&lt;sub&gt;220&lt;/sub&gt;7&lt;br&gt;5&lt;sub&gt;13&lt;/sub&gt;25&lt;br&gt;7&lt;sub&gt;12&lt;/sub&gt;1&lt;br&gt;77774&lt;sub&gt;6&lt;/sub&gt;1&lt;br&gt;74&lt;sub&gt;7&lt;/sub&gt;1&lt;br&gt;4&lt;sub&gt;8&lt;/sub&gt;1&lt;br&gt;5&lt;sub&gt;5&lt;/sub&gt;025&lt;br&gt;5550525&lt;br&gt;5500525&lt;br&gt;4&lt;sub&gt;5&lt;/sub&gt;77||221&lt;br&gt;15&lt;br&gt;13&lt;br&gt;11&lt;br&gt;9&lt;br&gt;9&lt;br&gt;8&lt;br&gt;7&lt;br&gt;7&lt;br&gt;7||200&lt;br&gt;14&lt;br&gt;12&lt;br&gt;10&lt;br&gt;9&lt;br&gt;8&lt;br&gt;8&lt;br&gt;7&lt;br&gt;7&lt;br&gt;7||(4×8&lt;sup&gt;221&lt;/sup&gt;+17)/7&lt;br&gt;(5×8&lt;sup&gt;15&lt;/sup&gt;−173)/7&lt;br&gt;8&lt;sup&gt;13&lt;/sup&gt;−7&lt;br&gt;(28669×8&lt;sup&gt;7&lt;/sup&gt;−25)/7&lt;br&gt;(53×8&lt;sup&gt;8&lt;/sup&gt;−25)/7&lt;br&gt;(4×8&lt;sup&gt;9&lt;/sup&gt;−25)/7&lt;br&gt;(5×8&lt;sup&gt;8&lt;/sup&gt;−2413)/7&lt;br&gt;1495381&lt;br&gt;1474901&lt;br&gt;(4×8&lt;sup&gt;7&lt;/sup&gt;+185)/7||http://factordb.com/index.php?id=1100000000416605822&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=25131694349141&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=549755813881&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=8589035809&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=127027489&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=76695841&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=11983381&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1495381&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1474901&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1198399&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000000416605822&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=25131694349141&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=549755813881&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=8589035809&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=127027489&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=76695841&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=11983381&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=1495381&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=1474901&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=1198399&amp;base=8&lt;nowiki/&gt;||0||–||
|-
||9||151||30&lt;sub&gt;1158&lt;/sub&gt;11&lt;br&gt;27&lt;sub&gt;686&lt;/sub&gt;07&lt;br&gt;76&lt;sub&gt;329&lt;/sub&gt;2&lt;br&gt;561&lt;sub&gt;36&lt;/sub&gt;&lt;br&gt;10&lt;sub&gt;25&lt;/sub&gt;57&lt;br&gt;30&lt;sub&gt;20&lt;/sub&gt;51&lt;br&gt;8&lt;sub&gt;19&lt;/sub&gt;335&lt;br&gt;727&lt;sub&gt;15&lt;/sub&gt;07&lt;br&gt;51&lt;sub&gt;13&lt;/sub&gt;61&lt;br&gt;10&lt;sub&gt;11&lt;/sub&gt;507||1161&lt;br&gt;689&lt;br&gt;331&lt;br&gt;38&lt;br&gt;28&lt;br&gt;23&lt;br&gt;22&lt;br&gt;19&lt;br&gt;16&lt;br&gt;15||1108&lt;br&gt;657&lt;br&gt;316&lt;br&gt;37&lt;br&gt;26&lt;br&gt;22&lt;br&gt;21&lt;br&gt;19&lt;br&gt;16&lt;br&gt;14||3×9&lt;sup&gt;1160&lt;/sup&gt;+10&lt;br&gt;(23×9&lt;sup&gt;688&lt;/sup&gt;−511)/8&lt;br&gt;(31×9&lt;sup&gt;330&lt;/sup&gt;−19)/4&lt;br&gt;(409×9&lt;sup&gt;36&lt;/sup&gt;−1)/8&lt;br&gt;9&lt;sup&gt;27&lt;/sup&gt;+52&lt;br&gt;3×9&lt;sup&gt;22&lt;/sup&gt;+46&lt;br&gt;9&lt;sup&gt;22&lt;/sup&gt;−454&lt;br&gt;(527×9&lt;sup&gt;17&lt;/sup&gt;−511)/8&lt;br&gt;(41×9&lt;sup&gt;15&lt;/sup&gt;+359)/8&lt;br&gt;9&lt;sup&gt;14&lt;/sup&gt;+412||http://factordb.com/index.php?id=1100000002376318423&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000002495467486&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000002359003642&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000001554010824&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1100000002512830927&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1100000000032261811&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1100000002495736583&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1100000003446800389&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1055192051985121&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=22876792455373&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000002376318423&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=1100000002495467486&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=1100000002359003642&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=1100000001554010824&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=1100000002512830927&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=1100000000032261811&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=1100000002495736583&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=1100000003446800389&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=1055192051985121&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=22876792455373&amp;base=9&lt;nowiki/&gt;||0||–||
|-
||10||77||50&lt;sub&gt;28&lt;/sub&gt;27&lt;br&gt;5&lt;sub&gt;11&lt;/sub&gt;1&lt;br&gt;805&lt;sub&gt;5&lt;/sub&gt;1&lt;br&gt;66600049&lt;br&gt;66000049&lt;br&gt;60&lt;sub&gt;5&lt;/sub&gt;49&lt;br&gt;220&lt;sub&gt;5&lt;/sub&gt;1&lt;br&gt;5200007&lt;br&gt;946669&lt;br&gt;666649||31&lt;br&gt;12&lt;br&gt;8&lt;br&gt;8&lt;br&gt;8&lt;br&gt;8&lt;br&gt;8&lt;br&gt;7&lt;br&gt;6&lt;br&gt;6||31&lt;br&gt;12&lt;br&gt;8&lt;br&gt;8&lt;br&gt;8&lt;br&gt;8&lt;br&gt;8&lt;br&gt;7&lt;br&gt;6&lt;br&gt;6||5×10&lt;sup&gt;30&lt;/sup&gt;+27&lt;br&gt;(5×10&lt;sup&gt;12&lt;/sup&gt;−41)/9&lt;br&gt;(725×10&lt;sup&gt;6&lt;/sup&gt;−41)/9&lt;br&gt;66600049&lt;br&gt;66000049&lt;br&gt;6×10&lt;sup&gt;7&lt;/sup&gt;+49&lt;br&gt;22×10&lt;sup&gt;6&lt;/sup&gt;+1&lt;br&gt;5200007&lt;br&gt;946669&lt;br&gt;666649||http://factordb.com/index.php?id=1100000000204142046&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=555555555551&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=80555551&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=66600049&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=66000049&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=60000049&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=22000001&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=5200007&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=946669&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=666649&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000000204142046&amp;base=10&lt;br&gt;http://factordb.com/index.php?showid=555555555551&amp;base=10&lt;br&gt;http://factordb.com/index.php?showid=80555551&amp;base=10&lt;br&gt;http://factordb.com/index.php?showid=66600049&amp;base=10&lt;br&gt;http://factordb.com/index.php?showid=66000049&amp;base=10&lt;br&gt;http://factordb.com/index.php?showid=60000049&amp;base=10&lt;br&gt;http://factordb.com/index.php?showid=22000001&amp;base=10&lt;br&gt;http://factordb.com/index.php?showid=5200007&amp;base=10&lt;br&gt;http://factordb.com/index.php?showid=946669&amp;base=10&lt;br&gt;http://factordb.com/index.php?showid=666649&amp;base=10&lt;nowiki/&gt;||0||–||
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||24||3409||N00N&lt;sub&gt;8129&lt;/sub&gt;LN&lt;br&gt;88N&lt;sub&gt;5951&lt;/sub&gt;&lt;br&gt;A0&lt;sub&gt;2951&lt;/sub&gt;8ID&lt;br&gt;D&lt;sub&gt;2698&lt;/sub&gt;LD&lt;br&gt;N&lt;sub&gt;2644&lt;/sub&gt;LLN&lt;br&gt;BC0&lt;sub&gt;331&lt;/sub&gt;B&lt;br&gt;20&lt;sub&gt;313&lt;/sub&gt;7&lt;br&gt;C7&lt;sub&gt;298&lt;/sub&gt;&lt;br&gt;D0&lt;sub&gt;259&lt;/sub&gt;KKD&lt;br&gt;I0&lt;sub&gt;241&lt;/sub&gt;I5||8134&lt;br&gt;5953&lt;br&gt;2955&lt;br&gt;2700&lt;br&gt;2647&lt;br&gt;334&lt;br&gt;315&lt;br&gt;299&lt;br&gt;263&lt;br&gt;244||11227&lt;br&gt;8216&lt;br&gt;4079&lt;br&gt;3727&lt;br&gt;3654&lt;br&gt;461&lt;br&gt;434&lt;br&gt;413&lt;br&gt;363&lt;br&gt;337||13249×24&lt;sup&gt;8131&lt;/sup&gt;−49&lt;br&gt;201×24&lt;sup&gt;5951&lt;/sup&gt;−1&lt;br&gt;10×24&lt;sup&gt;2954&lt;/sup&gt;+5053&lt;br&gt;(13×24&lt;sup&gt;2700&lt;/sup&gt;+4403)/23&lt;br&gt;24&lt;sup&gt;2647&lt;/sup&gt;−1201&lt;br&gt;276×24&lt;sup&gt;332&lt;/sup&gt;+11&lt;br&gt;2×24&lt;sup&gt;314&lt;/sup&gt;+7&lt;br&gt;(283×24&lt;sup&gt;298&lt;/sup&gt;−7)/23&lt;br&gt;13×24&lt;sup&gt;262&lt;/sup&gt;+12013&lt;br&gt;18×24&lt;sup&gt;243&lt;/sup&gt;+437||http://factordb.com/index.php?id=1100000003593391606&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003593275880&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003593269654&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003593269876&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003593270089&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000002633359842&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000002355610241&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000002326181235&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003593270725&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000002633360037&amp;open=prime&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000003593391606&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000003593275880&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000003593269654&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000003593269876&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000003593270089&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000002633359842&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000002355610241&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000002326181235&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000003593270725&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000002633360037&amp;base=24&lt;nowiki/&gt;||0||–||
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||36||35286~35290||P&lt;sub&gt;81993&lt;/sub&gt;SZ&lt;br&gt;S0&lt;sub&gt;75007&lt;/sub&gt;8H&lt;br&gt;7K&lt;sub&gt;26567&lt;/sub&gt;Z&lt;br&gt;J&lt;sub&gt;10117&lt;/sub&gt;LJ&lt;br&gt;VL0&lt;sub&gt;7258&lt;/sub&gt;J&lt;br&gt;EO0&lt;sub&gt;6177&lt;/sub&gt;V&lt;br&gt;FZ&lt;sub&gt;5777&lt;/sub&gt;3P&lt;br&gt;T09&lt;sub&gt;4618&lt;/sub&gt;1&lt;br&gt;RY&lt;sub&gt;4562&lt;/sub&gt;H&lt;br&gt;OZ&lt;sub&gt;3932&lt;/sub&gt;AZ||81995&lt;br&gt;75010&lt;br&gt;26569&lt;br&gt;10119&lt;br&gt;7261&lt;br&gt;6180&lt;br&gt;5780&lt;br&gt;4621&lt;br&gt;4564&lt;br&gt;3935||127609&lt;br&gt;116739&lt;br&gt;41349&lt;br&gt;15748&lt;br&gt;11301&lt;br&gt;9618&lt;br&gt;8996&lt;br&gt;7192&lt;br&gt;7103&lt;br&gt;6124||(5×36&lt;sup&gt;81995&lt;/sup&gt;+821)/7&lt;br&gt;28×36&lt;sup&gt;75009&lt;/sup&gt;+305&lt;br&gt;(53×36&lt;sup&gt;26568&lt;/sup&gt;+101)/7&lt;br&gt;(19×36&lt;sup&gt;10119&lt;/sup&gt;+2501)/35&lt;br&gt;1137×36&lt;sup&gt;7259&lt;/sup&gt;+19&lt;br&gt;528×36&lt;sup&gt;6178&lt;/sup&gt;+31&lt;br&gt;16×36&lt;sup&gt;5779&lt;/sup&gt;−1163&lt;br&gt;(36549×36&lt;sup&gt;4619&lt;/sup&gt;−289)/35&lt;br&gt;(979×36&lt;sup&gt;4563&lt;/sup&gt;−629)/35&lt;br&gt;25×36&lt;sup&gt;3934&lt;/sup&gt;−901||http://factordb.com/index.php?id=1100000002394962083&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004020085177&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003896952461&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362491&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362489&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362488&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362487&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362486&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362485&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000000840634476&amp;open=prime&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000002394962083&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000004020085177&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003896952461&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362491&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362489&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362488&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362487&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362486&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362485&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000000840634476&amp;base=36&lt;nowiki/&gt;||4||200000||
|}

== The fully proof of Athena problem in decimal (base ''b'' = 10) ==

'''Bold''' for the Athena primes, ''x'' ◁ ''y'' means ''x'' is a subsequence of ''y''.

Assume ''p'' is a prime &gt; 10, and the last digit of ''p'' must lie in {1,3,7,9}.

Case 1: ''p'' ends with 1.

In this case we can write ''p'' = ''x''1. If ''x'' contains 1, 3, 4, 6, or 7, then (respectively) '''11''' ◁ ''p'', '''31''' ◁ ''p'', '''41''' ◁ ''p'', '''61''' ◁ ''p'', or '''71''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 2, 5, 8, or 9.

Case 1.1: ''p'' begins with 2.

In this case we can write ''p'' = 2''y''1. If 5 ◁ ''y'', then '''251''' ◁ ''p''. If 8 ◁ ''y'', then '''281''' ◁ ''p''. If 9 ◁ ''y'', then 29 ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 2.

If 22 ◁ ''y'', then '''2221''' ◁ ''p''. Hence we may assume ''y'' contains zero or one 2's.

If ''y'' contains no 2's, then ''p'' ∈ 2{0}1. But then, since the sum of the digits of ''p'' is 3, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' contains exactly one 2, then we can write ''p'' = 2''z''2''w''1, where ''z'',''w'' ∈ {0}. If 0 ◁ ''z'' and 0 ◁ ''w'', then '''20201''' ◁ ''p''. Hence we may assume either ''z'' or ''w'' is empty.

If ''z'' is empty, then ''p'' ∈ 22{0}1, and the smallest prime ''p'' ∈ 22{0}1 is '''22000001'''.

If ''w'' is empty, then ''p'' ∈ 2{0}21, and the smallest prime ''p'' ∈ 2{0}21 is '''20021'''.

Case 1.2: ''p'' begins with 5.

In this case we can write ''p'' = 5''y''1. If 2 ◁ ''y'', then '''521''' ◁ ''p''. If 9 ◁ ''y'', then 59 ◁ ''p''. Hence we may assume all digits of ''y'' are 0, 5, or 8.

If 05 ◁ ''y'', then '''5051''' ◁ ''p''. If 08 ◁ ''y'', then '''5081''' ◁ ''p''. If 50 ◁ ''y'', then '''5501''' ◁ ''p''. If 58 ◁ ''y'', then '''5581''' ◁ ''p''. If 80 ◁ ''y'', then '''5801''' ◁ ''p''. If 85 ◁ ''y'', then '''5851''' ◁ ''p''. Hence we may assume ''y'' ∈ {0} ∪ {5} ∪ {8}.

If ''y'' ∈ {0}, then ''p'' ∈ 5{0}1. But then, since the sum of the digits of ''p'' is 6, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' ∈ {5}, then ''p'' ∈ 5{5}1, and the smallest prime ''p'' ∈ 5{5}1 is '''555555555551'''.

If ''y'' ∈ {8}, since if 88 ◁ ''y'', then 881 ◁ ''p'', hence we may assume ''y'' ∈ {''𝜆'',8}, and thus ''p'' ∈ {51,581}, but 51 and 581 are both composite.

Case 1.3: ''p'' begins with 8.

In this case we can write p = 8''y''1. If 2 ◁ ''y'', then '''821''' ◁ ''p''. If 8 ◁ ''y'', then '''881''' ◁ ''p''. If 9 ◁ ''y'', then 89 ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 5.

If 50 ◁ ''y'', then '''8501''' ◁ ''p''. Hence we may assume y ∈ {0}{5}.

If 005 ◁ ''y'', then '''80051''' ◁ p. Hence we may assume y ∈ {0} ∪ {5} ∪ 0{5}.

If y ∈ {0}, then ''p'' ∈ 8{0}1. But then, since the sum of the digits of ''p'' is 9, ''p'' is divisible by 3, so ''p'' cannot be prime.

If y ∈ {5}, since if 55555555555 ◁ ''y'', then 555555555551 ◁ ''p'', hence we may assume ''y'' ∈ {''𝜆'', 5, 55, 555, 5555, 55555, 555555, 5555555, 55555555, 555555555, 5555555555}, and thus ''p'' ∈ {81, 851, 8551, 85551, 855551, 8555551, 85555551, 855555551, 8555555551, 85555555551, 855555555551}, but all of these numbers are composite.

If y ∈ 0{5}, since if 55555555555 ◁ ''y'', then 555555555551 ◁ ''p'', hence we may assume ''y'' ∈ {0, 05, 055, 0555, 05555, 055555, 0555555, 05555555, 055555555, 0555555555, 05555555555}, and thus ''p'' ∈ {801, 8051, 80551, 805551, 8055551, 80555551, 805555551, 8055555551, 80555555551, 805555555551, 8055555555551}, and of these numbers only 80555551 and 8055555551 are primes, but 80555551 ◁ 8055555551, thus only '''80555551''' is a minimal element.

Case 1.4: ''p'' begins with 9.

In this case we can write p = 9''y''1. If 9 ◁ ''y'', then '''991''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0, 2, 5, or 8.

If 00 ◁ ''y'', then '''9001''' ◁ ''p''. If 22 ◁ ''y'', then '''9221''' ◁ ''p''. If 55 ◁ ''y'', then '''9551''' ◁ ''p''. If 88 ◁ ''y'', then 881 ◁ ''p''. Hence we may assume ''y'' contains at most one 0, at most one 2, at most one 5, and at most one 8.

If ''y'' only contains at most one 0 and does not contain any of {2,5,8}, then ''y'' ∈ {''𝜆'',0}, and thus ''p'' ∈ {91,901}, but 91 and 901 are both composite. If ''y'' only contains at most one 0 and only one of {2,5,8}, then the sum of the digits of ''p'' is divisible by 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Hence we may assume ''y'' contains at least two of {2,5,8}.

If 25 ◁ ''y'', then 251 ◁ ''p''. If 28 ◁ ''y'', then 281 ◁ ''p''. If 52 ◁ ''y'', then 521 ◁ ''p''. If 82 ◁ ''y'', then 821 ◁ ''p''. Hence we may assume ''y'' contains no 2's (since if ''y'' contains 2, then ''y'' cannot contain either 5's or 8's, which is a contradiction).

If 85 ◁ ''y'', then '''9851''' ◁ ''p''. Hence we may assume ''y'' ∈ {58,580,508,058}, and thus ''p'' ∈ {9581,95801,95081,90581}, and of these numbers only 95801 is prime, but 95801 is not a minimal element since 5801 ◁ 95801.

Case 2: ''p'' ends with 3.

In this case we can write p = ''x''3. If ''x'' contains 1, 2, 4, 5, 7, or 8, then (respectively) '''13''' ◁ ''p'', '''23''' ◁ ''p'', '''43''' ◁ ''p'', '''53''' ◁ ''p'', '''73''' ◁ ''p'', or '''83''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 3, 6, or 9, and thus all digits of ''p'' are 0, 3, 6, or 9. But then, since the digits of ''p'' all have a common factor 3, ''p'' is divisible by 3, so ''p'' cannot be prime.

Case 3: ''p'' ends with 7.

In this case we can write ''p'' = ''x''7. If ''x'' contains 1, 3, 4, 6, or 9, then (respectively) '''17''' ◁ ''p'', '''37''' ◁ ''p'', '''47''' ◁ ''p'', '''67''' ◁ ''p'', or '''97''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 2, 5, 7, or 8.

Case 3.1: ''p'' begins with 2.

In this case we can write ''p'' = 2''y''7. If 2 ◁ ''y'', then '''227''' ◁ ''p''. If 5 ◁ ''y'', then '''257''' ◁ ''p''. If 7 ◁ ''y'', then '''277''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 8.

If 08 ◁ ''y'', then '''2087''' ◁ ''p''. If 88 ◁ ''y'', then 887 ◁ ''p''. Hence we may assume ''y'' ∈ {0} ∪ 8{0}.

If ''y'' ∈ {0}, then ''p'' ∈ 2{0}7. But then, since the sum of the digits of ''p'' is 9, ''p'' is divisible by 3, so ''p'' cannot be prime.

If y ∈ 8{0}, then ''p'' ∈ 28{0}7. But then ''p'' is divisible by 7, since for ''n'' ≥ 0 we have 7 × 40&lt;sub&gt;''n''&lt;/sub&gt;1 = 280&lt;sub&gt;''n''&lt;/sub&gt;7.

Case 3.2: ''p'' begins with 5.

In this case we can write ''p'' = 5''y''7. If 5 ◁ ''y'', then '''557''' ◁ ''p''. If 7 ◁ ''y'', then '''577''' ◁ ''p''. If 8 ◁ ''y'', then '''587''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 2.

If 22 ◁ ''y'', then 227 ◁ ''p''. Hence we may assume ''y'' contains zero or one 2's.

If ''y'' contains no 2's, then ''p'' ∈ 5{0}7. But then, since the sum of the digits of ''p'' is 12, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' contains exactly one 2, then we can write ''p'' = 5''z''2''w''7, where ''z'',''w'' ∈ {0}. If 0 ◁ ''z'' and 0 ◁ ''w'', then '''50207''' ◁ ''p''. Hence we may assume either ''z'' or ''w'' is empty.

If ''z'' is empty, then ''p'' ∈ 52{0}7, and the smallest prime ''p'' ∈ 52{0}7 is '''5200007'''.

If ''w'' is empty, then ''p'' ∈ 5{0}27, and the smallest prime ''p'' ∈ 5{0}27 is '''5000000000000000000000000000027'''.

Case 3.3: ''p'' begins with 7.

In this case we can write ''p'' = 7''y''7. If 2 ◁ ''y'', then '''727''' ◁ ''p''. If 5 ◁ ''y'', then '''757''' ◁ ''p''. If 8 ◁ ''y'', then '''787''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 7, and thus all digits of ''p'' are 0 or 7. But then, since the digits of ''p'' all have a common factor 7, ''p'' is divisible by 7, so ''p'' cannot be prime.

Case 3.4: ''p'' begins with 8.

In this case we can write ''p'' = 8''y''7. If 2 ◁ ''y'', then '''827''' ◁ ''p''. If 5 ◁ ''y'', then '''857''' ◁ ''p''. If 7 ◁ ''y'', then '''877''' ◁ ''p''. If 8 ◁ ''y'', then '''887''' ◁ ''p''. Hence we may assume ''y'' ∈ {0}, and thus ''p'' ∈ 8{0}7. But then, since the sum of the digits of ''p'' is 15, ''p'' is divisible by 3, so ''p'' cannot be prime.

Case 4: ''p'' ends with 9.

In this case we can write ''p'' = ''x''9. If ''x'' contains 1, 2, 5, 7, or 8, then (respectively) '''19''' ◁ ''p'', '''29''' ◁ ''p'', '''59''' ◁ ''p'', '''79''' ◁ ''p'', or '''89''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 3, 4, 6, or 9.

If 44 ◁ ''x'', then '''449''' ◁ ''p''. Hence we may assume ''x'' contains zero or one 4's.

If x contains no 4's, then all digits of ''x'' are 0, 3, 6, or 9, and thus all digits of ''p'' are 0, 3, 6, or 9. But then, since the digits of ''p'' all have a common factor 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Hence we may assume that ''x'' contains exactly one 4.

Case 4.1: ''p'' begins with 3.

In this case we can write ''p'' = 3''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. We must have '''349''' ◁ ''p''.

Case 4.2: ''p'' begins with 4.

In this case we can write ''p'' = 4''y''9, where all digits of ''y'' are 0, 3, 6, or 9. If 0 ◁ ''y'', then '''409''' ◁ ''p''. If 3 ◁ ''y'', then 43 ◁ ''p''. If 9 ◁ ''y'', then '''499''' ◁ ''p''. Hence we may assume ''y'' ∈ {6}, and thus ''p'' ∈ 4{6}9. But then ''p'' is divisible by 7, since for ''n'' ≥ 0 we have 7 × 6&lt;sub&gt;''n''&lt;/sub&gt;7 = 46&lt;sub&gt;''n''&lt;/sub&gt;9.

Case 4.3: ''p'' begins with 6.

In this case we can write p = 6''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. If 0 ◁ ''z'', then 409 ◁ ''p''. If 3 ◁ ''z'', then 43 ◁ ''p''. If 6 ◁ ''z'', then '''6469''' ◁ ''p''. If 9 ◁ ''z'', then 499 ◁ ''p''. Hence we may assume ''z'' is empty.

If 3 ◁ ''y'', then 349 ◁ ''p''. If 9 ◁ ''y'', then '''6949''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 6.

If 06 ◁ ''y'', then '''60649''' ◁ ''p''. Hence we may assume ''y'' ∈ {6}{0}.

If 666 ◁ ''y'', then '''666649''' ◁ ''p''. If 00000 ◁ ''y'', then '''60000049''' ◁ ''p''. Hence we may assume ''y'' ∈ {''𝜆'', 0, 00, 000, 0000, 6, 60, 600, 6000, 60000, 66, 660, 6600, 66000, 660000}, and thus ''p'' ∈ {649, 6049, 60049, 600049, 6000049, 6649, 66049, 660049, 6600049, 66000049, 66649, 666049, 6660049, 66600049, 666000049}, and of these numbers only '''66000049''' and '''66600049''' are primes.

Case 4.4: ''p'' begins with 9.

In this case we can write p = 9''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. If 0 ◁ ''y'', then '''9049''' ◁ ''p''. If 3 ◁ ''y'', then 349 ◁ ''p''. If 6 ◁ ''y'', then '''9649''' ◁ ''p''. If 9 ◁ ''y'', then '''9949''' ◁ ''p''. Hence we may assume ''y'' is empty.

If 0 ◁ ''z'', then 409 ◁ ''p''. If 3 ◁ ''z'', then 43 ◁ ''p''. If 9 ◁ ''z'', then 499 ◁ ''p''. Hence we may assume ''z'' ∈ {6}, and thus ''p'' ∈ 94{6}9, and the smallest prime ''p'' ∈ 94{6}9 is 946669.

[[Category:Number theory]]</text>
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      <text bytes="128819" sha1="cre3vbee05poh7wawgv99lmld5cbd0m" xml:space="preserve">{{mathematics}}
'''Athena problem''' is an [[:w:List of unsolved problems in mathematics|unsolved problem]] in [[:w:Number theory|number theory]] and [[:w:Formal language theory|formal language theory]] and [[:w:Order theory|order theory]], this problem is named after the ancient Greek goddess [[:w:Athena|Athena]] (which is associated with [[:w:Wisdom|wisdom]]). Athena problem is: Give a [[:w:Natural number|natural number]] ''b'' &gt; 1, find the [[:w:Set (mathematics)|set]] of the [[:w:Minimal element|minimal element]]s of the set of the &quot;[[:w:Prime number|prime number]] [[:w:Greater than|&gt;]] ''b''&quot; [[:w:Numerical digit|digit]] [[:w:String (computer science)|string]]s in the [[:w:Positional numeral system|positional numeral system]] with [[:w:Radix|base]] ''b'' for the [[:w:Subsequence|subsequence]] [[:w:Partially ordered set|ordering]]. (A string ''x'' is a subsequence of another string ''y'', if ''x'' can be obtained from ''y'' by deleting zero or more of the [[:w:Character (computing)|character]]s in ''y''. For example, 514 is a subsequence of 352148, &quot;string&quot; is a subsequence of &quot;meistersinger&quot;. In contrast, 758 is not a subsequence of 378259, &quot;abc&quot; is not a subsequence of &quot;cbacacba&quot;, since the characters must be in the same order) (Unlike [[:w:Substring|substring]], subsequence is not required to occupy consecutive positions within the original sequences, e.g. the [[:w:Longest common subsequence|longest common subsequence problem]] is different from the [[:w:Longest common substring|longest common substring problem]])

Using [[:w:Formal language theory|formal language theory]] terminology, Athena problem is finding the [[:w:Set (mathematics)|set]] of the [[:w:Minimal element|minimal element]]s of the [[:w:Formal language|language]] of base-''b'' [[:w:Representation (mathematics)|representation]]s of the [[:w:Prime number|prime number]]s [[:w:Greater than|&gt;]] ''b'' (which is a set of [[:w:String (computer science)|string]]s of [[:w:Symbol|symbol]]s over the [[:w:Alphabet (formal languages)|alphabet]] ''Σ''&lt;sub&gt;''b''&lt;/sub&gt; := {0, 1, ..., ''b''−1}), under the subsequence ordering (i.e. the [[:w:Binary relation|binary relation]] &quot;is a subsequence of&quot;, which is a [[:w:Partially ordered set|partial ordering]]), for a given natural number ''b'' &gt; 1 (You can draw this partial ordering as a [[:w:Hasse diagram|Hasse diagram]] to find all [[:w:Minimal element|minimal element]]s), this set is called '''Athena set''', and the prime numbers in this set are called '''Athena primes'''.

By [[:w:Higman's lemma|Higman's lemma]], there are no [[:w:Infinite set|infinite]] [[:w:Antichain|antichain]]s for the subsequence ordering (i.e. the subsequence ordering is always a [[:w:Well-quasi-ordering|well quasi order]]) (i.e. under the subsequence ordering (i.e. the [[:w:Binary relation|binary relation]] &quot;is a subsequence of&quot;, which is a [[:w:Partially ordered set|partial ordering]]), every set of pairwise incomparable (i.e. not [[:w:Comparability|comparable]]) strings is finite), thus there must be only finitely many such minimal elements. In other words, the Athena set in every base ''b'' must be a [[:w:Finite set|finite set]], and every base ''b'' ≥ 2 has only finitely many Athena primes, e.g. in [[:w:Decimal|decimal]] (base ''b'' = 10), the Athena set has exactly 77 [[:w:Element of a set|element]]s (the Athena primes in decimal (base ''b'' = 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}.

For bases 2 ≤ ''b'' ≤ 36, Athena problem is fully solved in bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 24, and also solved in bases ''b'' = 11, 13, 16, 22, 30 if [[:w:Probable prime|probable prime]]s are allowed. For the unsolved bases ''b'' = 17, 19, 21, 23, 25, 26, 27, 28, 29, 31, 32, 34, 35, 36, Athena problem is solved (if probable primes are allowed) except 771 [[:w:Indexed family|families]] of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be [[:w:Empty string|empty]]) of digits in base ''b'', ''y'' is a digit in base ''b'') = sequence {''xz'', ''xyz'', ''xyyz'', ''xyyyz'', ''xyyyyz'', ''xyyyyyz'', ...} (i.e. &quot;''xy''&lt;sup&gt;+&lt;/sup&gt;''z''&quot; in [[:w:Regular expression|regular expression]]), all of these 771 families contain no primes &gt; ''b'' or probable primes &gt; ''b'' with length ≤ 100000.

== Solve the problem ==

To solve the Athena problem for a given base ''b'', we must [[:w:Computing|compute]] the elements up to families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''), and find the smallest prime &gt; ''b'' in all such families.

We call families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') &quot;linear&quot; families, and we reduce these families by removing all trailing digits ''y'' from ''x'', and removing all leading digits ''y'' from ''z'', to make the families be easier, e.g. family 12333{3}33345 in base ''b'' is reduced to family 12{3}45 in base ''b'', since they are in fact the same family. Our [[:w:Algorithm|algorithm]] then proceeds as follows:

* 1. ''M'' := {minimal primes in base ''b'' of length 2 or 3}, ''L'' := union of all ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'') such that ''x'' ≠ 0 and ''gcd''(''z'', ''b'') = 1 and ''Y'' is the set of digits ''y'' in base ''b'' such that ''xyz'' has no subsequence in ''M''.
* 2. While ''L'' contains nonlinear families (families which are not linear families): Explore each family of ''L'', and update ''L''. Examine each family of ''L'' by:
* 2.1. Let ''w'' be the shortest string in the family. If ''w'' has a subsequence in ''M'', then remove the family from ''L''. If ''w'' represents a prime, then add ''w'' to ''M'' and remove the family from ''L''.
* 2.2. If possible, simplify the family.
* 2.3. Using the techniques below (covering congruence, algebraic factorization, or combine of them), check if the family can be proven to only contain composites (only count the numbers &gt; ''b''), and if so then remove the family from ''L''.
* 3. Update ''L'', after each split examine the new families as in step 2.

e.g. in decimal (base ''b'' = 10):

''M'' := {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}

''L'' := {2{0,2}1, 2{0,8}7, 3{0,3,6,9}3, 3{0,3,6,9}9, 4{6}9, 5{0,5,8}1, 5{0,2}7, 6{0,3,6,9}3, 6{0,3,4,6,9}9, 7{0,7}7, 8{0,5}1, 8{0}7, 9{0,2,5,8}1, 9{0,3,6,9}3, 9{0,3,4,6,9}9}

and since 2221 is prime, it follows that the family 2{0,2}1 splits into the families 2{0}1 and 2{0}2{0}1

and since the family 2{0}1 can be proven to contain no primes &gt; base (since all numbers in this family are divisible by 3), it can be removed

and since 20201 is prime, it follows that the family 2{0}2{0}1 splits into the families 2{0}21 and 22{0}1

221 and 2021 are composites, but 20021 is prime, thus add 20021 to ''L''

none of 221, 2201, 22001, 220001, 2200001 are primes, but 22000001 is prime, thus add 22000001 to ''L''

and since the family 3{0,3,6,9}3 can be proven to contain no primes &gt; base (since all numbers in this family are divisible by 3), it can be removed

etc.

Since the number of possible (first digit,last digit) (also called (initial digit,final digit)) combos ([[:w:Ordered pair|ordered pair]]s) of a prime &gt; ''b'' in base ''b'' is (''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(''b'') (all digits except 0 can be the first digit of a prime &gt; ''b'' in base ''b'' (thus ''b''−1 possible digits), but only the digits coprime to ''b'' can be the last digit of a prime &gt; ''b'' in base ''b'' (thus ''eulerphi''(''b'') possible digits), and by the [[:w:Rule of product|rule of product]], there are (''b''−1)×''eulerphi''(''b'') choices of the (first digit,last digit) combo, also, both &quot;numbers of Athena primes in base ''b''&quot; and &quot;length of the largest Athena prime in base ''b''&quot; are [[:w:Asymptotic analysis|roughly]] ''[[:w:E (mathematical_constant)|e]]''&lt;sup&gt;''[[:w:Euler's constant|γ]]''×(''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(*b*)&lt;/sup&gt;.

Shrinking the family ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''Y'' is a set of digits in base ''b'')

* If ''y'' ∈ ''Y'' and the string ''xyyz'' represents a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or has a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''}''z'' ∪ ''x''{''Y'' \ ''y''}''y''{''Y'' \ ''y''}''z''.
* If ''y''&lt;sub&gt;1&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;2&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;1&lt;/sub&gt; ≠ ''y''&lt;sub&gt;2&lt;/sub&gt; and the string ''xy''&lt;sub&gt;1&lt;/sub&gt;''y''&lt;sub&gt;2&lt;/sub&gt;''z'' represents a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or has a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''&lt;sub&gt;1&lt;/sub&gt;}{''Y'' \ ''y''&lt;sub&gt;2&lt;/sub&gt;}''z''.
* If ''y''&lt;sub&gt;1&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;2&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;1&lt;/sub&gt; ≠ ''y''&lt;sub&gt;2&lt;/sub&gt; and both the strings ''xy''&lt;sub&gt;1&lt;/sub&gt;''y''&lt;sub&gt;2&lt;/sub&gt;''z'' and ''xy''&lt;sub&gt;2&lt;/sub&gt;''y''&lt;sub&gt;1&lt;/sub&gt;''z'' represent a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or have a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''&lt;sub&gt;1&lt;/sub&gt;}''z'' ∪ ''x''{''Y'' \ ''y''&lt;sub&gt;2&lt;/sub&gt;}''z''.

e.g. in decimal (base ''b'' = 10):

* 2221 is a prime &gt; 10, thus the family 2{0,2}1 splits into the two families 2{0}1 and 2{0}2{0}1.
* 227 is a prime &gt; 10, and it is a subsequence of 5227, thus the family 5{0,2}7 splits into the two families 5{0}7 and 5{0}2{0}7.
* 449 is a prime &gt; 10, and it is a subsequence of 6449, thus the family 6{0,3,4,6,9}9 splits into the two families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9.
* Both 5051 and 5501 are primes &gt; 10, thus the family 5{0,5}1 splits into the two families 5{0}1 and 5{5}1 = {5}1.
* 8501 is a prime &gt; 10, thus the family 8{0,5}1 splits into the family 8{0}{5}1.
* 887 is a prime &gt; 10, and it is a subsequence of 2887, also 2087 is a prime &gt; 10, thus the family 2{0,8}7 splits into the two families 2{0}7 and 28{0}7.
* 349 and 449 are primes &gt; 10, and they are subsequences of 9349 and 9449, respectively, also 9049, 9649, 9949 are primes &gt; 10, thus the family 9{0,3,4,6,9}9 splits into the two families 9{0,3,6,9}9 and 94{0,3,6,9}9.
* 251, 281, 521, 821, 881 are primes &gt; 10, and they are subsequences of 9251, 9281, 9521, 9821, 9881, respectively, also 9001, 9221, 9551, 9851 are primes &gt; 10, thus the family 9{0,2,5,8}1 splits into the numbers {91, 901, 921, 951, 981, 9021, 9051, 9081, 9201, 9501, 9581, 9801, 90581, 95081, 95801}.

If the methods we have discussed cannot be used to rule out or shrink ''x''{''Y''}''z'' where ''Y'' = {''y''&lt;sub&gt;1&lt;/sub&gt;, ''y''&lt;sub&gt;2&lt;/sub&gt;, ..., ''y''&lt;sub&gt;''n''&lt;/sub&gt;}, then we can replace ''x''{''Y''}''z'' by ''xy''&lt;sub&gt;1&lt;/sub&gt;{''Y''}''z'' ∪ ''xy''&lt;sub&gt;2&lt;/sub&gt;{''Y''}''z'' ∪ ... ∪ ''xy''&lt;sub&gt;''n''&lt;/sub&gt;{''Y''}''z'' and re-run the methods on this new [[:w:Formal language|language]].

If all remain families are linear families (i.e. of the form ''x''{''y''}''z'', where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''), then we search the smallest (probable) primes in these families and add these primes to the list.

e.g. in decimal (base ''b'' = 10):

* The smallest prime in the family 5{0}27 is 5000000000000000000000000000027.
* The smallest prime in the family {5}1 is 555555555551.
* The smallest prime in the family 8{5}1 is 8555555555555555555551, but 8555555555555555555551 is not a minimal element since 555555555551 is a subsequence of 8555555555555555555551.

There is no guarantee that the techniques discussed will ever terminate, but in practice they often do. They are able to determine the Athena set in base ''b'' for 2 ≤ ''b'' ≤ 16 and ''b'' = 18, 20, 22, 24, 30. The bases ''b'' = 17, 19, 21, 23, 25 ≤ ''b'' ≤ 29, 31 ≤ ''b'' ≤ 36 are solved with the exception of 771 families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'').

The following is a &quot;[[:w:Semi-algorithm|semi-algorithm]]&quot; that is guaranteed to solve the Athena problem for a given base ''b'', but it is not so easy to implement:

# ''M'' = ''[[:w:Empty string|∅]]''
# while (''L'' ≠ ''∅'') do
# choose ''x'', a shortest string in ''L''
# ''M'' := ''M'' ∪ {''x''}
# ''L'' := ''L'' − ''sup''({''x''})

In practice, for arbitrary ''L'', we cannot feasibly carry out step 5. Instead, we work with ''L''&amp;#39;, some regular overapproximation to ''L'', until we can show ''L''&amp;#39; = ''∅'' (which implies ''L'' = ''∅''). In practice, ''L''&amp;#39; is usually chosen to be a finite [[:w:Union (set theory)|union]] of sets of the form ''L''&lt;sub&gt;1&lt;/sub&gt;{''L''&lt;sub&gt;2&lt;/sub&gt;}''L''&lt;sub&gt;3&lt;/sub&gt;, where each of ''L''&lt;sub&gt;1&lt;/sub&gt;, ''L''&lt;sub&gt;2&lt;/sub&gt;, ''L''&lt;sub&gt;3&lt;/sub&gt; is finite. In the case we consider in this project, we then have to determine whether such a family contains a prime or not.

Thus, the [[:w:Time complexity|time complexity]] of the Athena problem in base ''b'' may be ''[[:w:Big O notation|O]]''(''[[:w:E (mathematical_constant)|e]]''&lt;sup&gt;''[[:w:Euler's constant|γ]]''×(''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(*b*)&lt;/sup&gt;), and the [[:w:CPU time|CPU time]] of the Athena problem in base ''b'' may be longer than [[:w:Age of the universe|the age of the universe]] for bases ''b'' = 19, 23, 25, 27, 29, 31, 32, 33, 34, 35, also, Athena problem in bases ''b'' around 500 may be [[:w:NP-complete|NP-complete]] or [[:w:NP-hard|NP-hard]], or an [[:w:Undecidable problem|undecidable problem]], or an example of [[:w:Gödel's incompleteness theorems|Gödel's incompleteness theorems]] (like the [[:w:Continuum hypothesis|continuum hypothesis]] and the [[:w:Halting problem|halting problem]]).

To solve the Athena problem (i.e. to [[:w:Computing|compute]] the Athena set), we need to determine whether a given family contains a prime. In practice, if family ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''Y'' is a set of digits in base ''b'') could not be ruled out as only containing composites and ''Y'' contains two or more digits, then a relatively small prime &gt; ''b'' could always be found in this family. Intuitively, this is because there are a large number of small strings in such a family, and at least one is likely to be prime (e.g. there are 2&lt;sup&gt;''n''−2&lt;/sup&gt; strings of length ''n'' in the family 1{3,7}9, and there are over a thousand strings of length 12 in the family 1{3,7}9, thus it is very impossible that these numbers are all composite). In the case ''Y'' contains only one digit, this family is of the form ''x''{''y''}''z'', and there is only a single string of each length &gt; (the length of ''x'' + the length of ''z''), and it is not known if the following [[:w:Decision problem|decision problem]] is recursively solvable (just like [[:w:Sierpiński number|Sierpiński problem]] and [[:w:Riesel number|Riesel problem]], Sierpiński problem and Riesel problem can be generalized to other bases ''b'' (references: http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm), in fact, Athena problem base ''b'' covers the Sierpiński problem base ''b'' and the Riesel problem base ''b'' with ''k'' &lt; ''b'', i.e. finding the smallest prime of the form ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 and ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (or prove such prime does not exist) with ''k'' &lt; ''b'' (specially, for bases ''b'' such that the conjectured smallest Sierpiński number or the conjectured smallest Riesel number is &lt; ''b'', Athena problem base ''b'' covers the Sierpiński problem base ''b'' or the Riesel problem base ''b'', respectively), since the smallest prime of the form ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 and ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (if exists) must be a minimal element in base ''b'', also, Athena problem base ''b'' covers finding the smallest prime of these forms in base ''b'' (or proving that such prime does not exist): (''b''&lt;sup&gt;''n''&lt;/sup&gt;−1)/(''b''−1) (for this form, ''n'' must be prime, and we want ''n'' ≥ 2) (references of this form: http://www.fermatquotient.com/PrimSerien/GenRepu.txt, https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html, http://www.primenumbers.net/Henri/us/MersFermus.htm, http://www.bitman.name/math/table/379, https://pzktupel.de/Primetables/TableRepunitGen.php, https://oeis.org/A084740, https://oeis.org/A084738, https://oeis.org/A128164, https://oeis.org/A285642; or for prime bases ''b'': https://oeis.org/A065854, https://oeis.org/A279068), ''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 (for this form, ''n'' must be power of 2, and we want ''n'' ≥ 1) (references of this form: http://jeppesn.dk/generalized-fermat.html, http://www.noprimeleftbehind.net/crus/GFN-primes.htm, https://web.archive.org/web/20231002190634/http://yves.gallot.pagesperso-orange.fr/primes/index.html, https://pzktupel.de/Primetables/TableFermatGFBB.php, https://oeis.org/A079706, https://oeis.org/A084712, https://oeis.org/A228101), (''b''&lt;sup&gt;''n''&lt;/sup&gt;+1)/2 (for odd ''b'') (for this form, ''n'' must be power of 2, and we want ''n'' ≥ 2) (reference of this form: http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt), (''sqrt''(''b'')×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1)/(''sqrt''(''b'')+1) (for square ''b'') (for this form, 2×''n''+1 must be prime, and we want ''n'' ≥ 2) (references of this form: http://www.fermatquotient.com/PrimSerien/GenRepuP.txt, http://www.primenumbers.net/Henri/us/MersFermus.htm, http://www.bitman.name/math/table/488, https://pzktupel.de/Primetables/TableWagstaffGen.php, https://oeis.org/A084742, https://oeis.org/A084741; or for bases ''b'' with ''sqrt''(''b'') prime: https://oeis.org/A065507), ((''b''−2)×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1)/(''b''−1) (''n'' ≥ 2) (reference of this form: https://oeis.org/A243404), 2×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 (''n'' ≥ 1) (references of this form: https://www.mersenneforum.org/showthread.php?t=6918, https://www.mersenneforum.org/showthread.php?t=19725, https://oeis.org/A119624), 2×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (''n'' ≥ 1) (references of this form: https://www.mersenneforum.org/showthread.php?t=24576, https://www.mersenneforum.org/attachment.php?attachmentid=20976&amp;d=1567314217, https://oeis.org/A119591), ''b''&lt;sup&gt;''n''&lt;/sup&gt;+2 (''n'' ≥ 1) (references of this form: https://oeis.org/A138066, https://oeis.org/A084713, https://oeis.org/A138067), ''b''&lt;sup&gt;''n''&lt;/sup&gt;−2 (''n'' ≥ 2) (references of this form: https://www.primepuzzles.net/puzzles/puzz_887.htm, https://oeis.org/A250200, https://oeis.org/A255707, https://oeis.org/A084714; or for prime bases ''b'': https://oeis.org/A292201), (''b''−1)×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 (''n'' ≥ 1) (references of this form: http://www.noprimeleftbehind.net/Williams-primes-MP.htm, http://www.bitman.name/math/table/477, https://pzktupel.de/Primetables/TableWilliams2.php, https://oeis.org/A305531; or for prime bases ''b'': https://oeis.org/A087139), (''b''−1)×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (''n'' ≥ 1) (references of this form: https://harvey563.tripod.com/wills.txt, http://www.noprimeleftbehind.net/Williams-primes-MM.htm, http://www.bitman.name/math/table/484, https://pzktupel.de/Primetables/TableWilliams1.php; or for prime bases ''b'': https://oeis.org/A122396), ''b''&lt;sup&gt;''n''&lt;/sup&gt;+(''b''−1) (''n'' ≥ 1) (references of this form: http://www.bitman.name/math/table/795, https://pzktupel.de/Primetables/TableWilliams6.php, https://oeis.org/A076845, https://oeis.org/A076846, https://oeis.org/A078178, https://oeis.org/A078179), ''b''&lt;sup&gt;''n''&lt;/sup&gt;−(''b''−1) (''n'' ≥ 2) (references of this form: http://www.bitman.name/math/table/792, https://pzktupel.de/Primetables/TableWilliams5.php, https://oeis.org/A113516, https://oeis.org/A343589; or for prime bases ''b'': https://cs.uwaterloo.ca/journals/JIS/VOL3/mccranie.html, http://www.bitman.name/math/table/435)):

Problem: Given strings ''x'', ''z'' (may be empty), a digit ''y'', and a base ''b'' (''x'' does not [[:w:Leading zero|start with the digit 0]], ''z'' ends with a digit which [[:w:Coprime integers|coprime]] to ''b'', ''y'' is not 0 if ''x'' is empty, ''y'' is coprime to ''b'' if ''z'' is empty), does there exist a prime number whose base-''b'' expansion is of the form ''xy''&lt;sub&gt;''n''&lt;/sub&gt;''z'' for some ''n'' ≥ 0?

Some families can be ruled out to contain no prime &gt; ''b'' by [[:w:Covering set|covering congruence]], [[:w:Factorization of polynomials|algebraic factorization]] (e.g. [[:w:Difference of two squares|difference of two squares]], [[:w:Sum of two cubes|sum of two cubes]], [[:w:Sophie Germain's identity|Sophie Germain's identity of ''x''&lt;sup&gt;4&lt;/sup&gt;+4×''y''&lt;sup&gt;4&lt;/sup&gt;]]), or combine of them, e.g.

* The base 9 family 2{7}: Always divisible by 2 or 5
* The base 11 family 2{5}: Always divisible by 2 or 3
* The base 14 family B{0}1: Always divisible by 3 or 5
* The base 13 family 95{0}3: Always divisible by 5, 7, or 17
* The base 16 family {4}D: Always divisible by 3, 7, or 13
* The base 16 family {8}F: Always divisible by 3, 7, or 13
* The base 21 family {7}D: Always divisible by 2, 13, or 17
* The base 23 family {D}GA: Always divisible by 2, 5, 7, 37, or 79
* The base 9 family {1}: Can be written as (9&lt;sup&gt;''n''&lt;/sup&gt;−1)/8 and can be factored as (3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (3&lt;sup&gt;''n''&lt;/sup&gt;+1) / 8
* The base 8 family 1{0}1: Can be written as 8&lt;sup&gt;''n''&lt;/sup&gt;+1 and can be factored as (2&lt;sup&gt;''n''&lt;/sup&gt;+1) × (4&lt;sup&gt;''n''&lt;/sup&gt;−2&lt;sup&gt;''n''&lt;/sup&gt;+1)
* The base 9 family 3{8}: Can be written as 4×9&lt;sup&gt;''n''&lt;/sup&gt;−1 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (2×3&lt;sup&gt;''n''&lt;/sup&gt;+1)
* The base 16 family 1{5}: Can be written as (4×16&lt;sup&gt;''n''&lt;/sup&gt;−1)/3 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (2×3&lt;sup&gt;''n''&lt;/sup&gt;+1) / 3
* The base 16 family {4}1: Can be written as (4×16&lt;sup&gt;''n''&lt;/sup&gt;−49)/15 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−7) × (2×3&lt;sup&gt;''n''&lt;/sup&gt;+7) / 15
* The base 27 family 7{Q}: Can be written as 8×27&lt;sup&gt;''n''&lt;/sup&gt;−1 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (4×9&lt;sup&gt;''n''&lt;/sup&gt;+2×3&lt;sup&gt;''n''&lt;/sup&gt;+1)
* The base 27 family 9{G}: Can be written as (125×27&lt;sup&gt;''n''&lt;/sup&gt;−8)/13 and can be factored as (5×3&lt;sup&gt;''n''&lt;/sup&gt;−2) × (25×9&lt;sup&gt;''n''&lt;/sup&gt;+10×3&lt;sup&gt;''n''&lt;/sup&gt;+4)
* The base 16 family {C}D: Can be written as (4×16&lt;sup&gt;''n''&lt;/sup&gt;+1)/5 and can be factored as (2×4&lt;sup&gt;''n''&lt;/sup&gt;−2×2&lt;sup&gt;''n''&lt;/sup&gt;+1) × (2×4&lt;sup&gt;''n''&lt;/sup&gt;+2×2&lt;sup&gt;''n''&lt;/sup&gt;+1) / 5
* The base 14 family 8{D}: Can be written as 9×14&lt;sup&gt;''n''&lt;/sup&gt;−1, it is divisible by 5 if ''n'' is odd and can be factored as (3×14&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (3×14&lt;sup&gt;''n''/2&lt;/sup&gt;+1) if ''n'' is even
* The base 12 family {B}9B: Can be written as 12&lt;sup&gt;''n''&lt;/sup&gt;−25, it is divisible by 13 if ''n'' is odd and can be factored as (12&lt;sup&gt;''n''/2&lt;/sup&gt;−5) × (12&lt;sup&gt;''n''/2&lt;/sup&gt;+5) if ''n'' is even
* The base 14 family {D}5: Can be written as 14&lt;sup&gt;''n''&lt;/sup&gt;−9, it is divisible by 5 if ''n'' is odd and can be factored as (14&lt;sup&gt;''n''/2&lt;/sup&gt;−3) × (14&lt;sup&gt;''n''/2&lt;/sup&gt;+3) if ''n'' is even
* The base 17 family 1{9}: Can be written as (25×17&lt;sup&gt;''n''&lt;/sup&gt;−9)/16, it is divisible by 2 if ''n'' is odd and can be factored as (5×17&lt;sup&gt;''n''/2&lt;/sup&gt;−3) × (5×17&lt;sup&gt;''n''/2&lt;/sup&gt;+3) / 16 if ''n'' is even
* The base 17 family 7{9}: Can be written as (121×17&lt;sup&gt;''n''&lt;/sup&gt;−9)/16, it is divisible by 2 if ''n'' is odd and can be factored as (11×17&lt;sup&gt;''n''/2&lt;/sup&gt;−3) × (11×17&lt;sup&gt;''n''/2&lt;/sup&gt;+3) / 16 if ''n'' is even
* The base 19 family 1{6}: Can be written as (4×19&lt;sup&gt;''n''&lt;/sup&gt;−1)/3, it is divisible by 5 if ''n'' is odd and can be factored as (2×19&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (2×19&lt;sup&gt;''n''/2&lt;/sup&gt;+1) / 3 if ''n'' is even
* The base 19 family 7{2}: Can be written as (64×19&lt;sup&gt;''n''&lt;/sup&gt;−1)/9, it is divisible by 5 if ''n'' is odd and can be factored as (8×19&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (8×19&lt;sup&gt;''n''/2&lt;/sup&gt;+1) / 9 if ''n'' is even
* The base 24 family 3{N}: Can be written as 4×24&lt;sup&gt;''n''&lt;/sup&gt;−1, it is divisible by 5 if ''n'' is odd and can be factored as (2×24&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (2×24&lt;sup&gt;''n''/2&lt;/sup&gt;+1) if ''n'' is even

By the [[:w:Prime number theorem|prime number theorem]], the [[:w:Probability|chance]] that a [[:w:Random number|random]] ''n''-digit base ''b'' number is prime is [[:w:Asymptotic analysis|approximately]] 1/''n'' (more accurately, the chance is approximately 1/(''n''×''ln''(''b'')), where ''ln'' is the [[:w:Natural logarithm|natural logarithm]]). If one conjectures the numbers ''x''{''y''}''z'' behave similarly (i.e. the numbers ''x''{''y''}''z'' is a [[:w:Pseudorandomness|pseudorandom sequence]]) you would expect [[:w:Harmonic_series (mathematics)|1/1 + 1/2 + 1/3 + 1/4 + ... = ∞]] primes of the form ''x''{''y''}''z'' (of course, this does not always happen, since some ''x''{''y''}''z'' families can be ruled out to contain no prime &gt; ''b'' (by covering congruence, algebraic factorization, or combine of them), but it is at least a reasonable conjecture in the absence of evidence to the contrary. Hence, the [[:w:Heuristic argument|heuristic argument]] suggests there are always infinitely many primes in family ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') if it cannot be ruled out to contain no prime or only contain finitely many primes, by covering congruence, algebraic factorization, or combine of them. However, some families ''x''{''y''}''z'' could not be proven to contain no primes &gt; ''b'' (by covering congruence, algebraic factorization, or combine of them) but no primes &gt; ''b'' could be found in the family, even after searching through numbers with over 100000 digits. In such a case, the only way to proceed is to [[:w:Primality test|test the primality]] of larger and larger numbers of such form and hope a prime is eventually discovered. e.g. the smallest (probable) prime in the family A{3}A in base ''b'' = 13 is A3&lt;sub&gt;592197&lt;/sub&gt;A, its algebraic form is (41×13&lt;sup&gt;592198&lt;/sup&gt;+27)/4, when written in decimal contains 659677 digits (it is only probable prime, i.e. not definitely prime, since technically, probable primality tests were used to show this (which have a ''very'' small chance of making an error, see https://t5k.org/notes/prp_prob.html) because all known primality tests run far too slowly to run on numbers of this size unless either [https://t5k.org/prove/prove3_1.html ''N''−1] or [https://t5k.org/prove/prove3_2.html ''N''+1] (or both) can be ≥ 1/3 factored).

The numbers in family ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') are of the form (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1) for some fixed ''a'', ''b'', ''c'' such that ''a'' ≥ 1, ''b'' ≥ 2 (''b'' is the base), ''c'' ≠ 0, ''gcd''(''a'',''c'') = 1, ''gcd''(''b'',''c'') = 1. Except in the [[:w:Special case|special case]] ''c'' = ±1 and ''gcd''(''a''+''c'',''b''−1) = 1, when ''n'' is large the known [[:w:Primality test|primality test]]s for such a number are too inefficient to run. In this case one must resort to a [[:w:Probabilistic algorithm|probable]] primality test such as a [[:w:Miller–Rabin primality test|Miller–Rabin primality test]] or a [[:w:Baillie–PSW primality test|Baillie–PSW primality test]], unless a divisor of the number can be found. Since we are testing many numbers in an [[:w:Exponential growth|exponential sequence]], it is possible to use a sieving process to find divisors rather than using [[:w:Trial division|trial division]].

To do this, we made use of Geoffrey Reynolds' ''srsieve'' software (download: https://pzktupel.de/Software/srsieve_1.1.4.7z). This program uses the [[:w:Baby-step giant-step|baby-step giant-step]] [[:w:Algorithm|algorithm]] to find all primes ''p'' which divide ''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'' where ''p'' and ''n'' lie in a [[:w:Interval_(mathematics)|specified range]]. Since this program cannot handle the general case (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1) when ''gcd''(''a''+''c'',''b''−1) &gt; 1 we only used it to sieve the sequence ''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'' for primes ''p'' not dividing ''gcd''(''a''+''c'',''b''−1), and initialized the list of candidates to not include ''n'' for which there is some prime ''p'' dividing ''gcd''(''a''+''c'',''b''−1) for which ''p'' dividing (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1). The program had to be modified slightly to remove a check which would prevent it from running in the case when ''a'', ''b'', and ''c'' were all odd (since then 2 divides ''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'', but 2 may not divide (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1)).

Once the numbers with small divisors had been removed, it remained to test the remaining numbers using a probable primality test. For this we used the software ''LLR'' by Jean Penné. (download: http://jpenne.free.fr/index2.html). Although undocumented, it is possible to run this program on numbers of the form (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1) when ''gcd''(''a''+''c'',''b''−1) &gt; 1, so this program required no modifications. A script was also written which allowed one to run ''srsieve'' while ''LLR'' was testing the remaining candidates, so that when a divisor was found by srsieve on a number which had not yet been tested by ''LLR'' it would be removed from the list of candidates.

For the primes &lt; 10&lt;sup&gt;25000&lt;/sup&gt; for the &quot;easy&quot; bases (bases ''b'' with ≤ 150 primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; (base ''b'' = 26 has 83 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 3 unsolved families, base ''b'' = 36 has 75 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 4 unsolved families, base ''b'' = 17 has 99 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 18 unsolved families, base ''b'' = 21 has 80 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 12 unsolved families, base ''b'' = 19 has 201 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 23 unsolved families), i.e. bases *b* = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 26, 28, 30, 36), we employed ''CM'' by Andreas Enge (download: https://www.multiprecision.org/cm/download.html), an elliptic curve primality proving implementation.

== Data ==

These are the results of the Athena problem in bases 2 ≤ ''b'' ≤ 36 (we stop at base 36 since this base is the maximum base for which it is possible to write the numbers with the [[:w:Symbol|symbol]]s 0, 1, 2, ..., 9 and A, B, C, ..., Z (i.e. the 10 [[:w:Arabic numerals|Arabic numerals]] and the 26 [[:w:Latin script|Latin letters]]): (some large Athena primes are only probable primes, i.e. not definitely primes, since they are too large to be [[:w:Elliptic curve primality|ECPP proved]] and [[:w:Pocklington primality test#Extensions and variants|neither ''N''−1 nor ''N''+1 can be ≥ 1/3 factored]], all of them pass the [[:w:Baillie–PSW primality test|Baillie–PSW primality test]] and the [[:w:Strong pseudoprime|strong primality test]] (i.e. the [[:w:Miller–Rabin primality test|Miller–Rabin primality test]]) with all prime bases ''p'' ≤ 61, however, all Athena primes &lt; 10&lt;sup&gt;25000&lt;/sup&gt; for bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 36 are definitely primes, most of them &gt; 10&lt;sup&gt;299&lt;/sup&gt; are proven primes with [[:w:Elliptic curve primality|ECPP proving]], others &gt; 10&lt;sup&gt;299&lt;/sup&gt; are proven primes with [[:w:Pocklington primality test#Extensions and variants|''N''−1 or ''N''+1 proving]])

All numbers are written in base ''b'', [[:w:Senary#Base 36 as senary compression|using A to Z to represent digit values 10 to 35]], &quot;{}&quot; means repeating, e.g. family 12{3}45 means the sequence {1245, 12345, 123345, 1233345, 12333345, 123333345, ...} (where the members are expressed as base ''b'' strings), subscripts are used to indicate repetitions of digits, e.g. 123&lt;sub&gt;4&lt;/sub&gt;567 means 123333567 (all subscripts are written in decimal).

Base 2: 1 Athena prime (the largest of which has 2 digits (it is 11, and its value is 3 in decimal)): {11}

Base 3: 3 Athena primes (the largest of which has 3 digits (it is 111, and its value is 13 in decimal)): {12, 21, 111}

Base 4: 5 Athena primes (the largest of which has 3 digits (it is 221, and its value is 41 in decimal)): {11, 13, 23, 31, 221}

Base 5: 22 Athena primes (the largest of which has 96 digits (it is 10&lt;sub&gt;93&lt;/sub&gt;13, and its algebraic form is 5&lt;sup&gt;95&lt;/sup&gt;+8)): {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 Athena primes (the largest of which has 5 digits (it is 40041, and its value is 5209 in decimal)): {11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041}

Base 7: 71 Athena primes (the largest of which has 17 digits (it is 3&lt;sub&gt;16&lt;/sub&gt;1, and its algebraic form is (7&lt;sup&gt;17&lt;/sup&gt;−5)/2)): {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: 75 Athena primes (the largest of which has 221 digits (it is 4&lt;sub&gt;220&lt;/sub&gt;7, and its algebraic form is (4×8&lt;sup&gt;221&lt;/sup&gt;+17)/7)): {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: 151 Athena primes (the largest of which has 1161 digits (it is 30&lt;sub&gt;1158&lt;/sub&gt;11, and its algebraic form is 3×9&lt;sup&gt;1160&lt;/sup&gt;+10)): {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: 77 Athena primes (the largest of which has 31 digits (it is 50&lt;sub&gt;28&lt;/sub&gt;27, and its algebraic form is 5×10&lt;sup&gt;30&lt;/sup&gt;+27)): {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: 1068 Athena primes (including 1 unproven probable prime: 57&lt;sub&gt;62668&lt;/sub&gt;), the largest of which has 62669 digits (it is 57&lt;sub&gt;62668&lt;/sub&gt;, and its algebraic form is (57×11&lt;sup&gt;62668&lt;/sup&gt;−7)/10), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel11 Data of Athena (probable) primes base 11]

Base 12: 106 Athena primes (the largest of which has 42 digits (it is 40&lt;sub&gt;39&lt;/sub&gt;77, and its algebraic form is 4×12&lt;sup&gt;41&lt;/sup&gt;+91)): {11, 15, 17, 1B, 25, 27, 31, 35, 37, 3B, 45, 4B, 51, 57, 5B, 61, 67, 6B, 75, 81, 85, 87, 8B, 91, 95, A7, AB, B5, B7, 221, 241, 2A1, 2B1, 2BB, 401, 421, 447, 471, 497, 565, 655, 665, 701, 70B, 721, 747, 771, 77B, 797, 7A1, 7BB, 907, 90B, 9BB, A41, B21, B2B, 2001, 200B, 202B, 222B, 229B, 292B, 299B, 4441, 4707, 4777, 6A05, 6AA5, 729B, 7441, 7B41, 929B, 9777, 992B, 9947, 997B, 9997, A0A1, A201, A605, A6A5, AA65, B001, B0B1, BB01, BB41, 600A5, 7999B, 9999B, AAAA1, B04A1, B0B9B, BAA01, BAAA1, BB09B, BBBB1, 44AAA1, A00065, BBBAA1, AAA0001, B00099B, AA000001, BBBBBB99B, B0000000000000000000000000009B, 400000000000000000000000000000000000000077}

Base 13: 3197 Athena primes (including 4 unproven probable primes: C5&lt;sub&gt;23755&lt;/sub&gt;C, 80&lt;sub&gt;32017&lt;/sub&gt;111, 95&lt;sub&gt;197420&lt;/sub&gt;, A3&lt;sub&gt;592197&lt;/sub&gt;A), the largest of which has 592199 digits (it is A3&lt;sub&gt;592197&lt;/sub&gt;A, and its algebraic form is (41×13&lt;sup&gt;592198&lt;/sup&gt;+27)/4), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel13 Data of Athena (probable) primes base 13]

Base 14: 650 Athena primes, the largest of which has 19699 digits (it is 4D&lt;sub&gt;19698&lt;/sub&gt;, and its algebraic form is 5×14&lt;sup&gt;19698&lt;/sup&gt;−1), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel14 Data of Athena primes base 14]

Base 15: 1284 Athena primes, the largest of which has 157 digits (it is 7&lt;sub&gt;155&lt;/sub&gt;97, and its algebraic form is (15&lt;sup&gt;157&lt;/sup&gt;+59)/2), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel15 Data of Athena primes base 15]

Base 16: 2347 Athena primes (including 3 unproven probable primes: DB&lt;sub&gt;32234&lt;/sub&gt;, 4&lt;sub&gt;72785&lt;/sub&gt;DD, 3&lt;sub&gt;116137&lt;/sub&gt;AF), the largest of which has 116139 digits (it is 3&lt;sub&gt;116137&lt;/sub&gt;AF, and its algebraic form is (16&lt;sup&gt;116139&lt;/sup&gt;+619)/5), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel16 Data of Athena (probable) primes base 16]

Base 17: 10415 known Athena primes (including many unproven probable primes) and 12 unsolved families (1{7}, 1F{0}7, 4{7}A, 70F{0}D, 8{B}9, 9{5}9, A{D}F, B{0}B3, {B}E9, {B}EE, F1{9}, FD0{D}, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel17 Data of known Athena (probable) primes base 17]

Base 18: 549 Athena primes, the largest of which has 6271 digits (it is C0&lt;sub&gt;6268&lt;/sub&gt;C5, and its algebraic form is 12×18&lt;sup&gt;6270&lt;/sup&gt;+221), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel18 Data of Athena primes base 18]

Base 19: 31417 known Athena primes (including many unproven probable primes) and 17 unsolved families (4B5{0}H, {5}3, 5{H}05, 5{H}0H, 5{H}5, 66{B}, 71{0}177, 7AF{0}H, 97{0}3, C{H}C, EE1{6}, F{7}5, F{B}G, F{D}F, H0F{0}7A, HB{0}5B5, II{D}, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel19 Data of known Athena (probable) primes base 19]

Base 20: 3314 Athena primes, the largest of which has 6271 digits (it is G0&lt;sub&gt;6269&lt;/sub&gt;D, and its algebraic form is 16×20&lt;sup&gt;6270&lt;/sup&gt;+13), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel20 Data of Athena primes base 20]

Base 21: 13386 known Athena primes (including many unproven probable primes) and 8 unsolved families (5{0}DJ, {9}D, B3{0}EB, B{H}6H, C{F}0K, {F}35, G{0}FK, H{0}7771, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel21 Data of known Athena (probable) primes base 21]

Base 22: 8003 Athena primes (including 1 unproven probable prime: BK&lt;sub&gt;22001&lt;/sub&gt;5), the largest of which has 22003 digits (it is BK&lt;sub&gt;22001&lt;/sub&gt;5, and its algebraic form is (251×22&lt;sup&gt;22002&lt;/sup&gt;−335)/21), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel22 Data of Athena (probable) primes base 22]

Base 23: 65178 known Athena primes (including many unproven probable primes) and 87 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel23 Data of known Athena (probable) primes base 23] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left23 Data of unsolved families for Athena problem base 23]

Base 24: 3409 Athena primes, the largest of which has 8134 digits (it is N00N&lt;sub&gt;8129&lt;/sub&gt;LN, and its algebraic form is 13249×24&lt;sup&gt;8131&lt;/sup&gt;−49), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel24 Data of Athena primes base 24]

Base 25: 133639 known Athena primes (including many unproven probable primes) and 85 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel25 Data of known Athena (probable) primes base 25] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left25 Data of unsolved families for Athena problem base 25]

Base 26: 25256 known Athena primes (including 7 unproven probable primes: 5&lt;sub&gt;19391&lt;/sub&gt;6F, 7&lt;sub&gt;20279&lt;/sub&gt;OL, LD0&lt;sub&gt;20975&lt;/sub&gt;7, 6K&lt;sub&gt;23300&lt;/sub&gt;5, J0&lt;sub&gt;44303&lt;/sub&gt;KCB, M0&lt;sub&gt;61186&lt;/sub&gt;2BB, 85M&lt;sub&gt;197060&lt;/sub&gt;B) and 3 unsolved families ({A}6F, {H}MH, {I}GL, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel26 Data of known Athena (probable) primes base 26]

Base 27: 102852 known Athena primes (including many unproven probable primes) and 44 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel27 Data of known Athena (probable) primes base 27] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left27 Data of unsolved families for Athena problem base 27]

Base 28: 25528 known Athena primes (including 3 unproven probable primes: N6&lt;sub&gt;24051&lt;/sub&gt;LR, 5OA&lt;sub&gt;31238&lt;/sub&gt;F, O4O&lt;sub&gt;94535&lt;/sub&gt;9) and 1 unsolved family (O{A}F, no primes or probable primes with length ≤ 900000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel28 Data of known Athena (probable) primes base 28]

Base 29: 355242 known Athena primes (including many unproven probable primes) and 125 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel29 Data of known Athena (probable) primes base 29] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left29 Data of unsolved families for Athena problem base 29]

Base 30: 2619 Athena primes (including 1 unproven probable prime: I0&lt;sub&gt;24608&lt;/sub&gt;D), the largest of which has 34206 digits (it is OT&lt;sub&gt;34205&lt;/sub&gt;, and its algebraic form is 25×30&lt;sup&gt;34205&lt;/sup&gt;−1), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel30 Data of Athena (probable) primes base 30]

Base 31: 569323 known Athena primes (including many unproven probable primes) and 77 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel31 Data of known Athena (probable) primes base 31] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left31 Data of unsolved families for Athena problem base 31]

Base 32: 168882 known Athena primes (including many unproven probable primes) and 120 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel32 Data of known Athena (probable) primes base 32] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left32 Data of unsolved families for Athena problem base 32]

Base 33: 280012 known Athena primes (including many unproven probable primes) and 81 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel33 Data of known Athena (probable) primes base 33] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left33 Data of unsolved families for Athena problem base 33]

Base 34: 184785 known Athena primes (including many unproven probable primes) and 47 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel34 Data of known Athena (probable) primes base 34] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left34 Data of unsolved families for Athena problem base 34]

Base 35: 720002 known Athena primes (including many unproven probable primes) and 60 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel35 Data of known Athena (probable) primes base 35] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left35 Data of unsolved families for Athena problem base 35]

Base 36: 35286 known Athena primes (including 3 unproven probable primes: 7K&lt;sub&gt;26567&lt;/sub&gt;Z, S0&lt;sub&gt;75007&lt;/sub&gt;8H, P&lt;sub&gt;81993&lt;/sub&gt;SZ) and 4 unsolved families (B{0}EUV, HM{0}N, N{0}YYN, O{L}Z, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel36 Data of known Athena (probable) primes base 36]

== Condensed table for bases 2 ≤ ''b'' ≤ 36 ==

{|class=&quot;wikitable&quot;
||''b''||number of Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b''||base-''b'' form of the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b'' (write &quot;''d''&lt;sub&gt;''n''&lt;/sub&gt;&quot; if there are 5 or more (''n'') consecutive same digits ''d'')||length of the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b''||length of the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b'' in decimal||algebraic ((''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1)) form of the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b''||''factordb'' entry of the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b''||the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b'' written in base ''b'' (use lower case letters instead of upper case letters)||number of unsolved families in the Athena problem in base ''b'' (all of these left families are linear families)||searching limit of length for the unsolved families in the Athena problem in base ''b'' (if there are different searching limits for the unsolved families in the Athena problem in base ''b'', choose the lowest searching limit)||
|-
||2||1||11||2||1||3||http://factordb.com/index.php?id=3&amp;open=ecm||http://factordb.com/index.php?showid=3&amp;base=2||0||–||
|-
||3||3||111&lt;br&gt;21&lt;br&gt;12||3&lt;br&gt;2&lt;br&gt;2||2&lt;br&gt;1&lt;br&gt;1||13&lt;br&gt;7&lt;br&gt;5||http://factordb.com/index.php?id=13&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=7&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=5&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=13&amp;base=3&lt;br&gt;http://factordb.com/index.php?showid=7&amp;base=3&lt;br&gt;http://factordb.com/index.php?showid=5&amp;base=3&lt;nowiki/&gt;||0||–||
|-
||4||5||221&lt;br&gt;31&lt;br&gt;23&lt;br&gt;13&lt;br&gt;11||3&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2||2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;1&lt;br&gt;1||41&lt;br&gt;13&lt;br&gt;11&lt;br&gt;7&lt;br&gt;5||http://factordb.com/index.php?id=41&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=13&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=11&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=7&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=5&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=41&amp;base=4&lt;br&gt;http://factordb.com/index.php?showid=13&amp;base=4&lt;br&gt;http://factordb.com/index.php?showid=11&amp;base=4&lt;br&gt;http://factordb.com/index.php?showid=7&amp;base=4&lt;br&gt;http://factordb.com/index.php?showid=5&amp;base=4&lt;nowiki/&gt;||0||–||
|-
||5||22||10&lt;sub&gt;93&lt;/sub&gt;13&lt;br&gt;300031&lt;br&gt;44441&lt;br&gt;33331&lt;br&gt;33001&lt;br&gt;30301&lt;br&gt;14444&lt;br&gt;10103&lt;br&gt;3101&lt;br&gt;414||96&lt;br&gt;6&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;4&lt;br&gt;3||67&lt;br&gt;4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;3&lt;br&gt;3&lt;br&gt;3||5&lt;sup&gt;95&lt;/sup&gt;+8&lt;br&gt;9391&lt;br&gt;3121&lt;br&gt;2341&lt;br&gt;2251&lt;br&gt;1951&lt;br&gt;1249&lt;br&gt;653&lt;br&gt;401&lt;br&gt;109||http://factordb.com/index.php?id=1100000000034686071&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=9391&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=3121&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=2341&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=2251&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1951&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1249&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=653&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=401&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=109&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000000034686071&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=9391&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=3121&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=2341&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=2251&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=1951&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=1249&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=653&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=401&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=109&amp;base=5&lt;nowiki/&gt;||0||–||
|-
||6||11||40041&lt;br&gt;4441&lt;br&gt;4401&lt;br&gt;51&lt;br&gt;45&lt;br&gt;35&lt;br&gt;31&lt;br&gt;25&lt;br&gt;21&lt;br&gt;15||5&lt;br&gt;4&lt;br&gt;4&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2||4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2||5209&lt;br&gt;1033&lt;br&gt;1009&lt;br&gt;31&lt;br&gt;29&lt;br&gt;23&lt;br&gt;19&lt;br&gt;17&lt;br&gt;13&lt;br&gt;11||http://factordb.com/index.php?id=5209&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1033&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1009&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=31&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=29&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=23&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=19&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=17&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=13&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=11&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=5209&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=1033&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=1009&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=31&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=29&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=23&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=19&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=17&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=13&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=11&amp;base=6&lt;nowiki/&gt;||0||–||
|-
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||33||280012~280093||DP&lt;sub&gt;95093&lt;/sub&gt;M5&lt;br&gt;HJ0&lt;sub&gt;94295&lt;/sub&gt;J&lt;br&gt;90&lt;sub&gt;93597&lt;/sub&gt;Q&lt;br&gt;9F0&lt;sub&gt;93157&lt;/sub&gt;N&lt;br&gt;7&lt;sub&gt;89449&lt;/sub&gt;333H&lt;br&gt;K3&lt;sub&gt;80751&lt;/sub&gt;6K&lt;br&gt;D&lt;sub&gt;80107&lt;/sub&gt;9UD&lt;br&gt;VFU&lt;sub&gt;72204&lt;/sub&gt;FK&lt;br&gt;J&lt;sub&gt;68715&lt;/sub&gt;2BJ&lt;br&gt;DF0&lt;sub&gt;68367&lt;/sub&gt;J||95096&lt;br&gt;94298&lt;br&gt;93599&lt;br&gt;93160&lt;br&gt;89453&lt;br&gt;80754&lt;br&gt;80110&lt;br&gt;72208&lt;br&gt;68718&lt;br&gt;68370||144405&lt;br&gt;143193&lt;br&gt;142131&lt;br&gt;141465&lt;br&gt;135835&lt;br&gt;122626&lt;br&gt;121648&lt;br&gt;109649&lt;br&gt;104350&lt;br&gt;103821||(441×33&lt;sup&gt;95095&lt;/sup&gt;−3833)/32&lt;br&gt;580×33&lt;sup&gt;94296&lt;/sup&gt;+19&lt;br&gt;9×33&lt;sup&gt;93598&lt;/sup&gt;+26&lt;br&gt;312×33&lt;sup&gt;93158&lt;/sup&gt;+23&lt;br&gt;(7×33&lt;sup&gt;89453&lt;/sup&gt;−4743239)/32&lt;br&gt;(643×33&lt;sup&gt;80753&lt;/sup&gt;+3709)/32&lt;br&gt;(13×33&lt;sup&gt;80110&lt;/sup&gt;−121453)/32&lt;br&gt;(16623×33&lt;sup&gt;72206&lt;/sup&gt;−8095)/16&lt;br&gt;(19×33&lt;sup&gt;68718&lt;/sup&gt;−600883)/32&lt;br&gt;444×33&lt;sup&gt;68368&lt;/sup&gt;+19||http://factordb.com/index.php?id=1100000005652348775&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000005652362811&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000005652375073&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000005652389776&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000005652430746&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000005652446200&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000005652461592&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004614764298&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004614770536&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004614784274&amp;open=prime&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000005652348775&amp;base=33&lt;br&gt;http://factordb.com/index.php?showid=1100000005652362811&amp;base=33&lt;br&gt;http://factordb.com/index.php?showid=1100000005652375073&amp;base=33&lt;br&gt;http://factordb.com/index.php?showid=1100000005652389776&amp;base=33&lt;br&gt;http://factordb.com/index.php?showid=1100000005652430746&amp;base=33&lt;br&gt;http://factordb.com/index.php?showid=1100000005652446200&amp;base=33&lt;br&gt;http://factordb.com/index.php?showid=1100000005652461592&amp;base=33&lt;br&gt;http://factordb.com/index.php?showid=1100000004614764298&amp;base=33&lt;br&gt;http://factordb.com/index.php?showid=1100000004614770536&amp;base=33&lt;br&gt;http://factordb.com/index.php?showid=1100000004614784274&amp;base=33&lt;nowiki/&gt;||81||100000||
|-
||34||184785~184832||GFGC&lt;sub&gt;99996&lt;/sub&gt;5&lt;br&gt;90&lt;sub&gt;97950&lt;/sub&gt;FJ&lt;br&gt;NM0&lt;sub&gt;85218&lt;/sub&gt;KX&lt;br&gt;F&lt;sub&gt;83189&lt;/sub&gt;H2HP&lt;br&gt;P&lt;sub&gt;79441&lt;/sub&gt;444P&lt;br&gt;6&lt;sub&gt;77027&lt;/sub&gt;8X&lt;br&gt;XQIQ&lt;sub&gt;72241&lt;/sub&gt;D&lt;br&gt;T&lt;sub&gt;66530&lt;/sub&gt;IF&lt;br&gt;4&lt;sub&gt;66152&lt;/sub&gt;B&lt;br&gt;2EEC&lt;sub&gt;66039&lt;/sub&gt;7||100000&lt;br&gt;97953&lt;br&gt;85222&lt;br&gt;83193&lt;br&gt;79445&lt;br&gt;77029&lt;br&gt;72245&lt;br&gt;66532&lt;br&gt;66153&lt;br&gt;66043||153148&lt;br&gt;150013&lt;br&gt;130516&lt;br&gt;127408&lt;br&gt;121669&lt;br&gt;117968&lt;br&gt;110642&lt;br&gt;101893&lt;br&gt;101312&lt;br&gt;101143||(209246×34&lt;sup&gt;99997&lt;/sup&gt;−81)/11&lt;br&gt;9×34&lt;sup&gt;97952&lt;/sup&gt;+529&lt;br&gt;804×34&lt;sup&gt;85220&lt;/sup&gt;+713&lt;br&gt;(5×34&lt;sup&gt;83193&lt;/sup&gt;+700233)/11&lt;br&gt;(25×34&lt;sup&gt;79445&lt;/sup&gt;−28062367)/33&lt;br&gt;(2×34&lt;sup&gt;77029&lt;/sup&gt;+1043)/11&lt;br&gt;(1288676×34&lt;sup&gt;72242&lt;/sup&gt;−455)/33&lt;br&gt;(29×34&lt;sup&gt;66532&lt;/sup&gt;−12833)/33&lt;br&gt;(4×34&lt;sup&gt;66153&lt;/sup&gt;+227)/33&lt;br&gt;(30826×34&lt;sup&gt;66040&lt;/sup&gt;−59)/11||http://factordb.com/index.php?id=1100000004702891268&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004702894713&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004702900996&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004702910376&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004702913746&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004702918600&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004399656529&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004399657696&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004399658651&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004399659716&amp;open=prime&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000004702891268&amp;base=34&lt;br&gt;http://factordb.com/index.php?showid=1100000004702894713&amp;base=34&lt;br&gt;http://factordb.com/index.php?showid=1100000004702900996&amp;base=34&lt;br&gt;http://factordb.com/index.php?showid=1100000004702910376&amp;base=34&lt;br&gt;http://factordb.com/index.php?showid=1100000004702913746&amp;base=34&lt;br&gt;http://factordb.com/index.php?showid=1100000004702918600&amp;base=34&lt;br&gt;http://factordb.com/index.php?showid=1100000004399656529&amp;base=34&lt;br&gt;http://factordb.com/index.php?showid=1100000004399657696&amp;base=34&lt;br&gt;http://factordb.com/index.php?showid=1100000004399658651&amp;base=34&lt;br&gt;http://factordb.com/index.php?showid=1100000004399659716&amp;base=34&lt;nowiki/&gt;||47||100000||
|-
||35||720002~720062||N0N&lt;sub&gt;99971&lt;/sub&gt;9&lt;br&gt;V0&lt;sub&gt;83669&lt;/sub&gt;E73&lt;br&gt;N&lt;sub&gt;81563&lt;/sub&gt;K7N&lt;br&gt;BJ0&lt;sub&gt;81279&lt;/sub&gt;N&lt;br&gt;J0&lt;sub&gt;80062&lt;/sub&gt;FUH&lt;br&gt;43V&lt;sub&gt;79754&lt;/sub&gt;&lt;br&gt;9&lt;sub&gt;76600&lt;/sub&gt;K3&lt;br&gt;LB&lt;sub&gt;71366&lt;/sub&gt;PB&lt;br&gt;Q&lt;sub&gt;64150&lt;/sub&gt;H&lt;br&gt;50&lt;sub&gt;63397&lt;/sub&gt;5R||99974&lt;br&gt;83673&lt;br&gt;81566&lt;br&gt;81282&lt;br&gt;80066&lt;br&gt;79756&lt;br&gt;76602&lt;br&gt;71369&lt;br&gt;64151&lt;br&gt;63400||154367&lt;br&gt;129197&lt;br&gt;125944&lt;br&gt;125505&lt;br&gt;123628&lt;br&gt;123148&lt;br&gt;118279&lt;br&gt;110199&lt;br&gt;99054&lt;br&gt;97894||(27393×35&lt;sup&gt;99972&lt;/sup&gt;−499)/34&lt;br&gt;31×35&lt;sup&gt;83672&lt;/sup&gt;+17398&lt;br&gt;(23×35&lt;sup&gt;81566&lt;/sup&gt;−144013)/34&lt;br&gt;404×35&lt;sup&gt;81280&lt;/sup&gt;+23&lt;br&gt;19×35&lt;sup&gt;80065&lt;/sup&gt;+19442&lt;br&gt;(4893×35&lt;sup&gt;79754&lt;/sup&gt;−31)/34&lt;br&gt;(9×35&lt;sup&gt;76602&lt;/sup&gt;+12877)/34&lt;br&gt;(725×35&lt;sup&gt;71368&lt;/sup&gt;+16649)/34&lt;br&gt;(13×35&lt;sup&gt;64151&lt;/sup&gt;−166)/17&lt;br&gt;5×35&lt;sup&gt;63399&lt;/sup&gt;+202||http://factordb.com/index.php?id=1100000008248342445&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008248353306&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008248375642&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008248397018&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008248412468&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008248418540&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008248423670&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008192119974&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008192126630&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008192129294&amp;open=prime&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000008248342445&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008248353306&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008248375642&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008248397018&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008248412468&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008248418540&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008248423670&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008192119974&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008192126630&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008192129294&amp;base=35&lt;nowiki/&gt;||60||100000||
|-
||36||35286~35290||P&lt;sub&gt;81993&lt;/sub&gt;SZ&lt;br&gt;S0&lt;sub&gt;75007&lt;/sub&gt;8H&lt;br&gt;7K&lt;sub&gt;26567&lt;/sub&gt;Z&lt;br&gt;J&lt;sub&gt;10117&lt;/sub&gt;LJ&lt;br&gt;VL0&lt;sub&gt;7258&lt;/sub&gt;J&lt;br&gt;EO0&lt;sub&gt;6177&lt;/sub&gt;V&lt;br&gt;FZ&lt;sub&gt;5777&lt;/sub&gt;3P&lt;br&gt;T09&lt;sub&gt;4618&lt;/sub&gt;1&lt;br&gt;RY&lt;sub&gt;4562&lt;/sub&gt;H&lt;br&gt;OZ&lt;sub&gt;3932&lt;/sub&gt;AZ||81995&lt;br&gt;75010&lt;br&gt;26569&lt;br&gt;10119&lt;br&gt;7261&lt;br&gt;6180&lt;br&gt;5780&lt;br&gt;4621&lt;br&gt;4564&lt;br&gt;3935||127609&lt;br&gt;116739&lt;br&gt;41349&lt;br&gt;15748&lt;br&gt;11301&lt;br&gt;9618&lt;br&gt;8996&lt;br&gt;7192&lt;br&gt;7103&lt;br&gt;6124||(5×36&lt;sup&gt;81995&lt;/sup&gt;+821)/7&lt;br&gt;28×36&lt;sup&gt;75009&lt;/sup&gt;+305&lt;br&gt;(53×36&lt;sup&gt;26568&lt;/sup&gt;+101)/7&lt;br&gt;(19×36&lt;sup&gt;10119&lt;/sup&gt;+2501)/35&lt;br&gt;1137×36&lt;sup&gt;7259&lt;/sup&gt;+19&lt;br&gt;528×36&lt;sup&gt;6178&lt;/sup&gt;+31&lt;br&gt;16×36&lt;sup&gt;5779&lt;/sup&gt;−1163&lt;br&gt;(36549×36&lt;sup&gt;4619&lt;/sup&gt;−289)/35&lt;br&gt;(979×36&lt;sup&gt;4563&lt;/sup&gt;−629)/35&lt;br&gt;25×36&lt;sup&gt;3934&lt;/sup&gt;−901||http://factordb.com/index.php?id=1100000002394962083&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004020085177&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003896952461&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362491&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362489&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362488&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362487&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362486&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362485&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000000840634476&amp;open=prime&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000002394962083&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000004020085177&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003896952461&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362491&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362489&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362488&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362487&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362486&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362485&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000000840634476&amp;base=36&lt;nowiki/&gt;||4||200000||
|}

== The fully proof of Athena problem in decimal (base ''b'' = 10) ==

'''Bold''' for the Athena primes, ''x'' ◁ ''y'' means ''x'' is a subsequence of ''y''.

Assume ''p'' is a prime &gt; 10, and the last digit of ''p'' must lie in {1,3,7,9}.

Case 1: ''p'' ends with 1.

In this case we can write ''p'' = ''x''1. If ''x'' contains 1, 3, 4, 6, or 7, then (respectively) '''11''' ◁ ''p'', '''31''' ◁ ''p'', '''41''' ◁ ''p'', '''61''' ◁ ''p'', or '''71''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 2, 5, 8, or 9.

Case 1.1: ''p'' begins with 2.

In this case we can write ''p'' = 2''y''1. If 5 ◁ ''y'', then '''251''' ◁ ''p''. If 8 ◁ ''y'', then '''281''' ◁ ''p''. If 9 ◁ ''y'', then 29 ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 2.

If 22 ◁ ''y'', then '''2221''' ◁ ''p''. Hence we may assume ''y'' contains zero or one 2's.

If ''y'' contains no 2's, then ''p'' ∈ 2{0}1. But then, since the sum of the digits of ''p'' is 3, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' contains exactly one 2, then we can write ''p'' = 2''z''2''w''1, where ''z'',''w'' ∈ {0}. If 0 ◁ ''z'' and 0 ◁ ''w'', then '''20201''' ◁ ''p''. Hence we may assume either ''z'' or ''w'' is empty.

If ''z'' is empty, then ''p'' ∈ 22{0}1, and the smallest prime ''p'' ∈ 22{0}1 is '''22000001'''.

If ''w'' is empty, then ''p'' ∈ 2{0}21, and the smallest prime ''p'' ∈ 2{0}21 is '''20021'''.

Case 1.2: ''p'' begins with 5.

In this case we can write ''p'' = 5''y''1. If 2 ◁ ''y'', then '''521''' ◁ ''p''. If 9 ◁ ''y'', then 59 ◁ ''p''. Hence we may assume all digits of ''y'' are 0, 5, or 8.

If 05 ◁ ''y'', then '''5051''' ◁ ''p''. If 08 ◁ ''y'', then '''5081''' ◁ ''p''. If 50 ◁ ''y'', then '''5501''' ◁ ''p''. If 58 ◁ ''y'', then '''5581''' ◁ ''p''. If 80 ◁ ''y'', then '''5801''' ◁ ''p''. If 85 ◁ ''y'', then '''5851''' ◁ ''p''. Hence we may assume ''y'' ∈ {0} ∪ {5} ∪ {8}.

If ''y'' ∈ {0}, then ''p'' ∈ 5{0}1. But then, since the sum of the digits of ''p'' is 6, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' ∈ {5}, then ''p'' ∈ 5{5}1, and the smallest prime ''p'' ∈ 5{5}1 is '''555555555551'''.

If ''y'' ∈ {8}, since if 88 ◁ ''y'', then 881 ◁ ''p'', hence we may assume ''y'' ∈ {''𝜆'',8}, and thus ''p'' ∈ {51,581}, but 51 and 581 are both composite.

Case 1.3: ''p'' begins with 8.

In this case we can write p = 8''y''1. If 2 ◁ ''y'', then '''821''' ◁ ''p''. If 8 ◁ ''y'', then '''881''' ◁ ''p''. If 9 ◁ ''y'', then 89 ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 5.

If 50 ◁ ''y'', then '''8501''' ◁ ''p''. Hence we may assume y ∈ {0}{5}.

If 005 ◁ ''y'', then '''80051''' ◁ p. Hence we may assume y ∈ {0} ∪ {5} ∪ 0{5}.

If y ∈ {0}, then ''p'' ∈ 8{0}1. But then, since the sum of the digits of ''p'' is 9, ''p'' is divisible by 3, so ''p'' cannot be prime.

If y ∈ {5}, since if 55555555555 ◁ ''y'', then 555555555551 ◁ ''p'', hence we may assume ''y'' ∈ {''𝜆'', 5, 55, 555, 5555, 55555, 555555, 5555555, 55555555, 555555555, 5555555555}, and thus ''p'' ∈ {81, 851, 8551, 85551, 855551, 8555551, 85555551, 855555551, 8555555551, 85555555551, 855555555551}, but all of these numbers are composite.

If y ∈ 0{5}, since if 55555555555 ◁ ''y'', then 555555555551 ◁ ''p'', hence we may assume ''y'' ∈ {0, 05, 055, 0555, 05555, 055555, 0555555, 05555555, 055555555, 0555555555, 05555555555}, and thus ''p'' ∈ {801, 8051, 80551, 805551, 8055551, 80555551, 805555551, 8055555551, 80555555551, 805555555551, 8055555555551}, and of these numbers only 80555551 and 8055555551 are primes, but 80555551 ◁ 8055555551, thus only '''80555551''' is a minimal element.

Case 1.4: ''p'' begins with 9.

In this case we can write p = 9''y''1. If 9 ◁ ''y'', then '''991''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0, 2, 5, or 8.

If 00 ◁ ''y'', then '''9001''' ◁ ''p''. If 22 ◁ ''y'', then '''9221''' ◁ ''p''. If 55 ◁ ''y'', then '''9551''' ◁ ''p''. If 88 ◁ ''y'', then 881 ◁ ''p''. Hence we may assume ''y'' contains at most one 0, at most one 2, at most one 5, and at most one 8.

If ''y'' only contains at most one 0 and does not contain any of {2,5,8}, then ''y'' ∈ {''𝜆'',0}, and thus ''p'' ∈ {91,901}, but 91 and 901 are both composite. If ''y'' only contains at most one 0 and only one of {2,5,8}, then the sum of the digits of ''p'' is divisible by 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Hence we may assume ''y'' contains at least two of {2,5,8}.

If 25 ◁ ''y'', then 251 ◁ ''p''. If 28 ◁ ''y'', then 281 ◁ ''p''. If 52 ◁ ''y'', then 521 ◁ ''p''. If 82 ◁ ''y'', then 821 ◁ ''p''. Hence we may assume ''y'' contains no 2's (since if ''y'' contains 2, then ''y'' cannot contain either 5's or 8's, which is a contradiction).

If 85 ◁ ''y'', then '''9851''' ◁ ''p''. Hence we may assume ''y'' ∈ {58,580,508,058}, and thus ''p'' ∈ {9581,95801,95081,90581}, and of these numbers only 95801 is prime, but 95801 is not a minimal element since 5801 ◁ 95801.

Case 2: ''p'' ends with 3.

In this case we can write p = ''x''3. If ''x'' contains 1, 2, 4, 5, 7, or 8, then (respectively) '''13''' ◁ ''p'', '''23''' ◁ ''p'', '''43''' ◁ ''p'', '''53''' ◁ ''p'', '''73''' ◁ ''p'', or '''83''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 3, 6, or 9, and thus all digits of ''p'' are 0, 3, 6, or 9. But then, since the digits of ''p'' all have a common factor 3, ''p'' is divisible by 3, so ''p'' cannot be prime.

Case 3: ''p'' ends with 7.

In this case we can write ''p'' = ''x''7. If ''x'' contains 1, 3, 4, 6, or 9, then (respectively) '''17''' ◁ ''p'', '''37''' ◁ ''p'', '''47''' ◁ ''p'', '''67''' ◁ ''p'', or '''97''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 2, 5, 7, or 8.

Case 3.1: ''p'' begins with 2.

In this case we can write ''p'' = 2''y''7. If 2 ◁ ''y'', then '''227''' ◁ ''p''. If 5 ◁ ''y'', then '''257''' ◁ ''p''. If 7 ◁ ''y'', then '''277''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 8.

If 08 ◁ ''y'', then '''2087''' ◁ ''p''. If 88 ◁ ''y'', then 887 ◁ ''p''. Hence we may assume ''y'' ∈ {0} ∪ 8{0}.

If ''y'' ∈ {0}, then ''p'' ∈ 2{0}7. But then, since the sum of the digits of ''p'' is 9, ''p'' is divisible by 3, so ''p'' cannot be prime.

If y ∈ 8{0}, then ''p'' ∈ 28{0}7. But then ''p'' is divisible by 7, since for ''n'' ≥ 0 we have 7 × 40&lt;sub&gt;''n''&lt;/sub&gt;1 = 280&lt;sub&gt;''n''&lt;/sub&gt;7.

Case 3.2: ''p'' begins with 5.

In this case we can write ''p'' = 5''y''7. If 5 ◁ ''y'', then '''557''' ◁ ''p''. If 7 ◁ ''y'', then '''577''' ◁ ''p''. If 8 ◁ ''y'', then '''587''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 2.

If 22 ◁ ''y'', then 227 ◁ ''p''. Hence we may assume ''y'' contains zero or one 2's.

If ''y'' contains no 2's, then ''p'' ∈ 5{0}7. But then, since the sum of the digits of ''p'' is 12, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' contains exactly one 2, then we can write ''p'' = 5''z''2''w''7, where ''z'',''w'' ∈ {0}. If 0 ◁ ''z'' and 0 ◁ ''w'', then '''50207''' ◁ ''p''. Hence we may assume either ''z'' or ''w'' is empty.

If ''z'' is empty, then ''p'' ∈ 52{0}7, and the smallest prime ''p'' ∈ 52{0}7 is '''5200007'''.

If ''w'' is empty, then ''p'' ∈ 5{0}27, and the smallest prime ''p'' ∈ 5{0}27 is '''5000000000000000000000000000027'''.

Case 3.3: ''p'' begins with 7.

In this case we can write ''p'' = 7''y''7. If 2 ◁ ''y'', then '''727''' ◁ ''p''. If 5 ◁ ''y'', then '''757''' ◁ ''p''. If 8 ◁ ''y'', then '''787''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 7, and thus all digits of ''p'' are 0 or 7. But then, since the digits of ''p'' all have a common factor 7, ''p'' is divisible by 7, so ''p'' cannot be prime.

Case 3.4: ''p'' begins with 8.

In this case we can write ''p'' = 8''y''7. If 2 ◁ ''y'', then '''827''' ◁ ''p''. If 5 ◁ ''y'', then '''857''' ◁ ''p''. If 7 ◁ ''y'', then '''877''' ◁ ''p''. If 8 ◁ ''y'', then '''887''' ◁ ''p''. Hence we may assume ''y'' ∈ {0}, and thus ''p'' ∈ 8{0}7. But then, since the sum of the digits of ''p'' is 15, ''p'' is divisible by 3, so ''p'' cannot be prime.

Case 4: ''p'' ends with 9.

In this case we can write ''p'' = ''x''9. If ''x'' contains 1, 2, 5, 7, or 8, then (respectively) '''19''' ◁ ''p'', '''29''' ◁ ''p'', '''59''' ◁ ''p'', '''79''' ◁ ''p'', or '''89''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 3, 4, 6, or 9.

If 44 ◁ ''x'', then '''449''' ◁ ''p''. Hence we may assume ''x'' contains zero or one 4's.

If x contains no 4's, then all digits of ''x'' are 0, 3, 6, or 9, and thus all digits of ''p'' are 0, 3, 6, or 9. But then, since the digits of ''p'' all have a common factor 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Hence we may assume that ''x'' contains exactly one 4.

Case 4.1: ''p'' begins with 3.

In this case we can write ''p'' = 3''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. We must have '''349''' ◁ ''p''.

Case 4.2: ''p'' begins with 4.

In this case we can write ''p'' = 4''y''9, where all digits of ''y'' are 0, 3, 6, or 9. If 0 ◁ ''y'', then '''409''' ◁ ''p''. If 3 ◁ ''y'', then 43 ◁ ''p''. If 9 ◁ ''y'', then '''499''' ◁ ''p''. Hence we may assume ''y'' ∈ {6}, and thus ''p'' ∈ 4{6}9. But then ''p'' is divisible by 7, since for ''n'' ≥ 0 we have 7 × 6&lt;sub&gt;''n''&lt;/sub&gt;7 = 46&lt;sub&gt;''n''&lt;/sub&gt;9.

Case 4.3: ''p'' begins with 6.

In this case we can write p = 6''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. If 0 ◁ ''z'', then 409 ◁ ''p''. If 3 ◁ ''z'', then 43 ◁ ''p''. If 6 ◁ ''z'', then '''6469''' ◁ ''p''. If 9 ◁ ''z'', then 499 ◁ ''p''. Hence we may assume ''z'' is empty.

If 3 ◁ ''y'', then 349 ◁ ''p''. If 9 ◁ ''y'', then '''6949''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 6.

If 06 ◁ ''y'', then '''60649''' ◁ ''p''. Hence we may assume ''y'' ∈ {6}{0}.

If 666 ◁ ''y'', then '''666649''' ◁ ''p''. If 00000 ◁ ''y'', then '''60000049''' ◁ ''p''. Hence we may assume ''y'' ∈ {''𝜆'', 0, 00, 000, 0000, 6, 60, 600, 6000, 60000, 66, 660, 6600, 66000, 660000}, and thus ''p'' ∈ {649, 6049, 60049, 600049, 6000049, 6649, 66049, 660049, 6600049, 66000049, 66649, 666049, 6660049, 66600049, 666000049}, and of these numbers only '''66000049''' and '''66600049''' are primes.

Case 4.4: ''p'' begins with 9.

In this case we can write p = 9''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. If 0 ◁ ''y'', then '''9049''' ◁ ''p''. If 3 ◁ ''y'', then 349 ◁ ''p''. If 6 ◁ ''y'', then '''9649''' ◁ ''p''. If 9 ◁ ''y'', then '''9949''' ◁ ''p''. Hence we may assume ''y'' is empty.

If 0 ◁ ''z'', then 409 ◁ ''p''. If 3 ◁ ''z'', then 43 ◁ ''p''. If 9 ◁ ''z'', then 499 ◁ ''p''. Hence we may assume ''z'' ∈ {6}, and thus ''p'' ∈ 94{6}9, and the smallest prime ''p'' ∈ 94{6}9 is 946669.

[[Category:Number theory]]</text>
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'''Athena problem''' is an [[:w:List of unsolved problems in mathematics|unsolved problem]] in [[:w:Number theory|number theory]] and [[:w:Formal language theory|formal language theory]] and [[:w:Order theory|order theory]], this problem is named after the ancient Greek goddess [[:w:Athena|Athena]] (which is associated with [[:w:Wisdom|wisdom]]). Athena problem is: Give a [[:w:Natural number|natural number]] ''b'' &gt; 1, find the [[:w:Set (mathematics)|set]] of the [[:w:Minimal element|minimal element]]s of the set of the &quot;[[:w:Prime number|prime number]] [[:w:Greater than|&gt;]] ''b''&quot; [[:w:Numerical digit|digit]] [[:w:String (computer science)|string]]s in the [[:w:Positional numeral system|positional numeral system]] with [[:w:Radix|base]] ''b'' for the [[:w:Subsequence|subsequence]] [[:w:Partially ordered set|ordering]]. (A string ''x'' is a subsequence of another string ''y'', if ''x'' can be obtained from ''y'' by deleting zero or more of the [[:w:Character (computing)|character]]s in ''y''. For example, 514 is a subsequence of 352148, &quot;string&quot; is a subsequence of &quot;meistersinger&quot;. In contrast, 758 is not a subsequence of 378259, &quot;abc&quot; is not a subsequence of &quot;cbacacba&quot;, since the characters must be in the same order) (Unlike [[:w:Substring|substring]], subsequence is not required to occupy consecutive positions within the original sequences, e.g. the [[:w:Longest common subsequence|longest common subsequence problem]] is different from the [[:w:Longest common substring|longest common substring problem]])

Using [[:w:Formal language theory|formal language theory]] terminology, Athena problem is finding the [[:w:Set (mathematics)|set]] of the [[:w:Minimal element|minimal element]]s of the [[:w:Formal language|language]] of base-''b'' [[:w:Representation (mathematics)|representation]]s of the [[:w:Prime number|prime number]]s [[:w:Greater than|&gt;]] ''b'' (which is a set of [[:w:String (computer science)|string]]s of [[:w:Symbol|symbol]]s over the [[:w:Alphabet (formal languages)|alphabet]] ''Σ''&lt;sub&gt;''b''&lt;/sub&gt; := {0, 1, ..., ''b''−1}), under the subsequence ordering (i.e. the [[:w:Binary relation|binary relation]] &quot;is a subsequence of&quot;, which is a [[:w:Partially ordered set|partial ordering]]), for a given natural number ''b'' &gt; 1 (You can draw this partial ordering as a [[:w:Hasse diagram|Hasse diagram]] to find all [[:w:Minimal element|minimal element]]s), this set is called '''Athena set''', and the prime numbers in this set are called '''Athena primes'''.

By [[:w:Higman's lemma|Higman's lemma]], there are no [[:w:Infinite set|infinite]] [[:w:Antichain|antichain]]s for the subsequence ordering (i.e. the subsequence ordering is always a [[:w:Well-quasi-ordering|well quasi order]]) (i.e. under the subsequence ordering (i.e. the [[:w:Binary relation|binary relation]] &quot;is a subsequence of&quot;, which is a [[:w:Partially ordered set|partial ordering]]), every set of pairwise incomparable (i.e. not [[:w:Comparability|comparable]]) strings is finite), thus there must be only finitely many such minimal elements. In other words, the Athena set in every base ''b'' must be a [[:w:Finite set|finite set]], and every base ''b'' ≥ 2 has only finitely many Athena primes, e.g. in [[:w:Decimal|decimal]] (base ''b'' = 10), the Athena set has exactly 77 [[:w:Element of a set|element]]s (they are exactly the Athena primes in decimal (base ''b'' = 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}.

For bases 2 ≤ ''b'' ≤ 36, Athena problem is fully solved in bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 24, and also solved in bases ''b'' = 11, 13, 16, 22, 30 if [[:w:Probable prime|probable prime]]s are allowed. For the unsolved bases ''b'' = 17, 19, 21, 23, 25, 26, 27, 28, 29, 31, 32, 34, 35, 36, Athena problem is solved (if probable primes are allowed) except 771 [[:w:Indexed family|families]] of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be [[:w:Empty string|empty]]) of digits in base ''b'', ''y'' is a digit in base ''b'') = sequence {''xz'', ''xyz'', ''xyyz'', ''xyyyz'', ''xyyyyz'', ''xyyyyyz'', ...} (i.e. &quot;''xy''&lt;sup&gt;+&lt;/sup&gt;''z''&quot; in [[:w:Regular expression|regular expression]]), all of these 771 families contain no primes &gt; ''b'' or probable primes &gt; ''b'' with length ≤ 100000.

== Solve the problem ==

To solve the Athena problem for a given base ''b'', we must [[:w:Computing|compute]] the elements up to families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''), and find the smallest prime &gt; ''b'' in all such families.

We call families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') &quot;linear&quot; families, and we reduce these families by removing all trailing digits ''y'' from ''x'', and removing all leading digits ''y'' from ''z'', to make the families be easier, e.g. family 12333{3}33345 in base ''b'' is reduced to family 12{3}45 in base ''b'', since they are in fact the same family. Our [[:w:Algorithm|algorithm]] then proceeds as follows:

* 1. ''M'' := {minimal primes in base ''b'' of length 2 or 3}, ''L'' := union of all ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'') such that ''x'' ≠ 0 and ''gcd''(''z'', ''b'') = 1 and ''Y'' is the set of digits ''y'' in base ''b'' such that ''xyz'' has no subsequence in ''M''.
* 2. While ''L'' contains nonlinear families (families which are not linear families): Explore each family of ''L'', and update ''L''. Examine each family of ''L'' by:
* 2.1. Let ''w'' be the shortest string in the family. If ''w'' has a subsequence in ''M'', then remove the family from ''L''. If ''w'' represents a prime, then add ''w'' to ''M'' and remove the family from ''L''.
* 2.2. If possible, simplify the family.
* 2.3. Using the techniques below (covering congruence, algebraic factorization, or combine of them), check if the family can be proven to only contain composites (only count the numbers &gt; ''b''), and if so then remove the family from ''L''.
* 3. Update ''L'', after each split examine the new families as in step 2.

e.g. in decimal (base ''b'' = 10):

''M'' := {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}

''L'' := {2{0,2}1, 2{0,8}7, 3{0,3,6,9}3, 3{0,3,6,9}9, 4{6}9, 5{0,5,8}1, 5{0,2}7, 6{0,3,6,9}3, 6{0,3,4,6,9}9, 7{0,7}7, 8{0,5}1, 8{0}7, 9{0,2,5,8}1, 9{0,3,6,9}3, 9{0,3,4,6,9}9}

and since 2221 is prime, it follows that the family 2{0,2}1 splits into the families 2{0}1 and 2{0}2{0}1

and since the family 2{0}1 can be proven to contain no primes &gt; base (since all numbers in this family are divisible by 3), it can be removed

and since 20201 is prime, it follows that the family 2{0}2{0}1 splits into the families 2{0}21 and 22{0}1

221 and 2021 are composites, but 20021 is prime, thus add 20021 to ''L''

none of 221, 2201, 22001, 220001, 2200001 are primes, but 22000001 is prime, thus add 22000001 to ''L''

and since the family 3{0,3,6,9}3 can be proven to contain no primes &gt; base (since all numbers in this family are divisible by 3), it can be removed

etc.

Since the number of possible (first digit,last digit) (also called (initial digit,final digit)) combos ([[:w:Ordered pair|ordered pair]]s) of a prime &gt; ''b'' in base ''b'' is (''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(''b'') (all digits except 0 can be the first digit of a prime &gt; ''b'' in base ''b'' (thus ''b''−1 possible digits), but only the digits coprime to ''b'' can be the last digit of a prime &gt; ''b'' in base ''b'' (thus ''eulerphi''(''b'') possible digits), and by the [[:w:Rule of product|rule of product]], there are (''b''−1)×''eulerphi''(''b'') choices of the (first digit,last digit) combo, also, both &quot;numbers of Athena primes in base ''b''&quot; and &quot;length of the largest Athena prime in base ''b''&quot; are [[:w:Asymptotic analysis|roughly]] ''[[:w:E (mathematical_constant)|e]]''&lt;sup&gt;''[[:w:Euler's constant|γ]]''×(''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(*b*)&lt;/sup&gt;.

Shrinking the family ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''Y'' is a set of digits in base ''b'')

* If ''y'' ∈ ''Y'' and the string ''xyyz'' represents a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or has a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''}''z'' ∪ ''x''{''Y'' \ ''y''}''y''{''Y'' \ ''y''}''z''.
* If ''y''&lt;sub&gt;1&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;2&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;1&lt;/sub&gt; ≠ ''y''&lt;sub&gt;2&lt;/sub&gt; and the string ''xy''&lt;sub&gt;1&lt;/sub&gt;''y''&lt;sub&gt;2&lt;/sub&gt;''z'' represents a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or has a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''&lt;sub&gt;1&lt;/sub&gt;}{''Y'' \ ''y''&lt;sub&gt;2&lt;/sub&gt;}''z''.
* If ''y''&lt;sub&gt;1&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;2&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;1&lt;/sub&gt; ≠ ''y''&lt;sub&gt;2&lt;/sub&gt; and both the strings ''xy''&lt;sub&gt;1&lt;/sub&gt;''y''&lt;sub&gt;2&lt;/sub&gt;''z'' and ''xy''&lt;sub&gt;2&lt;/sub&gt;''y''&lt;sub&gt;1&lt;/sub&gt;''z'' represent a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or have a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''&lt;sub&gt;1&lt;/sub&gt;}''z'' ∪ ''x''{''Y'' \ ''y''&lt;sub&gt;2&lt;/sub&gt;}''z''.

e.g. in decimal (base ''b'' = 10):

* 2221 is a prime &gt; 10, thus the family 2{0,2}1 splits into the two families 2{0}1 and 2{0}2{0}1.
* 227 is a prime &gt; 10, and it is a subsequence of 5227, thus the family 5{0,2}7 splits into the two families 5{0}7 and 5{0}2{0}7.
* 449 is a prime &gt; 10, and it is a subsequence of 6449, thus the family 6{0,3,4,6,9}9 splits into the two families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9.
* Both 5051 and 5501 are primes &gt; 10, thus the family 5{0,5}1 splits into the two families 5{0}1 and 5{5}1 = {5}1.
* 8501 is a prime &gt; 10, thus the family 8{0,5}1 splits into the family 8{0}{5}1.
* 887 is a prime &gt; 10, and it is a subsequence of 2887, also 2087 is a prime &gt; 10, thus the family 2{0,8}7 splits into the two families 2{0}7 and 28{0}7.
* 349 and 449 are primes &gt; 10, and they are subsequences of 9349 and 9449, respectively, also 9049, 9649, 9949 are primes &gt; 10, thus the family 9{0,3,4,6,9}9 splits into the two families 9{0,3,6,9}9 and 94{0,3,6,9}9.
* 251, 281, 521, 821, 881 are primes &gt; 10, and they are subsequences of 9251, 9281, 9521, 9821, 9881, respectively, also 9001, 9221, 9551, 9851 are primes &gt; 10, thus the family 9{0,2,5,8}1 splits into the numbers {91, 901, 921, 951, 981, 9021, 9051, 9081, 9201, 9501, 9581, 9801, 90581, 95081, 95801}.

If the methods we have discussed cannot be used to rule out or shrink ''x''{''Y''}''z'' where ''Y'' = {''y''&lt;sub&gt;1&lt;/sub&gt;, ''y''&lt;sub&gt;2&lt;/sub&gt;, ..., ''y''&lt;sub&gt;''n''&lt;/sub&gt;}, then we can replace ''x''{''Y''}''z'' by ''xy''&lt;sub&gt;1&lt;/sub&gt;{''Y''}''z'' ∪ ''xy''&lt;sub&gt;2&lt;/sub&gt;{''Y''}''z'' ∪ ... ∪ ''xy''&lt;sub&gt;''n''&lt;/sub&gt;{''Y''}''z'' and re-run the methods on this new [[:w:Formal language|language]].

If all remain families are linear families (i.e. of the form ''x''{''y''}''z'', where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''), then we search the smallest (probable) primes in these families and add these primes to the list.

e.g. in decimal (base ''b'' = 10):

* The smallest prime in the family 5{0}27 is 5000000000000000000000000000027.
* The smallest prime in the family {5}1 is 555555555551.
* The smallest prime in the family 8{5}1 is 8555555555555555555551, but 8555555555555555555551 is not a minimal element since 555555555551 is a subsequence of 8555555555555555555551.

There is no guarantee that the techniques discussed will ever terminate, but in practice they often do. They are able to determine the Athena set in base ''b'' for 2 ≤ ''b'' ≤ 16 and ''b'' = 18, 20, 22, 24, 30. The bases ''b'' = 17, 19, 21, 23, 25 ≤ ''b'' ≤ 29, 31 ≤ ''b'' ≤ 36 are solved with the exception of 771 families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'').

The following is a &quot;[[:w:Semi-algorithm|semi-algorithm]]&quot; that is guaranteed to solve the Athena problem for a given base ''b'', but it is not so easy to implement:

# ''M'' = ''[[:w:Empty string|∅]]''
# while (''L'' ≠ ''∅'') do
# choose ''x'', a shortest string in ''L''
# ''M'' := ''M'' ∪ {''x''}
# ''L'' := ''L'' − ''sup''({''x''})

In practice, for arbitrary ''L'', we cannot feasibly carry out step 5. Instead, we work with ''L''&amp;#39;, some regular overapproximation to ''L'', until we can show ''L''&amp;#39; = ''∅'' (which implies ''L'' = ''∅''). In practice, ''L''&amp;#39; is usually chosen to be a finite [[:w:Union (set theory)|union]] of sets of the form ''L''&lt;sub&gt;1&lt;/sub&gt;{''L''&lt;sub&gt;2&lt;/sub&gt;}''L''&lt;sub&gt;3&lt;/sub&gt;, where each of ''L''&lt;sub&gt;1&lt;/sub&gt;, ''L''&lt;sub&gt;2&lt;/sub&gt;, ''L''&lt;sub&gt;3&lt;/sub&gt; is finite. In the case we consider in this project, we then have to determine whether such a family contains a prime or not.

Thus, the [[:w:Time complexity|time complexity]] of the Athena problem in base ''b'' may be ''[[:w:Big O notation|O]]''(''[[:w:E (mathematical_constant)|e]]''&lt;sup&gt;''[[:w:Euler's constant|γ]]''×(''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(*b*)&lt;/sup&gt;), and the [[:w:CPU time|CPU time]] of the Athena problem in base ''b'' may be longer than [[:w:Age of the universe|the age of the universe]] for bases ''b'' = 19, 23, 25, 27, 29, 31, 32, 33, 34, 35, also, Athena problem in bases ''b'' around 500 may be [[:w:NP-complete|NP-complete]] or [[:w:NP-hard|NP-hard]], or an [[:w:Undecidable problem|undecidable problem]], or an example of [[:w:Gödel's incompleteness theorems|Gödel's incompleteness theorems]] (like the [[:w:Continuum hypothesis|continuum hypothesis]] and the [[:w:Halting problem|halting problem]]).

To solve the Athena problem (i.e. to [[:w:Computing|compute]] the Athena set), we need to determine whether a given family contains a prime. In practice, if family ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''Y'' is a set of digits in base ''b'') could not be ruled out as only containing composites and ''Y'' contains two or more digits, then a relatively small prime &gt; ''b'' could always be found in this family. Intuitively, this is because there are a large number of small strings in such a family, and at least one is likely to be prime (e.g. there are 2&lt;sup&gt;''n''−2&lt;/sup&gt; strings of length ''n'' in the family 1{3,7}9, and there are over a thousand strings of length 12 in the family 1{3,7}9, thus it is very impossible that these numbers are all composite). In the case ''Y'' contains only one digit, this family is of the form ''x''{''y''}''z'', and there is only a single string of each length &gt; (the length of ''x'' + the length of ''z''), and it is not known if the following [[:w:Decision problem|decision problem]] is recursively solvable (just like [[:w:Sierpiński number|Sierpiński problem]] and [[:w:Riesel number|Riesel problem]], Sierpiński problem and Riesel problem can be generalized to other bases ''b'' (references: http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm), in fact, Athena problem base ''b'' covers the Sierpiński problem base ''b'' and the Riesel problem base ''b'' with ''k'' &lt; ''b'', i.e. finding the smallest prime of the form ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 and ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (or prove such prime does not exist) with ''k'' &lt; ''b'' (specially, for bases ''b'' such that the conjectured smallest Sierpiński number or the conjectured smallest Riesel number is &lt; ''b'', Athena problem base ''b'' covers the Sierpiński problem base ''b'' or the Riesel problem base ''b'', respectively), since the smallest prime of the form ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 and ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (if exists) must be a minimal element in base ''b'', also, Athena problem base ''b'' covers finding the smallest prime of these forms in base ''b'' (or proving that such prime does not exist): (''b''&lt;sup&gt;''n''&lt;/sup&gt;−1)/(''b''−1) (for this form, ''n'' must be prime, and we want ''n'' ≥ 2) (references of this form: http://www.fermatquotient.com/PrimSerien/GenRepu.txt, https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html, http://www.primenumbers.net/Henri/us/MersFermus.htm, http://www.bitman.name/math/table/379, https://pzktupel.de/Primetables/TableRepunitGen.php, https://oeis.org/A084740, https://oeis.org/A084738, https://oeis.org/A128164, https://oeis.org/A285642; or for prime bases ''b'': https://oeis.org/A065854, https://oeis.org/A279068), ''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 (for this form, ''n'' must be power of 2, and we want ''n'' ≥ 1) (references of this form: http://jeppesn.dk/generalized-fermat.html, http://www.noprimeleftbehind.net/crus/GFN-primes.htm, https://web.archive.org/web/20231002190634/http://yves.gallot.pagesperso-orange.fr/primes/index.html, https://pzktupel.de/Primetables/TableFermatGFBB.php, https://oeis.org/A079706, https://oeis.org/A084712, https://oeis.org/A228101), (''b''&lt;sup&gt;''n''&lt;/sup&gt;+1)/2 (for odd ''b'') (for this form, ''n'' must be power of 2, and we want ''n'' ≥ 2) (reference of this form: http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt), (''sqrt''(''b'')×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1)/(''sqrt''(''b'')+1) (for square ''b'') (for this form, 2×''n''+1 must be prime, and we want ''n'' ≥ 2) (references of this form: http://www.fermatquotient.com/PrimSerien/GenRepuP.txt, http://www.primenumbers.net/Henri/us/MersFermus.htm, http://www.bitman.name/math/table/488, https://pzktupel.de/Primetables/TableWagstaffGen.php, https://oeis.org/A084742, https://oeis.org/A084741; or for bases ''b'' with ''sqrt''(''b'') prime: https://oeis.org/A065507), ((''b''−2)×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1)/(''b''−1) (''n'' ≥ 2) (reference of this form: https://oeis.org/A243404), 2×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 (''n'' ≥ 1) (references of this form: https://www.mersenneforum.org/showthread.php?t=6918, https://www.mersenneforum.org/showthread.php?t=19725, https://oeis.org/A119624), 2×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (''n'' ≥ 1) (references of this form: https://www.mersenneforum.org/showthread.php?t=24576, https://www.mersenneforum.org/attachment.php?attachmentid=20976&amp;d=1567314217, https://oeis.org/A119591), ''b''&lt;sup&gt;''n''&lt;/sup&gt;+2 (''n'' ≥ 1) (references of this form: https://oeis.org/A138066, https://oeis.org/A084713, https://oeis.org/A138067), ''b''&lt;sup&gt;''n''&lt;/sup&gt;−2 (''n'' ≥ 2) (references of this form: https://www.primepuzzles.net/puzzles/puzz_887.htm, https://oeis.org/A250200, https://oeis.org/A255707, https://oeis.org/A084714; or for prime bases ''b'': https://oeis.org/A292201), (''b''−1)×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 (''n'' ≥ 1) (references of this form: http://www.noprimeleftbehind.net/Williams-primes-MP.htm, http://www.bitman.name/math/table/477, https://pzktupel.de/Primetables/TableWilliams2.php, https://oeis.org/A305531; or for prime bases ''b'': https://oeis.org/A087139), (''b''−1)×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (''n'' ≥ 1) (references of this form: https://harvey563.tripod.com/wills.txt, http://www.noprimeleftbehind.net/Williams-primes-MM.htm, http://www.bitman.name/math/table/484, https://pzktupel.de/Primetables/TableWilliams1.php; or for prime bases ''b'': https://oeis.org/A122396), ''b''&lt;sup&gt;''n''&lt;/sup&gt;+(''b''−1) (''n'' ≥ 1) (references of this form: http://www.bitman.name/math/table/795, https://pzktupel.de/Primetables/TableWilliams6.php, https://oeis.org/A076845, https://oeis.org/A076846, https://oeis.org/A078178, https://oeis.org/A078179), ''b''&lt;sup&gt;''n''&lt;/sup&gt;−(''b''−1) (''n'' ≥ 2) (references of this form: http://www.bitman.name/math/table/792, https://pzktupel.de/Primetables/TableWilliams5.php, https://oeis.org/A113516, https://oeis.org/A343589; or for prime bases ''b'': https://cs.uwaterloo.ca/journals/JIS/VOL3/mccranie.html, http://www.bitman.name/math/table/435)):

Problem: Given strings ''x'', ''z'' (may be empty), a digit ''y'', and a base ''b'' (''x'' does not [[:w:Leading zero|start with the digit 0]], ''z'' ends with a digit which [[:w:Coprime integers|coprime]] to ''b'', ''y'' is not 0 if ''x'' is empty, ''y'' is coprime to ''b'' if ''z'' is empty), does there exist a prime number whose base-''b'' expansion is of the form ''xy''&lt;sub&gt;''n''&lt;/sub&gt;''z'' for some ''n'' ≥ 0?

Some families can be ruled out to contain no prime &gt; ''b'' by [[:w:Covering set|covering congruence]], [[:w:Factorization of polynomials|algebraic factorization]] (e.g. [[:w:Difference of two squares|difference of two squares]], [[:w:Sum of two cubes|sum of two cubes]], [[:w:Sophie Germain's identity|Sophie Germain's identity of ''x''&lt;sup&gt;4&lt;/sup&gt;+4×''y''&lt;sup&gt;4&lt;/sup&gt;]]), or combine of them, e.g.

* The base 9 family 2{7}: Always divisible by 2 or 5
* The base 11 family 2{5}: Always divisible by 2 or 3
* The base 14 family B{0}1: Always divisible by 3 or 5
* The base 13 family 95{0}3: Always divisible by 5, 7, or 17
* The base 16 family {4}D: Always divisible by 3, 7, or 13
* The base 16 family {8}F: Always divisible by 3, 7, or 13
* The base 21 family {7}D: Always divisible by 2, 13, or 17
* The base 23 family {D}GA: Always divisible by 2, 5, 7, 37, or 79
* The base 9 family {1}: Can be written as (9&lt;sup&gt;''n''&lt;/sup&gt;−1)/8 and can be factored as (3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (3&lt;sup&gt;''n''&lt;/sup&gt;+1) / 8
* The base 8 family 1{0}1: Can be written as 8&lt;sup&gt;''n''&lt;/sup&gt;+1 and can be factored as (2&lt;sup&gt;''n''&lt;/sup&gt;+1) × (4&lt;sup&gt;''n''&lt;/sup&gt;−2&lt;sup&gt;''n''&lt;/sup&gt;+1)
* The base 9 family 3{8}: Can be written as 4×9&lt;sup&gt;''n''&lt;/sup&gt;−1 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (2×3&lt;sup&gt;''n''&lt;/sup&gt;+1)
* The base 16 family 1{5}: Can be written as (4×16&lt;sup&gt;''n''&lt;/sup&gt;−1)/3 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (2×3&lt;sup&gt;''n''&lt;/sup&gt;+1) / 3
* The base 16 family {4}1: Can be written as (4×16&lt;sup&gt;''n''&lt;/sup&gt;−49)/15 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−7) × (2×3&lt;sup&gt;''n''&lt;/sup&gt;+7) / 15
* The base 27 family 7{Q}: Can be written as 8×27&lt;sup&gt;''n''&lt;/sup&gt;−1 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (4×9&lt;sup&gt;''n''&lt;/sup&gt;+2×3&lt;sup&gt;''n''&lt;/sup&gt;+1)
* The base 27 family 9{G}: Can be written as (125×27&lt;sup&gt;''n''&lt;/sup&gt;−8)/13 and can be factored as (5×3&lt;sup&gt;''n''&lt;/sup&gt;−2) × (25×9&lt;sup&gt;''n''&lt;/sup&gt;+10×3&lt;sup&gt;''n''&lt;/sup&gt;+4)
* The base 16 family {C}D: Can be written as (4×16&lt;sup&gt;''n''&lt;/sup&gt;+1)/5 and can be factored as (2×4&lt;sup&gt;''n''&lt;/sup&gt;−2×2&lt;sup&gt;''n''&lt;/sup&gt;+1) × (2×4&lt;sup&gt;''n''&lt;/sup&gt;+2×2&lt;sup&gt;''n''&lt;/sup&gt;+1) / 5
* The base 14 family 8{D}: Can be written as 9×14&lt;sup&gt;''n''&lt;/sup&gt;−1, it is divisible by 5 if ''n'' is odd and can be factored as (3×14&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (3×14&lt;sup&gt;''n''/2&lt;/sup&gt;+1) if ''n'' is even
* The base 12 family {B}9B: Can be written as 12&lt;sup&gt;''n''&lt;/sup&gt;−25, it is divisible by 13 if ''n'' is odd and can be factored as (12&lt;sup&gt;''n''/2&lt;/sup&gt;−5) × (12&lt;sup&gt;''n''/2&lt;/sup&gt;+5) if ''n'' is even
* The base 14 family {D}5: Can be written as 14&lt;sup&gt;''n''&lt;/sup&gt;−9, it is divisible by 5 if ''n'' is odd and can be factored as (14&lt;sup&gt;''n''/2&lt;/sup&gt;−3) × (14&lt;sup&gt;''n''/2&lt;/sup&gt;+3) if ''n'' is even
* The base 17 family 1{9}: Can be written as (25×17&lt;sup&gt;''n''&lt;/sup&gt;−9)/16, it is divisible by 2 if ''n'' is odd and can be factored as (5×17&lt;sup&gt;''n''/2&lt;/sup&gt;−3) × (5×17&lt;sup&gt;''n''/2&lt;/sup&gt;+3) / 16 if ''n'' is even
* The base 17 family 7{9}: Can be written as (121×17&lt;sup&gt;''n''&lt;/sup&gt;−9)/16, it is divisible by 2 if ''n'' is odd and can be factored as (11×17&lt;sup&gt;''n''/2&lt;/sup&gt;−3) × (11×17&lt;sup&gt;''n''/2&lt;/sup&gt;+3) / 16 if ''n'' is even
* The base 19 family 1{6}: Can be written as (4×19&lt;sup&gt;''n''&lt;/sup&gt;−1)/3, it is divisible by 5 if ''n'' is odd and can be factored as (2×19&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (2×19&lt;sup&gt;''n''/2&lt;/sup&gt;+1) / 3 if ''n'' is even
* The base 19 family 7{2}: Can be written as (64×19&lt;sup&gt;''n''&lt;/sup&gt;−1)/9, it is divisible by 5 if ''n'' is odd and can be factored as (8×19&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (8×19&lt;sup&gt;''n''/2&lt;/sup&gt;+1) / 9 if ''n'' is even
* The base 24 family 3{N}: Can be written as 4×24&lt;sup&gt;''n''&lt;/sup&gt;−1, it is divisible by 5 if ''n'' is odd and can be factored as (2×24&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (2×24&lt;sup&gt;''n''/2&lt;/sup&gt;+1) if ''n'' is even

By the [[:w:Prime number theorem|prime number theorem]], the [[:w:Probability|chance]] that a [[:w:Random number|random]] ''n''-digit base ''b'' number is prime is [[:w:Asymptotic analysis|approximately]] 1/''n'' (more accurately, the chance is approximately 1/(''n''×''ln''(''b'')), where ''ln'' is the [[:w:Natural logarithm|natural logarithm]]). If one conjectures the numbers ''x''{''y''}''z'' behave similarly (i.e. the numbers ''x''{''y''}''z'' is a [[:w:Pseudorandomness|pseudorandom sequence]]) you would expect [[:w:Harmonic_series (mathematics)|1/1 + 1/2 + 1/3 + 1/4 + ... = ∞]] primes of the form ''x''{''y''}''z'' (of course, this does not always happen, since some ''x''{''y''}''z'' families can be ruled out to contain no prime &gt; ''b'' (by covering congruence, algebraic factorization, or combine of them), but it is at least a reasonable conjecture in the absence of evidence to the contrary. Hence, the [[:w:Heuristic argument|heuristic argument]] suggests there are always infinitely many primes in family ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') if it cannot be ruled out to contain no prime or only contain finitely many primes, by covering congruence, algebraic factorization, or combine of them. However, some families ''x''{''y''}''z'' could not be proven to contain no primes &gt; ''b'' (by covering congruence, algebraic factorization, or combine of them) but no primes &gt; ''b'' could be found in the family, even after searching through numbers with over 100000 digits. In such a case, the only way to proceed is to [[:w:Primality test|test the primality]] of larger and larger numbers of such form and hope a prime is eventually discovered. e.g. the smallest (probable) prime in the family A{3}A in base ''b'' = 13 is A3&lt;sub&gt;592197&lt;/sub&gt;A, its algebraic form is (41×13&lt;sup&gt;592198&lt;/sup&gt;+27)/4, when written in decimal contains 659677 digits (it is only probable prime, i.e. not definitely prime, since technically, probable primality tests were used to show this (which have a ''very'' small chance of making an error, see https://t5k.org/notes/prp_prob.html) because all known primality tests run far too slowly to run on numbers of this size unless either [https://t5k.org/prove/prove3_1.html ''N''−1] or [https://t5k.org/prove/prove3_2.html ''N''+1] (or both) can be ≥ 1/3 factored).

The numbers in family ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') are of the form (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1) for some fixed ''a'', ''b'', ''c'' such that ''a'' ≥ 1, ''b'' ≥ 2 (''b'' is the base), ''c'' ≠ 0, ''gcd''(''a'',''c'') = 1, ''gcd''(''b'',''c'') = 1. Except in the [[:w:Special case|special case]] ''c'' = ±1 and ''gcd''(''a''+''c'',''b''−1) = 1, when ''n'' is large the known [[:w:Primality test|primality test]]s for such a number are too inefficient to run. In this case one must resort to a [[:w:Probabilistic algorithm|probable]] primality test such as a [[:w:Miller–Rabin primality test|Miller–Rabin primality test]] or a [[:w:Baillie–PSW primality test|Baillie–PSW primality test]], unless a divisor of the number can be found. Since we are testing many numbers in an [[:w:Exponential growth|exponential sequence]], it is possible to use a sieving process to find divisors rather than using [[:w:Trial division|trial division]].

To do this, we made use of Geoffrey Reynolds' ''srsieve'' software (download: https://pzktupel.de/Software/srsieve_1.1.4.7z). This program uses the [[:w:Baby-step giant-step|baby-step giant-step]] [[:w:Algorithm|algorithm]] to find all primes ''p'' which divide ''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'' where ''p'' and ''n'' lie in a [[:w:Interval_(mathematics)|specified range]]. Since this program cannot handle the general case (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1) when ''gcd''(''a''+''c'',''b''−1) &gt; 1 we only used it to sieve the sequence ''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'' for primes ''p'' not dividing ''gcd''(''a''+''c'',''b''−1), and initialized the list of candidates to not include ''n'' for which there is some prime ''p'' dividing ''gcd''(''a''+''c'',''b''−1) for which ''p'' dividing (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1). The program had to be modified slightly to remove a check which would prevent it from running in the case when ''a'', ''b'', and ''c'' were all odd (since then 2 divides ''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'', but 2 may not divide (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1)).

Once the numbers with small divisors had been removed, it remained to test the remaining numbers using a probable primality test. For this we used the software ''LLR'' by Jean Penné. (download: http://jpenne.free.fr/index2.html). Although undocumented, it is possible to run this program on numbers of the form (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1) when ''gcd''(''a''+''c'',''b''−1) &gt; 1, so this program required no modifications. A script was also written which allowed one to run ''srsieve'' while ''LLR'' was testing the remaining candidates, so that when a divisor was found by srsieve on a number which had not yet been tested by ''LLR'' it would be removed from the list of candidates.

For the primes &lt; 10&lt;sup&gt;25000&lt;/sup&gt; for the &quot;easy&quot; bases (bases ''b'' with ≤ 150 primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; (base ''b'' = 26 has 83 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 3 unsolved families, base ''b'' = 36 has 75 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 4 unsolved families, base ''b'' = 17 has 99 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 18 unsolved families, base ''b'' = 21 has 80 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 12 unsolved families, base ''b'' = 19 has 201 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 23 unsolved families), i.e. bases *b* = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 26, 28, 30, 36), we employed ''CM'' by Andreas Enge (download: https://www.multiprecision.org/cm/download.html), an elliptic curve primality proving implementation.

== Data ==

These are the results of the Athena problem in bases 2 ≤ ''b'' ≤ 36 (we stop at base 36 since this base is the maximum base for which it is possible to write the numbers with the [[:w:Symbol|symbol]]s 0, 1, 2, ..., 9 and A, B, C, ..., Z (i.e. the 10 [[:w:Arabic numerals|Arabic numerals]] and the 26 [[:w:Latin script|Latin letters]]): (some large Athena primes are only probable primes, i.e. not definitely primes, since they are too large to be [[:w:Elliptic curve primality|ECPP proved]] and [[:w:Pocklington primality test#Extensions and variants|neither ''N''−1 nor ''N''+1 can be ≥ 1/3 factored]], all of them pass the [[:w:Baillie–PSW primality test|Baillie–PSW primality test]] and the [[:w:Strong pseudoprime|strong primality test]] (i.e. the [[:w:Miller–Rabin primality test|Miller–Rabin primality test]]) with all prime bases ''p'' ≤ 61, however, all Athena primes &lt; 10&lt;sup&gt;25000&lt;/sup&gt; for bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 36 are definitely primes, most of them &gt; 10&lt;sup&gt;299&lt;/sup&gt; are proven primes with [[:w:Elliptic curve primality|ECPP proving]], others &gt; 10&lt;sup&gt;299&lt;/sup&gt; are proven primes with [[:w:Pocklington primality test#Extensions and variants|''N''−1 or ''N''+1 proving]])

All numbers are written in base ''b'', [[:w:Senary#Base 36 as senary compression|using A to Z to represent digit values 10 to 35]], &quot;{}&quot; means repeating, e.g. family 12{3}45 means the sequence {1245, 12345, 123345, 1233345, 12333345, 123333345, ...} (where the members are expressed as base ''b'' strings), subscripts are used to indicate repetitions of digits, e.g. 123&lt;sub&gt;4&lt;/sub&gt;567 means 123333567 (all subscripts are written in decimal).

Base 2: 1 Athena prime (the largest of which has 2 digits (it is 11, and its value is 3 in decimal)): {11}

Base 3: 3 Athena primes (the largest of which has 3 digits (it is 111, and its value is 13 in decimal)): {12, 21, 111}

Base 4: 5 Athena primes (the largest of which has 3 digits (it is 221, and its value is 41 in decimal)): {11, 13, 23, 31, 221}

Base 5: 22 Athena primes (the largest of which has 96 digits (it is 10&lt;sub&gt;93&lt;/sub&gt;13, and its algebraic form is 5&lt;sup&gt;95&lt;/sup&gt;+8)): {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 Athena primes (the largest of which has 5 digits (it is 40041, and its value is 5209 in decimal)): {11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041}

Base 7: 71 Athena primes (the largest of which has 17 digits (it is 3&lt;sub&gt;16&lt;/sub&gt;1, and its algebraic form is (7&lt;sup&gt;17&lt;/sup&gt;−5)/2)): {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: 75 Athena primes (the largest of which has 221 digits (it is 4&lt;sub&gt;220&lt;/sub&gt;7, and its algebraic form is (4×8&lt;sup&gt;221&lt;/sup&gt;+17)/7)): {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: 151 Athena primes (the largest of which has 1161 digits (it is 30&lt;sub&gt;1158&lt;/sub&gt;11, and its algebraic form is 3×9&lt;sup&gt;1160&lt;/sup&gt;+10)): {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: 77 Athena primes (the largest of which has 31 digits (it is 50&lt;sub&gt;28&lt;/sub&gt;27, and its algebraic form is 5×10&lt;sup&gt;30&lt;/sup&gt;+27)): {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: 1068 Athena primes (including 1 unproven probable prime: 57&lt;sub&gt;62668&lt;/sub&gt;), the largest of which has 62669 digits (it is 57&lt;sub&gt;62668&lt;/sub&gt;, and its algebraic form is (57×11&lt;sup&gt;62668&lt;/sup&gt;−7)/10), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel11 Data of Athena (probable) primes base 11]

Base 12: 106 Athena primes (the largest of which has 42 digits (it is 40&lt;sub&gt;39&lt;/sub&gt;77, and its algebraic form is 4×12&lt;sup&gt;41&lt;/sup&gt;+91)): {11, 15, 17, 1B, 25, 27, 31, 35, 37, 3B, 45, 4B, 51, 57, 5B, 61, 67, 6B, 75, 81, 85, 87, 8B, 91, 95, A7, AB, B5, B7, 221, 241, 2A1, 2B1, 2BB, 401, 421, 447, 471, 497, 565, 655, 665, 701, 70B, 721, 747, 771, 77B, 797, 7A1, 7BB, 907, 90B, 9BB, A41, B21, B2B, 2001, 200B, 202B, 222B, 229B, 292B, 299B, 4441, 4707, 4777, 6A05, 6AA5, 729B, 7441, 7B41, 929B, 9777, 992B, 9947, 997B, 9997, A0A1, A201, A605, A6A5, AA65, B001, B0B1, BB01, BB41, 600A5, 7999B, 9999B, AAAA1, B04A1, B0B9B, BAA01, BAAA1, BB09B, BBBB1, 44AAA1, A00065, BBBAA1, AAA0001, B00099B, AA000001, BBBBBB99B, B0000000000000000000000000009B, 400000000000000000000000000000000000000077}

Base 13: 3197 Athena primes (including 4 unproven probable primes: C5&lt;sub&gt;23755&lt;/sub&gt;C, 80&lt;sub&gt;32017&lt;/sub&gt;111, 95&lt;sub&gt;197420&lt;/sub&gt;, A3&lt;sub&gt;592197&lt;/sub&gt;A), the largest of which has 592199 digits (it is A3&lt;sub&gt;592197&lt;/sub&gt;A, and its algebraic form is (41×13&lt;sup&gt;592198&lt;/sup&gt;+27)/4), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel13 Data of Athena (probable) primes base 13]

Base 14: 650 Athena primes, the largest of which has 19699 digits (it is 4D&lt;sub&gt;19698&lt;/sub&gt;, and its algebraic form is 5×14&lt;sup&gt;19698&lt;/sup&gt;−1), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel14 Data of Athena primes base 14]

Base 15: 1284 Athena primes, the largest of which has 157 digits (it is 7&lt;sub&gt;155&lt;/sub&gt;97, and its algebraic form is (15&lt;sup&gt;157&lt;/sup&gt;+59)/2), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel15 Data of Athena primes base 15]

Base 16: 2347 Athena primes (including 3 unproven probable primes: DB&lt;sub&gt;32234&lt;/sub&gt;, 4&lt;sub&gt;72785&lt;/sub&gt;DD, 3&lt;sub&gt;116137&lt;/sub&gt;AF), the largest of which has 116139 digits (it is 3&lt;sub&gt;116137&lt;/sub&gt;AF, and its algebraic form is (16&lt;sup&gt;116139&lt;/sup&gt;+619)/5), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel16 Data of Athena (probable) primes base 16]

Base 17: 10415 known Athena primes (including many unproven probable primes) and 12 unsolved families (1{7}, 1F{0}7, 4{7}A, 70F{0}D, 8{B}9, 9{5}9, A{D}F, B{0}B3, {B}E9, {B}EE, F1{9}, FD0{D}, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel17 Data of known Athena (probable) primes base 17]

Base 18: 549 Athena primes, the largest of which has 6271 digits (it is C0&lt;sub&gt;6268&lt;/sub&gt;C5, and its algebraic form is 12×18&lt;sup&gt;6270&lt;/sup&gt;+221), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel18 Data of Athena primes base 18]

Base 19: 31417 known Athena primes (including many unproven probable primes) and 17 unsolved families (4B5{0}H, {5}3, 5{H}05, 5{H}0H, 5{H}5, 66{B}, 71{0}177, 7AF{0}H, 97{0}3, C{H}C, EE1{6}, F{7}5, F{B}G, F{D}F, H0F{0}7A, HB{0}5B5, II{D}, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel19 Data of known Athena (probable) primes base 19]

Base 20: 3314 Athena primes, the largest of which has 6271 digits (it is G0&lt;sub&gt;6269&lt;/sub&gt;D, and its algebraic form is 16×20&lt;sup&gt;6270&lt;/sup&gt;+13), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel20 Data of Athena primes base 20]

Base 21: 13386 known Athena primes (including many unproven probable primes) and 8 unsolved families (5{0}DJ, {9}D, B3{0}EB, B{H}6H, C{F}0K, {F}35, G{0}FK, H{0}7771, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel21 Data of known Athena (probable) primes base 21]

Base 22: 8003 Athena primes (including 1 unproven probable prime: BK&lt;sub&gt;22001&lt;/sub&gt;5), the largest of which has 22003 digits (it is BK&lt;sub&gt;22001&lt;/sub&gt;5, and its algebraic form is (251×22&lt;sup&gt;22002&lt;/sup&gt;−335)/21), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel22 Data of Athena (probable) primes base 22]

Base 23: 65178 known Athena primes (including many unproven probable primes) and 87 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel23 Data of known Athena (probable) primes base 23] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left23 Data of unsolved families for Athena problem base 23]

Base 24: 3409 Athena primes, the largest of which has 8134 digits (it is N00N&lt;sub&gt;8129&lt;/sub&gt;LN, and its algebraic form is 13249×24&lt;sup&gt;8131&lt;/sup&gt;−49), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel24 Data of Athena primes base 24]

Base 25: 133639 known Athena primes (including many unproven probable primes) and 85 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel25 Data of known Athena (probable) primes base 25] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left25 Data of unsolved families for Athena problem base 25]

Base 26: 25256 known Athena primes (including 7 unproven probable primes: 5&lt;sub&gt;19391&lt;/sub&gt;6F, 7&lt;sub&gt;20279&lt;/sub&gt;OL, LD0&lt;sub&gt;20975&lt;/sub&gt;7, 6K&lt;sub&gt;23300&lt;/sub&gt;5, J0&lt;sub&gt;44303&lt;/sub&gt;KCB, M0&lt;sub&gt;61186&lt;/sub&gt;2BB, 85M&lt;sub&gt;197060&lt;/sub&gt;B) and 3 unsolved families ({A}6F, {H}MH, {I}GL, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel26 Data of known Athena (probable) primes base 26]

Base 27: 102852 known Athena primes (including many unproven probable primes) and 44 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel27 Data of known Athena (probable) primes base 27] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left27 Data of unsolved families for Athena problem base 27]

Base 28: 25528 known Athena primes (including 3 unproven probable primes: N6&lt;sub&gt;24051&lt;/sub&gt;LR, 5OA&lt;sub&gt;31238&lt;/sub&gt;F, O4O&lt;sub&gt;94535&lt;/sub&gt;9) and 1 unsolved family (O{A}F, no primes or probable primes with length ≤ 900000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel28 Data of known Athena (probable) primes base 28]

Base 29: 355242 known Athena primes (including many unproven probable primes) and 125 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel29 Data of known Athena (probable) primes base 29] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left29 Data of unsolved families for Athena problem base 29]

Base 30: 2619 Athena primes (including 1 unproven probable prime: I0&lt;sub&gt;24608&lt;/sub&gt;D), the largest of which has 34206 digits (it is OT&lt;sub&gt;34205&lt;/sub&gt;, and its algebraic form is 25×30&lt;sup&gt;34205&lt;/sup&gt;−1), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel30 Data of Athena (probable) primes base 30]

Base 31: 569323 known Athena primes (including many unproven probable primes) and 77 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel31 Data of known Athena (probable) primes base 31] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left31 Data of unsolved families for Athena problem base 31]

Base 32: 168882 known Athena primes (including many unproven probable primes) and 120 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel32 Data of known Athena (probable) primes base 32] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left32 Data of unsolved families for Athena problem base 32]

Base 33: 280012 known Athena primes (including many unproven probable primes) and 81 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel33 Data of known Athena (probable) primes base 33] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left33 Data of unsolved families for Athena problem base 33]

Base 34: 184785 known Athena primes (including many unproven probable primes) and 47 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel34 Data of known Athena (probable) primes base 34] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left34 Data of unsolved families for Athena problem base 34]

Base 35: 720002 known Athena primes (including many unproven probable primes) and 60 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel35 Data of known Athena (probable) primes base 35] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left35 Data of unsolved families for Athena problem base 35]

Base 36: 35286 known Athena primes (including 3 unproven probable primes: 7K&lt;sub&gt;26567&lt;/sub&gt;Z, S0&lt;sub&gt;75007&lt;/sub&gt;8H, P&lt;sub&gt;81993&lt;/sub&gt;SZ) and 4 unsolved families (B{0}EUV, HM{0}N, N{0}YYN, O{L}Z, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel36 Data of known Athena (probable) primes base 36]

== Condensed table for bases 2 ≤ ''b'' ≤ 36 ==

{|class=&quot;wikitable&quot;
||''b''||number of Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b''||base-''b'' form of the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b'' (write &quot;''d''&lt;sub&gt;''n''&lt;/sub&gt;&quot; if there are 5 or more (''n'') consecutive same digits ''d'')||length of the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b''||length of the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b'' in decimal||algebraic ((''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1)) form of the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b''||''factordb'' entry of the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b''||the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b'' written in base ''b'' (use lower case letters instead of upper case letters)||number of unsolved families in the Athena problem in base ''b'' (all of these left families are linear families)||searching limit of length for the unsolved families in the Athena problem in base ''b'' (if there are different searching limits for the unsolved families in the Athena problem in base ''b'', choose the lowest searching limit)||
|-
||2||1||11||2||1||3||http://factordb.com/index.php?id=3&amp;open=ecm||http://factordb.com/index.php?showid=3&amp;base=2||0||–||
|-
||3||3||111&lt;br&gt;21&lt;br&gt;12||3&lt;br&gt;2&lt;br&gt;2||2&lt;br&gt;1&lt;br&gt;1||13&lt;br&gt;7&lt;br&gt;5||http://factordb.com/index.php?id=13&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=7&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=5&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=13&amp;base=3&lt;br&gt;http://factordb.com/index.php?showid=7&amp;base=3&lt;br&gt;http://factordb.com/index.php?showid=5&amp;base=3&lt;nowiki/&gt;||0||–||
|-
||4||5||221&lt;br&gt;31&lt;br&gt;23&lt;br&gt;13&lt;br&gt;11||3&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2||2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;1&lt;br&gt;1||41&lt;br&gt;13&lt;br&gt;11&lt;br&gt;7&lt;br&gt;5||http://factordb.com/index.php?id=41&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=13&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=11&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=7&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=5&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=41&amp;base=4&lt;br&gt;http://factordb.com/index.php?showid=13&amp;base=4&lt;br&gt;http://factordb.com/index.php?showid=11&amp;base=4&lt;br&gt;http://factordb.com/index.php?showid=7&amp;base=4&lt;br&gt;http://factordb.com/index.php?showid=5&amp;base=4&lt;nowiki/&gt;||0||–||
|-
||5||22||10&lt;sub&gt;93&lt;/sub&gt;13&lt;br&gt;300031&lt;br&gt;44441&lt;br&gt;33331&lt;br&gt;33001&lt;br&gt;30301&lt;br&gt;14444&lt;br&gt;10103&lt;br&gt;3101&lt;br&gt;414||96&lt;br&gt;6&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;4&lt;br&gt;3||67&lt;br&gt;4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;3&lt;br&gt;3&lt;br&gt;3||5&lt;sup&gt;95&lt;/sup&gt;+8&lt;br&gt;9391&lt;br&gt;3121&lt;br&gt;2341&lt;br&gt;2251&lt;br&gt;1951&lt;br&gt;1249&lt;br&gt;653&lt;br&gt;401&lt;br&gt;109||http://factordb.com/index.php?id=1100000000034686071&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=9391&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=3121&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=2341&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=2251&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1951&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1249&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=653&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=401&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=109&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000000034686071&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=9391&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=3121&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=2341&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=2251&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=1951&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=1249&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=653&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=401&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=109&amp;base=5&lt;nowiki/&gt;||0||–||
|-
||6||11||40041&lt;br&gt;4441&lt;br&gt;4401&lt;br&gt;51&lt;br&gt;45&lt;br&gt;35&lt;br&gt;31&lt;br&gt;25&lt;br&gt;21&lt;br&gt;15||5&lt;br&gt;4&lt;br&gt;4&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2||4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2||5209&lt;br&gt;1033&lt;br&gt;1009&lt;br&gt;31&lt;br&gt;29&lt;br&gt;23&lt;br&gt;19&lt;br&gt;17&lt;br&gt;13&lt;br&gt;11||http://factordb.com/index.php?id=5209&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1033&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1009&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=31&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=29&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=23&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=19&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=17&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=13&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=11&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=5209&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=1033&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=1009&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=31&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=29&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=23&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=19&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=17&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=13&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=11&amp;base=6&lt;nowiki/&gt;||0||–||
|-
||7||71||3&lt;sub&gt;16&lt;/sub&gt;1&lt;br&gt;510&lt;sub&gt;7&lt;/sub&gt;1&lt;br&gt;3&lt;sub&gt;6&lt;/sub&gt;01&lt;br&gt;1100021&lt;br&gt;531101&lt;br&gt;351101&lt;br&gt;300053&lt;br&gt;150001&lt;br&gt;100121&lt;br&gt;40054||17&lt;br&gt;10&lt;br&gt;8&lt;br&gt;7&lt;br&gt;6&lt;br&gt;6&lt;br&gt;6&lt;br&gt;6&lt;br&gt;6&lt;br&gt;5||15&lt;br&gt;9&lt;br&gt;7&lt;br&gt;6&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;4||(7&lt;sup&gt;17&lt;/sup&gt;−5)/2&lt;br&gt;36×7&lt;sup&gt;8&lt;/sup&gt;+1&lt;br&gt;(7&lt;sup&gt;8&lt;/sup&gt;−47)/2&lt;br&gt;134471&lt;br&gt;91631&lt;br&gt;62819&lt;br&gt;50459&lt;br&gt;28813&lt;br&gt;16871&lt;br&gt;9643||http://factordb.com/index.php?id=116315256993601&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=207532837&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=2882377&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=134471&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=91631&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=62819&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=50459&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=28813&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=16871&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=9643&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=116315256993601&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=207532837&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=2882377&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=134471&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=91631&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=62819&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=50459&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=28813&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=16871&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=9643&amp;base=7&lt;nowiki/&gt;||0||–||
|-
||8||75||4&lt;sub&gt;220&lt;/sub&gt;7&lt;br&gt;5&lt;sub&gt;13&lt;/sub&gt;25&lt;br&gt;7&lt;sub&gt;12&lt;/sub&gt;1&lt;br&gt;77774&lt;sub&gt;6&lt;/sub&gt;1&lt;br&gt;74&lt;sub&gt;7&lt;/sub&gt;1&lt;br&gt;4&lt;sub&gt;8&lt;/sub&gt;1&lt;br&gt;5&lt;sub&gt;5&lt;/sub&gt;025&lt;br&gt;5550525&lt;br&gt;5500525&lt;br&gt;4&lt;sub&gt;5&lt;/sub&gt;77||221&lt;br&gt;15&lt;br&gt;13&lt;br&gt;11&lt;br&gt;9&lt;br&gt;9&lt;br&gt;8&lt;br&gt;7&lt;br&gt;7&lt;br&gt;7||200&lt;br&gt;14&lt;br&gt;12&lt;br&gt;10&lt;br&gt;9&lt;br&gt;8&lt;br&gt;8&lt;br&gt;7&lt;br&gt;7&lt;br&gt;7||(4×8&lt;sup&gt;221&lt;/sup&gt;+17)/7&lt;br&gt;(5×8&lt;sup&gt;15&lt;/sup&gt;−173)/7&lt;br&gt;8&lt;sup&gt;13&lt;/sup&gt;−7&lt;br&gt;(28669×8&lt;sup&gt;7&lt;/sup&gt;−25)/7&lt;br&gt;(53×8&lt;sup&gt;8&lt;/sup&gt;−25)/7&lt;br&gt;(4×8&lt;sup&gt;9&lt;/sup&gt;−25)/7&lt;br&gt;(5×8&lt;sup&gt;8&lt;/sup&gt;−2413)/7&lt;br&gt;1495381&lt;br&gt;1474901&lt;br&gt;(4×8&lt;sup&gt;7&lt;/sup&gt;+185)/7||http://factordb.com/index.php?id=1100000000416605822&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=25131694349141&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=549755813881&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=8589035809&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=127027489&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=76695841&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=11983381&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1495381&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1474901&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1198399&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000000416605822&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=25131694349141&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=549755813881&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=8589035809&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=127027489&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=76695841&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=11983381&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=1495381&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=1474901&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=1198399&amp;base=8&lt;nowiki/&gt;||0||–||
|-
||9||151||30&lt;sub&gt;1158&lt;/sub&gt;11&lt;br&gt;27&lt;sub&gt;686&lt;/sub&gt;07&lt;br&gt;76&lt;sub&gt;329&lt;/sub&gt;2&lt;br&gt;561&lt;sub&gt;36&lt;/sub&gt;&lt;br&gt;10&lt;sub&gt;25&lt;/sub&gt;57&lt;br&gt;30&lt;sub&gt;20&lt;/sub&gt;51&lt;br&gt;8&lt;sub&gt;19&lt;/sub&gt;335&lt;br&gt;727&lt;sub&gt;15&lt;/sub&gt;07&lt;br&gt;51&lt;sub&gt;13&lt;/sub&gt;61&lt;br&gt;10&lt;sub&gt;11&lt;/sub&gt;507||1161&lt;br&gt;689&lt;br&gt;331&lt;br&gt;38&lt;br&gt;28&lt;br&gt;23&lt;br&gt;22&lt;br&gt;19&lt;br&gt;16&lt;br&gt;15||1108&lt;br&gt;657&lt;br&gt;316&lt;br&gt;37&lt;br&gt;26&lt;br&gt;22&lt;br&gt;21&lt;br&gt;19&lt;br&gt;16&lt;br&gt;14||3×9&lt;sup&gt;1160&lt;/sup&gt;+10&lt;br&gt;(23×9&lt;sup&gt;688&lt;/sup&gt;−511)/8&lt;br&gt;(31×9&lt;sup&gt;330&lt;/sup&gt;−19)/4&lt;br&gt;(409×9&lt;sup&gt;36&lt;/sup&gt;−1)/8&lt;br&gt;9&lt;sup&gt;27&lt;/sup&gt;+52&lt;br&gt;3×9&lt;sup&gt;22&lt;/sup&gt;+46&lt;br&gt;9&lt;sup&gt;22&lt;/sup&gt;−454&lt;br&gt;(527×9&lt;sup&gt;17&lt;/sup&gt;−511)/8&lt;br&gt;(41×9&lt;sup&gt;15&lt;/sup&gt;+359)/8&lt;br&gt;9&lt;sup&gt;14&lt;/sup&gt;+412||http://factordb.com/index.php?id=1100000002376318423&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000002495467486&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000002359003642&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000001554010824&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1100000002512830927&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1100000000032261811&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1100000002495736583&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1100000003446800389&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1055192051985121&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=22876792455373&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000002376318423&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=1100000002495467486&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=1100000002359003642&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=1100000001554010824&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=1100000002512830927&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=1100000000032261811&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=1100000002495736583&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=1100000003446800389&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=1055192051985121&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=22876792455373&amp;base=9&lt;nowiki/&gt;||0||–||
|-
||10||77||50&lt;sub&gt;28&lt;/sub&gt;27&lt;br&gt;5&lt;sub&gt;11&lt;/sub&gt;1&lt;br&gt;805&lt;sub&gt;5&lt;/sub&gt;1&lt;br&gt;66600049&lt;br&gt;66000049&lt;br&gt;60&lt;sub&gt;5&lt;/sub&gt;49&lt;br&gt;220&lt;sub&gt;5&lt;/sub&gt;1&lt;br&gt;5200007&lt;br&gt;946669&lt;br&gt;666649||31&lt;br&gt;12&lt;br&gt;8&lt;br&gt;8&lt;br&gt;8&lt;br&gt;8&lt;br&gt;8&lt;br&gt;7&lt;br&gt;6&lt;br&gt;6||31&lt;br&gt;12&lt;br&gt;8&lt;br&gt;8&lt;br&gt;8&lt;br&gt;8&lt;br&gt;8&lt;br&gt;7&lt;br&gt;6&lt;br&gt;6||5×10&lt;sup&gt;30&lt;/sup&gt;+27&lt;br&gt;(5×10&lt;sup&gt;12&lt;/sup&gt;−41)/9&lt;br&gt;(725×10&lt;sup&gt;6&lt;/sup&gt;−41)/9&lt;br&gt;66600049&lt;br&gt;66000049&lt;br&gt;6×10&lt;sup&gt;7&lt;/sup&gt;+49&lt;br&gt;22×10&lt;sup&gt;6&lt;/sup&gt;+1&lt;br&gt;5200007&lt;br&gt;946669&lt;br&gt;666649||http://factordb.com/index.php?id=1100000000204142046&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=555555555551&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=80555551&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=66600049&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=66000049&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=60000049&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=22000001&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=5200007&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=946669&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=666649&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000000204142046&amp;base=10&lt;br&gt;http://factordb.com/index.php?showid=555555555551&amp;base=10&lt;br&gt;http://factordb.com/index.php?showid=80555551&amp;base=10&lt;br&gt;http://factordb.com/index.php?showid=66600049&amp;base=10&lt;br&gt;http://factordb.com/index.php?showid=66000049&amp;base=10&lt;br&gt;http://factordb.com/index.php?showid=60000049&amp;base=10&lt;br&gt;http://factordb.com/index.php?showid=22000001&amp;base=10&lt;br&gt;http://factordb.com/index.php?showid=5200007&amp;base=10&lt;br&gt;http://factordb.com/index.php?showid=946669&amp;base=10&lt;br&gt;http://factordb.com/index.php?showid=666649&amp;base=10&lt;nowiki/&gt;||0||–||
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||24||3409||N00N&lt;sub&gt;8129&lt;/sub&gt;LN&lt;br&gt;88N&lt;sub&gt;5951&lt;/sub&gt;&lt;br&gt;A0&lt;sub&gt;2951&lt;/sub&gt;8ID&lt;br&gt;D&lt;sub&gt;2698&lt;/sub&gt;LD&lt;br&gt;N&lt;sub&gt;2644&lt;/sub&gt;LLN&lt;br&gt;BC0&lt;sub&gt;331&lt;/sub&gt;B&lt;br&gt;20&lt;sub&gt;313&lt;/sub&gt;7&lt;br&gt;C7&lt;sub&gt;298&lt;/sub&gt;&lt;br&gt;D0&lt;sub&gt;259&lt;/sub&gt;KKD&lt;br&gt;I0&lt;sub&gt;241&lt;/sub&gt;I5||8134&lt;br&gt;5953&lt;br&gt;2955&lt;br&gt;2700&lt;br&gt;2647&lt;br&gt;334&lt;br&gt;315&lt;br&gt;299&lt;br&gt;263&lt;br&gt;244||11227&lt;br&gt;8216&lt;br&gt;4079&lt;br&gt;3727&lt;br&gt;3654&lt;br&gt;461&lt;br&gt;434&lt;br&gt;413&lt;br&gt;363&lt;br&gt;337||13249×24&lt;sup&gt;8131&lt;/sup&gt;−49&lt;br&gt;201×24&lt;sup&gt;5951&lt;/sup&gt;−1&lt;br&gt;10×24&lt;sup&gt;2954&lt;/sup&gt;+5053&lt;br&gt;(13×24&lt;sup&gt;2700&lt;/sup&gt;+4403)/23&lt;br&gt;24&lt;sup&gt;2647&lt;/sup&gt;−1201&lt;br&gt;276×24&lt;sup&gt;332&lt;/sup&gt;+11&lt;br&gt;2×24&lt;sup&gt;314&lt;/sup&gt;+7&lt;br&gt;(283×24&lt;sup&gt;298&lt;/sup&gt;−7)/23&lt;br&gt;13×24&lt;sup&gt;262&lt;/sup&gt;+12013&lt;br&gt;18×24&lt;sup&gt;243&lt;/sup&gt;+437||http://factordb.com/index.php?id=1100000003593391606&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003593275880&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003593269654&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003593269876&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003593270089&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000002633359842&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000002355610241&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000002326181235&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003593270725&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000002633360037&amp;open=prime&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000003593391606&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000003593275880&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000003593269654&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000003593269876&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000003593270089&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000002633359842&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000002355610241&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000002326181235&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000003593270725&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000002633360037&amp;base=24&lt;nowiki/&gt;||0||–||
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|-
||36||35286~35290||P&lt;sub&gt;81993&lt;/sub&gt;SZ&lt;br&gt;S0&lt;sub&gt;75007&lt;/sub&gt;8H&lt;br&gt;7K&lt;sub&gt;26567&lt;/sub&gt;Z&lt;br&gt;J&lt;sub&gt;10117&lt;/sub&gt;LJ&lt;br&gt;VL0&lt;sub&gt;7258&lt;/sub&gt;J&lt;br&gt;EO0&lt;sub&gt;6177&lt;/sub&gt;V&lt;br&gt;FZ&lt;sub&gt;5777&lt;/sub&gt;3P&lt;br&gt;T09&lt;sub&gt;4618&lt;/sub&gt;1&lt;br&gt;RY&lt;sub&gt;4562&lt;/sub&gt;H&lt;br&gt;OZ&lt;sub&gt;3932&lt;/sub&gt;AZ||81995&lt;br&gt;75010&lt;br&gt;26569&lt;br&gt;10119&lt;br&gt;7261&lt;br&gt;6180&lt;br&gt;5780&lt;br&gt;4621&lt;br&gt;4564&lt;br&gt;3935||127609&lt;br&gt;116739&lt;br&gt;41349&lt;br&gt;15748&lt;br&gt;11301&lt;br&gt;9618&lt;br&gt;8996&lt;br&gt;7192&lt;br&gt;7103&lt;br&gt;6124||(5×36&lt;sup&gt;81995&lt;/sup&gt;+821)/7&lt;br&gt;28×36&lt;sup&gt;75009&lt;/sup&gt;+305&lt;br&gt;(53×36&lt;sup&gt;26568&lt;/sup&gt;+101)/7&lt;br&gt;(19×36&lt;sup&gt;10119&lt;/sup&gt;+2501)/35&lt;br&gt;1137×36&lt;sup&gt;7259&lt;/sup&gt;+19&lt;br&gt;528×36&lt;sup&gt;6178&lt;/sup&gt;+31&lt;br&gt;16×36&lt;sup&gt;5779&lt;/sup&gt;−1163&lt;br&gt;(36549×36&lt;sup&gt;4619&lt;/sup&gt;−289)/35&lt;br&gt;(979×36&lt;sup&gt;4563&lt;/sup&gt;−629)/35&lt;br&gt;25×36&lt;sup&gt;3934&lt;/sup&gt;−901||http://factordb.com/index.php?id=1100000002394962083&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004020085177&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003896952461&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362491&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362489&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362488&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362487&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362486&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362485&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000000840634476&amp;open=prime&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000002394962083&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000004020085177&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003896952461&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362491&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362489&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362488&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362487&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362486&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362485&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000000840634476&amp;base=36&lt;nowiki/&gt;||4||200000||
|}

== The fully proof of Athena problem in decimal (base ''b'' = 10) ==

'''Bold''' for the Athena primes, ''x'' ◁ ''y'' means ''x'' is a subsequence of ''y''.

Assume ''p'' is a prime &gt; 10, and the last digit of ''p'' must lie in {1,3,7,9}.

Case 1: ''p'' ends with 1.

In this case we can write ''p'' = ''x''1. If ''x'' contains 1, 3, 4, 6, or 7, then (respectively) '''11''' ◁ ''p'', '''31''' ◁ ''p'', '''41''' ◁ ''p'', '''61''' ◁ ''p'', or '''71''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 2, 5, 8, or 9.

Case 1.1: ''p'' begins with 2.

In this case we can write ''p'' = 2''y''1. If 5 ◁ ''y'', then '''251''' ◁ ''p''. If 8 ◁ ''y'', then '''281''' ◁ ''p''. If 9 ◁ ''y'', then 29 ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 2.

If 22 ◁ ''y'', then '''2221''' ◁ ''p''. Hence we may assume ''y'' contains zero or one 2's.

If ''y'' contains no 2's, then ''p'' ∈ 2{0}1. But then, since the sum of the digits of ''p'' is 3, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' contains exactly one 2, then we can write ''p'' = 2''z''2''w''1, where ''z'',''w'' ∈ {0}. If 0 ◁ ''z'' and 0 ◁ ''w'', then '''20201''' ◁ ''p''. Hence we may assume either ''z'' or ''w'' is empty.

If ''z'' is empty, then ''p'' ∈ 22{0}1, and the smallest prime ''p'' ∈ 22{0}1 is '''22000001'''.

If ''w'' is empty, then ''p'' ∈ 2{0}21, and the smallest prime ''p'' ∈ 2{0}21 is '''20021'''.

Case 1.2: ''p'' begins with 5.

In this case we can write ''p'' = 5''y''1. If 2 ◁ ''y'', then '''521''' ◁ ''p''. If 9 ◁ ''y'', then 59 ◁ ''p''. Hence we may assume all digits of ''y'' are 0, 5, or 8.

If 05 ◁ ''y'', then '''5051''' ◁ ''p''. If 08 ◁ ''y'', then '''5081''' ◁ ''p''. If 50 ◁ ''y'', then '''5501''' ◁ ''p''. If 58 ◁ ''y'', then '''5581''' ◁ ''p''. If 80 ◁ ''y'', then '''5801''' ◁ ''p''. If 85 ◁ ''y'', then '''5851''' ◁ ''p''. Hence we may assume ''y'' ∈ {0} ∪ {5} ∪ {8}.

If ''y'' ∈ {0}, then ''p'' ∈ 5{0}1. But then, since the sum of the digits of ''p'' is 6, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' ∈ {5}, then ''p'' ∈ 5{5}1, and the smallest prime ''p'' ∈ 5{5}1 is '''555555555551'''.

If ''y'' ∈ {8}, since if 88 ◁ ''y'', then 881 ◁ ''p'', hence we may assume ''y'' ∈ {''𝜆'',8}, and thus ''p'' ∈ {51,581}, but 51 and 581 are both composite.

Case 1.3: ''p'' begins with 8.

In this case we can write p = 8''y''1. If 2 ◁ ''y'', then '''821''' ◁ ''p''. If 8 ◁ ''y'', then '''881''' ◁ ''p''. If 9 ◁ ''y'', then 89 ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 5.

If 50 ◁ ''y'', then '''8501''' ◁ ''p''. Hence we may assume y ∈ {0}{5}.

If 005 ◁ ''y'', then '''80051''' ◁ p. Hence we may assume y ∈ {0} ∪ {5} ∪ 0{5}.

If y ∈ {0}, then ''p'' ∈ 8{0}1. But then, since the sum of the digits of ''p'' is 9, ''p'' is divisible by 3, so ''p'' cannot be prime.

If y ∈ {5}, since if 55555555555 ◁ ''y'', then 555555555551 ◁ ''p'', hence we may assume ''y'' ∈ {''𝜆'', 5, 55, 555, 5555, 55555, 555555, 5555555, 55555555, 555555555, 5555555555}, and thus ''p'' ∈ {81, 851, 8551, 85551, 855551, 8555551, 85555551, 855555551, 8555555551, 85555555551, 855555555551}, but all of these numbers are composite.

If y ∈ 0{5}, since if 55555555555 ◁ ''y'', then 555555555551 ◁ ''p'', hence we may assume ''y'' ∈ {0, 05, 055, 0555, 05555, 055555, 0555555, 05555555, 055555555, 0555555555, 05555555555}, and thus ''p'' ∈ {801, 8051, 80551, 805551, 8055551, 80555551, 805555551, 8055555551, 80555555551, 805555555551, 8055555555551}, and of these numbers only 80555551 and 8055555551 are primes, but 80555551 ◁ 8055555551, thus only '''80555551''' is a minimal element.

Case 1.4: ''p'' begins with 9.

In this case we can write p = 9''y''1. If 9 ◁ ''y'', then '''991''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0, 2, 5, or 8.

If 00 ◁ ''y'', then '''9001''' ◁ ''p''. If 22 ◁ ''y'', then '''9221''' ◁ ''p''. If 55 ◁ ''y'', then '''9551''' ◁ ''p''. If 88 ◁ ''y'', then 881 ◁ ''p''. Hence we may assume ''y'' contains at most one 0, at most one 2, at most one 5, and at most one 8.

If ''y'' only contains at most one 0 and does not contain any of {2,5,8}, then ''y'' ∈ {''𝜆'',0}, and thus ''p'' ∈ {91,901}, but 91 and 901 are both composite. If ''y'' only contains at most one 0 and only one of {2,5,8}, then the sum of the digits of ''p'' is divisible by 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Hence we may assume ''y'' contains at least two of {2,5,8}.

If 25 ◁ ''y'', then 251 ◁ ''p''. If 28 ◁ ''y'', then 281 ◁ ''p''. If 52 ◁ ''y'', then 521 ◁ ''p''. If 82 ◁ ''y'', then 821 ◁ ''p''. Hence we may assume ''y'' contains no 2's (since if ''y'' contains 2, then ''y'' cannot contain either 5's or 8's, which is a contradiction).

If 85 ◁ ''y'', then '''9851''' ◁ ''p''. Hence we may assume ''y'' ∈ {58,580,508,058}, and thus ''p'' ∈ {9581,95801,95081,90581}, and of these numbers only 95801 is prime, but 95801 is not a minimal element since 5801 ◁ 95801.

Case 2: ''p'' ends with 3.

In this case we can write p = ''x''3. If ''x'' contains 1, 2, 4, 5, 7, or 8, then (respectively) '''13''' ◁ ''p'', '''23''' ◁ ''p'', '''43''' ◁ ''p'', '''53''' ◁ ''p'', '''73''' ◁ ''p'', or '''83''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 3, 6, or 9, and thus all digits of ''p'' are 0, 3, 6, or 9. But then, since the digits of ''p'' all have a common factor 3, ''p'' is divisible by 3, so ''p'' cannot be prime.

Case 3: ''p'' ends with 7.

In this case we can write ''p'' = ''x''7. If ''x'' contains 1, 3, 4, 6, or 9, then (respectively) '''17''' ◁ ''p'', '''37''' ◁ ''p'', '''47''' ◁ ''p'', '''67''' ◁ ''p'', or '''97''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 2, 5, 7, or 8.

Case 3.1: ''p'' begins with 2.

In this case we can write ''p'' = 2''y''7. If 2 ◁ ''y'', then '''227''' ◁ ''p''. If 5 ◁ ''y'', then '''257''' ◁ ''p''. If 7 ◁ ''y'', then '''277''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 8.

If 08 ◁ ''y'', then '''2087''' ◁ ''p''. If 88 ◁ ''y'', then 887 ◁ ''p''. Hence we may assume ''y'' ∈ {0} ∪ 8{0}.

If ''y'' ∈ {0}, then ''p'' ∈ 2{0}7. But then, since the sum of the digits of ''p'' is 9, ''p'' is divisible by 3, so ''p'' cannot be prime.

If y ∈ 8{0}, then ''p'' ∈ 28{0}7. But then ''p'' is divisible by 7, since for ''n'' ≥ 0 we have 7 × 40&lt;sub&gt;''n''&lt;/sub&gt;1 = 280&lt;sub&gt;''n''&lt;/sub&gt;7.

Case 3.2: ''p'' begins with 5.

In this case we can write ''p'' = 5''y''7. If 5 ◁ ''y'', then '''557''' ◁ ''p''. If 7 ◁ ''y'', then '''577''' ◁ ''p''. If 8 ◁ ''y'', then '''587''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 2.

If 22 ◁ ''y'', then 227 ◁ ''p''. Hence we may assume ''y'' contains zero or one 2's.

If ''y'' contains no 2's, then ''p'' ∈ 5{0}7. But then, since the sum of the digits of ''p'' is 12, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' contains exactly one 2, then we can write ''p'' = 5''z''2''w''7, where ''z'',''w'' ∈ {0}. If 0 ◁ ''z'' and 0 ◁ ''w'', then '''50207''' ◁ ''p''. Hence we may assume either ''z'' or ''w'' is empty.

If ''z'' is empty, then ''p'' ∈ 52{0}7, and the smallest prime ''p'' ∈ 52{0}7 is '''5200007'''.

If ''w'' is empty, then ''p'' ∈ 5{0}27, and the smallest prime ''p'' ∈ 5{0}27 is '''5000000000000000000000000000027'''.

Case 3.3: ''p'' begins with 7.

In this case we can write ''p'' = 7''y''7. If 2 ◁ ''y'', then '''727''' ◁ ''p''. If 5 ◁ ''y'', then '''757''' ◁ ''p''. If 8 ◁ ''y'', then '''787''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 7, and thus all digits of ''p'' are 0 or 7. But then, since the digits of ''p'' all have a common factor 7, ''p'' is divisible by 7, so ''p'' cannot be prime.

Case 3.4: ''p'' begins with 8.

In this case we can write ''p'' = 8''y''7. If 2 ◁ ''y'', then '''827''' ◁ ''p''. If 5 ◁ ''y'', then '''857''' ◁ ''p''. If 7 ◁ ''y'', then '''877''' ◁ ''p''. If 8 ◁ ''y'', then '''887''' ◁ ''p''. Hence we may assume ''y'' ∈ {0}, and thus ''p'' ∈ 8{0}7. But then, since the sum of the digits of ''p'' is 15, ''p'' is divisible by 3, so ''p'' cannot be prime.

Case 4: ''p'' ends with 9.

In this case we can write ''p'' = ''x''9. If ''x'' contains 1, 2, 5, 7, or 8, then (respectively) '''19''' ◁ ''p'', '''29''' ◁ ''p'', '''59''' ◁ ''p'', '''79''' ◁ ''p'', or '''89''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 3, 4, 6, or 9.

If 44 ◁ ''x'', then '''449''' ◁ ''p''. Hence we may assume ''x'' contains zero or one 4's.

If x contains no 4's, then all digits of ''x'' are 0, 3, 6, or 9, and thus all digits of ''p'' are 0, 3, 6, or 9. But then, since the digits of ''p'' all have a common factor 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Hence we may assume that ''x'' contains exactly one 4.

Case 4.1: ''p'' begins with 3.

In this case we can write ''p'' = 3''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. We must have '''349''' ◁ ''p''.

Case 4.2: ''p'' begins with 4.

In this case we can write ''p'' = 4''y''9, where all digits of ''y'' are 0, 3, 6, or 9. If 0 ◁ ''y'', then '''409''' ◁ ''p''. If 3 ◁ ''y'', then 43 ◁ ''p''. If 9 ◁ ''y'', then '''499''' ◁ ''p''. Hence we may assume ''y'' ∈ {6}, and thus ''p'' ∈ 4{6}9. But then ''p'' is divisible by 7, since for ''n'' ≥ 0 we have 7 × 6&lt;sub&gt;''n''&lt;/sub&gt;7 = 46&lt;sub&gt;''n''&lt;/sub&gt;9.

Case 4.3: ''p'' begins with 6.

In this case we can write p = 6''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. If 0 ◁ ''z'', then 409 ◁ ''p''. If 3 ◁ ''z'', then 43 ◁ ''p''. If 6 ◁ ''z'', then '''6469''' ◁ ''p''. If 9 ◁ ''z'', then 499 ◁ ''p''. Hence we may assume ''z'' is empty.

If 3 ◁ ''y'', then 349 ◁ ''p''. If 9 ◁ ''y'', then '''6949''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 6.

If 06 ◁ ''y'', then '''60649''' ◁ ''p''. Hence we may assume ''y'' ∈ {6}{0}.

If 666 ◁ ''y'', then '''666649''' ◁ ''p''. If 00000 ◁ ''y'', then '''60000049''' ◁ ''p''. Hence we may assume ''y'' ∈ {''𝜆'', 0, 00, 000, 0000, 6, 60, 600, 6000, 60000, 66, 660, 6600, 66000, 660000}, and thus ''p'' ∈ {649, 6049, 60049, 600049, 6000049, 6649, 66049, 660049, 6600049, 66000049, 66649, 666049, 6660049, 66600049, 666000049}, and of these numbers only '''66000049''' and '''66600049''' are primes.

Case 4.4: ''p'' begins with 9.

In this case we can write p = 9''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. If 0 ◁ ''y'', then '''9049''' ◁ ''p''. If 3 ◁ ''y'', then 349 ◁ ''p''. If 6 ◁ ''y'', then '''9649''' ◁ ''p''. If 9 ◁ ''y'', then '''9949''' ◁ ''p''. Hence we may assume ''y'' is empty.

If 0 ◁ ''z'', then 409 ◁ ''p''. If 3 ◁ ''z'', then 43 ◁ ''p''. If 9 ◁ ''z'', then 499 ◁ ''p''. Hence we may assume ''z'' ∈ {6}, and thus ''p'' ∈ 94{6}9, and the smallest prime ''p'' ∈ 94{6}9 is 946669.

[[Category:Number theory]]</text>
      <sha1>74vk96ptgmjvni8u7c6aukl9mi2nmqj</sha1>
    </revision>
    <revision>
      <id>2819844</id>
      <parentid>2819843</parentid>
      <timestamp>2026-07-30T08:55:07Z</timestamp>
      <contributor>
        <username>Athene241</username>
        <id>3100061</id>
      </contributor>
      <comment>/* Solve the problem */</comment>
      <origin>2819844</origin>
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      <format>text/x-wiki</format>
      <text bytes="128819" sha1="lhk8bjrwop6ee31rsuoia9wex2yuh9x" xml:space="preserve">{{mathematics}}
'''Athena problem''' is an [[:w:List of unsolved problems in mathematics|unsolved problem]] in [[:w:Number theory|number theory]] and [[:w:Formal language theory|formal language theory]] and [[:w:Order theory|order theory]], this problem is named after the ancient Greek goddess [[:w:Athena|Athena]] (which is associated with [[:w:Wisdom|wisdom]]). Athena problem is: Give a [[:w:Natural number|natural number]] ''b'' &gt; 1, find the [[:w:Set (mathematics)|set]] of the [[:w:Minimal element|minimal element]]s of the set of the &quot;[[:w:Prime number|prime number]] [[:w:Greater than|&gt;]] ''b''&quot; [[:w:Numerical digit|digit]] [[:w:String (computer science)|string]]s in the [[:w:Positional numeral system|positional numeral system]] with [[:w:Radix|base]] ''b'' for the [[:w:Subsequence|subsequence]] [[:w:Partially ordered set|ordering]]. (A string ''x'' is a subsequence of another string ''y'', if ''x'' can be obtained from ''y'' by deleting zero or more of the [[:w:Character (computing)|character]]s in ''y''. For example, 514 is a subsequence of 352148, &quot;string&quot; is a subsequence of &quot;meistersinger&quot;. In contrast, 758 is not a subsequence of 378259, &quot;abc&quot; is not a subsequence of &quot;cbacacba&quot;, since the characters must be in the same order) (Unlike [[:w:Substring|substring]], subsequence is not required to occupy consecutive positions within the original sequences, e.g. the [[:w:Longest common subsequence|longest common subsequence problem]] is different from the [[:w:Longest common substring|longest common substring problem]])

Using [[:w:Formal language theory|formal language theory]] terminology, Athena problem is finding the [[:w:Set (mathematics)|set]] of the [[:w:Minimal element|minimal element]]s of the [[:w:Formal language|language]] of base-''b'' [[:w:Representation (mathematics)|representation]]s of the [[:w:Prime number|prime number]]s [[:w:Greater than|&gt;]] ''b'' (which is a set of [[:w:String (computer science)|string]]s of [[:w:Symbol|symbol]]s over the [[:w:Alphabet (formal languages)|alphabet]] ''Σ''&lt;sub&gt;''b''&lt;/sub&gt; := {0, 1, ..., ''b''−1}), under the subsequence ordering (i.e. the [[:w:Binary relation|binary relation]] &quot;is a subsequence of&quot;, which is a [[:w:Partially ordered set|partial ordering]]), for a given natural number ''b'' &gt; 1 (You can draw this partial ordering as a [[:w:Hasse diagram|Hasse diagram]] to find all [[:w:Minimal element|minimal element]]s), this set is called '''Athena set''', and the prime numbers in this set are called '''Athena primes'''.

By [[:w:Higman's lemma|Higman's lemma]], there are no [[:w:Infinite set|infinite]] [[:w:Antichain|antichain]]s for the subsequence ordering (i.e. the subsequence ordering is always a [[:w:Well-quasi-ordering|well quasi order]]) (i.e. under the subsequence ordering (i.e. the [[:w:Binary relation|binary relation]] &quot;is a subsequence of&quot;, which is a [[:w:Partially ordered set|partial ordering]]), every set of pairwise incomparable (i.e. not [[:w:Comparability|comparable]]) strings is finite), thus there must be only finitely many such minimal elements. In other words, the Athena set in every base ''b'' must be a [[:w:Finite set|finite set]], and every base ''b'' ≥ 2 has only finitely many Athena primes, e.g. in [[:w:Decimal|decimal]] (base ''b'' = 10), the Athena set has exactly 77 [[:w:Element of a set|element]]s (they are exactly the Athena primes in decimal (base ''b'' = 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}.

For bases 2 ≤ ''b'' ≤ 36, Athena problem is fully solved in bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 24, and also solved in bases ''b'' = 11, 13, 16, 22, 30 if [[:w:Probable prime|probable prime]]s are allowed. For the unsolved bases ''b'' = 17, 19, 21, 23, 25, 26, 27, 28, 29, 31, 32, 34, 35, 36, Athena problem is solved (if probable primes are allowed) except 771 [[:w:Indexed family|families]] of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be [[:w:Empty string|empty]]) of digits in base ''b'', ''y'' is a digit in base ''b'') = sequence {''xz'', ''xyz'', ''xyyz'', ''xyyyz'', ''xyyyyz'', ''xyyyyyz'', ...} (i.e. &quot;''xy''&lt;sup&gt;+&lt;/sup&gt;''z''&quot; in [[:w:Regular expression|regular expression]]), all of these 771 families contain no primes &gt; ''b'' or probable primes &gt; ''b'' with length ≤ 100000.

== Solve the problem ==

To solve the Athena problem for a given base ''b'', we must [[:w:Computing|compute]] the elements up to families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''), and find the smallest prime &gt; ''b'' in all such families.

We call families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') &quot;linear&quot; families, and we reduce these families by removing all trailing digits ''y'' from ''x'', and removing all leading digits ''y'' from ''z'', to make the families be easier, e.g. family 12333{3}33345 in base ''b'' is reduced to family 12{3}45 in base ''b'', since they are in fact the same family. Our [[:w:Algorithm|algorithm]] then proceeds as follows:

* 1. ''M'' := {minimal primes in base ''b'' of length 2 or 3}, ''L'' := union of all ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'') such that ''x'' ≠ 0 and ''gcd''(''z'', ''b'') = 1 and ''Y'' is the set of digits ''y'' in base ''b'' such that ''xyz'' has no subsequence in ''M''.
* 2. While ''L'' contains nonlinear families (families which are not linear families): Explore each family of ''L'', and update ''L''. Examine each family of ''L'' by:
* 2.1. Let ''w'' be the shortest string in the family. If ''w'' has a subsequence in ''M'', then remove the family from ''L''. If ''w'' represents a prime, then add ''w'' to ''M'' and remove the family from ''L''.
* 2.2. If possible, simplify the family.
* 2.3. Using the techniques below (covering congruence, algebraic factorization, or combine of them), check if the family can be proven to only contain composites (only count the numbers &gt; ''b''), and if so then remove the family from ''L''.
* 3. Update ''L'', after each split examine the new families as in step 2.

e.g. in decimal (base ''b'' = 10):

''M'' := {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}

''L'' := {2{0,2}1, 2{0,8}7, 3{0,3,6,9}3, 3{0,3,6,9}9, 4{6}9, 5{0,5,8}1, 5{0,2}7, 6{0,3,6,9}3, 6{0,3,4,6,9}9, 7{0,7}7, 8{0,5}1, 8{0}7, 9{0,2,5,8}1, 9{0,3,6,9}3, 9{0,3,4,6,9}9}

and since 2221 is prime, it follows that the family 2{0,2}1 splits into the families 2{0}1 and 2{0}2{0}1

and since the family 2{0}1 can be proven to contain no primes &gt; base (since all numbers in this family are divisible by 3), it can be removed

and since 20201 is prime, it follows that the family 2{0}2{0}1 splits into the families 2{0}21 and 22{0}1

221 and 2021 are composites, but 20021 is prime, thus add 20021 to ''L''

none of 221, 2201, 22001, 220001, 2200001 are primes, but 22000001 is prime, thus add 22000001 to ''L''

and since the family 3{0,3,6,9}3 can be proven to contain no primes &gt; base (since all numbers in this family are divisible by 3), it can be removed

etc.

Since the number of possible (first digit,last digit) (also called (initial digit,final digit)) combos ([[:w:Ordered pair|ordered pair]]s) of a prime &gt; ''b'' in base ''b'' is (''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(''b'') (all digits except 0 can be the first digit of a prime &gt; ''b'' in base ''b'' (thus ''b''−1 possible digits), but only the digits coprime to ''b'' can be the last digit of a prime &gt; ''b'' in base ''b'' (thus ''eulerphi''(''b'') possible digits), and by the [[:w:Rule of product|rule of product]], there are (''b''−1)×''eulerphi''(''b'') choices of the (first digit,last digit) combo, also, both &quot;numbers of Athena primes in base ''b''&quot; and &quot;length of the largest Athena prime in base ''b''&quot; are [[:w:Asymptotic analysis|roughly]] ''[[:w:E (mathematical_constant)|e]]''&lt;sup&gt;''[[:w:Euler's constant|γ]]''×(''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(*b*)&lt;/sup&gt;.

Shrinking the family ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''Y'' is a set of digits in base ''b'')

* If ''y'' ∈ ''Y'' and the string ''xyyz'' represents a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or has a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''}''z'' ∪ ''x''{''Y'' \ ''y''}''y''{''Y'' \ ''y''}''z''.
* If ''y''&lt;sub&gt;1&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;2&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;1&lt;/sub&gt; ≠ ''y''&lt;sub&gt;2&lt;/sub&gt; and the string ''xy''&lt;sub&gt;1&lt;/sub&gt;''y''&lt;sub&gt;2&lt;/sub&gt;''z'' represents a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or has a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''&lt;sub&gt;1&lt;/sub&gt;}{''Y'' \ ''y''&lt;sub&gt;2&lt;/sub&gt;}''z''.
* If ''y''&lt;sub&gt;1&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;2&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;1&lt;/sub&gt; ≠ ''y''&lt;sub&gt;2&lt;/sub&gt; and both the strings ''xy''&lt;sub&gt;1&lt;/sub&gt;''y''&lt;sub&gt;2&lt;/sub&gt;''z'' and ''xy''&lt;sub&gt;2&lt;/sub&gt;''y''&lt;sub&gt;1&lt;/sub&gt;''z'' represent a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or have a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''&lt;sub&gt;1&lt;/sub&gt;}''z'' ∪ ''x''{''Y'' \ ''y''&lt;sub&gt;2&lt;/sub&gt;}''z''.

e.g. in decimal (base ''b'' = 10):

* 2221 is a prime &gt; 10, thus the family 2{0,2}1 splits into the two families 2{0}1 and 2{0}2{0}1.
* 227 is a prime &gt; 10, and it is a subsequence of 5227, thus the family 5{0,2}7 splits into the two families 5{0}7 and 5{0}2{0}7.
* 449 is a prime &gt; 10, and it is a subsequence of 6449, thus the family 6{0,3,4,6,9}9 splits into the two families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9.
* Both 5051 and 5501 are primes &gt; 10, thus the family 5{0,5}1 splits into the two families 5{0}1 and 5{5}1 = {5}1.
* 8501 is a prime &gt; 10, thus the family 8{0,5}1 splits into the family 8{0}{5}1.
* 887 is a prime &gt; 10, and it is a subsequence of 2887, also 2087 is a prime &gt; 10, thus the family 2{0,8}7 splits into the two families 2{0}7 and 28{0}7.
* 349 and 449 are primes &gt; 10, and they are subsequences of 9349 and 9449, respectively, also 9049, 9649, 9949 are primes &gt; 10, thus the family 9{0,3,4,6,9}9 splits into the two families 9{0,3,6,9}9 and 94{0,3,6,9}9.
* 251, 281, 521, 821, 881 are primes &gt; 10, and they are subsequences of 9251, 9281, 9521, 9821, 9881, respectively, also 9001, 9221, 9551, 9851 are primes &gt; 10, thus the family 9{0,2,5,8}1 splits into the numbers {91, 901, 921, 951, 981, 9021, 9051, 9081, 9201, 9501, 9581, 9801, 90581, 95081, 95801}.

If the methods we have discussed cannot be used to rule out or shrink ''x''{''Y''}''z'' where ''Y'' = {''y''&lt;sub&gt;1&lt;/sub&gt;, ''y''&lt;sub&gt;2&lt;/sub&gt;, ..., ''y''&lt;sub&gt;''n''&lt;/sub&gt;}, then we can replace ''x''{''Y''}''z'' by ''xy''&lt;sub&gt;1&lt;/sub&gt;{''Y''}''z'' ∪ ''xy''&lt;sub&gt;2&lt;/sub&gt;{''Y''}''z'' ∪ ... ∪ ''xy''&lt;sub&gt;''n''&lt;/sub&gt;{''Y''}''z'' and re-run the methods on this new [[:w:Formal language|language]].

If all remain families are linear families (i.e. of the form ''x''{''y''}''z'', where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''), then we search the smallest (probable) primes in these families and add these primes to the list.

e.g. in decimal (base ''b'' = 10):

* The smallest prime in the family 5{0}27 is 5000000000000000000000000000027.
* The smallest prime in the family {5}1 is 555555555551.
* The smallest prime in the family 8{5}1 is 8555555555555555555551, but 8555555555555555555551 is not a minimal element since 555555555551 is a subsequence of 8555555555555555555551.

There is no guarantee that the techniques discussed will ever terminate, but in practice they often do. They are able to determine the Athena set in base ''b'' for 2 ≤ ''b'' ≤ 16 and ''b'' = 18, 20, 22, 24, 30. The bases ''b'' = 17, 19, 21, 23, 25 ≤ ''b'' ≤ 29, 31 ≤ ''b'' ≤ 36 are solved with the exception of 771 families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'').

The following is a &quot;[[:w:Semi-algorithm|semi-algorithm]]&quot; that is guaranteed to solve the Athena problem for a given base ''b'', but it is not so easy to implement:

# ''M'' = ''[[:w:Empty string|∅]]''
# while (''L'' ≠ ''∅'') do
# choose ''x'', a shortest string in ''L''
# ''M'' := ''M'' ∪ {''x''}
# ''L'' := ''L'' − ''sup''({''x''})

In practice, for arbitrary ''L'', we cannot feasibly carry out step 5. Instead, we work with ''L''&amp;#39;, some regular overapproximation to ''L'', until we can show ''L''&amp;#39; = ''∅'' (which implies ''L'' = ''∅''). In practice, ''L''&amp;#39; is usually chosen to be a finite [[:w:Union (set theory)|union]] of sets of the form ''L''&lt;sub&gt;1&lt;/sub&gt;{''L''&lt;sub&gt;2&lt;/sub&gt;}''L''&lt;sub&gt;3&lt;/sub&gt;, where each of ''L''&lt;sub&gt;1&lt;/sub&gt;, ''L''&lt;sub&gt;2&lt;/sub&gt;, ''L''&lt;sub&gt;3&lt;/sub&gt; is finite. In the case we consider in this project, we then have to determine whether such a family contains a prime or not.

Thus, the [[:w:Time complexity|time complexity]] of the Athena problem in base ''b'' may be ''[[:w:Big O notation|O]]''(''[[:w:E (mathematical_constant)|e]]''&lt;sup&gt;''[[:w:Euler's constant|γ]]''×(''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(*b*)&lt;/sup&gt;), and the [[:w:CPU time|CPU time]] of the Athena problem in base ''b'' may be longer than [[:w:Age of the universe|the age of the universe]] for bases ''b'' = 19, 23, 25, 27, 29, 31, 32, 33, 34, 35, also, Athena problem in bases ''b'' around 500 may be [[:w:NP-complete|NP-complete]] or [[:w:NP-hard|NP-hard]], or an [[:w:Undecidable problem|undecidable problem]], or an example of [[:w:Gödel's incompleteness theorems|Gödel's incompleteness theorems]] (like the [[:w:Continuum hypothesis|continuum hypothesis]] and the [[:w:Halting problem|halting problem]]).

To solve the Athena problem (i.e. to compute the Athena set), we need to determine whether a given family contains a prime. In practice, if family ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''Y'' is a set of digits in base ''b'') could not be ruled out as only containing composites and ''Y'' contains two or more digits, then a relatively small prime &gt; ''b'' could always be found in this family. Intuitively, this is because there are a large number of small strings in such a family, and at least one is likely to be prime (e.g. there are 2&lt;sup&gt;''n''−2&lt;/sup&gt; strings of length ''n'' in the family 1{3,7}9, and there are over a thousand strings of length 12 in the family 1{3,7}9, thus it is very impossible that these numbers are all composite). In the case ''Y'' contains only one digit, this family is of the form ''x''{''y''}''z'', and there is only a single string of each length &gt; (the length of ''x'' + the length of ''z''), and it is not known if the following [[:w:Decision problem|decision problem]] is recursively solvable (just like [[:w:Sierpiński number|Sierpiński problem]] and [[:w:Riesel number|Riesel problem]], Sierpiński problem and Riesel problem can be generalized to other bases ''b'' (references: http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm), in fact, Athena problem base ''b'' covers the Sierpiński problem base ''b'' and the Riesel problem base ''b'' with ''k'' &lt; ''b'', i.e. finding the smallest prime of the form ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 and ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (or prove such prime does not exist) with ''k'' &lt; ''b'' (specially, for bases ''b'' such that the conjectured smallest Sierpiński number or the conjectured smallest Riesel number is &lt; ''b'', Athena problem base ''b'' covers the Sierpiński problem base ''b'' or the Riesel problem base ''b'', respectively), since the smallest prime of the form ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 and ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (if exists) must be a minimal element in base ''b'', also, Athena problem base ''b'' covers finding the smallest prime of these forms in base ''b'' (or proving that such prime does not exist): (''b''&lt;sup&gt;''n''&lt;/sup&gt;−1)/(''b''−1) (for this form, ''n'' must be prime, and we want ''n'' ≥ 2) (references of this form: http://www.fermatquotient.com/PrimSerien/GenRepu.txt, https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html, http://www.primenumbers.net/Henri/us/MersFermus.htm, http://www.bitman.name/math/table/379, https://pzktupel.de/Primetables/TableRepunitGen.php, https://oeis.org/A084740, https://oeis.org/A084738, https://oeis.org/A128164, https://oeis.org/A285642; or for prime bases ''b'': https://oeis.org/A065854, https://oeis.org/A279068), ''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 (for this form, ''n'' must be power of 2, and we want ''n'' ≥ 1) (references of this form: http://jeppesn.dk/generalized-fermat.html, http://www.noprimeleftbehind.net/crus/GFN-primes.htm, https://web.archive.org/web/20231002190634/http://yves.gallot.pagesperso-orange.fr/primes/index.html, https://pzktupel.de/Primetables/TableFermatGFBB.php, https://oeis.org/A079706, https://oeis.org/A084712, https://oeis.org/A228101), (''b''&lt;sup&gt;''n''&lt;/sup&gt;+1)/2 (for odd ''b'') (for this form, ''n'' must be power of 2, and we want ''n'' ≥ 2) (reference of this form: http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt), (''sqrt''(''b'')×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1)/(''sqrt''(''b'')+1) (for square ''b'') (for this form, 2×''n''+1 must be prime, and we want ''n'' ≥ 2) (references of this form: http://www.fermatquotient.com/PrimSerien/GenRepuP.txt, http://www.primenumbers.net/Henri/us/MersFermus.htm, http://www.bitman.name/math/table/488, https://pzktupel.de/Primetables/TableWagstaffGen.php, https://oeis.org/A084742, https://oeis.org/A084741; or for bases ''b'' with ''sqrt''(''b'') prime: https://oeis.org/A065507), ((''b''−2)×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1)/(''b''−1) (''n'' ≥ 2) (reference of this form: https://oeis.org/A243404), 2×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 (''n'' ≥ 1) (references of this form: https://www.mersenneforum.org/showthread.php?t=6918, https://www.mersenneforum.org/showthread.php?t=19725, https://oeis.org/A119624), 2×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (''n'' ≥ 1) (references of this form: https://www.mersenneforum.org/showthread.php?t=24576, https://www.mersenneforum.org/attachment.php?attachmentid=20976&amp;d=1567314217, https://oeis.org/A119591), ''b''&lt;sup&gt;''n''&lt;/sup&gt;+2 (''n'' ≥ 1) (references of this form: https://oeis.org/A138066, https://oeis.org/A084713, https://oeis.org/A138067), ''b''&lt;sup&gt;''n''&lt;/sup&gt;−2 (''n'' ≥ 2) (references of this form: https://www.primepuzzles.net/puzzles/puzz_887.htm, https://oeis.org/A250200, https://oeis.org/A255707, https://oeis.org/A084714; or for prime bases ''b'': https://oeis.org/A292201), (''b''−1)×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 (''n'' ≥ 1) (references of this form: http://www.noprimeleftbehind.net/Williams-primes-MP.htm, http://www.bitman.name/math/table/477, https://pzktupel.de/Primetables/TableWilliams2.php, https://oeis.org/A305531; or for prime bases ''b'': https://oeis.org/A087139), (''b''−1)×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (''n'' ≥ 1) (references of this form: https://harvey563.tripod.com/wills.txt, http://www.noprimeleftbehind.net/Williams-primes-MM.htm, http://www.bitman.name/math/table/484, https://pzktupel.de/Primetables/TableWilliams1.php; or for prime bases ''b'': https://oeis.org/A122396), ''b''&lt;sup&gt;''n''&lt;/sup&gt;+(''b''−1) (''n'' ≥ 1) (references of this form: http://www.bitman.name/math/table/795, https://pzktupel.de/Primetables/TableWilliams6.php, https://oeis.org/A076845, https://oeis.org/A076846, https://oeis.org/A078178, https://oeis.org/A078179), ''b''&lt;sup&gt;''n''&lt;/sup&gt;−(''b''−1) (''n'' ≥ 2) (references of this form: http://www.bitman.name/math/table/792, https://pzktupel.de/Primetables/TableWilliams5.php, https://oeis.org/A113516, https://oeis.org/A343589; or for prime bases ''b'': https://cs.uwaterloo.ca/journals/JIS/VOL3/mccranie.html, http://www.bitman.name/math/table/435)):

Problem: Given strings ''x'', ''z'' (may be empty), a digit ''y'', and a base ''b'' (''x'' does not [[:w:Leading zero|start with the digit 0]], ''z'' ends with a digit which [[:w:Coprime integers|coprime]] to ''b'', ''y'' is not 0 if ''x'' is empty, ''y'' is coprime to ''b'' if ''z'' is empty), does there exist a prime number whose base-''b'' expansion is of the form ''xy''&lt;sub&gt;''n''&lt;/sub&gt;''z'' for some ''n'' ≥ 0?

Some families can be ruled out to contain no prime &gt; ''b'' by [[:w:Covering set|covering congruence]], [[:w:Factorization of polynomials|algebraic factorization]] (e.g. [[:w:Difference of two squares|difference of two squares]], [[:w:Sum of two cubes|sum of two cubes]], [[:w:Sophie Germain's identity|Sophie Germain's identity of ''x''&lt;sup&gt;4&lt;/sup&gt;+4×''y''&lt;sup&gt;4&lt;/sup&gt;]]), or combine of them, e.g.

* The base 9 family 2{7}: Always divisible by 2 or 5
* The base 11 family 2{5}: Always divisible by 2 or 3
* The base 14 family B{0}1: Always divisible by 3 or 5
* The base 13 family 95{0}3: Always divisible by 5, 7, or 17
* The base 16 family {4}D: Always divisible by 3, 7, or 13
* The base 16 family {8}F: Always divisible by 3, 7, or 13
* The base 21 family {7}D: Always divisible by 2, 13, or 17
* The base 23 family {D}GA: Always divisible by 2, 5, 7, 37, or 79
* The base 9 family {1}: Can be written as (9&lt;sup&gt;''n''&lt;/sup&gt;−1)/8 and can be factored as (3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (3&lt;sup&gt;''n''&lt;/sup&gt;+1) / 8
* The base 8 family 1{0}1: Can be written as 8&lt;sup&gt;''n''&lt;/sup&gt;+1 and can be factored as (2&lt;sup&gt;''n''&lt;/sup&gt;+1) × (4&lt;sup&gt;''n''&lt;/sup&gt;−2&lt;sup&gt;''n''&lt;/sup&gt;+1)
* The base 9 family 3{8}: Can be written as 4×9&lt;sup&gt;''n''&lt;/sup&gt;−1 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (2×3&lt;sup&gt;''n''&lt;/sup&gt;+1)
* The base 16 family 1{5}: Can be written as (4×16&lt;sup&gt;''n''&lt;/sup&gt;−1)/3 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (2×3&lt;sup&gt;''n''&lt;/sup&gt;+1) / 3
* The base 16 family {4}1: Can be written as (4×16&lt;sup&gt;''n''&lt;/sup&gt;−49)/15 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−7) × (2×3&lt;sup&gt;''n''&lt;/sup&gt;+7) / 15
* The base 27 family 7{Q}: Can be written as 8×27&lt;sup&gt;''n''&lt;/sup&gt;−1 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (4×9&lt;sup&gt;''n''&lt;/sup&gt;+2×3&lt;sup&gt;''n''&lt;/sup&gt;+1)
* The base 27 family 9{G}: Can be written as (125×27&lt;sup&gt;''n''&lt;/sup&gt;−8)/13 and can be factored as (5×3&lt;sup&gt;''n''&lt;/sup&gt;−2) × (25×9&lt;sup&gt;''n''&lt;/sup&gt;+10×3&lt;sup&gt;''n''&lt;/sup&gt;+4)
* The base 16 family {C}D: Can be written as (4×16&lt;sup&gt;''n''&lt;/sup&gt;+1)/5 and can be factored as (2×4&lt;sup&gt;''n''&lt;/sup&gt;−2×2&lt;sup&gt;''n''&lt;/sup&gt;+1) × (2×4&lt;sup&gt;''n''&lt;/sup&gt;+2×2&lt;sup&gt;''n''&lt;/sup&gt;+1) / 5
* The base 14 family 8{D}: Can be written as 9×14&lt;sup&gt;''n''&lt;/sup&gt;−1, it is divisible by 5 if ''n'' is odd and can be factored as (3×14&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (3×14&lt;sup&gt;''n''/2&lt;/sup&gt;+1) if ''n'' is even
* The base 12 family {B}9B: Can be written as 12&lt;sup&gt;''n''&lt;/sup&gt;−25, it is divisible by 13 if ''n'' is odd and can be factored as (12&lt;sup&gt;''n''/2&lt;/sup&gt;−5) × (12&lt;sup&gt;''n''/2&lt;/sup&gt;+5) if ''n'' is even
* The base 14 family {D}5: Can be written as 14&lt;sup&gt;''n''&lt;/sup&gt;−9, it is divisible by 5 if ''n'' is odd and can be factored as (14&lt;sup&gt;''n''/2&lt;/sup&gt;−3) × (14&lt;sup&gt;''n''/2&lt;/sup&gt;+3) if ''n'' is even
* The base 17 family 1{9}: Can be written as (25×17&lt;sup&gt;''n''&lt;/sup&gt;−9)/16, it is divisible by 2 if ''n'' is odd and can be factored as (5×17&lt;sup&gt;''n''/2&lt;/sup&gt;−3) × (5×17&lt;sup&gt;''n''/2&lt;/sup&gt;+3) / 16 if ''n'' is even
* The base 17 family 7{9}: Can be written as (121×17&lt;sup&gt;''n''&lt;/sup&gt;−9)/16, it is divisible by 2 if ''n'' is odd and can be factored as (11×17&lt;sup&gt;''n''/2&lt;/sup&gt;−3) × (11×17&lt;sup&gt;''n''/2&lt;/sup&gt;+3) / 16 if ''n'' is even
* The base 19 family 1{6}: Can be written as (4×19&lt;sup&gt;''n''&lt;/sup&gt;−1)/3, it is divisible by 5 if ''n'' is odd and can be factored as (2×19&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (2×19&lt;sup&gt;''n''/2&lt;/sup&gt;+1) / 3 if ''n'' is even
* The base 19 family 7{2}: Can be written as (64×19&lt;sup&gt;''n''&lt;/sup&gt;−1)/9, it is divisible by 5 if ''n'' is odd and can be factored as (8×19&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (8×19&lt;sup&gt;''n''/2&lt;/sup&gt;+1) / 9 if ''n'' is even
* The base 24 family 3{N}: Can be written as 4×24&lt;sup&gt;''n''&lt;/sup&gt;−1, it is divisible by 5 if ''n'' is odd and can be factored as (2×24&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (2×24&lt;sup&gt;''n''/2&lt;/sup&gt;+1) if ''n'' is even

By the [[:w:Prime number theorem|prime number theorem]], the [[:w:Probability|chance]] that a [[:w:Random number|random]] ''n''-digit base ''b'' number is prime is [[:w:Asymptotic analysis|approximately]] 1/''n'' (more accurately, the chance is approximately 1/(''n''×''ln''(''b'')), where ''ln'' is the [[:w:Natural logarithm|natural logarithm]]). If one conjectures the numbers ''x''{''y''}''z'' behave similarly (i.e. the numbers ''x''{''y''}''z'' is a [[:w:Pseudorandomness|pseudorandom sequence]]) you would expect [[:w:Harmonic_series (mathematics)|1/1 + 1/2 + 1/3 + 1/4 + ... = ∞]] primes of the form ''x''{''y''}''z'' (of course, this does not always happen, since some ''x''{''y''}''z'' families can be ruled out to contain no prime &gt; ''b'' (by covering congruence, algebraic factorization, or combine of them), but it is at least a reasonable conjecture in the absence of evidence to the contrary. Hence, the [[:w:Heuristic argument|heuristic argument]] suggests there are always infinitely many primes in family ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') if it cannot be ruled out to contain no prime or only contain finitely many primes, by covering congruence, algebraic factorization, or combine of them. However, some families ''x''{''y''}''z'' could not be proven to contain no primes &gt; ''b'' (by covering congruence, algebraic factorization, or combine of them) but no primes &gt; ''b'' could be found in the family, even after searching through numbers with over 100000 digits. In such a case, the only way to proceed is to [[:w:Primality test|test the primality]] of larger and larger numbers of such form and hope a prime is eventually discovered. e.g. the smallest (probable) prime in the family A{3}A in base ''b'' = 13 is A3&lt;sub&gt;592197&lt;/sub&gt;A, its algebraic form is (41×13&lt;sup&gt;592198&lt;/sup&gt;+27)/4, when written in decimal contains 659677 digits (it is only probable prime, i.e. not definitely prime, since technically, probable primality tests were used to show this (which have a ''very'' small chance of making an error, see https://t5k.org/notes/prp_prob.html) because all known primality tests run far too slowly to run on numbers of this size unless either [https://t5k.org/prove/prove3_1.html ''N''−1] or [https://t5k.org/prove/prove3_2.html ''N''+1] (or both) can be ≥ 1/3 factored).

The numbers in family ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') are of the form (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1) for some fixed ''a'', ''b'', ''c'' such that ''a'' ≥ 1, ''b'' ≥ 2 (''b'' is the base), ''c'' ≠ 0, ''gcd''(''a'',''c'') = 1, ''gcd''(''b'',''c'') = 1. Except in the [[:w:Special case|special case]] ''c'' = ±1 and ''gcd''(''a''+''c'',''b''−1) = 1, when ''n'' is large the known [[:w:Primality test|primality test]]s for such a number are too inefficient to run. In this case one must resort to a [[:w:Probabilistic algorithm|probable]] primality test such as a [[:w:Miller–Rabin primality test|Miller–Rabin primality test]] or a [[:w:Baillie–PSW primality test|Baillie–PSW primality test]], unless a divisor of the number can be found. Since we are testing many numbers in an [[:w:Exponential growth|exponential sequence]], it is possible to use a sieving process to find divisors rather than using [[:w:Trial division|trial division]].

To do this, we made use of Geoffrey Reynolds' ''srsieve'' software (download: https://pzktupel.de/Software/srsieve_1.1.4.7z). This program uses the [[:w:Baby-step giant-step|baby-step giant-step]] [[:w:Algorithm|algorithm]] to find all primes ''p'' which divide ''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'' where ''p'' and ''n'' lie in a [[:w:Interval_(mathematics)|specified range]]. Since this program cannot handle the general case (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1) when ''gcd''(''a''+''c'',''b''−1) &gt; 1 we only used it to sieve the sequence ''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'' for primes ''p'' not dividing ''gcd''(''a''+''c'',''b''−1), and initialized the list of candidates to not include ''n'' for which there is some prime ''p'' dividing ''gcd''(''a''+''c'',''b''−1) for which ''p'' dividing (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1). The program had to be modified slightly to remove a check which would prevent it from running in the case when ''a'', ''b'', and ''c'' were all odd (since then 2 divides ''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'', but 2 may not divide (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1)).

Once the numbers with small divisors had been removed, it remained to test the remaining numbers using a probable primality test. For this we used the software ''LLR'' by Jean Penné. (download: http://jpenne.free.fr/index2.html). Although undocumented, it is possible to run this program on numbers of the form (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1) when ''gcd''(''a''+''c'',''b''−1) &gt; 1, so this program required no modifications. A script was also written which allowed one to run ''srsieve'' while ''LLR'' was testing the remaining candidates, so that when a divisor was found by srsieve on a number which had not yet been tested by ''LLR'' it would be removed from the list of candidates.

For the primes &lt; 10&lt;sup&gt;25000&lt;/sup&gt; for the &quot;easy&quot; bases (bases ''b'' with ≤ 150 primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; (base ''b'' = 26 has 83 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 3 unsolved families, base ''b'' = 36 has 75 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 4 unsolved families, base ''b'' = 17 has 99 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 18 unsolved families, base ''b'' = 21 has 80 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 12 unsolved families, base ''b'' = 19 has 201 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 23 unsolved families), i.e. bases *b* = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 26, 28, 30, 36), we employed ''CM'' by Andreas Enge (download: https://www.multiprecision.org/cm/download.html), an elliptic curve primality proving implementation.

== Data ==

These are the results of the Athena problem in bases 2 ≤ ''b'' ≤ 36 (we stop at base 36 since this base is the maximum base for which it is possible to write the numbers with the [[:w:Symbol|symbol]]s 0, 1, 2, ..., 9 and A, B, C, ..., Z (i.e. the 10 [[:w:Arabic numerals|Arabic numerals]] and the 26 [[:w:Latin script|Latin letters]]): (some large Athena primes are only probable primes, i.e. not definitely primes, since they are too large to be [[:w:Elliptic curve primality|ECPP proved]] and [[:w:Pocklington primality test#Extensions and variants|neither ''N''−1 nor ''N''+1 can be ≥ 1/3 factored]], all of them pass the [[:w:Baillie–PSW primality test|Baillie–PSW primality test]] and the [[:w:Strong pseudoprime|strong primality test]] (i.e. the [[:w:Miller–Rabin primality test|Miller–Rabin primality test]]) with all prime bases ''p'' ≤ 61, however, all Athena primes &lt; 10&lt;sup&gt;25000&lt;/sup&gt; for bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 36 are definitely primes, most of them &gt; 10&lt;sup&gt;299&lt;/sup&gt; are proven primes with [[:w:Elliptic curve primality|ECPP proving]], others &gt; 10&lt;sup&gt;299&lt;/sup&gt; are proven primes with [[:w:Pocklington primality test#Extensions and variants|''N''−1 or ''N''+1 proving]])

All numbers are written in base ''b'', [[:w:Senary#Base 36 as senary compression|using A to Z to represent digit values 10 to 35]], &quot;{}&quot; means repeating, e.g. family 12{3}45 means the sequence {1245, 12345, 123345, 1233345, 12333345, 123333345, ...} (where the members are expressed as base ''b'' strings), subscripts are used to indicate repetitions of digits, e.g. 123&lt;sub&gt;4&lt;/sub&gt;567 means 123333567 (all subscripts are written in decimal).

Base 2: 1 Athena prime (the largest of which has 2 digits (it is 11, and its value is 3 in decimal)): {11}

Base 3: 3 Athena primes (the largest of which has 3 digits (it is 111, and its value is 13 in decimal)): {12, 21, 111}

Base 4: 5 Athena primes (the largest of which has 3 digits (it is 221, and its value is 41 in decimal)): {11, 13, 23, 31, 221}

Base 5: 22 Athena primes (the largest of which has 96 digits (it is 10&lt;sub&gt;93&lt;/sub&gt;13, and its algebraic form is 5&lt;sup&gt;95&lt;/sup&gt;+8)): {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 Athena primes (the largest of which has 5 digits (it is 40041, and its value is 5209 in decimal)): {11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041}

Base 7: 71 Athena primes (the largest of which has 17 digits (it is 3&lt;sub&gt;16&lt;/sub&gt;1, and its algebraic form is (7&lt;sup&gt;17&lt;/sup&gt;−5)/2)): {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: 75 Athena primes (the largest of which has 221 digits (it is 4&lt;sub&gt;220&lt;/sub&gt;7, and its algebraic form is (4×8&lt;sup&gt;221&lt;/sup&gt;+17)/7)): {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: 151 Athena primes (the largest of which has 1161 digits (it is 30&lt;sub&gt;1158&lt;/sub&gt;11, and its algebraic form is 3×9&lt;sup&gt;1160&lt;/sup&gt;+10)): {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: 77 Athena primes (the largest of which has 31 digits (it is 50&lt;sub&gt;28&lt;/sub&gt;27, and its algebraic form is 5×10&lt;sup&gt;30&lt;/sup&gt;+27)): {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: 1068 Athena primes (including 1 unproven probable prime: 57&lt;sub&gt;62668&lt;/sub&gt;), the largest of which has 62669 digits (it is 57&lt;sub&gt;62668&lt;/sub&gt;, and its algebraic form is (57×11&lt;sup&gt;62668&lt;/sup&gt;−7)/10), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel11 Data of Athena (probable) primes base 11]

Base 12: 106 Athena primes (the largest of which has 42 digits (it is 40&lt;sub&gt;39&lt;/sub&gt;77, and its algebraic form is 4×12&lt;sup&gt;41&lt;/sup&gt;+91)): {11, 15, 17, 1B, 25, 27, 31, 35, 37, 3B, 45, 4B, 51, 57, 5B, 61, 67, 6B, 75, 81, 85, 87, 8B, 91, 95, A7, AB, B5, B7, 221, 241, 2A1, 2B1, 2BB, 401, 421, 447, 471, 497, 565, 655, 665, 701, 70B, 721, 747, 771, 77B, 797, 7A1, 7BB, 907, 90B, 9BB, A41, B21, B2B, 2001, 200B, 202B, 222B, 229B, 292B, 299B, 4441, 4707, 4777, 6A05, 6AA5, 729B, 7441, 7B41, 929B, 9777, 992B, 9947, 997B, 9997, A0A1, A201, A605, A6A5, AA65, B001, B0B1, BB01, BB41, 600A5, 7999B, 9999B, AAAA1, B04A1, B0B9B, BAA01, BAAA1, BB09B, BBBB1, 44AAA1, A00065, BBBAA1, AAA0001, B00099B, AA000001, BBBBBB99B, B0000000000000000000000000009B, 400000000000000000000000000000000000000077}

Base 13: 3197 Athena primes (including 4 unproven probable primes: C5&lt;sub&gt;23755&lt;/sub&gt;C, 80&lt;sub&gt;32017&lt;/sub&gt;111, 95&lt;sub&gt;197420&lt;/sub&gt;, A3&lt;sub&gt;592197&lt;/sub&gt;A), the largest of which has 592199 digits (it is A3&lt;sub&gt;592197&lt;/sub&gt;A, and its algebraic form is (41×13&lt;sup&gt;592198&lt;/sup&gt;+27)/4), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel13 Data of Athena (probable) primes base 13]

Base 14: 650 Athena primes, the largest of which has 19699 digits (it is 4D&lt;sub&gt;19698&lt;/sub&gt;, and its algebraic form is 5×14&lt;sup&gt;19698&lt;/sup&gt;−1), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel14 Data of Athena primes base 14]

Base 15: 1284 Athena primes, the largest of which has 157 digits (it is 7&lt;sub&gt;155&lt;/sub&gt;97, and its algebraic form is (15&lt;sup&gt;157&lt;/sup&gt;+59)/2), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel15 Data of Athena primes base 15]

Base 16: 2347 Athena primes (including 3 unproven probable primes: DB&lt;sub&gt;32234&lt;/sub&gt;, 4&lt;sub&gt;72785&lt;/sub&gt;DD, 3&lt;sub&gt;116137&lt;/sub&gt;AF), the largest of which has 116139 digits (it is 3&lt;sub&gt;116137&lt;/sub&gt;AF, and its algebraic form is (16&lt;sup&gt;116139&lt;/sup&gt;+619)/5), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel16 Data of Athena (probable) primes base 16]

Base 17: 10415 known Athena primes (including many unproven probable primes) and 12 unsolved families (1{7}, 1F{0}7, 4{7}A, 70F{0}D, 8{B}9, 9{5}9, A{D}F, B{0}B3, {B}E9, {B}EE, F1{9}, FD0{D}, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel17 Data of known Athena (probable) primes base 17]

Base 18: 549 Athena primes, the largest of which has 6271 digits (it is C0&lt;sub&gt;6268&lt;/sub&gt;C5, and its algebraic form is 12×18&lt;sup&gt;6270&lt;/sup&gt;+221), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel18 Data of Athena primes base 18]

Base 19: 31417 known Athena primes (including many unproven probable primes) and 17 unsolved families (4B5{0}H, {5}3, 5{H}05, 5{H}0H, 5{H}5, 66{B}, 71{0}177, 7AF{0}H, 97{0}3, C{H}C, EE1{6}, F{7}5, F{B}G, F{D}F, H0F{0}7A, HB{0}5B5, II{D}, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel19 Data of known Athena (probable) primes base 19]

Base 20: 3314 Athena primes, the largest of which has 6271 digits (it is G0&lt;sub&gt;6269&lt;/sub&gt;D, and its algebraic form is 16×20&lt;sup&gt;6270&lt;/sup&gt;+13), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel20 Data of Athena primes base 20]

Base 21: 13386 known Athena primes (including many unproven probable primes) and 8 unsolved families (5{0}DJ, {9}D, B3{0}EB, B{H}6H, C{F}0K, {F}35, G{0}FK, H{0}7771, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel21 Data of known Athena (probable) primes base 21]

Base 22: 8003 Athena primes (including 1 unproven probable prime: BK&lt;sub&gt;22001&lt;/sub&gt;5), the largest of which has 22003 digits (it is BK&lt;sub&gt;22001&lt;/sub&gt;5, and its algebraic form is (251×22&lt;sup&gt;22002&lt;/sup&gt;−335)/21), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel22 Data of Athena (probable) primes base 22]

Base 23: 65178 known Athena primes (including many unproven probable primes) and 87 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel23 Data of known Athena (probable) primes base 23] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left23 Data of unsolved families for Athena problem base 23]

Base 24: 3409 Athena primes, the largest of which has 8134 digits (it is N00N&lt;sub&gt;8129&lt;/sub&gt;LN, and its algebraic form is 13249×24&lt;sup&gt;8131&lt;/sup&gt;−49), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel24 Data of Athena primes base 24]

Base 25: 133639 known Athena primes (including many unproven probable primes) and 85 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel25 Data of known Athena (probable) primes base 25] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left25 Data of unsolved families for Athena problem base 25]

Base 26: 25256 known Athena primes (including 7 unproven probable primes: 5&lt;sub&gt;19391&lt;/sub&gt;6F, 7&lt;sub&gt;20279&lt;/sub&gt;OL, LD0&lt;sub&gt;20975&lt;/sub&gt;7, 6K&lt;sub&gt;23300&lt;/sub&gt;5, J0&lt;sub&gt;44303&lt;/sub&gt;KCB, M0&lt;sub&gt;61186&lt;/sub&gt;2BB, 85M&lt;sub&gt;197060&lt;/sub&gt;B) and 3 unsolved families ({A}6F, {H}MH, {I}GL, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel26 Data of known Athena (probable) primes base 26]

Base 27: 102852 known Athena primes (including many unproven probable primes) and 44 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel27 Data of known Athena (probable) primes base 27] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left27 Data of unsolved families for Athena problem base 27]

Base 28: 25528 known Athena primes (including 3 unproven probable primes: N6&lt;sub&gt;24051&lt;/sub&gt;LR, 5OA&lt;sub&gt;31238&lt;/sub&gt;F, O4O&lt;sub&gt;94535&lt;/sub&gt;9) and 1 unsolved family (O{A}F, no primes or probable primes with length ≤ 900000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel28 Data of known Athena (probable) primes base 28]

Base 29: 355242 known Athena primes (including many unproven probable primes) and 125 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel29 Data of known Athena (probable) primes base 29] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left29 Data of unsolved families for Athena problem base 29]

Base 30: 2619 Athena primes (including 1 unproven probable prime: I0&lt;sub&gt;24608&lt;/sub&gt;D), the largest of which has 34206 digits (it is OT&lt;sub&gt;34205&lt;/sub&gt;, and its algebraic form is 25×30&lt;sup&gt;34205&lt;/sup&gt;−1), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel30 Data of Athena (probable) primes base 30]

Base 31: 569323 known Athena primes (including many unproven probable primes) and 77 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel31 Data of known Athena (probable) primes base 31] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left31 Data of unsolved families for Athena problem base 31]

Base 32: 168882 known Athena primes (including many unproven probable primes) and 120 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel32 Data of known Athena (probable) primes base 32] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left32 Data of unsolved families for Athena problem base 32]

Base 33: 280012 known Athena primes (including many unproven probable primes) and 81 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel33 Data of known Athena (probable) primes base 33] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left33 Data of unsolved families for Athena problem base 33]

Base 34: 184785 known Athena primes (including many unproven probable primes) and 47 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel34 Data of known Athena (probable) primes base 34] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left34 Data of unsolved families for Athena problem base 34]

Base 35: 720002 known Athena primes (including many unproven probable primes) and 60 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel35 Data of known Athena (probable) primes base 35] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left35 Data of unsolved families for Athena problem base 35]

Base 36: 35286 known Athena primes (including 3 unproven probable primes: 7K&lt;sub&gt;26567&lt;/sub&gt;Z, S0&lt;sub&gt;75007&lt;/sub&gt;8H, P&lt;sub&gt;81993&lt;/sub&gt;SZ) and 4 unsolved families (B{0}EUV, HM{0}N, N{0}YYN, O{L}Z, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel36 Data of known Athena (probable) primes base 36]

== Condensed table for bases 2 ≤ ''b'' ≤ 36 ==

{|class=&quot;wikitable&quot;
||''b''||number of Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b''||base-''b'' form of the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b'' (write &quot;''d''&lt;sub&gt;''n''&lt;/sub&gt;&quot; if there are 5 or more (''n'') consecutive same digits ''d'')||length of the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b''||length of the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b'' in decimal||algebraic ((''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1)) form of the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b''||''factordb'' entry of the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b''||the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b'' written in base ''b'' (use lower case letters instead of upper case letters)||number of unsolved families in the Athena problem in base ''b'' (all of these left families are linear families)||searching limit of length for the unsolved families in the Athena problem in base ''b'' (if there are different searching limits for the unsolved families in the Athena problem in base ''b'', choose the lowest searching limit)||
|-
||2||1||11||2||1||3||http://factordb.com/index.php?id=3&amp;open=ecm||http://factordb.com/index.php?showid=3&amp;base=2||0||–||
|-
||3||3||111&lt;br&gt;21&lt;br&gt;12||3&lt;br&gt;2&lt;br&gt;2||2&lt;br&gt;1&lt;br&gt;1||13&lt;br&gt;7&lt;br&gt;5||http://factordb.com/index.php?id=13&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=7&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=5&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=13&amp;base=3&lt;br&gt;http://factordb.com/index.php?showid=7&amp;base=3&lt;br&gt;http://factordb.com/index.php?showid=5&amp;base=3&lt;nowiki/&gt;||0||–||
|-
||4||5||221&lt;br&gt;31&lt;br&gt;23&lt;br&gt;13&lt;br&gt;11||3&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2||2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;1&lt;br&gt;1||41&lt;br&gt;13&lt;br&gt;11&lt;br&gt;7&lt;br&gt;5||http://factordb.com/index.php?id=41&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=13&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=11&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=7&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=5&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=41&amp;base=4&lt;br&gt;http://factordb.com/index.php?showid=13&amp;base=4&lt;br&gt;http://factordb.com/index.php?showid=11&amp;base=4&lt;br&gt;http://factordb.com/index.php?showid=7&amp;base=4&lt;br&gt;http://factordb.com/index.php?showid=5&amp;base=4&lt;nowiki/&gt;||0||–||
|-
||5||22||10&lt;sub&gt;93&lt;/sub&gt;13&lt;br&gt;300031&lt;br&gt;44441&lt;br&gt;33331&lt;br&gt;33001&lt;br&gt;30301&lt;br&gt;14444&lt;br&gt;10103&lt;br&gt;3101&lt;br&gt;414||96&lt;br&gt;6&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;4&lt;br&gt;3||67&lt;br&gt;4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;3&lt;br&gt;3&lt;br&gt;3||5&lt;sup&gt;95&lt;/sup&gt;+8&lt;br&gt;9391&lt;br&gt;3121&lt;br&gt;2341&lt;br&gt;2251&lt;br&gt;1951&lt;br&gt;1249&lt;br&gt;653&lt;br&gt;401&lt;br&gt;109||http://factordb.com/index.php?id=1100000000034686071&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=9391&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=3121&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=2341&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=2251&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1951&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1249&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=653&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=401&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=109&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000000034686071&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=9391&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=3121&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=2341&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=2251&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=1951&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=1249&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=653&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=401&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=109&amp;base=5&lt;nowiki/&gt;||0||–||
|-
||6||11||40041&lt;br&gt;4441&lt;br&gt;4401&lt;br&gt;51&lt;br&gt;45&lt;br&gt;35&lt;br&gt;31&lt;br&gt;25&lt;br&gt;21&lt;br&gt;15||5&lt;br&gt;4&lt;br&gt;4&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2||4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2||5209&lt;br&gt;1033&lt;br&gt;1009&lt;br&gt;31&lt;br&gt;29&lt;br&gt;23&lt;br&gt;19&lt;br&gt;17&lt;br&gt;13&lt;br&gt;11||http://factordb.com/index.php?id=5209&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1033&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1009&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=31&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=29&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=23&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=19&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=17&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=13&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=11&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=5209&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=1033&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=1009&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=31&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=29&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=23&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=19&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=17&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=13&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=11&amp;base=6&lt;nowiki/&gt;||0||–||
|-
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||33||280012~280093||DP&lt;sub&gt;95093&lt;/sub&gt;M5&lt;br&gt;HJ0&lt;sub&gt;94295&lt;/sub&gt;J&lt;br&gt;90&lt;sub&gt;93597&lt;/sub&gt;Q&lt;br&gt;9F0&lt;sub&gt;93157&lt;/sub&gt;N&lt;br&gt;7&lt;sub&gt;89449&lt;/sub&gt;333H&lt;br&gt;K3&lt;sub&gt;80751&lt;/sub&gt;6K&lt;br&gt;D&lt;sub&gt;80107&lt;/sub&gt;9UD&lt;br&gt;VFU&lt;sub&gt;72204&lt;/sub&gt;FK&lt;br&gt;J&lt;sub&gt;68715&lt;/sub&gt;2BJ&lt;br&gt;DF0&lt;sub&gt;68367&lt;/sub&gt;J||95096&lt;br&gt;94298&lt;br&gt;93599&lt;br&gt;93160&lt;br&gt;89453&lt;br&gt;80754&lt;br&gt;80110&lt;br&gt;72208&lt;br&gt;68718&lt;br&gt;68370||144405&lt;br&gt;143193&lt;br&gt;142131&lt;br&gt;141465&lt;br&gt;135835&lt;br&gt;122626&lt;br&gt;121648&lt;br&gt;109649&lt;br&gt;104350&lt;br&gt;103821||(441×33&lt;sup&gt;95095&lt;/sup&gt;−3833)/32&lt;br&gt;580×33&lt;sup&gt;94296&lt;/sup&gt;+19&lt;br&gt;9×33&lt;sup&gt;93598&lt;/sup&gt;+26&lt;br&gt;312×33&lt;sup&gt;93158&lt;/sup&gt;+23&lt;br&gt;(7×33&lt;sup&gt;89453&lt;/sup&gt;−4743239)/32&lt;br&gt;(643×33&lt;sup&gt;80753&lt;/sup&gt;+3709)/32&lt;br&gt;(13×33&lt;sup&gt;80110&lt;/sup&gt;−121453)/32&lt;br&gt;(16623×33&lt;sup&gt;72206&lt;/sup&gt;−8095)/16&lt;br&gt;(19×33&lt;sup&gt;68718&lt;/sup&gt;−600883)/32&lt;br&gt;444×33&lt;sup&gt;68368&lt;/sup&gt;+19||http://factordb.com/index.php?id=1100000005652348775&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000005652362811&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000005652375073&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000005652389776&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000005652430746&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000005652446200&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000005652461592&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004614764298&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004614770536&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004614784274&amp;open=prime&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000005652348775&amp;base=33&lt;br&gt;http://factordb.com/index.php?showid=1100000005652362811&amp;base=33&lt;br&gt;http://factordb.com/index.php?showid=1100000005652375073&amp;base=33&lt;br&gt;http://factordb.com/index.php?showid=1100000005652389776&amp;base=33&lt;br&gt;http://factordb.com/index.php?showid=1100000005652430746&amp;base=33&lt;br&gt;http://factordb.com/index.php?showid=1100000005652446200&amp;base=33&lt;br&gt;http://factordb.com/index.php?showid=1100000005652461592&amp;base=33&lt;br&gt;http://factordb.com/index.php?showid=1100000004614764298&amp;base=33&lt;br&gt;http://factordb.com/index.php?showid=1100000004614770536&amp;base=33&lt;br&gt;http://factordb.com/index.php?showid=1100000004614784274&amp;base=33&lt;nowiki/&gt;||81||100000||
|-
||34||184785~184832||GFGC&lt;sub&gt;99996&lt;/sub&gt;5&lt;br&gt;90&lt;sub&gt;97950&lt;/sub&gt;FJ&lt;br&gt;NM0&lt;sub&gt;85218&lt;/sub&gt;KX&lt;br&gt;F&lt;sub&gt;83189&lt;/sub&gt;H2HP&lt;br&gt;P&lt;sub&gt;79441&lt;/sub&gt;444P&lt;br&gt;6&lt;sub&gt;77027&lt;/sub&gt;8X&lt;br&gt;XQIQ&lt;sub&gt;72241&lt;/sub&gt;D&lt;br&gt;T&lt;sub&gt;66530&lt;/sub&gt;IF&lt;br&gt;4&lt;sub&gt;66152&lt;/sub&gt;B&lt;br&gt;2EEC&lt;sub&gt;66039&lt;/sub&gt;7||100000&lt;br&gt;97953&lt;br&gt;85222&lt;br&gt;83193&lt;br&gt;79445&lt;br&gt;77029&lt;br&gt;72245&lt;br&gt;66532&lt;br&gt;66153&lt;br&gt;66043||153148&lt;br&gt;150013&lt;br&gt;130516&lt;br&gt;127408&lt;br&gt;121669&lt;br&gt;117968&lt;br&gt;110642&lt;br&gt;101893&lt;br&gt;101312&lt;br&gt;101143||(209246×34&lt;sup&gt;99997&lt;/sup&gt;−81)/11&lt;br&gt;9×34&lt;sup&gt;97952&lt;/sup&gt;+529&lt;br&gt;804×34&lt;sup&gt;85220&lt;/sup&gt;+713&lt;br&gt;(5×34&lt;sup&gt;83193&lt;/sup&gt;+700233)/11&lt;br&gt;(25×34&lt;sup&gt;79445&lt;/sup&gt;−28062367)/33&lt;br&gt;(2×34&lt;sup&gt;77029&lt;/sup&gt;+1043)/11&lt;br&gt;(1288676×34&lt;sup&gt;72242&lt;/sup&gt;−455)/33&lt;br&gt;(29×34&lt;sup&gt;66532&lt;/sup&gt;−12833)/33&lt;br&gt;(4×34&lt;sup&gt;66153&lt;/sup&gt;+227)/33&lt;br&gt;(30826×34&lt;sup&gt;66040&lt;/sup&gt;−59)/11||http://factordb.com/index.php?id=1100000004702891268&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004702894713&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004702900996&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004702910376&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004702913746&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004702918600&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004399656529&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004399657696&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004399658651&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004399659716&amp;open=prime&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000004702891268&amp;base=34&lt;br&gt;http://factordb.com/index.php?showid=1100000004702894713&amp;base=34&lt;br&gt;http://factordb.com/index.php?showid=1100000004702900996&amp;base=34&lt;br&gt;http://factordb.com/index.php?showid=1100000004702910376&amp;base=34&lt;br&gt;http://factordb.com/index.php?showid=1100000004702913746&amp;base=34&lt;br&gt;http://factordb.com/index.php?showid=1100000004702918600&amp;base=34&lt;br&gt;http://factordb.com/index.php?showid=1100000004399656529&amp;base=34&lt;br&gt;http://factordb.com/index.php?showid=1100000004399657696&amp;base=34&lt;br&gt;http://factordb.com/index.php?showid=1100000004399658651&amp;base=34&lt;br&gt;http://factordb.com/index.php?showid=1100000004399659716&amp;base=34&lt;nowiki/&gt;||47||100000||
|-
||35||720002~720062||N0N&lt;sub&gt;99971&lt;/sub&gt;9&lt;br&gt;V0&lt;sub&gt;83669&lt;/sub&gt;E73&lt;br&gt;N&lt;sub&gt;81563&lt;/sub&gt;K7N&lt;br&gt;BJ0&lt;sub&gt;81279&lt;/sub&gt;N&lt;br&gt;J0&lt;sub&gt;80062&lt;/sub&gt;FUH&lt;br&gt;43V&lt;sub&gt;79754&lt;/sub&gt;&lt;br&gt;9&lt;sub&gt;76600&lt;/sub&gt;K3&lt;br&gt;LB&lt;sub&gt;71366&lt;/sub&gt;PB&lt;br&gt;Q&lt;sub&gt;64150&lt;/sub&gt;H&lt;br&gt;50&lt;sub&gt;63397&lt;/sub&gt;5R||99974&lt;br&gt;83673&lt;br&gt;81566&lt;br&gt;81282&lt;br&gt;80066&lt;br&gt;79756&lt;br&gt;76602&lt;br&gt;71369&lt;br&gt;64151&lt;br&gt;63400||154367&lt;br&gt;129197&lt;br&gt;125944&lt;br&gt;125505&lt;br&gt;123628&lt;br&gt;123148&lt;br&gt;118279&lt;br&gt;110199&lt;br&gt;99054&lt;br&gt;97894||(27393×35&lt;sup&gt;99972&lt;/sup&gt;−499)/34&lt;br&gt;31×35&lt;sup&gt;83672&lt;/sup&gt;+17398&lt;br&gt;(23×35&lt;sup&gt;81566&lt;/sup&gt;−144013)/34&lt;br&gt;404×35&lt;sup&gt;81280&lt;/sup&gt;+23&lt;br&gt;19×35&lt;sup&gt;80065&lt;/sup&gt;+19442&lt;br&gt;(4893×35&lt;sup&gt;79754&lt;/sup&gt;−31)/34&lt;br&gt;(9×35&lt;sup&gt;76602&lt;/sup&gt;+12877)/34&lt;br&gt;(725×35&lt;sup&gt;71368&lt;/sup&gt;+16649)/34&lt;br&gt;(13×35&lt;sup&gt;64151&lt;/sup&gt;−166)/17&lt;br&gt;5×35&lt;sup&gt;63399&lt;/sup&gt;+202||http://factordb.com/index.php?id=1100000008248342445&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008248353306&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008248375642&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008248397018&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008248412468&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008248418540&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008248423670&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008192119974&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008192126630&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008192129294&amp;open=prime&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000008248342445&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008248353306&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008248375642&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008248397018&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008248412468&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008248418540&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008248423670&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008192119974&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008192126630&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008192129294&amp;base=35&lt;nowiki/&gt;||60||100000||
|-
||36||35286~35290||P&lt;sub&gt;81993&lt;/sub&gt;SZ&lt;br&gt;S0&lt;sub&gt;75007&lt;/sub&gt;8H&lt;br&gt;7K&lt;sub&gt;26567&lt;/sub&gt;Z&lt;br&gt;J&lt;sub&gt;10117&lt;/sub&gt;LJ&lt;br&gt;VL0&lt;sub&gt;7258&lt;/sub&gt;J&lt;br&gt;EO0&lt;sub&gt;6177&lt;/sub&gt;V&lt;br&gt;FZ&lt;sub&gt;5777&lt;/sub&gt;3P&lt;br&gt;T09&lt;sub&gt;4618&lt;/sub&gt;1&lt;br&gt;RY&lt;sub&gt;4562&lt;/sub&gt;H&lt;br&gt;OZ&lt;sub&gt;3932&lt;/sub&gt;AZ||81995&lt;br&gt;75010&lt;br&gt;26569&lt;br&gt;10119&lt;br&gt;7261&lt;br&gt;6180&lt;br&gt;5780&lt;br&gt;4621&lt;br&gt;4564&lt;br&gt;3935||127609&lt;br&gt;116739&lt;br&gt;41349&lt;br&gt;15748&lt;br&gt;11301&lt;br&gt;9618&lt;br&gt;8996&lt;br&gt;7192&lt;br&gt;7103&lt;br&gt;6124||(5×36&lt;sup&gt;81995&lt;/sup&gt;+821)/7&lt;br&gt;28×36&lt;sup&gt;75009&lt;/sup&gt;+305&lt;br&gt;(53×36&lt;sup&gt;26568&lt;/sup&gt;+101)/7&lt;br&gt;(19×36&lt;sup&gt;10119&lt;/sup&gt;+2501)/35&lt;br&gt;1137×36&lt;sup&gt;7259&lt;/sup&gt;+19&lt;br&gt;528×36&lt;sup&gt;6178&lt;/sup&gt;+31&lt;br&gt;16×36&lt;sup&gt;5779&lt;/sup&gt;−1163&lt;br&gt;(36549×36&lt;sup&gt;4619&lt;/sup&gt;−289)/35&lt;br&gt;(979×36&lt;sup&gt;4563&lt;/sup&gt;−629)/35&lt;br&gt;25×36&lt;sup&gt;3934&lt;/sup&gt;−901||http://factordb.com/index.php?id=1100000002394962083&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004020085177&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003896952461&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362491&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362489&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362488&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362487&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362486&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362485&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000000840634476&amp;open=prime&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000002394962083&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000004020085177&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003896952461&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362491&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362489&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362488&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362487&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362486&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362485&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000000840634476&amp;base=36&lt;nowiki/&gt;||4||200000||
|}

== The fully proof of Athena problem in decimal (base ''b'' = 10) ==

'''Bold''' for the Athena primes, ''x'' ◁ ''y'' means ''x'' is a subsequence of ''y''.

Assume ''p'' is a prime &gt; 10, and the last digit of ''p'' must lie in {1,3,7,9}.

Case 1: ''p'' ends with 1.

In this case we can write ''p'' = ''x''1. If ''x'' contains 1, 3, 4, 6, or 7, then (respectively) '''11''' ◁ ''p'', '''31''' ◁ ''p'', '''41''' ◁ ''p'', '''61''' ◁ ''p'', or '''71''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 2, 5, 8, or 9.

Case 1.1: ''p'' begins with 2.

In this case we can write ''p'' = 2''y''1. If 5 ◁ ''y'', then '''251''' ◁ ''p''. If 8 ◁ ''y'', then '''281''' ◁ ''p''. If 9 ◁ ''y'', then 29 ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 2.

If 22 ◁ ''y'', then '''2221''' ◁ ''p''. Hence we may assume ''y'' contains zero or one 2's.

If ''y'' contains no 2's, then ''p'' ∈ 2{0}1. But then, since the sum of the digits of ''p'' is 3, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' contains exactly one 2, then we can write ''p'' = 2''z''2''w''1, where ''z'',''w'' ∈ {0}. If 0 ◁ ''z'' and 0 ◁ ''w'', then '''20201''' ◁ ''p''. Hence we may assume either ''z'' or ''w'' is empty.

If ''z'' is empty, then ''p'' ∈ 22{0}1, and the smallest prime ''p'' ∈ 22{0}1 is '''22000001'''.

If ''w'' is empty, then ''p'' ∈ 2{0}21, and the smallest prime ''p'' ∈ 2{0}21 is '''20021'''.

Case 1.2: ''p'' begins with 5.

In this case we can write ''p'' = 5''y''1. If 2 ◁ ''y'', then '''521''' ◁ ''p''. If 9 ◁ ''y'', then 59 ◁ ''p''. Hence we may assume all digits of ''y'' are 0, 5, or 8.

If 05 ◁ ''y'', then '''5051''' ◁ ''p''. If 08 ◁ ''y'', then '''5081''' ◁ ''p''. If 50 ◁ ''y'', then '''5501''' ◁ ''p''. If 58 ◁ ''y'', then '''5581''' ◁ ''p''. If 80 ◁ ''y'', then '''5801''' ◁ ''p''. If 85 ◁ ''y'', then '''5851''' ◁ ''p''. Hence we may assume ''y'' ∈ {0} ∪ {5} ∪ {8}.

If ''y'' ∈ {0}, then ''p'' ∈ 5{0}1. But then, since the sum of the digits of ''p'' is 6, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' ∈ {5}, then ''p'' ∈ 5{5}1, and the smallest prime ''p'' ∈ 5{5}1 is '''555555555551'''.

If ''y'' ∈ {8}, since if 88 ◁ ''y'', then 881 ◁ ''p'', hence we may assume ''y'' ∈ {''𝜆'',8}, and thus ''p'' ∈ {51,581}, but 51 and 581 are both composite.

Case 1.3: ''p'' begins with 8.

In this case we can write p = 8''y''1. If 2 ◁ ''y'', then '''821''' ◁ ''p''. If 8 ◁ ''y'', then '''881''' ◁ ''p''. If 9 ◁ ''y'', then 89 ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 5.

If 50 ◁ ''y'', then '''8501''' ◁ ''p''. Hence we may assume y ∈ {0}{5}.

If 005 ◁ ''y'', then '''80051''' ◁ p. Hence we may assume y ∈ {0} ∪ {5} ∪ 0{5}.

If y ∈ {0}, then ''p'' ∈ 8{0}1. But then, since the sum of the digits of ''p'' is 9, ''p'' is divisible by 3, so ''p'' cannot be prime.

If y ∈ {5}, since if 55555555555 ◁ ''y'', then 555555555551 ◁ ''p'', hence we may assume ''y'' ∈ {''𝜆'', 5, 55, 555, 5555, 55555, 555555, 5555555, 55555555, 555555555, 5555555555}, and thus ''p'' ∈ {81, 851, 8551, 85551, 855551, 8555551, 85555551, 855555551, 8555555551, 85555555551, 855555555551}, but all of these numbers are composite.

If y ∈ 0{5}, since if 55555555555 ◁ ''y'', then 555555555551 ◁ ''p'', hence we may assume ''y'' ∈ {0, 05, 055, 0555, 05555, 055555, 0555555, 05555555, 055555555, 0555555555, 05555555555}, and thus ''p'' ∈ {801, 8051, 80551, 805551, 8055551, 80555551, 805555551, 8055555551, 80555555551, 805555555551, 8055555555551}, and of these numbers only 80555551 and 8055555551 are primes, but 80555551 ◁ 8055555551, thus only '''80555551''' is a minimal element.

Case 1.4: ''p'' begins with 9.

In this case we can write p = 9''y''1. If 9 ◁ ''y'', then '''991''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0, 2, 5, or 8.

If 00 ◁ ''y'', then '''9001''' ◁ ''p''. If 22 ◁ ''y'', then '''9221''' ◁ ''p''. If 55 ◁ ''y'', then '''9551''' ◁ ''p''. If 88 ◁ ''y'', then 881 ◁ ''p''. Hence we may assume ''y'' contains at most one 0, at most one 2, at most one 5, and at most one 8.

If ''y'' only contains at most one 0 and does not contain any of {2,5,8}, then ''y'' ∈ {''𝜆'',0}, and thus ''p'' ∈ {91,901}, but 91 and 901 are both composite. If ''y'' only contains at most one 0 and only one of {2,5,8}, then the sum of the digits of ''p'' is divisible by 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Hence we may assume ''y'' contains at least two of {2,5,8}.

If 25 ◁ ''y'', then 251 ◁ ''p''. If 28 ◁ ''y'', then 281 ◁ ''p''. If 52 ◁ ''y'', then 521 ◁ ''p''. If 82 ◁ ''y'', then 821 ◁ ''p''. Hence we may assume ''y'' contains no 2's (since if ''y'' contains 2, then ''y'' cannot contain either 5's or 8's, which is a contradiction).

If 85 ◁ ''y'', then '''9851''' ◁ ''p''. Hence we may assume ''y'' ∈ {58,580,508,058}, and thus ''p'' ∈ {9581,95801,95081,90581}, and of these numbers only 95801 is prime, but 95801 is not a minimal element since 5801 ◁ 95801.

Case 2: ''p'' ends with 3.

In this case we can write p = ''x''3. If ''x'' contains 1, 2, 4, 5, 7, or 8, then (respectively) '''13''' ◁ ''p'', '''23''' ◁ ''p'', '''43''' ◁ ''p'', '''53''' ◁ ''p'', '''73''' ◁ ''p'', or '''83''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 3, 6, or 9, and thus all digits of ''p'' are 0, 3, 6, or 9. But then, since the digits of ''p'' all have a common factor 3, ''p'' is divisible by 3, so ''p'' cannot be prime.

Case 3: ''p'' ends with 7.

In this case we can write ''p'' = ''x''7. If ''x'' contains 1, 3, 4, 6, or 9, then (respectively) '''17''' ◁ ''p'', '''37''' ◁ ''p'', '''47''' ◁ ''p'', '''67''' ◁ ''p'', or '''97''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 2, 5, 7, or 8.

Case 3.1: ''p'' begins with 2.

In this case we can write ''p'' = 2''y''7. If 2 ◁ ''y'', then '''227''' ◁ ''p''. If 5 ◁ ''y'', then '''257''' ◁ ''p''. If 7 ◁ ''y'', then '''277''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 8.

If 08 ◁ ''y'', then '''2087''' ◁ ''p''. If 88 ◁ ''y'', then 887 ◁ ''p''. Hence we may assume ''y'' ∈ {0} ∪ 8{0}.

If ''y'' ∈ {0}, then ''p'' ∈ 2{0}7. But then, since the sum of the digits of ''p'' is 9, ''p'' is divisible by 3, so ''p'' cannot be prime.

If y ∈ 8{0}, then ''p'' ∈ 28{0}7. But then ''p'' is divisible by 7, since for ''n'' ≥ 0 we have 7 × 40&lt;sub&gt;''n''&lt;/sub&gt;1 = 280&lt;sub&gt;''n''&lt;/sub&gt;7.

Case 3.2: ''p'' begins with 5.

In this case we can write ''p'' = 5''y''7. If 5 ◁ ''y'', then '''557''' ◁ ''p''. If 7 ◁ ''y'', then '''577''' ◁ ''p''. If 8 ◁ ''y'', then '''587''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 2.

If 22 ◁ ''y'', then 227 ◁ ''p''. Hence we may assume ''y'' contains zero or one 2's.

If ''y'' contains no 2's, then ''p'' ∈ 5{0}7. But then, since the sum of the digits of ''p'' is 12, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' contains exactly one 2, then we can write ''p'' = 5''z''2''w''7, where ''z'',''w'' ∈ {0}. If 0 ◁ ''z'' and 0 ◁ ''w'', then '''50207''' ◁ ''p''. Hence we may assume either ''z'' or ''w'' is empty.

If ''z'' is empty, then ''p'' ∈ 52{0}7, and the smallest prime ''p'' ∈ 52{0}7 is '''5200007'''.

If ''w'' is empty, then ''p'' ∈ 5{0}27, and the smallest prime ''p'' ∈ 5{0}27 is '''5000000000000000000000000000027'''.

Case 3.3: ''p'' begins with 7.

In this case we can write ''p'' = 7''y''7. If 2 ◁ ''y'', then '''727''' ◁ ''p''. If 5 ◁ ''y'', then '''757''' ◁ ''p''. If 8 ◁ ''y'', then '''787''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 7, and thus all digits of ''p'' are 0 or 7. But then, since the digits of ''p'' all have a common factor 7, ''p'' is divisible by 7, so ''p'' cannot be prime.

Case 3.4: ''p'' begins with 8.

In this case we can write ''p'' = 8''y''7. If 2 ◁ ''y'', then '''827''' ◁ ''p''. If 5 ◁ ''y'', then '''857''' ◁ ''p''. If 7 ◁ ''y'', then '''877''' ◁ ''p''. If 8 ◁ ''y'', then '''887''' ◁ ''p''. Hence we may assume ''y'' ∈ {0}, and thus ''p'' ∈ 8{0}7. But then, since the sum of the digits of ''p'' is 15, ''p'' is divisible by 3, so ''p'' cannot be prime.

Case 4: ''p'' ends with 9.

In this case we can write ''p'' = ''x''9. If ''x'' contains 1, 2, 5, 7, or 8, then (respectively) '''19''' ◁ ''p'', '''29''' ◁ ''p'', '''59''' ◁ ''p'', '''79''' ◁ ''p'', or '''89''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 3, 4, 6, or 9.

If 44 ◁ ''x'', then '''449''' ◁ ''p''. Hence we may assume ''x'' contains zero or one 4's.

If x contains no 4's, then all digits of ''x'' are 0, 3, 6, or 9, and thus all digits of ''p'' are 0, 3, 6, or 9. But then, since the digits of ''p'' all have a common factor 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Hence we may assume that ''x'' contains exactly one 4.

Case 4.1: ''p'' begins with 3.

In this case we can write ''p'' = 3''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. We must have '''349''' ◁ ''p''.

Case 4.2: ''p'' begins with 4.

In this case we can write ''p'' = 4''y''9, where all digits of ''y'' are 0, 3, 6, or 9. If 0 ◁ ''y'', then '''409''' ◁ ''p''. If 3 ◁ ''y'', then 43 ◁ ''p''. If 9 ◁ ''y'', then '''499''' ◁ ''p''. Hence we may assume ''y'' ∈ {6}, and thus ''p'' ∈ 4{6}9. But then ''p'' is divisible by 7, since for ''n'' ≥ 0 we have 7 × 6&lt;sub&gt;''n''&lt;/sub&gt;7 = 46&lt;sub&gt;''n''&lt;/sub&gt;9.

Case 4.3: ''p'' begins with 6.

In this case we can write p = 6''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. If 0 ◁ ''z'', then 409 ◁ ''p''. If 3 ◁ ''z'', then 43 ◁ ''p''. If 6 ◁ ''z'', then '''6469''' ◁ ''p''. If 9 ◁ ''z'', then 499 ◁ ''p''. Hence we may assume ''z'' is empty.

If 3 ◁ ''y'', then 349 ◁ ''p''. If 9 ◁ ''y'', then '''6949''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 6.

If 06 ◁ ''y'', then '''60649''' ◁ ''p''. Hence we may assume ''y'' ∈ {6}{0}.

If 666 ◁ ''y'', then '''666649''' ◁ ''p''. If 00000 ◁ ''y'', then '''60000049''' ◁ ''p''. Hence we may assume ''y'' ∈ {''𝜆'', 0, 00, 000, 0000, 6, 60, 600, 6000, 60000, 66, 660, 6600, 66000, 660000}, and thus ''p'' ∈ {649, 6049, 60049, 600049, 6000049, 6649, 66049, 660049, 6600049, 66000049, 66649, 666049, 6660049, 66600049, 666000049}, and of these numbers only '''66000049''' and '''66600049''' are primes.

Case 4.4: ''p'' begins with 9.

In this case we can write p = 9''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. If 0 ◁ ''y'', then '''9049''' ◁ ''p''. If 3 ◁ ''y'', then 349 ◁ ''p''. If 6 ◁ ''y'', then '''9649''' ◁ ''p''. If 9 ◁ ''y'', then '''9949''' ◁ ''p''. Hence we may assume ''y'' is empty.

If 0 ◁ ''z'', then 409 ◁ ''p''. If 3 ◁ ''z'', then 43 ◁ ''p''. If 9 ◁ ''z'', then 499 ◁ ''p''. Hence we may assume ''z'' ∈ {6}, and thus ''p'' ∈ 94{6}9, and the smallest prime ''p'' ∈ 94{6}9 is 946669.

[[Category:Number theory]]</text>
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    <revision>
      <id>2819846</id>
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      <text bytes="129919" sha1="dbgole6aw9ltyo9fkp54k0kcgwoucqz" xml:space="preserve">{{mathematics}}
'''Athena problem''' is an [[:w:List of unsolved problems in mathematics|unsolved problem]] in [[:w:Number theory|number theory]] and [[:w:Formal language theory|formal language theory]] and [[:w:Order theory|order theory]], this problem is named after the ancient Greek goddess [[:w:Athena|Athena]] (which is associated with [[:w:Wisdom|wisdom]]). Athena problem is: Give a [[:w:Natural number|natural number]] ''b'' &gt; 1, find the [[:w:Set (mathematics)|set]] of the [[:w:Minimal element|minimal element]]s of the set of the &quot;[[:w:Prime number|prime number]] [[:w:Greater than|&gt;]] ''b''&quot; [[:w:Numerical digit|digit]] [[:w:String (computer science)|string]]s in the [[:w:Positional numeral system|positional numeral system]] with [[:w:Radix|base]] ''b'' for the [[:w:Subsequence|subsequence]] [[:w:Partially ordered set|ordering]]. (A string ''x'' is a subsequence of another string ''y'', if ''x'' can be obtained from ''y'' by deleting zero or more of the [[:w:Character (computing)|character]]s in ''y''. For example, 514 is a subsequence of 352148, &quot;string&quot; is a subsequence of &quot;meistersinger&quot;. In contrast, 758 is not a subsequence of 378259, &quot;abc&quot; is not a subsequence of &quot;cbacacba&quot;, since the characters must be in the same order) (Unlike [[:w:Substring|substring]], subsequence is not required to occupy consecutive positions within the original sequences, e.g. the [[:w:Longest common subsequence|longest common subsequence problem]] is different from the [[:w:Longest common substring|longest common substring problem]])

Using [[:w:Formal language theory|formal language theory]] terminology, Athena problem is finding the [[:w:Set (mathematics)|set]] of the [[:w:Minimal element|minimal element]]s of the [[:w:Formal language|language]] of base-''b'' [[:w:Representation (mathematics)|representation]]s of the [[:w:Prime number|prime number]]s [[:w:Greater than|&gt;]] ''b'' (which is a set of [[:w:String (computer science)|string]]s of [[:w:Symbol|symbol]]s over the [[:w:Alphabet (formal languages)|alphabet]] ''Σ''&lt;sub&gt;''b''&lt;/sub&gt; := {0, 1, ..., ''b''−1}), under the subsequence ordering (i.e. the [[:w:Binary relation|binary relation]] &quot;is a subsequence of&quot;, which is a [[:w:Partially ordered set|partial ordering]]), for a given natural number ''b'' &gt; 1 (You can draw this partial ordering as a [[:w:Hasse diagram|Hasse diagram]] to find all [[:w:Minimal element|minimal element]]s), this set is called '''Athena set''', and the prime numbers in this set are called '''Athena primes'''.

By [[:w:Higman's lemma|Higman's lemma]], there are no [[:w:Infinite set|infinite]] [[:w:Antichain|antichain]]s for the subsequence ordering (i.e. the subsequence ordering is always a [[:w:Well-quasi-ordering|well quasi order]]) (i.e. under the subsequence ordering (i.e. the [[:w:Binary relation|binary relation]] &quot;is a subsequence of&quot;, which is a [[:w:Partially ordered set|partial ordering]]), every set of pairwise incomparable (i.e. not [[:w:Comparability|comparable]]) strings is finite), thus there must be only finitely many such minimal elements. In other words, the Athena set in every base ''b'' must be a [[:w:Finite set|finite set]], and every base ''b'' ≥ 2 has only finitely many Athena primes, e.g. in [[:w:Decimal|decimal]] (base ''b'' = 10), the Athena set has exactly 77 [[:w:Element of a set|element]]s (they are exactly the Athena primes in decimal (base ''b'' = 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}.

For bases 2 ≤ ''b'' ≤ 36, Athena problem is fully solved in bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 24, and also solved in bases ''b'' = 11, 13, 16, 22, 30 if [[:w:Probable prime|probable prime]]s are allowed. For the unsolved bases ''b'' = 17, 19, 21, 23, 25, 26, 27, 28, 29, 31, 32, 34, 35, 36, Athena problem is solved (if probable primes are allowed) except 771 [[:w:Indexed family|families]] of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be [[:w:Empty string|empty]]) of digits in base ''b'', ''y'' is a digit in base ''b'') = sequence {''xz'', ''xyz'', ''xyyz'', ''xyyyz'', ''xyyyyz'', ''xyyyyyz'', ...} (i.e. &quot;''xy''&lt;sup&gt;+&lt;/sup&gt;''z''&quot; in [[:w:Regular expression|regular expression]]), all of these 771 families contain no primes &gt; ''b'' or probable primes &gt; ''b'' with length ≤ 100000.

== Solve the problem ==

To solve the Athena problem for a given base ''b'', we must [[:w:Computing|compute]] the elements up to families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''), and find the smallest prime &gt; ''b'' in all such families.

We call families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') &quot;linear&quot; families, and we reduce these families by removing all trailing digits ''y'' from ''x'', and removing all leading digits ''y'' from ''z'', to make the families be easier, e.g. family 12333{3}33345 in base ''b'' is reduced to family 12{3}45 in base ''b'', since they are in fact the same family. Our [[:w:Algorithm|algorithm]] then proceeds as follows:

* 1. ''M'' := {minimal primes in base ''b'' of length 2 or 3}, ''L'' := union of all ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'') such that ''x'' ≠ 0 and ''gcd''(''z'', ''b'') = 1 and ''Y'' is the set of digits ''y'' in base ''b'' such that ''xyz'' has no subsequence in ''M''.
* 2. While ''L'' contains nonlinear families (families which are not linear families): Explore each family of ''L'', and update ''L''. Examine each family of ''L'' by:
* 2.1. Let ''w'' be the shortest string in the family. If ''w'' has a subsequence in ''M'', then remove the family from ''L''. If ''w'' represents a prime, then add ''w'' to ''M'' and remove the family from ''L''.
* 2.2. If possible, simplify the family.
* 2.3. Using the techniques below (covering congruence, algebraic factorization, or combine of them), check if the family can be proven to only contain composites (only count the numbers &gt; ''b''), and if so then remove the family from ''L''.
* 3. Update ''L'', after each split examine the new families as in step 2.

e.g. in decimal (base ''b'' = 10):

''M'' := {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}

''L'' := {2{0,2}1, 2{0,8}7, 3{0,3,6,9}3, 3{0,3,6,9}9, 4{6}9, 5{0,5,8}1, 5{0,2}7, 6{0,3,6,9}3, 6{0,3,4,6,9}9, 7{0,7}7, 8{0,5}1, 8{0}7, 9{0,2,5,8}1, 9{0,3,6,9}3, 9{0,3,4,6,9}9}

and since 2221 is prime, it follows that the family 2{0,2}1 splits into the families 2{0}1 and 2{0}2{0}1

and since the family 2{0}1 can be proven to contain no primes &gt; base (since all numbers in this family are divisible by 3), it can be removed

and since 20201 is prime, it follows that the family 2{0}2{0}1 splits into the families 2{0}21 and 22{0}1

221 and 2021 are composites, but 20021 is prime, thus add 20021 to ''L''

none of 221, 2201, 22001, 220001, 2200001 are primes, but 22000001 is prime, thus add 22000001 to ''L''

and since the family 3{0,3,6,9}3 can be proven to contain no primes &gt; base (since all numbers in this family are divisible by 3), it can be removed

etc.

Since the number of possible (first digit,last digit) (also called (initial digit,final digit)) combos ([[:w:Ordered pair|ordered pair]]s) of a prime &gt; ''b'' in base ''b'' is (''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(''b'') (all digits except 0 can be the first digit of a prime &gt; ''b'' in base ''b'' (thus ''b''−1 possible digits), but only the digits coprime to ''b'' can be the last digit of a prime &gt; ''b'' in base ''b'' (thus ''eulerphi''(''b'') possible digits), and by the [[:w:Rule of product|rule of product]], there are (''b''−1)×''eulerphi''(''b'') choices of the (first digit,last digit) combo, also, both &quot;numbers of Athena primes in base ''b''&quot; and &quot;length of the largest Athena prime in base ''b''&quot; are [[:w:Asymptotic analysis|roughly]] ''[[:w:E (mathematical_constant)|e]]''&lt;sup&gt;''[[:w:Euler's constant|γ]]''×(''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(*b*)&lt;/sup&gt;.

Shrinking the family ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''Y'' is a set of digits in base ''b'')

* If ''y'' ∈ ''Y'' and the string ''xyyz'' represents a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or has a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''}''z'' ∪ ''x''{''Y'' \ ''y''}''y''{''Y'' \ ''y''}''z''.
* If ''y''&lt;sub&gt;1&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;2&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;1&lt;/sub&gt; ≠ ''y''&lt;sub&gt;2&lt;/sub&gt; and the string ''xy''&lt;sub&gt;1&lt;/sub&gt;''y''&lt;sub&gt;2&lt;/sub&gt;''z'' represents a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or has a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''&lt;sub&gt;1&lt;/sub&gt;}{''Y'' \ ''y''&lt;sub&gt;2&lt;/sub&gt;}''z''.
* If ''y''&lt;sub&gt;1&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;2&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;1&lt;/sub&gt; ≠ ''y''&lt;sub&gt;2&lt;/sub&gt; and both the strings ''xy''&lt;sub&gt;1&lt;/sub&gt;''y''&lt;sub&gt;2&lt;/sub&gt;''z'' and ''xy''&lt;sub&gt;2&lt;/sub&gt;''y''&lt;sub&gt;1&lt;/sub&gt;''z'' represent a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or have a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''&lt;sub&gt;1&lt;/sub&gt;}''z'' ∪ ''x''{''Y'' \ ''y''&lt;sub&gt;2&lt;/sub&gt;}''z''.

e.g. in decimal (base ''b'' = 10):

* 2221 is a prime &gt; 10, thus the family 2{0,2}1 splits into the two families 2{0}1 and 2{0}2{0}1.
* 227 is a prime &gt; 10, and it is a subsequence of 5227, thus the family 5{0,2}7 splits into the two families 5{0}7 and 5{0}2{0}7.
* 449 is a prime &gt; 10, and it is a subsequence of 6449, thus the family 6{0,3,4,6,9}9 splits into the two families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9.
* Both 5051 and 5501 are primes &gt; 10, thus the family 5{0,5}1 splits into the two families 5{0}1 and 5{5}1 = {5}1.
* 8501 is a prime &gt; 10, thus the family 8{0,5}1 splits into the family 8{0}{5}1.
* 887 is a prime &gt; 10, and it is a subsequence of 2887, also 2087 is a prime &gt; 10, thus the family 2{0,8}7 splits into the two families 2{0}7 and 28{0}7.
* 349 and 449 are primes &gt; 10, and they are subsequences of 9349 and 9449, respectively, also 9049, 9649, 9949 are primes &gt; 10, thus the family 9{0,3,4,6,9}9 splits into the two families 9{0,3,6,9}9 and 94{0,3,6,9}9.
* 251, 281, 521, 821, 881 are primes &gt; 10, and they are subsequences of 9251, 9281, 9521, 9821, 9881, respectively, also 9001, 9221, 9551, 9851 are primes &gt; 10, thus the family 9{0,2,5,8}1 splits into the numbers {91, 901, 921, 951, 981, 9021, 9051, 9081, 9201, 9501, 9581, 9801, 90581, 95081, 95801}.

If the methods we have discussed cannot be used to rule out or shrink ''x''{''Y''}''z'' where ''Y'' = {''y''&lt;sub&gt;1&lt;/sub&gt;, ''y''&lt;sub&gt;2&lt;/sub&gt;, ..., ''y''&lt;sub&gt;''n''&lt;/sub&gt;}, then we can replace ''x''{''Y''}''z'' by ''xy''&lt;sub&gt;1&lt;/sub&gt;{''Y''}''z'' ∪ ''xy''&lt;sub&gt;2&lt;/sub&gt;{''Y''}''z'' ∪ ... ∪ ''xy''&lt;sub&gt;''n''&lt;/sub&gt;{''Y''}''z'' and re-run the methods on this new [[:w:Formal language|language]].

If all remain families are linear families (i.e. of the form ''x''{''y''}''z'', where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''), then we search the smallest (probable) primes in these families and add these primes to the list.

e.g. in decimal (base ''b'' = 10):

* The smallest prime in the family 5{0}27 is 5000000000000000000000000000027.
* The smallest prime in the family {5}1 is 555555555551.
* The smallest prime in the family 8{5}1 is 8555555555555555555551, but 8555555555555555555551 is not a minimal element since 555555555551 is a subsequence of 8555555555555555555551.

There is no guarantee that the techniques discussed will ever terminate, but in practice they often do. They are able to determine the Athena set in base ''b'' for 2 ≤ ''b'' ≤ 16 and ''b'' = 18, 20, 22, 24, 30. The bases ''b'' = 17, 19, 21, 23, 25 ≤ ''b'' ≤ 29, 31 ≤ ''b'' ≤ 36 are solved with the exception of 771 families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'').

The following is a &quot;[[:w:Semi-algorithm|semi-algorithm]]&quot; that is guaranteed to solve the Athena problem for a given base ''b'', but it is not so easy to implement:

# ''M'' = ''[[:w:Empty string|∅]]''
# while (''L'' ≠ ''∅'') do
# choose ''x'', a shortest string in ''L''
# ''M'' := ''M'' ∪ {''x''}
# ''L'' := ''L'' − ''sup''({''x''})

In practice, for arbitrary ''L'', we cannot feasibly carry out step 5. Instead, we work with ''L''&amp;#39;, some regular overapproximation to ''L'', until we can show ''L''&amp;#39; = ''∅'' (which implies ''L'' = ''∅''). In practice, ''L''&amp;#39; is usually chosen to be a finite [[:w:Union (set theory)|union]] of sets of the form ''L''&lt;sub&gt;1&lt;/sub&gt;{''L''&lt;sub&gt;2&lt;/sub&gt;}''L''&lt;sub&gt;3&lt;/sub&gt;, where each of ''L''&lt;sub&gt;1&lt;/sub&gt;, ''L''&lt;sub&gt;2&lt;/sub&gt;, ''L''&lt;sub&gt;3&lt;/sub&gt; is finite. In the case we consider in this project, we then have to determine whether such a family contains a prime or not.

Thus, the [[:w:Time complexity|time complexity]] of the Athena problem in base ''b'' may be ''[[:w:Big O notation|O]]''(''[[:w:E (mathematical_constant)|e]]''&lt;sup&gt;''[[:w:Euler's constant|γ]]''×(''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(*b*)&lt;/sup&gt;), and the [[:w:CPU time|CPU time]] of the Athena problem in base ''b'' may be longer than [[:w:Age of the universe|the age of the universe]] for bases ''b'' = 19, 23, 25, 27, 29, 31, 32, 33, 34, 35, also, Athena problem in bases ''b'' around 500 may be [[:w:NP-complete|NP-complete]] or [[:w:NP-hard|NP-hard]], or an [[:w:Undecidable problem|undecidable problem]], or an example of [[:w:Gödel's incompleteness theorems|Gödel's incompleteness theorems]] (like the [[:w:Continuum hypothesis|continuum hypothesis]] and the [[:w:Halting problem|halting problem]]).

To solve the Athena problem (i.e. to compute the Athena set), we need to determine whether a given family contains a prime. In practice, if family ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''Y'' is a set of digits in base ''b'') could not be ruled out as only containing composites and ''Y'' contains two or more digits, then a relatively small prime &gt; ''b'' could always be found in this family. Intuitively, this is because there are a large number of small strings in such a family, and at least one is likely to be prime (e.g. there are 2&lt;sup&gt;''n''−2&lt;/sup&gt; strings of length ''n'' in the family 1{3,7}9, and there are over a thousand strings of length 12 in the family 1{3,7}9, thus it is very impossible that these numbers are all composite). In the case ''Y'' contains only one digit, this family is of the form ''x''{''y''}''z'', and there is only a single string of each length &gt; (the length of ''x'' + the length of ''z''), and it is not known if the following [[:w:Decision problem|decision problem]] is recursively solvable (just like [[:w:Sierpiński number|Sierpiński problem]] and [[:w:Riesel number|Riesel problem]], Sierpiński problem and Riesel problem can be generalized to other bases ''b'' (references: http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm), in fact, Athena problem base ''b'' covers the Sierpiński problem base ''b'' and the Riesel problem base ''b'' with ''k'' &lt; ''b'', i.e. finding the smallest prime of the form ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 and ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (or prove such prime does not exist) with ''k'' &lt; ''b'' (specially, for bases ''b'' such that the conjectured smallest Sierpiński number or the conjectured smallest Riesel number is &lt; ''b'', Athena problem base ''b'' covers the Sierpiński problem base ''b'' or the Riesel problem base ''b'', respectively), since the smallest prime of the form ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 and ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (if exists) must be a minimal element in base ''b'', also, Athena problem base ''b'' covers finding the smallest prime of these forms in base ''b'' (or proving that such prime does not exist): (''b''&lt;sup&gt;''n''&lt;/sup&gt;−1)/(''b''−1) (for this form, ''n'' must be prime, and we want ''n'' ≥ 2) (references of this form: http://www.fermatquotient.com/PrimSerien/GenRepu.txt, https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html, http://www.primenumbers.net/Henri/us/MersFermus.htm, http://www.bitman.name/math/table/379, https://pzktupel.de/Primetables/TableRepunitGen.php, https://oeis.org/A084740, https://oeis.org/A084738, https://oeis.org/A128164, https://oeis.org/A285642; or for prime bases ''b'': https://oeis.org/A065854, https://oeis.org/A279068), ''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 (for this form, ''n'' must be power of 2, and we want ''n'' ≥ 1) (references of this form: http://jeppesn.dk/generalized-fermat.html, http://www.noprimeleftbehind.net/crus/GFN-primes.htm, https://web.archive.org/web/20231002190634/http://yves.gallot.pagesperso-orange.fr/primes/index.html, https://pzktupel.de/Primetables/TableFermatGFBB.php, https://oeis.org/A079706, https://oeis.org/A084712, https://oeis.org/A228101), (''b''&lt;sup&gt;''n''&lt;/sup&gt;+1)/2 (for odd ''b'') (for this form, ''n'' must be power of 2, and we want ''n'' ≥ 2) (reference of this form: http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt), (''sqrt''(''b'')×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1)/(''sqrt''(''b'')+1) (for square ''b'') (for this form, 2×''n''+1 must be prime, and we want ''n'' ≥ 2) (references of this form: http://www.fermatquotient.com/PrimSerien/GenRepuP.txt, http://www.primenumbers.net/Henri/us/MersFermus.htm, http://www.bitman.name/math/table/488, https://pzktupel.de/Primetables/TableWagstaffGen.php, https://oeis.org/A084742, https://oeis.org/A084741; or for bases ''b'' with ''sqrt''(''b'') prime: https://oeis.org/A065507), ((''b''−2)×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1)/(''b''−1) (''n'' ≥ 2) (reference of this form: https://oeis.org/A243404), 2×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 (''n'' ≥ 1) (references of this form: https://www.mersenneforum.org/showthread.php?t=6918, https://www.mersenneforum.org/showthread.php?t=19725, https://oeis.org/A119624), 2×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (''n'' ≥ 1) (references of this form: https://www.mersenneforum.org/showthread.php?t=24576, https://www.mersenneforum.org/attachment.php?attachmentid=20976&amp;d=1567314217, https://oeis.org/A119591), ''b''&lt;sup&gt;''n''&lt;/sup&gt;+2 (''n'' ≥ 1) (references of this form: https://oeis.org/A138066, https://oeis.org/A084713, https://oeis.org/A138067), ''b''&lt;sup&gt;''n''&lt;/sup&gt;−2 (''n'' ≥ 2) (references of this form: https://www.primepuzzles.net/puzzles/puzz_887.htm, https://oeis.org/A250200, https://oeis.org/A255707, https://oeis.org/A084714; or for prime bases ''b'': https://oeis.org/A292201), (''b''−1)×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 (''n'' ≥ 1) (references of this form: http://www.noprimeleftbehind.net/Williams-primes-MP.htm, http://www.bitman.name/math/table/477, https://pzktupel.de/Primetables/TableWilliams2.php, https://oeis.org/A305531; or for prime bases ''b'': https://oeis.org/A087139), (''b''−1)×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (''n'' ≥ 1) (references of this form: https://harvey563.tripod.com/wills.txt, http://www.noprimeleftbehind.net/Williams-primes-MM.htm, http://www.bitman.name/math/table/484, https://pzktupel.de/Primetables/TableWilliams1.php; or for prime bases ''b'': https://oeis.org/A122396), ''b''&lt;sup&gt;''n''&lt;/sup&gt;+(''b''−1) (''n'' ≥ 1) (references of this form: http://www.bitman.name/math/table/795, https://pzktupel.de/Primetables/TableWilliams6.php, https://oeis.org/A076845, https://oeis.org/A076846, https://oeis.org/A078178, https://oeis.org/A078179), ''b''&lt;sup&gt;''n''&lt;/sup&gt;−(''b''−1) (''n'' ≥ 2) (references of this form: http://www.bitman.name/math/table/792, https://pzktupel.de/Primetables/TableWilliams5.php, https://oeis.org/A113516, https://oeis.org/A343589; or for prime bases ''b'': https://cs.uwaterloo.ca/journals/JIS/VOL3/mccranie.html, http://www.bitman.name/math/table/435)):

Problem: Given strings ''x'', ''z'' (may be empty), a digit ''y'', and a base ''b'' (''x'' does not [[:w:Leading zero|start with the digit 0]], ''z'' ends with a digit which [[:w:Coprime integers|coprime]] to ''b'', ''y'' is not 0 if ''x'' is empty, ''y'' is coprime to ''b'' if ''z'' is empty), does there exist a prime number whose base-''b'' expansion is of the form ''xy''&lt;sub&gt;''n''&lt;/sub&gt;''z'' for some ''n'' ≥ 0?

Some families can be ruled out to contain no prime &gt; ''b'' by [[:w:Covering set|covering congruence]], [[:w:Factorization of polynomials|algebraic factorization]] (e.g. [[:w:Difference of two squares|difference of two squares]], [[:w:Sum of two cubes|sum of two cubes]], [[:w:Sophie Germain's identity|Sophie Germain's identity of ''x''&lt;sup&gt;4&lt;/sup&gt;+4×''y''&lt;sup&gt;4&lt;/sup&gt;]]), or combine of them, e.g.

* The base 9 family 2{7}: Always divisible by 2 or 5
* The base 11 family 2{5}: Always divisible by 2 or 3
* The base 14 family B{0}1: Always divisible by 3 or 5
* The base 13 family 95{0}3: Always divisible by 5, 7, or 17
* The base 16 family {4}D: Always divisible by 3, 7, or 13
* The base 16 family {8}F: Always divisible by 3, 7, or 13
* The base 21 family {7}D: Always divisible by 2, 13, or 17
* The base 23 family {D}GA: Always divisible by 2, 5, 7, 37, or 79
* The base 9 family {1}: Can be written as (9&lt;sup&gt;''n''&lt;/sup&gt;−1)/8 and can be factored as (3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (3&lt;sup&gt;''n''&lt;/sup&gt;+1) / 8
* The base 8 family 1{0}1: Can be written as 8&lt;sup&gt;''n''&lt;/sup&gt;+1 and can be factored as (2&lt;sup&gt;''n''&lt;/sup&gt;+1) × (4&lt;sup&gt;''n''&lt;/sup&gt;−2&lt;sup&gt;''n''&lt;/sup&gt;+1)
* The base 9 family 3{8}: Can be written as 4×9&lt;sup&gt;''n''&lt;/sup&gt;−1 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (2×3&lt;sup&gt;''n''&lt;/sup&gt;+1)
* The base 16 family 1{5}: Can be written as (4×16&lt;sup&gt;''n''&lt;/sup&gt;−1)/3 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (2×3&lt;sup&gt;''n''&lt;/sup&gt;+1) / 3
* The base 16 family {4}1: Can be written as (4×16&lt;sup&gt;''n''&lt;/sup&gt;−49)/15 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−7) × (2×3&lt;sup&gt;''n''&lt;/sup&gt;+7) / 15
* The base 27 family 7{Q}: Can be written as 8×27&lt;sup&gt;''n''&lt;/sup&gt;−1 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (4×9&lt;sup&gt;''n''&lt;/sup&gt;+2×3&lt;sup&gt;''n''&lt;/sup&gt;+1)
* The base 27 family 9{G}: Can be written as (125×27&lt;sup&gt;''n''&lt;/sup&gt;−8)/13 and can be factored as (5×3&lt;sup&gt;''n''&lt;/sup&gt;−2) × (25×9&lt;sup&gt;''n''&lt;/sup&gt;+10×3&lt;sup&gt;''n''&lt;/sup&gt;+4)
* The base 16 family {C}D: Can be written as (4×16&lt;sup&gt;''n''&lt;/sup&gt;+1)/5 and can be factored as (2×4&lt;sup&gt;''n''&lt;/sup&gt;−2×2&lt;sup&gt;''n''&lt;/sup&gt;+1) × (2×4&lt;sup&gt;''n''&lt;/sup&gt;+2×2&lt;sup&gt;''n''&lt;/sup&gt;+1) / 5
* The base 14 family 8{D}: Can be written as 9×14&lt;sup&gt;''n''&lt;/sup&gt;−1, it is divisible by 5 if ''n'' is odd and can be factored as (3×14&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (3×14&lt;sup&gt;''n''/2&lt;/sup&gt;+1) if ''n'' is even
* The base 12 family {B}9B: Can be written as 12&lt;sup&gt;''n''&lt;/sup&gt;−25, it is divisible by 13 if ''n'' is odd and can be factored as (12&lt;sup&gt;''n''/2&lt;/sup&gt;−5) × (12&lt;sup&gt;''n''/2&lt;/sup&gt;+5) if ''n'' is even
* The base 14 family {D}5: Can be written as 14&lt;sup&gt;''n''&lt;/sup&gt;−9, it is divisible by 5 if ''n'' is odd and can be factored as (14&lt;sup&gt;''n''/2&lt;/sup&gt;−3) × (14&lt;sup&gt;''n''/2&lt;/sup&gt;+3) if ''n'' is even
* The base 17 family 1{9}: Can be written as (25×17&lt;sup&gt;''n''&lt;/sup&gt;−9)/16, it is divisible by 2 if ''n'' is odd and can be factored as (5×17&lt;sup&gt;''n''/2&lt;/sup&gt;−3) × (5×17&lt;sup&gt;''n''/2&lt;/sup&gt;+3) / 16 if ''n'' is even
* The base 17 family 7{9}: Can be written as (121×17&lt;sup&gt;''n''&lt;/sup&gt;−9)/16, it is divisible by 2 if ''n'' is odd and can be factored as (11×17&lt;sup&gt;''n''/2&lt;/sup&gt;−3) × (11×17&lt;sup&gt;''n''/2&lt;/sup&gt;+3) / 16 if ''n'' is even
* The base 19 family 1{6}: Can be written as (4×19&lt;sup&gt;''n''&lt;/sup&gt;−1)/3, it is divisible by 5 if ''n'' is odd and can be factored as (2×19&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (2×19&lt;sup&gt;''n''/2&lt;/sup&gt;+1) / 3 if ''n'' is even
* The base 19 family 7{2}: Can be written as (64×19&lt;sup&gt;''n''&lt;/sup&gt;−1)/9, it is divisible by 5 if ''n'' is odd and can be factored as (8×19&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (8×19&lt;sup&gt;''n''/2&lt;/sup&gt;+1) / 9 if ''n'' is even
* The base 24 family 3{N}: Can be written as 4×24&lt;sup&gt;''n''&lt;/sup&gt;−1, it is divisible by 5 if ''n'' is odd and can be factored as (2×24&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (2×24&lt;sup&gt;''n''/2&lt;/sup&gt;+1) if ''n'' is even

By the [[:w:Prime number theorem|prime number theorem]], the [[:w:Probability|chance]] that a [[:w:Random number|random]] ''n''-digit base ''b'' number is prime is [[:w:Asymptotic analysis|approximately]] 1/''n'' (more accurately, the chance is approximately 1/(''n''×''ln''(''b'')), where ''ln'' is the [[:w:Natural logarithm|natural logarithm]]). If one conjectures the numbers ''x''{''y''}''z'' behave similarly (i.e. the numbers ''x''{''y''}''z'' is a [[:w:Pseudorandomness|pseudorandom sequence]]) you would expect [[:w:Harmonic_series (mathematics)|1/1 + 1/2 + 1/3 + 1/4 + ... = ∞]] primes of the form ''x''{''y''}''z'' (of course, this does not always happen, since some ''x''{''y''}''z'' families can be ruled out to contain no prime &gt; ''b'' (by covering congruence, algebraic factorization, or combine of them), but it is at least a reasonable conjecture in the absence of evidence to the contrary. Hence, the [[:w:Heuristic argument|heuristic argument]] suggests there are always infinitely many primes in family ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') if it cannot be ruled out to contain no prime or only contain finitely many primes, by covering congruence, algebraic factorization, or combine of them. However, some families ''x''{''y''}''z'' could not be proven to contain no primes &gt; ''b'' (by covering congruence, algebraic factorization, or combine of them) but no primes &gt; ''b'' could be found in the family, even after searching through numbers with over 100000 digits. In such a case, the only way to proceed is to [[:w:Primality test|test the primality]] of larger and larger numbers of such form and hope a prime is eventually discovered. e.g. the smallest (probable) prime in the family A{3}A in base ''b'' = 13 is A3&lt;sub&gt;592197&lt;/sub&gt;A, its algebraic form is (41×13&lt;sup&gt;592198&lt;/sup&gt;+27)/4, when written in decimal contains 659677 digits (it is only probable prime, i.e. not definitely prime, since technically, probable primality tests were used to show this (which have a ''very'' small chance of making an error, see https://t5k.org/notes/prp_prob.html) because all known primality tests run far too slowly to run on numbers of this size unless either [https://t5k.org/prove/prove3_1.html ''N''−1] or [https://t5k.org/prove/prove3_2.html ''N''+1] (or both) can be ≥ 1/3 factored).

'''Athena conjecture''': If family ''xy''&lt;sub&gt;''n''&lt;/sub&gt;''z'' (with fixed strings ''x'', ''z'' (may be empty), fixed digit ''y'', and variable ''n'') in base ''b'' (with fixed ''b'' ≥ 2) (''x'' does not start with the digit 0, ''z'' ends with a digit which coprime to ''b'', ''y'' is not 0 if ''x'' is empty, ''y'' is coprime to ''b'' if ''z'' is empty) cannot be proven to only contain composites or only contain finitely many primes (by covering congruence, algebraic factorization, or combine of them), then family ''xy''&lt;sub&gt;''n''&lt;/sub&gt;''z'' in base ''b'' contains infinitely many primes (this is equivalent to: If form (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1) (with fixed integers ''a'' ≥ 1, ''b'' ≥ 2, ''c'' ≠ 0 (with ''gcd''(''a'',''c'') = 1 and ''gcd''(''b'',''c'') = 1), and variable ''n'') cannot be proven to only contain composites or only contain finitely many primes (by covering congruence, algebraic factorization, or combine of them), then form (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1) contains infinitely many primes)

The numbers in family ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') are of the form (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1) for some fixed ''a'', ''b'', ''c'' such that ''a'' ≥ 1, ''b'' ≥ 2 (''b'' is the base), ''c'' ≠ 0, ''gcd''(''a'',''c'') = 1, ''gcd''(''b'',''c'') = 1. Except in the [[:w:Special case|special case]] ''c'' = ±1 and ''gcd''(''a''+''c'',''b''−1) = 1, when ''n'' is large the known [[:w:Primality test|primality test]]s for such a number are too inefficient to run. In this case one must resort to a [[:w:Probabilistic algorithm|probable]] primality test such as a [[:w:Miller–Rabin primality test|Miller–Rabin primality test]] or a [[:w:Baillie–PSW primality test|Baillie–PSW primality test]], unless a divisor of the number can be found. Since we are testing many numbers in an [[:w:Exponential growth|exponential sequence]], it is possible to use a sieving process to find divisors rather than using [[:w:Trial division|trial division]].

To do this, we made use of Geoffrey Reynolds' ''srsieve'' software (download: https://pzktupel.de/Software/srsieve_1.1.4.7z). This program uses the [[:w:Baby-step giant-step|baby-step giant-step]] [[:w:Algorithm|algorithm]] to find all primes ''p'' which divide ''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'' where ''p'' and ''n'' lie in a [[:w:Interval_(mathematics)|specified range]]. Since this program cannot handle the general case (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1) when ''gcd''(''a''+''c'',''b''−1) &gt; 1 we only used it to sieve the sequence ''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'' for primes ''p'' not dividing ''gcd''(''a''+''c'',''b''−1), and initialized the list of candidates to not include ''n'' for which there is some prime ''p'' dividing ''gcd''(''a''+''c'',''b''−1) for which ''p'' dividing (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1). The program had to be modified slightly to remove a check which would prevent it from running in the case when ''a'', ''b'', and ''c'' were all odd (since then 2 divides ''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'', but 2 may not divide (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1)).

Once the numbers with small divisors had been removed, it remained to test the remaining numbers using a probable primality test. For this we used the software ''LLR'' by Jean Penné. (download: http://jpenne.free.fr/index2.html). Although undocumented, it is possible to run this program on numbers of the form (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1) when ''gcd''(''a''+''c'',''b''−1) &gt; 1, so this program required no modifications. A script was also written which allowed one to run ''srsieve'' while ''LLR'' was testing the remaining candidates, so that when a divisor was found by srsieve on a number which had not yet been tested by ''LLR'' it would be removed from the list of candidates.

For the primes &lt; 10&lt;sup&gt;25000&lt;/sup&gt; for the &quot;easy&quot; bases (bases ''b'' with ≤ 150 primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; (base ''b'' = 26 has 83 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 3 unsolved families, base ''b'' = 36 has 75 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 4 unsolved families, base ''b'' = 17 has 99 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 18 unsolved families, base ''b'' = 21 has 80 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 12 unsolved families, base ''b'' = 19 has 201 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 23 unsolved families), i.e. bases *b* = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 26, 28, 30, 36), we employed ''CM'' by Andreas Enge (download: https://www.multiprecision.org/cm/download.html), an elliptic curve primality proving implementation.

== Data ==

These are the results of the Athena problem in bases 2 ≤ ''b'' ≤ 36 (we stop at base 36 since this base is the maximum base for which it is possible to write the numbers with the [[:w:Symbol|symbol]]s 0, 1, 2, ..., 9 and A, B, C, ..., Z (i.e. the 10 [[:w:Arabic numerals|Arabic numerals]] and the 26 [[:w:Latin script|Latin letters]]): (some large Athena primes are only probable primes, i.e. not definitely primes, since they are too large to be [[:w:Elliptic curve primality|ECPP proved]] and [[:w:Pocklington primality test#Extensions and variants|neither ''N''−1 nor ''N''+1 can be ≥ 1/3 factored]], all of them pass the [[:w:Baillie–PSW primality test|Baillie–PSW primality test]] and the [[:w:Strong pseudoprime|strong primality test]] (i.e. the [[:w:Miller–Rabin primality test|Miller–Rabin primality test]]) with all prime bases ''p'' ≤ 61, however, all Athena primes &lt; 10&lt;sup&gt;25000&lt;/sup&gt; for bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 36 are definitely primes, most of them &gt; 10&lt;sup&gt;299&lt;/sup&gt; are proven primes with [[:w:Elliptic curve primality|ECPP proving]], others &gt; 10&lt;sup&gt;299&lt;/sup&gt; are proven primes with [[:w:Pocklington primality test#Extensions and variants|''N''−1 or ''N''+1 proving]])

All numbers are written in base ''b'', [[:w:Senary#Base 36 as senary compression|using A to Z to represent digit values 10 to 35]], &quot;{}&quot; means repeating, e.g. family 12{3}45 means the sequence {1245, 12345, 123345, 1233345, 12333345, 123333345, ...} (where the members are expressed as base ''b'' strings), subscripts are used to indicate repetitions of digits, e.g. 123&lt;sub&gt;4&lt;/sub&gt;567 means 123333567 (all subscripts are written in decimal).

Base 2: 1 Athena prime (the largest of which has 2 digits (it is 11, and its value is 3 in decimal)): {11}

Base 3: 3 Athena primes (the largest of which has 3 digits (it is 111, and its value is 13 in decimal)): {12, 21, 111}

Base 4: 5 Athena primes (the largest of which has 3 digits (it is 221, and its value is 41 in decimal)): {11, 13, 23, 31, 221}

Base 5: 22 Athena primes (the largest of which has 96 digits (it is 10&lt;sub&gt;93&lt;/sub&gt;13, and its algebraic form is 5&lt;sup&gt;95&lt;/sup&gt;+8)): {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 Athena primes (the largest of which has 5 digits (it is 40041, and its value is 5209 in decimal)): {11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041}

Base 7: 71 Athena primes (the largest of which has 17 digits (it is 3&lt;sub&gt;16&lt;/sub&gt;1, and its algebraic form is (7&lt;sup&gt;17&lt;/sup&gt;−5)/2)): {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: 75 Athena primes (the largest of which has 221 digits (it is 4&lt;sub&gt;220&lt;/sub&gt;7, and its algebraic form is (4×8&lt;sup&gt;221&lt;/sup&gt;+17)/7)): {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: 151 Athena primes (the largest of which has 1161 digits (it is 30&lt;sub&gt;1158&lt;/sub&gt;11, and its algebraic form is 3×9&lt;sup&gt;1160&lt;/sup&gt;+10)): {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: 77 Athena primes (the largest of which has 31 digits (it is 50&lt;sub&gt;28&lt;/sub&gt;27, and its algebraic form is 5×10&lt;sup&gt;30&lt;/sup&gt;+27)): {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: 1068 Athena primes (including 1 unproven probable prime: 57&lt;sub&gt;62668&lt;/sub&gt;), the largest of which has 62669 digits (it is 57&lt;sub&gt;62668&lt;/sub&gt;, and its algebraic form is (57×11&lt;sup&gt;62668&lt;/sup&gt;−7)/10), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel11 Data of Athena (probable) primes base 11]

Base 12: 106 Athena primes (the largest of which has 42 digits (it is 40&lt;sub&gt;39&lt;/sub&gt;77, and its algebraic form is 4×12&lt;sup&gt;41&lt;/sup&gt;+91)): {11, 15, 17, 1B, 25, 27, 31, 35, 37, 3B, 45, 4B, 51, 57, 5B, 61, 67, 6B, 75, 81, 85, 87, 8B, 91, 95, A7, AB, B5, B7, 221, 241, 2A1, 2B1, 2BB, 401, 421, 447, 471, 497, 565, 655, 665, 701, 70B, 721, 747, 771, 77B, 797, 7A1, 7BB, 907, 90B, 9BB, A41, B21, B2B, 2001, 200B, 202B, 222B, 229B, 292B, 299B, 4441, 4707, 4777, 6A05, 6AA5, 729B, 7441, 7B41, 929B, 9777, 992B, 9947, 997B, 9997, A0A1, A201, A605, A6A5, AA65, B001, B0B1, BB01, BB41, 600A5, 7999B, 9999B, AAAA1, B04A1, B0B9B, BAA01, BAAA1, BB09B, BBBB1, 44AAA1, A00065, BBBAA1, AAA0001, B00099B, AA000001, BBBBBB99B, B0000000000000000000000000009B, 400000000000000000000000000000000000000077}

Base 13: 3197 Athena primes (including 4 unproven probable primes: C5&lt;sub&gt;23755&lt;/sub&gt;C, 80&lt;sub&gt;32017&lt;/sub&gt;111, 95&lt;sub&gt;197420&lt;/sub&gt;, A3&lt;sub&gt;592197&lt;/sub&gt;A), the largest of which has 592199 digits (it is A3&lt;sub&gt;592197&lt;/sub&gt;A, and its algebraic form is (41×13&lt;sup&gt;592198&lt;/sup&gt;+27)/4), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel13 Data of Athena (probable) primes base 13]

Base 14: 650 Athena primes, the largest of which has 19699 digits (it is 4D&lt;sub&gt;19698&lt;/sub&gt;, and its algebraic form is 5×14&lt;sup&gt;19698&lt;/sup&gt;−1), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel14 Data of Athena primes base 14]

Base 15: 1284 Athena primes, the largest of which has 157 digits (it is 7&lt;sub&gt;155&lt;/sub&gt;97, and its algebraic form is (15&lt;sup&gt;157&lt;/sup&gt;+59)/2), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel15 Data of Athena primes base 15]

Base 16: 2347 Athena primes (including 3 unproven probable primes: DB&lt;sub&gt;32234&lt;/sub&gt;, 4&lt;sub&gt;72785&lt;/sub&gt;DD, 3&lt;sub&gt;116137&lt;/sub&gt;AF), the largest of which has 116139 digits (it is 3&lt;sub&gt;116137&lt;/sub&gt;AF, and its algebraic form is (16&lt;sup&gt;116139&lt;/sup&gt;+619)/5), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel16 Data of Athena (probable) primes base 16]

Base 17: 10415 known Athena primes (including many unproven probable primes) and 12 unsolved families (1{7}, 1F{0}7, 4{7}A, 70F{0}D, 8{B}9, 9{5}9, A{D}F, B{0}B3, {B}E9, {B}EE, F1{9}, FD0{D}, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel17 Data of known Athena (probable) primes base 17]

Base 18: 549 Athena primes, the largest of which has 6271 digits (it is C0&lt;sub&gt;6268&lt;/sub&gt;C5, and its algebraic form is 12×18&lt;sup&gt;6270&lt;/sup&gt;+221), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel18 Data of Athena primes base 18]

Base 19: 31417 known Athena primes (including many unproven probable primes) and 17 unsolved families (4B5{0}H, {5}3, 5{H}05, 5{H}0H, 5{H}5, 66{B}, 71{0}177, 7AF{0}H, 97{0}3, C{H}C, EE1{6}, F{7}5, F{B}G, F{D}F, H0F{0}7A, HB{0}5B5, II{D}, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel19 Data of known Athena (probable) primes base 19]

Base 20: 3314 Athena primes, the largest of which has 6271 digits (it is G0&lt;sub&gt;6269&lt;/sub&gt;D, and its algebraic form is 16×20&lt;sup&gt;6270&lt;/sup&gt;+13), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel20 Data of Athena primes base 20]

Base 21: 13386 known Athena primes (including many unproven probable primes) and 8 unsolved families (5{0}DJ, {9}D, B3{0}EB, B{H}6H, C{F}0K, {F}35, G{0}FK, H{0}7771, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel21 Data of known Athena (probable) primes base 21]

Base 22: 8003 Athena primes (including 1 unproven probable prime: BK&lt;sub&gt;22001&lt;/sub&gt;5), the largest of which has 22003 digits (it is BK&lt;sub&gt;22001&lt;/sub&gt;5, and its algebraic form is (251×22&lt;sup&gt;22002&lt;/sup&gt;−335)/21), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel22 Data of Athena (probable) primes base 22]

Base 23: 65178 known Athena primes (including many unproven probable primes) and 87 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel23 Data of known Athena (probable) primes base 23] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left23 Data of unsolved families for Athena problem base 23]

Base 24: 3409 Athena primes, the largest of which has 8134 digits (it is N00N&lt;sub&gt;8129&lt;/sub&gt;LN, and its algebraic form is 13249×24&lt;sup&gt;8131&lt;/sup&gt;−49), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel24 Data of Athena primes base 24]

Base 25: 133639 known Athena primes (including many unproven probable primes) and 85 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel25 Data of known Athena (probable) primes base 25] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left25 Data of unsolved families for Athena problem base 25]

Base 26: 25256 known Athena primes (including 7 unproven probable primes: 5&lt;sub&gt;19391&lt;/sub&gt;6F, 7&lt;sub&gt;20279&lt;/sub&gt;OL, LD0&lt;sub&gt;20975&lt;/sub&gt;7, 6K&lt;sub&gt;23300&lt;/sub&gt;5, J0&lt;sub&gt;44303&lt;/sub&gt;KCB, M0&lt;sub&gt;61186&lt;/sub&gt;2BB, 85M&lt;sub&gt;197060&lt;/sub&gt;B) and 3 unsolved families ({A}6F, {H}MH, {I}GL, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel26 Data of known Athena (probable) primes base 26]

Base 27: 102852 known Athena primes (including many unproven probable primes) and 44 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel27 Data of known Athena (probable) primes base 27] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left27 Data of unsolved families for Athena problem base 27]

Base 28: 25528 known Athena primes (including 3 unproven probable primes: N6&lt;sub&gt;24051&lt;/sub&gt;LR, 5OA&lt;sub&gt;31238&lt;/sub&gt;F, O4O&lt;sub&gt;94535&lt;/sub&gt;9) and 1 unsolved family (O{A}F, no primes or probable primes with length ≤ 900000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel28 Data of known Athena (probable) primes base 28]

Base 29: 355242 known Athena primes (including many unproven probable primes) and 125 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel29 Data of known Athena (probable) primes base 29] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left29 Data of unsolved families for Athena problem base 29]

Base 30: 2619 Athena primes (including 1 unproven probable prime: I0&lt;sub&gt;24608&lt;/sub&gt;D), the largest of which has 34206 digits (it is OT&lt;sub&gt;34205&lt;/sub&gt;, and its algebraic form is 25×30&lt;sup&gt;34205&lt;/sup&gt;−1), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel30 Data of Athena (probable) primes base 30]

Base 31: 569323 known Athena primes (including many unproven probable primes) and 77 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel31 Data of known Athena (probable) primes base 31] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left31 Data of unsolved families for Athena problem base 31]

Base 32: 168882 known Athena primes (including many unproven probable primes) and 120 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel32 Data of known Athena (probable) primes base 32] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left32 Data of unsolved families for Athena problem base 32]

Base 33: 280012 known Athena primes (including many unproven probable primes) and 81 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel33 Data of known Athena (probable) primes base 33] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left33 Data of unsolved families for Athena problem base 33]

Base 34: 184785 known Athena primes (including many unproven probable primes) and 47 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel34 Data of known Athena (probable) primes base 34] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left34 Data of unsolved families for Athena problem base 34]

Base 35: 720002 known Athena primes (including many unproven probable primes) and 60 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel35 Data of known Athena (probable) primes base 35] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left35 Data of unsolved families for Athena problem base 35]

Base 36: 35286 known Athena primes (including 3 unproven probable primes: 7K&lt;sub&gt;26567&lt;/sub&gt;Z, S0&lt;sub&gt;75007&lt;/sub&gt;8H, P&lt;sub&gt;81993&lt;/sub&gt;SZ) and 4 unsolved families (B{0}EUV, HM{0}N, N{0}YYN, O{L}Z, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel36 Data of known Athena (probable) primes base 36]

== Condensed table for bases 2 ≤ ''b'' ≤ 36 ==

{|class=&quot;wikitable&quot;
||''b''||number of Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b''||base-''b'' form of the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b'' (write &quot;''d''&lt;sub&gt;''n''&lt;/sub&gt;&quot; if there are 5 or more (''n'') consecutive same digits ''d'')||length of the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b''||length of the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b'' in decimal||algebraic ((''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1)) form of the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b''||''factordb'' entry of the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b''||the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b'' written in base ''b'' (use lower case letters instead of upper case letters)||number of unsolved families in the Athena problem in base ''b'' (all of these left families are linear families)||searching limit of length for the unsolved families in the Athena problem in base ''b'' (if there are different searching limits for the unsolved families in the Athena problem in base ''b'', choose the lowest searching limit)||
|-
||2||1||11||2||1||3||http://factordb.com/index.php?id=3&amp;open=ecm||http://factordb.com/index.php?showid=3&amp;base=2||0||–||
|-
||3||3||111&lt;br&gt;21&lt;br&gt;12||3&lt;br&gt;2&lt;br&gt;2||2&lt;br&gt;1&lt;br&gt;1||13&lt;br&gt;7&lt;br&gt;5||http://factordb.com/index.php?id=13&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=7&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=5&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=13&amp;base=3&lt;br&gt;http://factordb.com/index.php?showid=7&amp;base=3&lt;br&gt;http://factordb.com/index.php?showid=5&amp;base=3&lt;nowiki/&gt;||0||–||
|-
||4||5||221&lt;br&gt;31&lt;br&gt;23&lt;br&gt;13&lt;br&gt;11||3&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2||2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;1&lt;br&gt;1||41&lt;br&gt;13&lt;br&gt;11&lt;br&gt;7&lt;br&gt;5||http://factordb.com/index.php?id=41&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=13&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=11&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=7&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=5&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=41&amp;base=4&lt;br&gt;http://factordb.com/index.php?showid=13&amp;base=4&lt;br&gt;http://factordb.com/index.php?showid=11&amp;base=4&lt;br&gt;http://factordb.com/index.php?showid=7&amp;base=4&lt;br&gt;http://factordb.com/index.php?showid=5&amp;base=4&lt;nowiki/&gt;||0||–||
|-
||5||22||10&lt;sub&gt;93&lt;/sub&gt;13&lt;br&gt;300031&lt;br&gt;44441&lt;br&gt;33331&lt;br&gt;33001&lt;br&gt;30301&lt;br&gt;14444&lt;br&gt;10103&lt;br&gt;3101&lt;br&gt;414||96&lt;br&gt;6&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;4&lt;br&gt;3||67&lt;br&gt;4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;3&lt;br&gt;3&lt;br&gt;3||5&lt;sup&gt;95&lt;/sup&gt;+8&lt;br&gt;9391&lt;br&gt;3121&lt;br&gt;2341&lt;br&gt;2251&lt;br&gt;1951&lt;br&gt;1249&lt;br&gt;653&lt;br&gt;401&lt;br&gt;109||http://factordb.com/index.php?id=1100000000034686071&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=9391&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=3121&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=2341&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=2251&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1951&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1249&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=653&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=401&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=109&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000000034686071&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=9391&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=3121&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=2341&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=2251&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=1951&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=1249&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=653&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=401&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=109&amp;base=5&lt;nowiki/&gt;||0||–||
|-
||6||11||40041&lt;br&gt;4441&lt;br&gt;4401&lt;br&gt;51&lt;br&gt;45&lt;br&gt;35&lt;br&gt;31&lt;br&gt;25&lt;br&gt;21&lt;br&gt;15||5&lt;br&gt;4&lt;br&gt;4&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2||4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2||5209&lt;br&gt;1033&lt;br&gt;1009&lt;br&gt;31&lt;br&gt;29&lt;br&gt;23&lt;br&gt;19&lt;br&gt;17&lt;br&gt;13&lt;br&gt;11||http://factordb.com/index.php?id=5209&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1033&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1009&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=31&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=29&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=23&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=19&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=17&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=13&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=11&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=5209&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=1033&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=1009&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=31&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=29&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=23&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=19&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=17&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=13&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=11&amp;base=6&lt;nowiki/&gt;||0||–||
|-
||7||71||3&lt;sub&gt;16&lt;/sub&gt;1&lt;br&gt;510&lt;sub&gt;7&lt;/sub&gt;1&lt;br&gt;3&lt;sub&gt;6&lt;/sub&gt;01&lt;br&gt;1100021&lt;br&gt;531101&lt;br&gt;351101&lt;br&gt;300053&lt;br&gt;150001&lt;br&gt;100121&lt;br&gt;40054||17&lt;br&gt;10&lt;br&gt;8&lt;br&gt;7&lt;br&gt;6&lt;br&gt;6&lt;br&gt;6&lt;br&gt;6&lt;br&gt;6&lt;br&gt;5||15&lt;br&gt;9&lt;br&gt;7&lt;br&gt;6&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;4||(7&lt;sup&gt;17&lt;/sup&gt;−5)/2&lt;br&gt;36×7&lt;sup&gt;8&lt;/sup&gt;+1&lt;br&gt;(7&lt;sup&gt;8&lt;/sup&gt;−47)/2&lt;br&gt;134471&lt;br&gt;91631&lt;br&gt;62819&lt;br&gt;50459&lt;br&gt;28813&lt;br&gt;16871&lt;br&gt;9643||http://factordb.com/index.php?id=116315256993601&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=207532837&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=2882377&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=134471&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=91631&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=62819&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=50459&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=28813&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=16871&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=9643&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=116315256993601&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=207532837&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=2882377&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=134471&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=91631&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=62819&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=50459&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=28813&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=16871&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=9643&amp;base=7&lt;nowiki/&gt;||0||–||
|-
||8||75||4&lt;sub&gt;220&lt;/sub&gt;7&lt;br&gt;5&lt;sub&gt;13&lt;/sub&gt;25&lt;br&gt;7&lt;sub&gt;12&lt;/sub&gt;1&lt;br&gt;77774&lt;sub&gt;6&lt;/sub&gt;1&lt;br&gt;74&lt;sub&gt;7&lt;/sub&gt;1&lt;br&gt;4&lt;sub&gt;8&lt;/sub&gt;1&lt;br&gt;5&lt;sub&gt;5&lt;/sub&gt;025&lt;br&gt;5550525&lt;br&gt;5500525&lt;br&gt;4&lt;sub&gt;5&lt;/sub&gt;77||221&lt;br&gt;15&lt;br&gt;13&lt;br&gt;11&lt;br&gt;9&lt;br&gt;9&lt;br&gt;8&lt;br&gt;7&lt;br&gt;7&lt;br&gt;7||200&lt;br&gt;14&lt;br&gt;12&lt;br&gt;10&lt;br&gt;9&lt;br&gt;8&lt;br&gt;8&lt;br&gt;7&lt;br&gt;7&lt;br&gt;7||(4×8&lt;sup&gt;221&lt;/sup&gt;+17)/7&lt;br&gt;(5×8&lt;sup&gt;15&lt;/sup&gt;−173)/7&lt;br&gt;8&lt;sup&gt;13&lt;/sup&gt;−7&lt;br&gt;(28669×8&lt;sup&gt;7&lt;/sup&gt;−25)/7&lt;br&gt;(53×8&lt;sup&gt;8&lt;/sup&gt;−25)/7&lt;br&gt;(4×8&lt;sup&gt;9&lt;/sup&gt;−25)/7&lt;br&gt;(5×8&lt;sup&gt;8&lt;/sup&gt;−2413)/7&lt;br&gt;1495381&lt;br&gt;1474901&lt;br&gt;(4×8&lt;sup&gt;7&lt;/sup&gt;+185)/7||http://factordb.com/index.php?id=1100000000416605822&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=25131694349141&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=549755813881&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=8589035809&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=127027489&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=76695841&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=11983381&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1495381&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1474901&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1198399&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000000416605822&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=25131694349141&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=549755813881&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=8589035809&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=127027489&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=76695841&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=11983381&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=1495381&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=1474901&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=1198399&amp;base=8&lt;nowiki/&gt;||0||–||
|-
||9||151||30&lt;sub&gt;1158&lt;/sub&gt;11&lt;br&gt;27&lt;sub&gt;686&lt;/sub&gt;07&lt;br&gt;76&lt;sub&gt;329&lt;/sub&gt;2&lt;br&gt;561&lt;sub&gt;36&lt;/sub&gt;&lt;br&gt;10&lt;sub&gt;25&lt;/sub&gt;57&lt;br&gt;30&lt;sub&gt;20&lt;/sub&gt;51&lt;br&gt;8&lt;sub&gt;19&lt;/sub&gt;335&lt;br&gt;727&lt;sub&gt;15&lt;/sub&gt;07&lt;br&gt;51&lt;sub&gt;13&lt;/sub&gt;61&lt;br&gt;10&lt;sub&gt;11&lt;/sub&gt;507||1161&lt;br&gt;689&lt;br&gt;331&lt;br&gt;38&lt;br&gt;28&lt;br&gt;23&lt;br&gt;22&lt;br&gt;19&lt;br&gt;16&lt;br&gt;15||1108&lt;br&gt;657&lt;br&gt;316&lt;br&gt;37&lt;br&gt;26&lt;br&gt;22&lt;br&gt;21&lt;br&gt;19&lt;br&gt;16&lt;br&gt;14||3×9&lt;sup&gt;1160&lt;/sup&gt;+10&lt;br&gt;(23×9&lt;sup&gt;688&lt;/sup&gt;−511)/8&lt;br&gt;(31×9&lt;sup&gt;330&lt;/sup&gt;−19)/4&lt;br&gt;(409×9&lt;sup&gt;36&lt;/sup&gt;−1)/8&lt;br&gt;9&lt;sup&gt;27&lt;/sup&gt;+52&lt;br&gt;3×9&lt;sup&gt;22&lt;/sup&gt;+46&lt;br&gt;9&lt;sup&gt;22&lt;/sup&gt;−454&lt;br&gt;(527×9&lt;sup&gt;17&lt;/sup&gt;−511)/8&lt;br&gt;(41×9&lt;sup&gt;15&lt;/sup&gt;+359)/8&lt;br&gt;9&lt;sup&gt;14&lt;/sup&gt;+412||http://factordb.com/index.php?id=1100000002376318423&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000002495467486&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000002359003642&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000001554010824&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1100000002512830927&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1100000000032261811&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1100000002495736583&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1100000003446800389&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1055192051985121&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=22876792455373&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000002376318423&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=1100000002495467486&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=1100000002359003642&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=1100000001554010824&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=1100000002512830927&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=1100000000032261811&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=1100000002495736583&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=1100000003446800389&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=1055192051985121&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=22876792455373&amp;base=9&lt;nowiki/&gt;||0||–||
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||23||65178~65265||B0&lt;sub&gt;93046&lt;/sub&gt;FB&lt;br&gt;L&lt;sub&gt;86444&lt;/sub&gt;D&lt;br&gt;AJ&lt;sub&gt;81065&lt;/sub&gt;4&lt;br&gt;20&lt;sub&gt;73560&lt;/sub&gt;98&lt;br&gt;J&lt;sub&gt;68217&lt;/sub&gt;G4&lt;br&gt;D70&lt;sub&gt;66770&lt;/sub&gt;B&lt;br&gt;5F&lt;sub&gt;62340&lt;/sub&gt;6&lt;br&gt;A7M7&lt;sub&gt;61532&lt;/sub&gt;&lt;br&gt;B30&lt;sub&gt;61136&lt;/sub&gt;5&lt;br&gt;EJ&lt;sub&gt;52169&lt;/sub&gt;||93049&lt;br&gt;86445&lt;br&gt;81067&lt;br&gt;73563&lt;br&gt;68219&lt;br&gt;66773&lt;br&gt;62342&lt;br&gt;61535&lt;br&gt;61139&lt;br&gt;52170||126708&lt;br&gt;117715&lt;br&gt;110391&lt;br&gt;100172&lt;br&gt;92896&lt;br&gt;90927&lt;br&gt;84893&lt;br&gt;83794&lt;br&gt;83255&lt;br&gt;71042||11×23&lt;sup&gt;93048&lt;/sup&gt;+356&lt;br&gt;(21×23&lt;sup&gt;86445&lt;/sup&gt;−197)/22&lt;br&gt;(239×23&lt;sup&gt;81066&lt;/sup&gt;−349)/22&lt;br&gt;2×23&lt;sup&gt;73562&lt;/sup&gt;+215&lt;br&gt;(19×23&lt;sup&gt;68219&lt;/sup&gt;−1867)/22&lt;br&gt;306×23&lt;sup&gt;66771&lt;/sup&gt;+11&lt;br&gt;(125×23&lt;sup&gt;62341&lt;/sup&gt;−213)/22&lt;br&gt;(120413×23&lt;sup&gt;61532&lt;/sup&gt;−7)/22&lt;br&gt;256×23&lt;sup&gt;61137&lt;/sup&gt;+5&lt;br&gt;(327×23&lt;sup&gt;52169&lt;/sup&gt;−19)/22||http://factordb.com/index.php?id=1100000004691540361&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004691546739&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004691548070&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004691548569&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004691549462&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004691549803&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004691551005&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004691556967&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004691557254&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004691557548&amp;open=prime&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000004691540361&amp;base=23&lt;br&gt;http://factordb.com/index.php?showid=1100000004691546739&amp;base=23&lt;br&gt;http://factordb.com/index.php?showid=1100000004691548070&amp;base=23&lt;br&gt;http://factordb.com/index.php?showid=1100000004691548569&amp;base=23&lt;br&gt;http://factordb.com/index.php?showid=1100000004691549462&amp;base=23&lt;br&gt;http://factordb.com/index.php?showid=1100000004691549803&amp;base=23&lt;br&gt;http://factordb.com/index.php?showid=1100000004691551005&amp;base=23&lt;br&gt;http://factordb.com/index.php?showid=1100000004691556967&amp;base=23&lt;br&gt;http://factordb.com/index.php?showid=1100000004691557254&amp;base=23&lt;br&gt;http://factordb.com/index.php?showid=1100000004691557548&amp;base=23&lt;nowiki/&gt;||87||100000||
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||24||3409||N00N&lt;sub&gt;8129&lt;/sub&gt;LN&lt;br&gt;88N&lt;sub&gt;5951&lt;/sub&gt;&lt;br&gt;A0&lt;sub&gt;2951&lt;/sub&gt;8ID&lt;br&gt;D&lt;sub&gt;2698&lt;/sub&gt;LD&lt;br&gt;N&lt;sub&gt;2644&lt;/sub&gt;LLN&lt;br&gt;BC0&lt;sub&gt;331&lt;/sub&gt;B&lt;br&gt;20&lt;sub&gt;313&lt;/sub&gt;7&lt;br&gt;C7&lt;sub&gt;298&lt;/sub&gt;&lt;br&gt;D0&lt;sub&gt;259&lt;/sub&gt;KKD&lt;br&gt;I0&lt;sub&gt;241&lt;/sub&gt;I5||8134&lt;br&gt;5953&lt;br&gt;2955&lt;br&gt;2700&lt;br&gt;2647&lt;br&gt;334&lt;br&gt;315&lt;br&gt;299&lt;br&gt;263&lt;br&gt;244||11227&lt;br&gt;8216&lt;br&gt;4079&lt;br&gt;3727&lt;br&gt;3654&lt;br&gt;461&lt;br&gt;434&lt;br&gt;413&lt;br&gt;363&lt;br&gt;337||13249×24&lt;sup&gt;8131&lt;/sup&gt;−49&lt;br&gt;201×24&lt;sup&gt;5951&lt;/sup&gt;−1&lt;br&gt;10×24&lt;sup&gt;2954&lt;/sup&gt;+5053&lt;br&gt;(13×24&lt;sup&gt;2700&lt;/sup&gt;+4403)/23&lt;br&gt;24&lt;sup&gt;2647&lt;/sup&gt;−1201&lt;br&gt;276×24&lt;sup&gt;332&lt;/sup&gt;+11&lt;br&gt;2×24&lt;sup&gt;314&lt;/sup&gt;+7&lt;br&gt;(283×24&lt;sup&gt;298&lt;/sup&gt;−7)/23&lt;br&gt;13×24&lt;sup&gt;262&lt;/sup&gt;+12013&lt;br&gt;18×24&lt;sup&gt;243&lt;/sup&gt;+437||http://factordb.com/index.php?id=1100000003593391606&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003593275880&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003593269654&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003593269876&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003593270089&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000002633359842&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000002355610241&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000002326181235&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003593270725&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000002633360037&amp;open=prime&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000003593391606&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000003593275880&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000003593269654&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000003593269876&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000003593270089&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000002633359842&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000002355610241&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000002326181235&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000003593270725&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000002633360037&amp;base=24&lt;nowiki/&gt;||0||–||
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|-
||35||720002~720062||N0N&lt;sub&gt;99971&lt;/sub&gt;9&lt;br&gt;V0&lt;sub&gt;83669&lt;/sub&gt;E73&lt;br&gt;N&lt;sub&gt;81563&lt;/sub&gt;K7N&lt;br&gt;BJ0&lt;sub&gt;81279&lt;/sub&gt;N&lt;br&gt;J0&lt;sub&gt;80062&lt;/sub&gt;FUH&lt;br&gt;43V&lt;sub&gt;79754&lt;/sub&gt;&lt;br&gt;9&lt;sub&gt;76600&lt;/sub&gt;K3&lt;br&gt;LB&lt;sub&gt;71366&lt;/sub&gt;PB&lt;br&gt;Q&lt;sub&gt;64150&lt;/sub&gt;H&lt;br&gt;50&lt;sub&gt;63397&lt;/sub&gt;5R||99974&lt;br&gt;83673&lt;br&gt;81566&lt;br&gt;81282&lt;br&gt;80066&lt;br&gt;79756&lt;br&gt;76602&lt;br&gt;71369&lt;br&gt;64151&lt;br&gt;63400||154367&lt;br&gt;129197&lt;br&gt;125944&lt;br&gt;125505&lt;br&gt;123628&lt;br&gt;123148&lt;br&gt;118279&lt;br&gt;110199&lt;br&gt;99054&lt;br&gt;97894||(27393×35&lt;sup&gt;99972&lt;/sup&gt;−499)/34&lt;br&gt;31×35&lt;sup&gt;83672&lt;/sup&gt;+17398&lt;br&gt;(23×35&lt;sup&gt;81566&lt;/sup&gt;−144013)/34&lt;br&gt;404×35&lt;sup&gt;81280&lt;/sup&gt;+23&lt;br&gt;19×35&lt;sup&gt;80065&lt;/sup&gt;+19442&lt;br&gt;(4893×35&lt;sup&gt;79754&lt;/sup&gt;−31)/34&lt;br&gt;(9×35&lt;sup&gt;76602&lt;/sup&gt;+12877)/34&lt;br&gt;(725×35&lt;sup&gt;71368&lt;/sup&gt;+16649)/34&lt;br&gt;(13×35&lt;sup&gt;64151&lt;/sup&gt;−166)/17&lt;br&gt;5×35&lt;sup&gt;63399&lt;/sup&gt;+202||http://factordb.com/index.php?id=1100000008248342445&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008248353306&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008248375642&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008248397018&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008248412468&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008248418540&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008248423670&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008192119974&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008192126630&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008192129294&amp;open=prime&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000008248342445&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008248353306&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008248375642&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008248397018&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008248412468&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008248418540&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008248423670&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008192119974&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008192126630&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008192129294&amp;base=35&lt;nowiki/&gt;||60||100000||
|-
||36||35286~35290||P&lt;sub&gt;81993&lt;/sub&gt;SZ&lt;br&gt;S0&lt;sub&gt;75007&lt;/sub&gt;8H&lt;br&gt;7K&lt;sub&gt;26567&lt;/sub&gt;Z&lt;br&gt;J&lt;sub&gt;10117&lt;/sub&gt;LJ&lt;br&gt;VL0&lt;sub&gt;7258&lt;/sub&gt;J&lt;br&gt;EO0&lt;sub&gt;6177&lt;/sub&gt;V&lt;br&gt;FZ&lt;sub&gt;5777&lt;/sub&gt;3P&lt;br&gt;T09&lt;sub&gt;4618&lt;/sub&gt;1&lt;br&gt;RY&lt;sub&gt;4562&lt;/sub&gt;H&lt;br&gt;OZ&lt;sub&gt;3932&lt;/sub&gt;AZ||81995&lt;br&gt;75010&lt;br&gt;26569&lt;br&gt;10119&lt;br&gt;7261&lt;br&gt;6180&lt;br&gt;5780&lt;br&gt;4621&lt;br&gt;4564&lt;br&gt;3935||127609&lt;br&gt;116739&lt;br&gt;41349&lt;br&gt;15748&lt;br&gt;11301&lt;br&gt;9618&lt;br&gt;8996&lt;br&gt;7192&lt;br&gt;7103&lt;br&gt;6124||(5×36&lt;sup&gt;81995&lt;/sup&gt;+821)/7&lt;br&gt;28×36&lt;sup&gt;75009&lt;/sup&gt;+305&lt;br&gt;(53×36&lt;sup&gt;26568&lt;/sup&gt;+101)/7&lt;br&gt;(19×36&lt;sup&gt;10119&lt;/sup&gt;+2501)/35&lt;br&gt;1137×36&lt;sup&gt;7259&lt;/sup&gt;+19&lt;br&gt;528×36&lt;sup&gt;6178&lt;/sup&gt;+31&lt;br&gt;16×36&lt;sup&gt;5779&lt;/sup&gt;−1163&lt;br&gt;(36549×36&lt;sup&gt;4619&lt;/sup&gt;−289)/35&lt;br&gt;(979×36&lt;sup&gt;4563&lt;/sup&gt;−629)/35&lt;br&gt;25×36&lt;sup&gt;3934&lt;/sup&gt;−901||http://factordb.com/index.php?id=1100000002394962083&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004020085177&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003896952461&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362491&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362489&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362488&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362487&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362486&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362485&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000000840634476&amp;open=prime&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000002394962083&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000004020085177&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003896952461&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362491&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362489&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362488&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362487&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362486&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362485&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000000840634476&amp;base=36&lt;nowiki/&gt;||4||200000||
|}

== The fully proof of Athena problem in decimal (base ''b'' = 10) ==

'''Bold''' for the Athena primes, ''x'' ◁ ''y'' means ''x'' is a subsequence of ''y''.

Assume ''p'' is a prime &gt; 10, and the last digit of ''p'' must lie in {1,3,7,9}.

Case 1: ''p'' ends with 1.

In this case we can write ''p'' = ''x''1. If ''x'' contains 1, 3, 4, 6, or 7, then (respectively) '''11''' ◁ ''p'', '''31''' ◁ ''p'', '''41''' ◁ ''p'', '''61''' ◁ ''p'', or '''71''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 2, 5, 8, or 9.

Case 1.1: ''p'' begins with 2.

In this case we can write ''p'' = 2''y''1. If 5 ◁ ''y'', then '''251''' ◁ ''p''. If 8 ◁ ''y'', then '''281''' ◁ ''p''. If 9 ◁ ''y'', then 29 ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 2.

If 22 ◁ ''y'', then '''2221''' ◁ ''p''. Hence we may assume ''y'' contains zero or one 2's.

If ''y'' contains no 2's, then ''p'' ∈ 2{0}1. But then, since the sum of the digits of ''p'' is 3, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' contains exactly one 2, then we can write ''p'' = 2''z''2''w''1, where ''z'',''w'' ∈ {0}. If 0 ◁ ''z'' and 0 ◁ ''w'', then '''20201''' ◁ ''p''. Hence we may assume either ''z'' or ''w'' is empty.

If ''z'' is empty, then ''p'' ∈ 22{0}1, and the smallest prime ''p'' ∈ 22{0}1 is '''22000001'''.

If ''w'' is empty, then ''p'' ∈ 2{0}21, and the smallest prime ''p'' ∈ 2{0}21 is '''20021'''.

Case 1.2: ''p'' begins with 5.

In this case we can write ''p'' = 5''y''1. If 2 ◁ ''y'', then '''521''' ◁ ''p''. If 9 ◁ ''y'', then 59 ◁ ''p''. Hence we may assume all digits of ''y'' are 0, 5, or 8.

If 05 ◁ ''y'', then '''5051''' ◁ ''p''. If 08 ◁ ''y'', then '''5081''' ◁ ''p''. If 50 ◁ ''y'', then '''5501''' ◁ ''p''. If 58 ◁ ''y'', then '''5581''' ◁ ''p''. If 80 ◁ ''y'', then '''5801''' ◁ ''p''. If 85 ◁ ''y'', then '''5851''' ◁ ''p''. Hence we may assume ''y'' ∈ {0} ∪ {5} ∪ {8}.

If ''y'' ∈ {0}, then ''p'' ∈ 5{0}1. But then, since the sum of the digits of ''p'' is 6, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' ∈ {5}, then ''p'' ∈ 5{5}1, and the smallest prime ''p'' ∈ 5{5}1 is '''555555555551'''.

If ''y'' ∈ {8}, since if 88 ◁ ''y'', then 881 ◁ ''p'', hence we may assume ''y'' ∈ {''𝜆'',8}, and thus ''p'' ∈ {51,581}, but 51 and 581 are both composite.

Case 1.3: ''p'' begins with 8.

In this case we can write p = 8''y''1. If 2 ◁ ''y'', then '''821''' ◁ ''p''. If 8 ◁ ''y'', then '''881''' ◁ ''p''. If 9 ◁ ''y'', then 89 ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 5.

If 50 ◁ ''y'', then '''8501''' ◁ ''p''. Hence we may assume y ∈ {0}{5}.

If 005 ◁ ''y'', then '''80051''' ◁ p. Hence we may assume y ∈ {0} ∪ {5} ∪ 0{5}.

If y ∈ {0}, then ''p'' ∈ 8{0}1. But then, since the sum of the digits of ''p'' is 9, ''p'' is divisible by 3, so ''p'' cannot be prime.

If y ∈ {5}, since if 55555555555 ◁ ''y'', then 555555555551 ◁ ''p'', hence we may assume ''y'' ∈ {''𝜆'', 5, 55, 555, 5555, 55555, 555555, 5555555, 55555555, 555555555, 5555555555}, and thus ''p'' ∈ {81, 851, 8551, 85551, 855551, 8555551, 85555551, 855555551, 8555555551, 85555555551, 855555555551}, but all of these numbers are composite.

If y ∈ 0{5}, since if 55555555555 ◁ ''y'', then 555555555551 ◁ ''p'', hence we may assume ''y'' ∈ {0, 05, 055, 0555, 05555, 055555, 0555555, 05555555, 055555555, 0555555555, 05555555555}, and thus ''p'' ∈ {801, 8051, 80551, 805551, 8055551, 80555551, 805555551, 8055555551, 80555555551, 805555555551, 8055555555551}, and of these numbers only 80555551 and 8055555551 are primes, but 80555551 ◁ 8055555551, thus only '''80555551''' is a minimal element.

Case 1.4: ''p'' begins with 9.

In this case we can write p = 9''y''1. If 9 ◁ ''y'', then '''991''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0, 2, 5, or 8.

If 00 ◁ ''y'', then '''9001''' ◁ ''p''. If 22 ◁ ''y'', then '''9221''' ◁ ''p''. If 55 ◁ ''y'', then '''9551''' ◁ ''p''. If 88 ◁ ''y'', then 881 ◁ ''p''. Hence we may assume ''y'' contains at most one 0, at most one 2, at most one 5, and at most one 8.

If ''y'' only contains at most one 0 and does not contain any of {2,5,8}, then ''y'' ∈ {''𝜆'',0}, and thus ''p'' ∈ {91,901}, but 91 and 901 are both composite. If ''y'' only contains at most one 0 and only one of {2,5,8}, then the sum of the digits of ''p'' is divisible by 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Hence we may assume ''y'' contains at least two of {2,5,8}.

If 25 ◁ ''y'', then 251 ◁ ''p''. If 28 ◁ ''y'', then 281 ◁ ''p''. If 52 ◁ ''y'', then 521 ◁ ''p''. If 82 ◁ ''y'', then 821 ◁ ''p''. Hence we may assume ''y'' contains no 2's (since if ''y'' contains 2, then ''y'' cannot contain either 5's or 8's, which is a contradiction).

If 85 ◁ ''y'', then '''9851''' ◁ ''p''. Hence we may assume ''y'' ∈ {58,580,508,058}, and thus ''p'' ∈ {9581,95801,95081,90581}, and of these numbers only 95801 is prime, but 95801 is not a minimal element since 5801 ◁ 95801.

Case 2: ''p'' ends with 3.

In this case we can write p = ''x''3. If ''x'' contains 1, 2, 4, 5, 7, or 8, then (respectively) '''13''' ◁ ''p'', '''23''' ◁ ''p'', '''43''' ◁ ''p'', '''53''' ◁ ''p'', '''73''' ◁ ''p'', or '''83''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 3, 6, or 9, and thus all digits of ''p'' are 0, 3, 6, or 9. But then, since the digits of ''p'' all have a common factor 3, ''p'' is divisible by 3, so ''p'' cannot be prime.

Case 3: ''p'' ends with 7.

In this case we can write ''p'' = ''x''7. If ''x'' contains 1, 3, 4, 6, or 9, then (respectively) '''17''' ◁ ''p'', '''37''' ◁ ''p'', '''47''' ◁ ''p'', '''67''' ◁ ''p'', or '''97''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 2, 5, 7, or 8.

Case 3.1: ''p'' begins with 2.

In this case we can write ''p'' = 2''y''7. If 2 ◁ ''y'', then '''227''' ◁ ''p''. If 5 ◁ ''y'', then '''257''' ◁ ''p''. If 7 ◁ ''y'', then '''277''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 8.

If 08 ◁ ''y'', then '''2087''' ◁ ''p''. If 88 ◁ ''y'', then 887 ◁ ''p''. Hence we may assume ''y'' ∈ {0} ∪ 8{0}.

If ''y'' ∈ {0}, then ''p'' ∈ 2{0}7. But then, since the sum of the digits of ''p'' is 9, ''p'' is divisible by 3, so ''p'' cannot be prime.

If y ∈ 8{0}, then ''p'' ∈ 28{0}7. But then ''p'' is divisible by 7, since for ''n'' ≥ 0 we have 7 × 40&lt;sub&gt;''n''&lt;/sub&gt;1 = 280&lt;sub&gt;''n''&lt;/sub&gt;7.

Case 3.2: ''p'' begins with 5.

In this case we can write ''p'' = 5''y''7. If 5 ◁ ''y'', then '''557''' ◁ ''p''. If 7 ◁ ''y'', then '''577''' ◁ ''p''. If 8 ◁ ''y'', then '''587''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 2.

If 22 ◁ ''y'', then 227 ◁ ''p''. Hence we may assume ''y'' contains zero or one 2's.

If ''y'' contains no 2's, then ''p'' ∈ 5{0}7. But then, since the sum of the digits of ''p'' is 12, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' contains exactly one 2, then we can write ''p'' = 5''z''2''w''7, where ''z'',''w'' ∈ {0}. If 0 ◁ ''z'' and 0 ◁ ''w'', then '''50207''' ◁ ''p''. Hence we may assume either ''z'' or ''w'' is empty.

If ''z'' is empty, then ''p'' ∈ 52{0}7, and the smallest prime ''p'' ∈ 52{0}7 is '''5200007'''.

If ''w'' is empty, then ''p'' ∈ 5{0}27, and the smallest prime ''p'' ∈ 5{0}27 is '''5000000000000000000000000000027'''.

Case 3.3: ''p'' begins with 7.

In this case we can write ''p'' = 7''y''7. If 2 ◁ ''y'', then '''727''' ◁ ''p''. If 5 ◁ ''y'', then '''757''' ◁ ''p''. If 8 ◁ ''y'', then '''787''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 7, and thus all digits of ''p'' are 0 or 7. But then, since the digits of ''p'' all have a common factor 7, ''p'' is divisible by 7, so ''p'' cannot be prime.

Case 3.4: ''p'' begins with 8.

In this case we can write ''p'' = 8''y''7. If 2 ◁ ''y'', then '''827''' ◁ ''p''. If 5 ◁ ''y'', then '''857''' ◁ ''p''. If 7 ◁ ''y'', then '''877''' ◁ ''p''. If 8 ◁ ''y'', then '''887''' ◁ ''p''. Hence we may assume ''y'' ∈ {0}, and thus ''p'' ∈ 8{0}7. But then, since the sum of the digits of ''p'' is 15, ''p'' is divisible by 3, so ''p'' cannot be prime.

Case 4: ''p'' ends with 9.

In this case we can write ''p'' = ''x''9. If ''x'' contains 1, 2, 5, 7, or 8, then (respectively) '''19''' ◁ ''p'', '''29''' ◁ ''p'', '''59''' ◁ ''p'', '''79''' ◁ ''p'', or '''89''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 3, 4, 6, or 9.

If 44 ◁ ''x'', then '''449''' ◁ ''p''. Hence we may assume ''x'' contains zero or one 4's.

If x contains no 4's, then all digits of ''x'' are 0, 3, 6, or 9, and thus all digits of ''p'' are 0, 3, 6, or 9. But then, since the digits of ''p'' all have a common factor 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Hence we may assume that ''x'' contains exactly one 4.

Case 4.1: ''p'' begins with 3.

In this case we can write ''p'' = 3''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. We must have '''349''' ◁ ''p''.

Case 4.2: ''p'' begins with 4.

In this case we can write ''p'' = 4''y''9, where all digits of ''y'' are 0, 3, 6, or 9. If 0 ◁ ''y'', then '''409''' ◁ ''p''. If 3 ◁ ''y'', then 43 ◁ ''p''. If 9 ◁ ''y'', then '''499''' ◁ ''p''. Hence we may assume ''y'' ∈ {6}, and thus ''p'' ∈ 4{6}9. But then ''p'' is divisible by 7, since for ''n'' ≥ 0 we have 7 × 6&lt;sub&gt;''n''&lt;/sub&gt;7 = 46&lt;sub&gt;''n''&lt;/sub&gt;9.

Case 4.3: ''p'' begins with 6.

In this case we can write p = 6''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. If 0 ◁ ''z'', then 409 ◁ ''p''. If 3 ◁ ''z'', then 43 ◁ ''p''. If 6 ◁ ''z'', then '''6469''' ◁ ''p''. If 9 ◁ ''z'', then 499 ◁ ''p''. Hence we may assume ''z'' is empty.

If 3 ◁ ''y'', then 349 ◁ ''p''. If 9 ◁ ''y'', then '''6949''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 6.

If 06 ◁ ''y'', then '''60649''' ◁ ''p''. Hence we may assume ''y'' ∈ {6}{0}.

If 666 ◁ ''y'', then '''666649''' ◁ ''p''. If 00000 ◁ ''y'', then '''60000049''' ◁ ''p''. Hence we may assume ''y'' ∈ {''𝜆'', 0, 00, 000, 0000, 6, 60, 600, 6000, 60000, 66, 660, 6600, 66000, 660000}, and thus ''p'' ∈ {649, 6049, 60049, 600049, 6000049, 6649, 66049, 660049, 6600049, 66000049, 66649, 666049, 6660049, 66600049, 666000049}, and of these numbers only '''66000049''' and '''66600049''' are primes.

Case 4.4: ''p'' begins with 9.

In this case we can write p = 9''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. If 0 ◁ ''y'', then '''9049''' ◁ ''p''. If 3 ◁ ''y'', then 349 ◁ ''p''. If 6 ◁ ''y'', then '''9649''' ◁ ''p''. If 9 ◁ ''y'', then '''9949''' ◁ ''p''. Hence we may assume ''y'' is empty.

If 0 ◁ ''z'', then 409 ◁ ''p''. If 3 ◁ ''z'', then 43 ◁ ''p''. If 9 ◁ ''z'', then 499 ◁ ''p''. Hence we may assume ''z'' ∈ {6}, and thus ''p'' ∈ 94{6}9, and the smallest prime ''p'' ∈ 94{6}9 is 946669.

[[Category:Number theory]]</text>
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      <text bytes="72134" sha1="h3e0aryt9m2decotpkfsnlj10ipzz0f" xml:space="preserve">Hi ! I'm a librarian that occasionally edits on Wikipedia and Wikiversity.

== Things I'm working on ==
[[Creating Wikiversity Courses]]

=== Bartending Wikiversity Course ===

==== Notes to Myself ====

# Course with several lesson subpages - could use the box template to organize.
# Finish/Build up the Hospitality Lesson
# Add an Advanced Bartending lesson with more recipes/mixology/significant bartenders/freepouring?
# Move stirring photo in skills for mobile friendly placement

==== Course Outline ====

# Bartending Basics
## Sources of alcohol
## Types of alcohol
## Tools of Bartending
## Safety
# Mixing Drinks
## Glassware
## Typology of Cocktails
## Preparation and Skills
## Essential Cocktails
# Hospitality
## What is Hospitality?
## Steps of Service
## Internal Hospitality
## Developing Taste
# Industry 
## History of pubs/bartenders/mixologists
## Wages + tips
## Hours + Working conditions
## Unions
# Advanced Topics
## Free-pouring
## Other recipes to know
## Round building
## Mixology
# Recommended Reading List
## Mr. Boston's Official Bartender's Guide - Available on Internet Archive
##

= Bartending for Beginners =

=== Introduction ===
Mixing drinks, talking to people, and making money: that's bartending! If you find that definition to be less-than-enough, this course will guide you through the essential skills of a bartender. This course is intended for adults of legal drinking age where they live who want to start bartending in a professional establishment either as a side gig or a career. Adults wanting to learn about bartending in general or how to mix drinks at home can also benefit from this course. No prior experience is required. By the end of the course, you should know: how to mix common drinks ordered at different types of bars and how they are related to one another; the liquors, spirits, ales, wines and other drinks used at the bar; the tools of the trade; the basics of safety as a bartender; how to create a hospitable environment for bar patrons; and details on the bartending industry, customs and history.

This course makes use of the [[wikipedia:Wikimedia_Foundation#Projects_and_initiatives|Wikimedia ecosystem]] of projects. Throughout the course, there will be links to Wikipedia pages, Wiktionary entries, and images from Wikimedia Commons. When a link is casually included in a lesson, I encourage you to browse its contents. Occasionally, clicking a link and digesting its contents will be formally assigned as part of the course.

=== Structure of the Course ===
This course is comprised of lessons which each focus on a component of bartending. It is recommended to complete them in order, as each lesson will build on the previous. The lessons can be found below:

# Bartending Basics
# Mixing Drinks
# Hospitality 
# Industry (Under Construction)

=== Course Objectives ===
By the of the course, you should be able to:

* Share information about and identify the different types of alcohol
* Identify and use the tools of a bartender to complete common techniques
* Mix the 20 essential cocktails
* Serve bar patrons safely and hospitably

=== Recommended Materials ===
Bartending is a physical practice, and theory alone is not enough to prepare yourself. There will be activities that you complete at home as part of the course. The materials required to complete these activities are:
# Boston shaker (Preferably not a Cobbler shaker or a Parisian shaker, though they may be easier to find.)
# Hawthorn Strainer
# Bar Spoon
# Jigger
#A glass for mixing, and a glass to pour drinks into (Don't worry about the style of glass, it just needs to be large enough to hold ice and 10 oz of liquid.)
[[File:Bartender_Photo.jpg|left|thumb|Bartender with Boston Shaker]]
[[File:Cocktail-strainer.jpg|thumb|Hawthorne Strainer]]
[[File:Jigger.jpg|center|thumb|160x160px|Jigger]]
In addition to these materials, you will also need to ingredients to the recipes we'll prepare. Those will be discussed when we talk about mixing drinks. Many of these materials can bought online or found at secondhand or thrift shops depending on your area. There may also be a specialty store near you catering to restaurants and bars. Having these materials at home are an invaluable way to practice mixing drinks, with the added benefit of allowing you to entertain guests at home! If you cannot acquire these materials, the course is still completable in theory, but you will miss out on the practical aspects of the course. No materials are needed for lesson one.

In addition to these materials, you will also need to ingredients to the recipes we'll prepare. Those will be discussed when we talk about mixing drinks. Many of these materials can bought online or found at secondhand or thrift shops depending on your area. There may also be a specialty store near you catering to restaurants and bars. Having these materials at home are an invaluable way to practice mixing drinks, with the added benefit of allowing you to entertain guests at home! If you cannot acquire these materials, the course is still completable in theory, but you will miss out on the practical aspects of the course. No materials are needed for lesson one.

In addition to these materials, you will also need to ingredients to the recipes we'll prepare. This will be discussed when we talk about mixing drinks.

Start here with Lesson One: Bartending Basics.

= Bartending Basics =
What is a [[wikipedia:Bartender|bartender]]? At its simplest, a bartender serves drinks in a bar. In reality, a bartender selects, mixes, pours, and serves drinks while hosting patrons and creating a hospitable atmosphere at a bar, pub, restaurant, nightclub, living room or [[wikipedia:Parking_lot|parking lot]]. A bartender must have a variety of skills, both technical and interpersonal in order to be successful. A bartender can work in many locations, but this course will focus on bartenders that tend to tend in bars.

This lesson covers: 

#Sources of Alcohol
#Types of Alcohol
# Tools of Bartending
# Safety

== Sources of Alcohol ==
Behind the bar, you are the expert in the room on alcohol. While no one can claim to know everything on alcohol, it's important to have the basics. This and the next lesson focus on learning what alcohol is and what kind of drinks and liquors are made with it.

Every alcoholic drink you've ever had has included the same basic type of alcohol: [[wikipedia:Ethanol|Ethanol]]. Ethanol is one of three types of alcohol and the only alcohol humans can safely drink. All alcoholic beverages whether it be beer, wine or a spirit all contain Ethanol.

There are hundreds of different beverages and liquors that are served at bars around the world.  The method of refining raw ingredients into a final product with Ethanol is what distinguishes each type of beverage. Fortunately they're all related to each other (since they all contain Ethanol) and have some major categories to guide us in understanding what they are. The first step to make any type of alcohol is [[wikipedia:fermentation|fermentation]].

=== Fermentation ===
[[File:40168_2022_1274_Fig6.webp|thumb|People get very scientific with it.]]
Using a process that has existed for thousands of years, we can employ a fungus called [[wikipedia:Yeast#Uses|yeast]] to create alcohol. This process is called fermentation. At it base, fermentation for alcohol production is the process of using yeast to convert sugars to ethanol. 

&quot;Sugars&quot; is intentionally plural. There are many places you can find sugar suitable for fermentation, and you can derive sugar from grains, fruits and vegetables. The source of the sugar is often what defines what type of final alcoholic beverage you get. We'll talk more on how fermenting different materials yields different drinks in a moment. 

Fermentation happens in a couple of steps.

# '''Mashing''': Grains like barley or rice are milled into a coarse flour and fruits are mashed into a pulp. These are sometimes mixed with hot water, where enzymes convert starches to sugars. For beer, this mash is then often boiled.
# '''Fermentation''': The mash is transferred to a fermentation vessel. Yeast is added, and fermentation begins. Over a varying period of time (often 5-10 days), yeast consumes the sugars, producing alcohol and carbon dioxide. This is called primary fermentation.
# '''Conditioning and Packaging''': After primary fermentation, there is often secondary fermentation or conditioning to add or mature flavors and to clarify. It’s then sometimes carbonated and packaged.

Now fermented, there is some amount of alcohol in your drink. But what if ''some'' isn't enough? A secondary process called [[wikipedia:Distillation|distillation]] allowed for fermented drinks and solutions to be concentrated into spirits. 

=== Distillation ===
[[File:Alambins_industrials_per_a_la_destil·lació_de_licors_a_Catalunya.jpg|left|thumb|Industrial stills for liquor production in Catalonia]]
Distillation is a process by which brewers can concentrate the level of alcohol in a drink. Fermentation alone cannot make alcohol concentrations like that of vodka or gin. To reach that level of concentration, [[wiktionary:distiller|distillers]] (referring to both the apparatus that distills and the person that runs it) can boil off and collect the alcohol produced by fermentation. Distillation happens in a few steps:

# The fermented substance is placed into a vessel called a [[wikipedia:Still|still]] (this is where di''still''ation gets its name).
# The ferment is heated slowly from room temperature until it reaches the boiling point of ethanol. The boiling point of ethanol is lower than the boiling point of water, so the ethanol will vaporize, leaving the water behind.
# The ethanol vapor rises through a column until it reaches the condenser. In the condenser, the vapor makes contact with the a cold surface that cools it back into a liquid state.
# The ethanol is collected, and the process is often repeated to further concentrate and purify the result. Sometimes the process is done without having to stop and restart in a process called continuous distillation.

The result of distillation is a liquor with a higher amount of alcohol by volume (ABV) than before. 

=== Alcohol By Volume ===
The amount of alcohol in a drink is calculated based on the percent of ethanol compared to non-ethanol in the total beverage, which is called Alcohol by Volume or ABV for short. For example, if a beverage has an ABV of 50%, half of liquid in the drink will be ethanol and the other half will be everything else.

ABV is important to understand, because it is the basis of drinking and serving drinks safely as well as how many drinks are categorized. While all alcohol is made through fermentation, the process of fermentation alone can only create a concentration of alcohol of about 5% to 20% ABV. Spirits are generally distilled liquids that have a higher ABV (20% or more, as high as 95%). This distinction lower-ABV drinks like wine, beer and cider versus higher-ABV spirits like vodka, gin and whiskey. 

== Types of Alcohol ==
Fermentation and distilling are the core of all alcohol production, but there many more ways that brewers and distillers affect their final product. This incredible flow chart shows many of the processes of fermentation, distillation, carbonation, ageing and processing that make different alcoholic drinks. Take a look at it and compare the different starting ingredients with each other, and then compare the starting ingredients with their final products.
[[File:Alcohol_Flow_Chart.svg|center|frame|Alcohol Flow Chart]]
There is so much to learn about alcohol, and while you should be knowledgeable on the different types, you by no means need to be an expert (see: sommelier) to be a bartender. Below follows an overview of each family of alcohol, but if you want to learn more you can click on the links to Wikipedia pages in each subsection below, or I recommend browsing the [[wikipedia:Portal:Liquor|Liquor Portal]] to view all that Wikipedia has to offer.

Hopefully it's easier to see now how beverages and spirits can both be categorized based on what they're fermented from and how concentrated their alcohol content is. Let's use these categories to talk more about each major type of alcohol.

=== Wine, Beer and Cider (Low-ABV Drinks) ===

==== Wine ====
We call fermented grape juice [[wikipedia:wine|wine]]. Yeast is added to pressed grapes to develop it into one of four types: [[wikipedia:Red_wine|red]], [[wikipedia:white_wine|white]], [[wikipedia:rose_wine|rosé]], and [[wikipedia:Orange_wine|orange]]. The type of grape and the duration of the contact with the skins of the grapes help to define which type of wine is produced. The table below is from the Wikipedia page on wine.
{| class=&quot;wikitable&quot;
|+Colors of wine
!
!Long contact with grape skins
!Short contact with grape skins
|-
!Red grapes
|'''''Red wine''''', made from dark-colored red grape varieties. The actual color of the wine can range from dark pink to almost black. The juice from red grapes is actually pale gray; the color of red wine and some of its flavor (notably tannins) comes from phenolics in the skin, seeds and stem fragments of the grape, extracted by allowing the grapes to soak in the juice.
|'''''Rosé wine''''', which gains color from red grape skins, but not enough to qualify it as a red wine. The color can range from a very pale pink to pale red.
There are two primary ways to produce rosé wine. The preferred technique is allowing a short period of maceration after crushing red grapes, which extracts a certain amount of color. The juice is then fermented like a white wine. An alternative is blending a small amount of finished red wine into finished white wine.
|-
!White grapes
|'''''Orange wine''''', sometimes called amber wine, is made with white grapes but with the skins allowed to macerate during and beyond fermentation, similar to red wine production. This results in their darker color compared to white wines, and produces a deliberately astringent result.
|'''''White wine''''', typically made from white grape varieties (those with yellow or green skins), and range from practically colorless to golden. When skin contact is used, to improve the flavor or to increase the body or aging potential, it is usually limited to between four and 24 hours; any longer leads to bitterness.
|}
''Common names: There are thousands of wine varietals. Red wines like Cabernet Sauvignon, Merlot, Pinot Noir, Malbec, Zinfandel; white wines like Chardonnay, Sauvignon Blanc, Pinot Grigio, Riesling; and rosé wines like Provence, White Zinfandel and Pink Moscato.'' 

In addition to the colors of wine, wine can be [[wikipedia:Sparkling_wine|sparkling]] or still. Wine can also be [[wikipedia:Fortified_wine|fortified]]. Fortifying a wine is the process of adding a distilled spirit to the fermented wine.  

''Common names: Sparkling wine includes Champagne, Prosecco and other Brut wines. Fortified wines include Port, Sherry and Vermouth.''
==== Fruit Wine, Cider and Perry ====
[[wikipedia:Fruit_wine|Fruit wine]] is mostly what it sounds like: wine made from fruits other than grapes. A variety of fruits can be used, some common ones include cherry, plum, dandelion and pineapple. Fruit wine is far less common than traditional grape wines, and is often called by the fruit it's fermented from (ex. &quot;cherry wine&quot; or &quot;dandelion wine&quot;). 

[[wikipedia:Cider|Cider]] is a beverage made specifically from fermenting apples. Despite being a sort of wine fermented from fruit, it isn't considered a fruit wine due to its unique cultural history as a beverage. Its name can cause some confusion particularly in the United States and Canada, where &quot;cider&quot; also refers to unfiltered and sometimes spiced apple juice that has not been fermented. The term &quot;hard cider&quot; is sometimes used to distinguish the alcoholic beverage due to this overlap. Cider can also be carbonated to make a sparkling cider. 

[[wikipedia:Perry|Perry]], or Pear Cider, is another fruit wine exception. Fermented pears create a drink called perry, that though technically is a wine made from fruit, is considered its own thing. 

''Common names: There are thousands of brands of fruit wine, cider and perry. You may have some on tap or bottled; it's mostly important to know the names of what you have.''
[[File:Beer_in_glasses_and_steins_on_a_table_with_bottles_in_the_background_and_a_brick_wall_(15700131777).jpg|thumb|264x264px|There are many varieties of beer.]]

==== Beer ====
[[wikipedia:beer|Beer]] is an ancient drink that has been brewed around the world for hundreds of years. Most beers have four basic ingredients: grain, hops, yeast and water. [[wikipedia:Hops|Hops]] are a type of dried flower that give beer flavor. While there are thousands of brands of beer, there are only a few main types:  

* '''Lagers''' - One of the two main types of beer, with Ales. Lighter and crisp, refreshing, less bitter than Ale.
** Amber Lager - A medium, amber color with a slight bitterness. ''Common Names: Yuengling''
** Pale Lager - Light in color and flavor, not very bitter. Easy drinking and very popular. ''Common Names: Red Stripe, Heineken, Pabst Blue Ribbon (PBR), Hamms''
** Pilsner - Medium amber in color. Hoppy and flavorful. ''Common Names: Modelo Especial, Stella Artois, Pilsner Urquell''

* '''Ales''' - The other main type of beer, with Lager. Richer and flavorful, more bitter than Lager.
** Pale Ale - A diverse group of beer that various depending on its country of origin. Generally balanced taste that pairs with many foods. ''Common Names: Sierra Nevada, Boulevard''
** Indian Pale Ale (IPA) - Often very bitter and fruity. A unique tasting beer with bite. ''Common Names: Voodoo Ranger, many other large brands have an IPA''
** Stouts and Porters - Dark beers that have rich, roasted flavors often imitating chocolate or coffee. These beers can feel like a meal. ''Common Names: Guinness, Imperial Stouts, Breakfast Stouts''
** Wheat Beer - Ales that have some wheat added to them. It can give them a fruity, almost banana like flavor. ''Common Names: Blue Moon''
* Radlers and Shandys - Beers that have been mixed with a fruit juice. Radlers can be any type of citrus while Shandys are always lemon juice. Very refreshing, summary drinks.
* Sour Beer - Beer that's, well, sour. Tangy and punchy, usually served in a smaller &quot;sour beer&quot; glass since they pack a punch.
* Light Beer - Any beer with an especially low ABV is a light beer (less than 4% ABV), though the term is sometimes also used for low-calorie beers. ''Common Names: Bud Light, Coors Light''

==== Sake ====
[[wikipedia:sake|Sake]] is a drink brewed from rice, and is sometimes also called rice wine though it is brewed more similarly to beer. Like beer and wine, sake has a wide range of flavor profiles though it is often fruity and light. It is sometimes served lightly warmed in ceramic cups depending on the variety of sake. 

''Common Names: Dassai, Hakutsuru, Kubota''

=== Spirits (High-ABV Drinks) ===
[[File:Alcoholic beverages.jpg|thumb|High-ABV spirits like vodka and brandy.]]

==== Vodka ====
[[wikipedia:vodka|Vodka]] is a spirit that can be distilled from many sources including grains, potatoes and sugarcane before the pure ethanol is diluted with water. It is generally a neutral spirit, meaning it has little taste and is mostly pure ethanol and water. This isn't entirely true though, as each vodka has its own impurities and subtle flavor profiles. Vodka is also commonly flavored, as its neutral taste allows it to take on flavor easily. It can served up, often freezer chilled, or mixed into many cocktails.

''Common Names: Absolut, Smirnoff, Kettle One''

==== Gin ====
[[wikipedia:Gin|Gin]] is a spirit flavored with [[wikipedia:juniper_berries|juniper berries]] and other botanicals. It has a distinct herbal flavor, and different brands will have their own flavor profile, and perceived dryness.
[[File:Agave tequilana 2.jpg|thumb|Agave is a plant native to Mexico.]]
''Common Names: Beefeater, Tanqueray, Hendrick's, Plymouth''

==== Mezcal (Tequila) ====
[[wikipedia:mezcal|Mezcal]] is a spirit distilled from Agave. The most common type of mezcal is [[wikipedia:tequila|Tequila]], which is distilled only from [[wikipedia:blue_agave|blue agave]]. Many people mistake mezcal for a type of tequila due to tequila's popularity, but it is the other way around. Mezcal and tequila are often aged, which impart a darker, more amber color the longer it sits. Mezcal often has a smokey flavor and can also be sweet, fruity or earthy, and can be enjoyed chilled, up or in many popular cocktails.

''Common Names: Casamigos, Patrón, Jose Cuervo''

==== Rum ====
[[wikipedia:rum|Rum]] is a spirit distilled from [[wikipedia:sugarcane|sugarcane]] then aged in barrels. The less-aged rum has a lighter color, and is called &quot;light rum&quot;. It is most commonly used in cocktails, like the Mojito and the Daiquiri. &quot;Aged&quot; or &quot;Dark&quot; rum has a much deeper flavor is often drank straight or on the rocks. Rum originates from and maintains strong ties to the history of the people of the [[wikipedia:caribbean|Caribbean]] and appears commonly in cultural cuisines as well as drinks. I encourage you to learn about the history of rum, as it can teach a lot about sugarcane as a [[wikipedia:Sugar_plantations_in_the_Caribbean|plantation crop]] and the[[wikipedia:Atlantic_slave_trade|Transatlantic Slave Trade]]. 

''Common Names: Baccardi, Captain Morgan, Malibu'' 

==== Brandy ====
[[wikipedia:Brandy|Brandy]] is made by distilling wine. While most brandy is made from grapes, it can be made from any fruit to yield a [[wikipedia:Fruit_brandy|fruit brandy]]. 

''Common names: Cognac and Armagnac''

==== Whiskey ====
[[wikipedia:whiskey|Whiskey]] is made by fermenting, distilling then aging in barrels various grains. Whiskey drinkers can be particular, as there are many types of whiskey. Some whiskey is &quot;malted&quot;, meaning it uses grains that have been malted. Malting is the process of allowing grains to germinate (begin to sprout), before heating them to halt the germination. This yields &quot;malt whiskey&quot;. The specific grain used also yields different types of whiskey, for example fermenting rye yields &quot;rye whiskey&quot;. &quot;[[wikipedia:Scotch_whisky|Scotch]]&quot; is simply whiskey from Scotland and is also called Scotch Whiskey.

''Common Names: Jameson, Jack Daniel's, Crown Royale''

==== Soju ====
[[wikipedia:soju|Soju]] is a made by distilling rice or other grains. Similarly to vodka, it is a neutral spirit that is most flavorless. It is often flavored with fruits and florals. It can be served up or mixed into cocktails.

''Common Names: Jinro''

==== Liqueurs ====
[[wikipedia:liqueur|Liqueurs]] are spirits that have been heavily flavored and sweetened. They are used very frequently in cocktails to impart their particular flavor. Liqueurs can be used to add a flavor that would otherwise be troublesome, like coffee, liquorish, or almond. Sometimes they're served up as a dessert or a [[wikipedia:digestif|digestif]]. 

''Common Names: Kahlua'' 

== Tools of Bartending ==
You'll be using both common and specialty tools as a bartender.  A collection of such tools are below. Start by trying to name as many of them as you can. Note how many are familiar to you, how many you've seen but cannot name, and how many are unfamiliar. A key to the image follows.
[[File:Bartools2.jpg|border|center]]
Answer key: (1) champagne bottle stopper, (2) kitchen knife, (3) ice tongs, (4) ice scoop, (5) ice bucket, (6) small bar spoon, (7) cocktail-pick, (8) jigger, (9) mesh strainer, (10) boston shaker (metal bottom), (11) bar spoon, (12) lime/lemon squeezer, (13) hawthorn strainer, (14) zester, (15) boston shaker (mixing glass), (16) muddler, (17) citrus reamer, (18) fine grater, (19) Y-peeler, (20) wine key.

How'd you do? Some of these items you'll see more often than others. I want to draw special attention these crucial pieces:

* '''Jigger''' (8) - A small double sided measuring device used to quickly portion spirits and other ingredients. Though the exact measurements may change depending on where you are and the manufacturer, in the US most jiggers measure 1.5 oz on one end and 0.75 oz on the other.
* '''Juicer and Peeler''' (12 &amp; 19) - Many cocktails rely on citrus for flavor and fragrance. The handheld juicer and the peeler are commonly used both at the start of the shift to prep the citrus for the day, collecting fresh citrus juice and peels for adding to cocktails.
* '''The Boston Shaker''' (10 &amp; 15) - This is the industry standard for fast, versatile and accurate cocktail mixing. Coming in two parts, often one side is glass and the other is metal, though often both are metal. The two parts are sealed together allowing for a cocktail to be shaker vigorously inside before being cracked apart without spilling.
* '''The Hawthorn Strainer''' (23) - The final piece to any Boston Shaker. This uniquely shaped strainer allows for cocktails to be poured into a glass while leaving the ice in the shaker.

These are tools you will see every time you step in to bar to make a cocktail.

== Safety ==
A good bartender is the difference between a safe time for your bar patrons and an unsafe environment. It is your responsibility to keep both yourself and your bar patrons safe. It is important first to understand how alcohol affects the body in order to understand how to properly make and serve drinks, so first let's review ABV.

Alcohol by Volume is a percentage of ethanol to all other contents of a drink. However, ABV does not tell you everything about drink. A standard beer has about 5% ABV while a shot of whiskey has about 40% ABV, but because you would only drink about 1.5 oz of whiskey, and would drink 12 oz beer, the total amount of alcohol ingested would actually be about the same. This is the idea behind a &quot;standard drink&quot;. A standard drink contains one &quot;unit&quot; of alcohol regardless of how much liquid it contains in total. The chart below shows some examples of standard drink equivalents. Each of the below drinks contain one unit of alcohol.
[[File:NIH standard drink comparison.jpg|center|frame|National Institute of Health's &quot;standard drink&quot;.]]
Some general best practices when consuming alcohol:

* One unit of alcohol per hour is around what a liver can process. This is generally a safer cadence of drinking.
* Drink a glass of water for every unit of alcohol consumed to stay hydrated.
* Avoid drinking on an empty stomach. Eating while consuming alcohol slows the alcohol's absorption into the blood and curbs its effects.
* Don't order a double. Ordering and serving drinks with more than one standard unit of alcohol in it can cause you or others to become more impaired more quickly than intended. Try to keep it to the cadence of one standard drink per hour.

Standards should be upheld both with your own drinking and the drinking of the patrons at your bar. Some standards to uphold in your bar are:

* '''Never drink and drive'''. Alcohol impairs motor coordination and clouds judgement. Drinking and then driving puts your own life and the lives of everyone around you in immediate danger. Even if you are under the legal limit in your area, you are impaired after one drink. This should be enforced both for yourself and for your bar patrons.
* '''You can say no'''. It's always okay to say no to a drink. Ordering something without alcohol is always an option. Good friends don't pressure you to drink when you don't want to. You can leave a situation when you are uncomfortable.
* '''Alcohol is not medicine'''. Addiction is real and treatable. Help is available.
* '''Abuse isn't acceptable'''. You are not a punching bag. It is not bad hospitality to stand up for yourself. 

Sometimes people drink more than they intended and may become too inebriated to be served. Some signs that this is the case are:

* Slurred speech
* Stumbling
* Glazed over or unfocused eyes
* Speaking too loudly or two softly
* Repeating questions or ordering from multiple bartenders

If you meet a bar patron who has had too much to drink, you can help by offering water, tea, coffee or food, helping them find their group, or asking another member of the staff at your bar for help. It may be that what's needed is to no longer serve the bar patron. This is called &quot;cutting them off&quot;. When a bar patron is cut off, they will no longer be served alcohol for the rest of the service. This can be embarrassing, both for the bar patron and the bartender, so it's best practice to be kind, clear and discrete. If a patron is aggressive however, all bets are off and they should be removed from the bar.

Bartenders have legal obligations depending on where you work. This can include checking the legal age of the patrons at your bar and monitoring the alcohol intake of the patrons at your bar. Sometimes, a bartender can be held individually liable for the failure to meet these obligations. Take the time to find a reputable source to discover what the legal obligation of bartenders in your area are now.

This concludes Lesson 1: Bartending Basics. You can move on now to Lesson 2: Mixing Drinks. 

= Mixing Drinks =
Congratulations! You've made it to the fun part of the course. Mixing and serving drinks are fundamental to bartending, and they're fun to do. In this lesson we'll focus on the most common and most important recipes you'll encounter working at a bar along with the basic techniques and know-how needed to complete them.

In this lesson we'll discuss:

# Glassware
# Typology of cocktails
# Preparation and Skills
# Essential Cocktails
This is the part of the course that really benefits from hands-on practice. When we talk about preparing fruit and mixing cocktails, follow along! It isn't necessary to build an entire bar in your home in order to practice, but having some of the basics and making substitutions when needed will go a long way to making these actions feel natural and easy. If you have a bartending gig in your future, even doing each of these recipes once will put you leagues ahead of your first try being on the clock.

== Glassware ==
Before you can mix a drink, you have to know what glass you're gonna pour it in. Wikipedians have done an excellent job of creating a near comprehensive list of the glasses commonly used to serve alcohol. I've transcluded a navigation box below to all of their pages. Take the time to familiarize yourself with these glasses and their uses by clicking on the pages and digesting the contained information. You don't have to know the history and origin of every glass, just be familiar their shape and general use. The most important ones for this course are:

* Collins
* Highball
* Old fashioned (or Lowball or Rocks)
* Champaign Coupe (or just Coupe)
* Champaign Flute
* Cocktail Glass (or Martini glass)
* Hurricane Glass
* Wine Glass (Red and White)

When we get to cocktails, each recipe will place the drink in a specific type of glass, and you can return here as a reference tool if you need it.

{{Navbox
| name     =  Glassware
| title    = [[:wikipedia:List of glassware|Glassware]]
|listclass = hlist
|state = expanded
|group1 = [[:wikipedia:Tumbler (glass)|Tumblers]]
|list1  =
* {{Nowrap|[[File:Collins glass silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Collins glass|Collins glass]]}}
* {{Nowrap|[[File:Granyonyi Stakan Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Faceted glass|Faceted glass]]}}
* {{Nowrap|[[File:Highball glass silhouette.svg|16x28px|link=|class=skin-invert]] [[:wikipedia:Highball glass|Highball glass]]}}
* {{Nowrap|[[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Old fashioned glass|Old fashioned glass]]}}
* {{Nowrap|[[File:Shot glass.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Shot glass|Shot glass]]}}

|group2 = [[:wikipedia:Beer glassware|Beer glassware]]
|list2  =
* {{Nowrap|[[File:beer stein.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Beer stein|Beer stein]]}}
* {{Nowrap|[[File:Pilsner glass silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Beer glassware#Pilsner glass|Pilsner glass]]}}
* {{Nowrap|[[File:Pint glass.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Pint glass|Pint Glass]]}}
* {{Nowrap|[[File:Pony Glass Silhouette.svg|14x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Pony glass|Pony glass]]}}
* {{Nowrap|[[File:Tankard Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Tankard|Tankard]]}}
* {{Nowrap|[[File:Wheat beer glass silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Wheat beer glass|Wheat beer glass]]}}
* {{Nowrap|[[File:yardglass.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Yard of ale|Yard glass]]}}

|group3 = [[:wikipedia:Stemware|Stemware]]
|list3  =
* {{Nowrap|[[File:Absinthe Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Absinthiana#Absinthe glass|Absinthe glass]]}}
* {{Nowrap|[[File:Chalice Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Chalice|Chalice]]}}
* {{Nowrap|[[File:Coppa Asti sagoma.svg|24x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Champagne glass#Coupe|Champagne coupe]]}}
* {{Nowrap|[[File:flutesil.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Champagne glass#Flute|Champagne flute]]}}
* {{Nowrap|[[File:Coctail glass.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Cocktail glass|Cocktail glass]]}}
* {{Nowrap|[[File:Glencairn Whisky Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Glencairn whisky glass|Glencairn whisky glass]]}}
* {{Nowrap|[[File:Hurricane Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Hurricane glass|Hurricane glass]]}}
* {{Nowrap|[[File:Margarita Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Margarita#Glass|Margarita glass]]}}
* {{Nowrap|[[File:Alcohol glass nick and nora.svg|24x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Nick &amp; Nora (glass)|Nick &amp; Nora]]}}
* [[:wikipedia:Rummer|Rummer]]
* {{Nowrap|[[File:Sherry Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Wine glass#Sherry glass|Sherry glass]]}}
* {{Nowrap|[[File:brandysnifsi.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Snifter|Snifter]]}}
* {{Nowrap|[[File:wineglass.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Wine glass|Wine glass]]}}

}}

== Typology of Cocktails (Cocktail Families) ==
One last thing before we starting preparing garnishes and mixing drinks! I promise! Cocktails and their recipes didn't just appear from nothing in a vacuum. They were developed over time as tastes changed and different ingredients became available or popular. Because of this process, we can group cocktails into broad categories called Cocktail Families. Not all bartenders and mixologists agree on how cocktails should be grouped or how many families there are, but by understanding generally how cocktails relate to each other you can prioritize your learning and remember what goes in which cocktail easier. 

The families we're going to talk about are: 

* Sour Cocktails
* Ancestral/Aromatic Cocktails
* Lengthened Cocktails
* Frozen Cocktails

This isn't by any means comprehensive, but they'll cover most the cocktails we'll explore shortly.

=== Sour Cocktails ===
[[wikipedia:Sour_(cocktail)|Sour cocktails]] are an old family of cocktails who's drinks that are all, well sour. The template for a sour cocktail is simple:

&lt;u&gt;Shake&lt;/u&gt; with ice: Spirit + Sweetener + Something Sour + Egg (sometimes)

The &quot;Something Sour&quot; is most often a citrus juice like lemon or lime. Many sour cocktails will have &quot;sour&quot; in the name, like a Gin Sour, Rum Sour, Whiskey Sour, or Amaretto Sour. Some other notable sour cocktails are the Daiquiri, Cosmopolitan and Margarita. Note that it doesn't matter what base spirit you use, any base spirit can be used in a sour. A sub-family of sour cocktails replaces the traditional sweetener of simple syrup with a liqueur. These cocktails are called a &quot;Daisy&quot;.

=== Ancestral or Aromatic Cocktails ===
Ancestral or Aromatic Cocktails are spirit-forward drinks that use bitters and sometimes liqueur to give an aromatic taste. The template to an aromatic cocktail is:

&lt;u&gt;Stir&lt;/u&gt; with ice: Spirit + Sweetener + Bitters 

These cocktails let the taste of the spirit shine while deepening the flavor with bitters. Some common aromatic cocktails are the Martini, Manhattan, Old Fashioned, Sazerac and Negroni.

=== Lengthened Cocktails ===
Lengthened Cocktails are drinks that have been &quot;lengthened&quot; by adding a carbonated beverage. Two major sub-families of lengthened cocktails are &quot;Highballs&quot; and &quot;Spritz&quot; cocktails. A Highball is just a spirit lengthened with non-alcoholic mixer, think Gin and Tonic or Rum and Coke. A Spritz is a wine lengthened with a mixer, most often soda or sparkling wine. An example you'll find more in Spain is the Kalimotxo (KAL-EE-MO-KO), a drink with Red Wine and Coca Cola. Don't knock it 'till you try it! There are hundreds of combinations to make Lengthened Cocktails, Highballs and Spritz are just the major subtypes.

=== Frozen Cocktails ===
Frozen drinks are frozen! They are just so unlike a non-frozen drink that they get their own family. Frozen drinks are either blended with ice or churned over frozen metal to make a slushy consistency. Many frozen cocktails were adapted from classic cocktails, like a Frozen Daiquiri, but others are always frozen, like a Piña Colada. These are often &quot;batched&quot; or made ahead of time, especially if they're going to come from a slushy machine.

== Preparation and Skills ==
In a bar, preparation is all the steps that are taken before a service starts to get ready in advance. Sometimes this is things that are done at close the night before, and sometimes these are things done in the hours before a service. Some examples of prep are:

* Stocking fridges with beer and wine
* Stocking the bar's backstock of spirits
* Juicing citrus
* Cutting garnishes for drinks
* Making syrups
* Topping up bottles
* Cleaning and preparing tools for the service

All of these make a good bartender. We'll talk more about how good prep is good hospitality, and how it defines a good bartender later. Some of these are self explanatory, but some merit further investigation to make you a more effective and efficient bartender.

While you go through this section, I recommend you actually follow along and prepare the ingredients listed ''especially the lemons and limes''. Immediately following prep, we'll start mixing cocktails that use these ingredients. Store them in an air tight, labeled and dated container if you plan on mixing some drinks within the next day or two. Cut citrus doesn't last that long, so don't overkeep it. You can always get more practice by preparing more garnishes. A lemon wedge can be used for a dozen other things anyway (I like one with my Coke).
[[File:Hellingshoek_antoineren_02.png|left|thumb|241x241px|This isn't me. I don't know who this man is. Thanks to him for uploading this picture though. Curl your fingers away from the knife.]]

=== Before you prep ===
Most preparation can and should be done with a small paring knife. A larger chef knife can be used for large fruits but is harder to maneuver especially when working with small, round fruits and fragile herbs. When cutting something with a knife, curl the fingers of your non-dominant hand away from the blade. This will help protect your fingertips and cut faster with less risk.

You should always have clean hands when you handle food. Before touching food, wash your hands with soap and water for at least 20 seconds, and dry them with a clean towel. Rewash to your hands whenever you step away from your task or touch something dirty like your phone.

=== Juicing Citrus ===
[[File:Citrus × aurantium - fruits cut.jpg|thumb|The lengthwise cut (left) is unsuitable for juicing. The widthwise cut (right) is perfect to juice.]]
Most bars will have one or two types of juicer: a handheld citrus squeezer and/or a countertop citrus reamer. The handheld squeezer is faster and more efficient but cannot fit larger citrus like grapefruit. There are few wrongs ways to juice citrus, but doing it well will save you a lot of time during the prep. There are a few things to keep in mind when juicing citrus at the bar:

# Start with room temperature citrus. Cold citrus will yield less juice. If you have a particularly hard lime or lemon, place it on the counter and put your palm on top. Roll the citrus back and forth while applying firm pressure. This can free up some juice.
# Cut the fruit widthwise down the middle, so that you have two even sections. The cut face of the citrus should look like a wagon wheel with dot of pith in the center.
# Place your fruit cut-side down in the squeezer. Squeeze the citrus through a mesh strainer to remove any seeds or pulp. Squeeze with strength from your arms, not your wrists to avoid injury over time.
# Always label and date fresh juice and store in a sealed container in a refrigerator (Ex. Lime Juice Date: 2/27).

=== Making Simple Syrup ===
Simple syrup is an appropriately named combination of sugar and water made by combining a ratio of 1 parts granulated sugar and 1 parts water. It's a common ingredient used to sweeten cocktails by pre-dissolving the sugar. Here's the recipe:

# Place your sealable storage container on a scale and tare the scale so that it shows zero.
# Add a quantity of white granulated sugar to your container and note the measurement (ex. 200 grams of sugar).
# Add an equal amount of room temperature water (ex. 200 grams of water, making a total measurement of 400 grams).
# Seal the container and shake the mixture until the sugar has dissolved. It may be cloudy, but it with clarify as the mixture settles.
# Label and date your syrup (ex. Simple Syrup Date: 8/29).
If you're preparing this at home and don't have a scale (you're probably American), you can measure by volume, but it will be less accurate and may affect the final result. In a bar, always measure by weight.

=== Preparing Garnishes ===
There are so many things that can be put in drinks to improve their overall look and taste. There are a few, however, that you will find at most bars. If your bar has a specialty garnish, they should train you on how to prepare the house garnishes. For fruit garnishes, they should be used the same day they're cut and discarded at the end of the service. Let's go over some garnishes you'll find just about anywhere.
[[File:Knife_by_Bob_Kramer_(17970647591).jpg|thumb|How beautiful [[File:Face-kiss.svg|20x20px]] ]]

==== Cutting Citrus Half-Moons ====

# Cut the fruit in half lengthwise pole-to-pole. The cut-side of the fruit should have a stripe of pith down the center.
# Place the halves cut-side down, so that the nubs on each end are facing to your left and right.
# Cutting straight ahead, make slices about a quarter inch thick.
# Store in a sealed, labeled and dated container in a refrigerator until needed.

==== Cutting Citrus Wedges ====

# Trim the top and bottom of the fruit to create two flat surfaces. The flat surfaces should look like a wagon wheel with a white dot of pith in the center, or may be entirely pith. This step can be skipped for grapefruit, as they are large enough to rest on a side without additional cuts.
# Rest the fruit on its end, and cut it in half lengthwise. This should reveal a stripe of pith down the center of the cut face.
# Looking at the cut-side, make a shallow cut perpendicular to the pith extending about a half inch either side. Cut through about half of the meat of the fruit, but not all the way through to the peel.
# Placing the fruit cut-side down, cut the half further into quarters, then angling your knife, cut those fourths in half as well. Smaller fruits may only yield three wedges per half fruit.
# Store in a sealed, labeled and dated container in a refrigerator until needed.

==== Cutting Citrus Wheels ====

# Trim the nub off one end of the fruit. The cut should be deep enough to reveal a small wagon wheel shape with a dot of pith in the center.
# With the cut side facing to your left or right depending on your dominant hand, slice straight away from you creating about quarter inch thick circles until you no longer have enough fruit to safely hold while cutting.
# You many way to score a slit in the wheels depending on if you are putting them on the rim of a glass.
# Store in a sealed, labeled and dated container in a refrigerator until needed.

==== Cutting Citrus Peels and Twists ====

# Holding the fruit in your non-dominant hand, drag a y-peeler starting away from you and pulling towards you.
# If peeling an orange or grapefruit, using a paring knife, trim away excess pith from the back of the peel.
# Optionally, using a paring knife, trim the edges of the peel to make straight even sides.

==== Cutting Grapefruit &quot;Horses Neck&quot; ====

# Holding the grapefruit in your non-dominant hand, use a y-peeler to peel around the circumference of the fruit creating a long peel.
# Optionally, trim the ends with a paring knife for a cleaner look.

==== Preparing Pineapple Wedges ====

# Using a large chef knife, cut off the top of the pineapple.
# Resting on its bottom with the cut-side up, cut the pineapple in half from top to bottom.
# Resting the pineapple on its cut-side, angle your knife and cut the fruit into three equal parts creating triangle shapes.
# Take each section and slice into about inch thick wedges.
# You may way to score the point of each wedge to be able to hang it on a glass.
# Store in a sealed, labeled and dated container in a refrigerator until needed.

==== Preparing Mint ====

# Place a damp paper towel in the bottom of a sealable storage container.
# Take a sprig of mint in one hand, and pick the leaves at the bottom of the stem. Leave the ones at the top of the sprig attached to the stem. Place the leaves in the prepared container.
# Bunch the stems of the picked sprigs together and trim them to be only a couple inches long.

=== Essential Skills ===
There are a couple techniques that are used over and over in bartending. The two most essential to mix cocktails are shaking and stirring. Refer back to these steps the first few times you mix a cocktail in the next section. 
[[File:Bartender Photo.jpg|thumb|Shaking a sealed Boston shaker horizontally.]]

==== Using a Boston Shaker ====
A Boston shaker comes in two parts. One will be a large metal mixing cup, and the other will either be a smaller metal mixing cup or a glass mixing cup. Using a shaker comes in a few steps:

# We will always start by building a shaken cocktail in the small or glass mixing cup. Add the liquid ingredients (including egg, if being used), then add the ice after. The longer the liquids are in contact with the ice, the more they will be diluted as the ice melts. 
# The large mixing cup is placed over top the smaller mixing cup with enough force to create a seal. Feel free to whack the back of the large tin with the palm of your hand to ensure a seal.
# With one hand on either cup holding them together, the assembled cups are then flipped over so that the drink is primarily in the larger mixing cup. 
# The most efficient way to shake is horizontally, so that the drinks sloshes vigorously from left to right. Shaking vertically will achieve the same result, but it is more energy intensive and slower as it throws the drink upwards before slamming it back down. You will feel the metal become cold, and it may frost as the drink is chilled. These are signs that you can stop shaking.
# At this point, the cold from the ice has caused the metal tins to contract and shrink slightly which will bind them together tightly. To unseal the shaker, hold the larger cup or place it on a counter. Identify which was the smaller cup is leaning. With the palm of your hand, strike the smaller cup away from the direction it's leaning. There should be a crack as the seal is broken (this is called &quot;cracking&quot; a shaker open).
# To strain, place the Hawthorn strainer over the mouth of the larger cup, and pour the drink through the strainer.
[[File:Stirring with a bar spoon.jpg|thumb|230x230px|Stirring with a Bar Spoon]]
==== Stirring a Cocktail ====
It sounds simple, but there's a right way and a wrong way to stir a cocktail. The goal is to incorporate and chill the ingredients without agitating, decarbonating or introducing air. Here are a few simple steps:

# Insert a bar spoon to the bottom of the mixing glass containing you ingredients and ice. Push the spoon all the way to the bottom edge of the mixing glass.
# Swirl the spoon so that it stays in contact with the edge of the glass, rotating the ice in a circle.

==== Serving Sparking Wine ====

# Use the tab to peel away the foil.
# Keep one hand on top of the cage and cork. Lift the handle and twist it open.
# Keep the cage on the cork. Hold the cage and the cork with your dominant hand.
# Holding the bottle in your non-dominant hand, twist the bottle back and forth. The goal is to gradually release the pressure in the bottle, not to make a loud &quot;pop&quot;. Ideally, you should open the bottle almost silently.
# Holding the bottle at the base with one hand (if possible), pour slowly into the glass as the bubbles will climb.

[[File:Pouring wine into a decanter.jpg|left|thumb|Pouring wine|296x296px]]

==== Uncorking Still Wine ====

# Find a wine key with a double-hinge.
# Using the knife, cut the foil around the lip of the bottle. Using the bottle opener end of the key, peel back to foil to expose the cork.
# Insert the corkscrew into the center of the cork by twisting until most of the screw is embedded.
# Place the first hinge on the lip of the bottle, and use it as leverage to lift up on the wine key (lifting the cork partially).
# Place the second hinge of the lip of the bottle, and completely remove the cork.
'''Serving Beer'''

# Whether pouring from a bottle or a tap, tilt the glass 45° and pour the beer down the side of glass to prevent too much foam, or &quot;head&quot;, from developing.
# At the end of the pour, straighten the glass to allow for some head to form.

== Essential Cocktails ==
Here we are, finally mixing some drinks! It can be overwhelming to open a book of cocktail recipes and see the hundreds of drinks with unique names and recipes. Take a deep breath, because there isn't a bartender in the world that knows all of those cocktails by heart, and you aren't expected to either. Over time, you'll pick up on more drinks and their makeup, but that takes time! That's why it's important to focus on the essential cocktails that every bartender knows and that any bar patron might order. What follows are 20 cocktail recipes for classic cocktails that you might make every day at a bar. 

I highly encourage you to make these drinks at home. If you intend to become a bartender, the difference between your first attempt at a recipe being in the privacy of your home and on your first day on the job is ''enormous''. That raises the question, &quot;How am I supposed to supply all the ingredients needed to make all these drinks‽&quot; Start with things you can buy at the grocery store:

* Lemons, Limes and Oranges
* Club Soda
* White Sugar

You may already have these things from trying out the preparation techniques above. If you don't, go out and grab these ingredients, cut some garnishes, and prepare some simple syrup.

The most expensive part of any cocktail is the spirit. I've color coded the following recipes by which spirit they use. The list generally starts with easy recipes and becomes more complicated as you descend. I recommend staring with one type of spirit and trying a few recipes that use that spirit. For example, with only vodka, you can make a Highball, Cosmopolitan, Moscow Mule, Bloody Mary and a (Vodka) Martini. That's one fifth of the list done, with a spirit you likely already have at home.

I don't recommend buying fancy or expensive spirits for this. Buy something cheap or middle-of-the-road to practice with. That's like what your bar patrons are going to be ordering most often anyhow.

All that being said, 20 recipes is a lot to learn. Take your time, maybe make one or two a day. The recipes aren't changing, and some are very simple. You by no means have to drink what you make. You DO have to TASTE what you make. Bartenders, when trying a drink on the job, will smell a drink, swirl it in their mouth and spit it out. Do this at least twice for each drink if you don't intend to drink them. Take note of the taste, the balance of the drink and your personal preference for or against it. Please don't make all 20 in one night and get hammered. A daiquiri will be a daiquiri tomorrow, and it'll be sweeter for the waiting.

Well, get started!

=== Spirit Color Key ===
{{RoundBoxLeft|theme=1|width=50}}
Vodka
{{RoundBoxRight|theme=2|width=50}}
Gin
{{LeftRightBoxClose}}
{{RoundBoxLeft|theme=3|width=50}}
Rum
{{RoundBoxRight|theme=4|width=50}}
Tequila
{{LeftRightBoxClose}}
{{RoundBoxLeft|theme=5|width=50}}
Whiskey
{{RoundBoxRight|theme=11|width=50}}
Other/Mixed
{{LeftRightBoxClose}}
=== Highball ===
Highballs are a class of cocktail sometimes called &quot;plus one&quot; cocktails because they're a blend of a spirit plus a mixer. Some common highballs are Gin and Tonic, Rum and Coke, Whiskey and Ginger Ale, and Scotch and Soda.

{{RoundBoxTop|theme=11}} 
'''Glass:''' Highball [[File:Highball glass silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Wedges with soda, Lime Wedges with Tonic, Cola and Ginger Beer/Ale

'''Ingredients'''

2 oz of Spirit

Mixer of choice, to top

'''Combine''' Spirit and ice. Stir. Add Mixer to fill. Drag spoon through drink a few times to gently combine. Garnish
{{RoundBoxBottom}}

=== Cosmopolitan ===
{{RoundBoxTop|theme=1}} 
'''Glass:''' Coupe or Martini Nowrap|[[File:Coctail glass.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Twist

'''Ingredients'''

1 ½ oz Vodka (Preferably Citron Vodka)

⅓ oz Lime Juice

⅓ oz Cointreau (or other orange liqueur)

⅓ oz Sweet Cranberry Juice

'''Shake''' all ingredients with ice until chilled. Strain into glass. Garnish. 
{{RoundBoxBottom}}

=== Margarita ===
{{RoundBoxTop|theme=4}} 
'''Glass:''' Rocks [[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lime Wedge and ask preference for Salt, Sugar, or Tajin rim

'''Ingredients'''

2 oz Tequila

½ oz Agave Nectar

1 oz Lime Juice

½ oz Triple Sec

'''Moisten''' the rim of the glass with a lime wedge, then roll exterior of glass in rimming material careful to avoid getting any inside the glass. Combine all ingredients in shaker with ice. Shake. Add ice to glass. Strain. Garnish.
{{RoundBoxBottom}}

=== Paloma ===
{{RoundBoxTop|theme=4}} 
'''Glass:''' Highball [[File:Highball glass silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Grapefruit Wedge or Lime Wheel

'''Ingredients'''

2 oz Tequila

½ oz Lime Juice

Dash of Salt

Grapefruit Soda, to top

'''Combine''' Tequila, Sale and Lime Juice in glass with ice. Stir to chill. Add Grapefruit Soda to fill. Drag spoon through drink a few times to gently combine. Garnish.
{{RoundBoxBottom}}

=== Whiskey Sour ===
{{RoundBoxTop|theme=5}} 
'''Glass:''' Sour or Coupe [[File:Coppa Asti sagoma.svg|24x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Wheel and Cherry

'''Ingredients'''

2 oz Whiskey

1 oz Lemon Juice

½ tsp. Sugar or Simple Syrup

'''Shake''' ingredients with ice. Strain. Garnish.
{{RoundBoxBottom}}

=== Moscow Mule ===
{{RoundBoxTop|theme=1}} 
'''Glass:''' Mule Mug [[File:Alcohol_glass_copper_mug.svg|24x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lime wheel (optional)

'''Ingredients'''

2 oz Vodka

½ oz Lime Juice

3 oz Ginger Beer

'''Combine''' ingredients in a copper mug filled with ice. Stir. Garnish, optionally.
{{RoundBoxBottom}}

=== Spritz ===
Spritz are a class of cocktails that follow the same general 3-2-1 recipe of 3 oz Prosecco, 2 oz liqueur, 1 oz club soda. The most common is the Aperol Spritz.

{{RoundBoxTop|theme=11}} 
'''Glass:''' Highball or Wine Glass [[File:wineglass.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Orange Wheel

'''Ingredients'''

3 oz Prosecco

2 oz Liqueur (Aperol, for an Aperol Spritz)

1 oz Club Soda

'''Combine''' Prosecco and Liqueur with ice in glass. Stir gently. Top with Club Soda. Garnish.
{{RoundBoxBottom}}

=== Mojito ===
{{RoundBoxTop|theme=3}} 
'''Glass:''' Highball [[File:Highball glass silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Mint Sprig

'''Ingredients'''

1 ½ oz White Rum

5 to 6 mint leaves

1 oz Lime Juice

2 tsp Sugar

Soda water

'''Muddle''' the Mint, Lime Juice and Sugar until well mixed but not mushy. Add the Rum and fill the glass with ice. Top with Soda Water. Garnish.
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=== Daiquiri ===
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'''Glass:''' Coupe [[File:Coppa Asti sagoma.svg|24x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lime wheel (optional)

'''Ingredients'''

1 ½ oz White Rum

2 tsp Sugar

1 oz Fresh Lime Juice

'''Shake''' ingredients with ice until chilled and frothy. Strain into a coupe glass. Garnish, optionally.
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=== French 75 ===
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'''Glass:''' Champagne Flute [[File:flutesil.svg|24x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Twist

'''Ingredients'''

1 oz Gin

½ oz Lemon Juice

½ oz Simple Syrup

3 oz Champagne (or other sparkling wine)

'''Shake''' Gin, Lemon Juice, and Simple Syrup with ice. Strain into glass. Top with Champagne. Garnish.
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=== Gimlet ===
{{RoundBoxTop|theme=2}} 
'''Glass:''' Coupe [[File:Coppa Asti sagoma.svg|24x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lime Wheel

'''Ingredients'''

2 ½ oz Gin

½ oz Lime Juice

½ oz Simple Syrup

'''Shake''' Gin, Lime Juice, and Simple Syrup with ice. Strain into glass. Garnish.
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=== Manhattan ===
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'''Glass:''' Rocks [[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Twist and/or Cherry

'''Ingredients'''

1 ½ oz Rye Whiskey

½ oz Sweet Vermouth

1 Dash Angostura Bitters

'''Chill''' rocks glass by combining ice and water, or use pre-chilled glass. Combine ingredients in a mixer with ice. Stir. Strain. Garnish.
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=== Mai Tai ===
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'''Glass:''' Rocks [[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Pineapple Wedge, Cherry, and Mint Spring

'''Ingredients'''

1 oz Light Rum

1 oz Gold Rum

½ oz Orange Curaçao

½ oz Orgeat

½ oz Lime Juice

'''Shake''' ingredients with ice. Strain into ice filled glass. Garnish.
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=== Pina Colada ===
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'''Glass:''' Hurricane [[File:Hurricane Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Pineapple Wedge

'''Ingredients'''

1 ½ oz Rum (Preferably Coconut, like Malibu)

1 ½ oz Coconut Cream

1 oz Pineapple Juice

½ oz Lime Juice

'''Combine''' ingredients with ice in a blender. Blend on high for 20-30 seconds or until a flowing slushy consistency. If too stiff add water, if too thin add ice and re-blend for 2-3 seconds. Pour into glass. Garnish.
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=== Bloody Mary ===
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'''Glass:''' Pint or Highball [[File:Highball glass silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Wheel or Celery Stalk (or anything savory)

'''Ingredients'''

2 oz Vodka

6 dashes Worcestershire sauce

3 dashes Tobasco Sauce

Pinch of Salt

Pinch of Ground Black Pepper

½ Lemon, juiced

5 oz Tomato Juice

'''Combine''' all ingredients in a glass. Add ice and stir. Garnish.
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=== Negroni ===
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'''Glass:''' Rocks [[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Orange Twist or Orange Wheel

'''Ingredients'''

1 oz Dry Gin

1 oz Sweet Vermouth

1 oz Compari

1 oz Sparkling Water

'''Combine''' all ingredients in a glass. Add ice, preferably 2 large rocks. Stir. Garnish.

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=== Old Fashioned ===
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'''Glass:''' Rocks [[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Orange twist

'''Ingredients'''

2 oz Whiskey

1 Sugar Cube

4 dashes Aromatic Bitters

1 dash Soda Water

'''Muddle''' Sugar, Bitters and Soda Water in a glass until a uniform syrup is made. Add Whiskey. Add Ice. Stir. Express Orange Twist over drink then add to glass.
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=== Tom Collins ===
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'''Glass:''' Collins or Highball [[File:Collins glass silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Wheel and Cherry (optional)

'''Ingredients'''

2 oz Gin

1 oz Lemon Juice

½ oz Simple Syrup

Club Soda, to top

'''Combine''' Gin, Lemon Juice, and Simple Syrup in glass. Add ice to top. Fill with Club Soda. Stir gently. Garnish.
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=== Martini ===
Martinis are kind of their own thing, and there's some vocab to go over. There are many drinks that call themselves a martini, and the Classic Martini itself has many variations to suit each bar patron's taste. To make a Martini to order, first you have to have a conversation with your bar patron. 

# First, ask if they prefer Gin or Vodka. Gin is traditional, but it can be made with either.
# Second determine how much Vermouth they would like. The amount of Vermouth will determine how &quot;Dry&quot; or &quot;Wet&quot; the Martini is.

&quot;Extra Dry&quot; or &quot;In and Out&quot; = Vermouth is swirled in the glass then poured out, then the drink is entirely Gin or Vodka

&quot;Dry&quot; = 1 part Vermouth to 5 parts Gin or Vodka

&quot;Wet&quot; = 1 part Vermouth to 2 parts Gin or Vodka

&quot;50/50&quot; = 1 part Vermouth to 1 part Gin or Vodka

&quot;Dirty&quot; = 1 part Olive Brine to 5 parts Gin or Vodka, instead of Vermouth. Garnish with Olives instead of lemon.

The following recipe is for a Dry Martini. You can modify it to make it a Dirty Martini or to suit your taste of Vermouth. Martinis are often stirred, but they can be shaken on request.

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'''Glass:''' Martini [[File:Coctail glass.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon twist

'''Ingredients'''

2 oz Gin or Vodka (patron preference)

½ oz Dry Vermouth

'''Chill''' glassware by pouring water and ice in a glass, or use a pre-chilled glass. Combine ingredients with ice in a mixing glass. Stir until chilled. Strain into chilled glass. Garnish.
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=== Espresso Martini ===
The Espresso Martini is really not much like a Classic Martini at all. The word Martini really gets thrown around. Nonetheless, it's delicious. If your bar has an espresso machine, you'll be trained on how to pull an espresso shot. Otherwise, your bar may use cold brew concentrate made in house or from a can.

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'''Glass:''' Martini [[File:Coctail glass.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' 3 Coffee Beans

'''Ingredients'''

2 oz Vodka

½ oz Coffee Liqueur (often Kahlúa)

1 oz Espresso, or cold brew concentrate

1/4 oz Simple Syrup

'''Chill''' glass with water and ice, or use pre-chilled glass. Shake ingredients with ice. Strain into glass. Garnish by place 3 coffee beans in center.
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Hey! Congrats! You've made it to the bottom of the list. These 20 recipes are just the beginning of the possibilities when it comes to mixing drinks, but if you can nail these down you can get through just about any shift at any bar. That was an undertaking, and you did a great job.

This concludes Lesson 2: Mixing Drinks. You can move on now to Lesson 3: Hospitality.

= Hospitality =
In this lesson, we'll be talking about some points of service and hospitality that are as essential to bartending as alcohol. The sections in this lesson are:
# What is Hospitality?
# Steps of Service
# Internal Hospitality
# Developing Taste

== What is Hospitality? ==
Mixing drinks is only half the job, maybe even less than half. The real job is [[wiktionary:hospitality|hospitality]].&lt;blockquote&gt;The act or service of welcoming, receiving, hosting, or entertaining guests; an appropriate attitude of openness, respect, and generosity toward guests. &lt;/blockquote&gt;In German, the word for hospitality is ''gastfreundschaft'' or guest-friendship. This encapsulates the charge of all bartenders to make the patrons of our bars feel welcome from the moment the walk through our doors to the moment they leave. Ideally, they still feel welcome long after they've left—they might just come back!

'''If you want to be there, so will they.''' People can immediately tell if you want to be there or not. If your vibe's a drag, people won't want to talk to you. There's the catch: ''people need to talk to you to get drinks''. If people don't want to talk to you, they'll leave, and they won't come back. The entire purpose of a bar is not only pushed forward by you, it's ''gate-kept'' by you. If there's no bartender, there's no bar; if there's a poor bartender, there's a poor bar. If you can find time to smile while behind the bar, you'll find your job is actually easier.

== Steps of Service ==
Service and hospitality are related, but separate ideas. Service is the mechanics of bartending: mixing drinks, closing tabs, cleaning and clearing. Hospitality is how you make your patrons feel. Keeping your service up will go a long ways towards making an environment hospitable. If the drink service is slow and the bar is messy, patrons are not gonna feel good. That's an example of service impacting hospitality.

When a patron comes to a bar, there's steps to how the they should be served that blend hospitality and service techniques. Let's go over the steps of providing good service.

# &lt;u&gt;Greet the Patron&lt;/u&gt; - Make eye contact with each person as they enter or approach your bar. Try to keep a relaxed, or at least welcoming, posture. Greet them warmly, making it clear that you see them and that you're happy they're here. Offer them a menu and pour them a water depending on your bar.
# &lt;u&gt;Take the Order&lt;/u&gt; - Once they've had a moment, or if they look immediately ready, ask what they would like to order or if they have any questions. Repeat what they've order back to them so that they can let you know if you misheard.
# &lt;u&gt;Mix the Drinks&lt;/u&gt; - Maybe they've ordered beer or wine that just need to be poured, or maybe they've ordered several shaken cocktails. Now's the time to make them. While you're making drinks, be sure to look up at the bar and greet people as they arrive.
# &lt;u&gt;Drop the Drinks&lt;/u&gt; - Bring finished drinks to guests as soon as they're finished. Mixed drinks have a short shelf life and should be  delivered promptly.
# &lt;u&gt;Check in&lt;/u&gt; - Give them a few moments to taste their drinks. Check in with an open ended question. &quot;How is everything?&quot; Many people won't tell you if they don't like the drink, so feel free to try to make them comfortable with giving honest feedback. You can always remake a drink!
# &lt;u&gt;Clear and Ask&lt;/u&gt; - When they're done with their drinks, clear any unneeded dishes and give the bar a wipe (avoiding pushing any crumbs onto your patron's lap). Then ask if they'd like another round. It's best to keep this question open and neutral. We don't need to push people to get another drink. A simple, &quot;What's next?&quot; gives patrons the floor to tell you what they need.
# &lt;u&gt;Close Out&lt;/u&gt; - When they've asked for a check, close out and bring the check quickly. The dead time when a patron wants to leave but can't yet is a vibe killer. You should prioritize closing them out and letting them leave quickly. When you drop the check, make eye contact, smile and thank them for coming.

== Internal Hospitality ==
Hospitality isn't something reserved for the patrons of your bar. Hospitality is something that should be shown to your teammates and everyone you work with. Think of it like this: &lt;blockquote&gt;If you're willing to give 100% attention, kindness and respect to complete strangers who walk up to your bar, shouldn't you be willing to give even more to the people you work with and see every day?&lt;/blockquote&gt;The people you work with at your bar will be the most valuable and important resource you have, making them feel good when they're around you is equally if not more important than patron hospitality. They will give you invaluable tips, unlock opportunities you've never imagined, and maybe become lifelong friends, but only if you treat them with the same, or more, hospitality you give to bar patrons.

One step further, hospitality is something you can show yourself. If we return to the definition of hospitality ''as an appropriate attitude of openness, respect, and generosity'', you can absolutely show hospitality to yourself. Respect your needs, be generous with time away from work, and be open to making mistakes and learning. You'll find that bartending will be more pleasurable and that positive feeling will make your job even easier.

== Developing Taste ==
Oftentimes patrons of your bar will ask your opinion on drinks, the menu and even the area around the bar. It's good hospitality to be able to answer their questions to the best of your ability. Here's some tips on each of those topics.

=== Tasting Alcohol ===
Tasting alcohol, including straight spirits, is a unique experience that is often undervalued. Skilled sommeliers can detect subtle notes at different moments when drinking a wine. Something similar can be done with spirits. Just a reminder that you don't have to swallow a drink to taste it. It's common for bartenders to spit a drink out when tasting it, especially on the job. To effectively taste a spirit:

# Smell the spirit. Don't stick your nose straight into it! Hover your nose above the spirit so that you can get a natural whiff of the aroma.
# Take a sip and swirl it around your mouth. Your mouth can hold on to tastes for quite a while, and this first swirl will clear the taste of whatever you've eaten and drank earlier in the day.
# Take another sip and appreciate the flavor. This is the part that takes time, patience, and skill. There are several ways to break up flavors. What did you taste the second it hit your tongue? What about after spitting it out or swallowing? You can try to pick up on the biggest notes first. Is it smokey? Sour? Bitter? Then try honing in on something more subtle. Are there herbal notes? Is it fruity?

=== Giving Recommendations ===
One of the most common responses you'll get when taking someone's order is some variation of &quot;I don't know&quot;, &quot;What you recommend?&quot;, and &quot;What do you like?&quot;. All of these mean basically the same thing: &quot;Talk to me about what's on your menu and what you can make.&quot; 

If you bar has a menu, this should be an easier task as the first thing you'll do after you arrive is memorize the menu. Giving recommendations isn't as easy as knowing the menu though. In these moments, it's important to engage with the patron to find out what kind of drink would suit their taste.

My preferred way of navigating a drink recommendation conversation is treating it like a &quot;choose your own adventure&quot; story. Offer a series of choices that bring both of you closer to a good drink choice. Start with broad strokes of flavor: &quot;Do you like a drink that's sweet, smokey, or sour?&quot; Oftentimes, that is enough to select off a menu. If they say something smokey, you might think about drinks that have a whiskey or mezcal base. Sweet or sour could have all kinds of spirits. You can ask if they have a preferred alcohol. You can also ask if they've had any drinks in the past that they've really liked.

== Conclusion ==
Well, that's the basics.

''If you're reading this, this lesson is still under construction! Feel free to add, edit, or [[User:Sabbier|message me]].''</text>
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      <text bytes="72374" sha1="arc8wa0yl79x0t046jedgw17590heia" xml:space="preserve">Hi ! I'm a librarian that occasionally edits on Wikipedia and Wikiversity.

== Things I'm working on ==
[[Creating Wikiversity Courses]]

=== Bartending Wikiversity Course ===

==== Notes to Myself ====

# Course with several lesson subpages - could use the box template to organize.
# Finish/Build up the Hospitality Lesson
# Add an Advanced Bartending lesson with more recipes/mixology/significant bartenders/freepouring?
# Move stirring photo in skills for mobile friendly placement

==== Course Outline ====

# Bartending Basics
## Sources of alcohol
## Types of alcohol
## Tools of Bartending
## Safety
# Mixing Drinks
## Glassware
## Typology of Cocktails
## Preparation and Skills
## Essential Cocktails
# Hospitality
## What is Hospitality?
## Steps of Service
## Internal Hospitality
## Developing Taste
# Industry 
## History of pubs/bartenders/mixologists
## Wages + tips
## Hours + Working conditions
## Unions
# Advanced Topics
## Free-pouring
## Other recipes to know
## Round building
## Mixology
# Recommended Reading List
## Mr. Boston's Official Bartender's Guide - Available on Internet Archive
##

= Bartending for Beginners =

=== Introduction ===
Mixing drinks, talking to people, and making money: that's bartending! If you find that definition to be less-than-enough, this course will guide you through the essential skills of a bartender. This course is intended for adults of legal drinking age where they live who want to start bartending in a professional establishment either as a side gig or a career. Adults wanting to learn about bartending in general or how to mix drinks at home can also benefit from this course. No prior experience is required. By the end of the course, you should know: how to mix common drinks ordered at different types of bars and how they are related to one another; the liquors, spirits, ales, wines and other drinks used at the bar; the tools of the trade; the basics of safety as a bartender; how to create a hospitable environment for bar patrons; and details on the bartending industry, customs and history.

This course makes use of the [[wikipedia:Wikimedia_Foundation#Projects_and_initiatives|Wikimedia ecosystem]] of projects. Throughout the course, there will be links to Wikipedia pages, Wiktionary entries, and images from Wikimedia Commons. When a link is casually included in a lesson, I encourage you to browse its contents. Occasionally, clicking a link and digesting its contents will be formally assigned as part of the course.

=== Structure of the Course ===
This course is comprised of lessons which each focus on a component of bartending. It is recommended to complete them in order, as each lesson will build on the previous. The lessons can be found below:

# Bartending Basics
# Mixing Drinks
# Hospitality 
# Industry (Under Construction)

=== Course Objectives ===
By the of the course, you should be able to:

* Share information about and identify the different types of alcohol
* Identify and use the tools of a bartender to complete common techniques
* Mix the 20 essential cocktails
* Serve bar patrons safely and hospitably

=== Recommended Materials ===
Bartending is a physical practice, and theory alone is not enough to prepare yourself. There will be activities that you complete at home as part of the course. The materials required to complete these activities are:
# Boston shaker (Preferably not a Cobbler shaker or a Parisian shaker, though they may be easier to find.)
# Hawthorn Strainer
# Bar Spoon
# Jigger
#A glass for mixing, and a glass to pour drinks into (Don't worry about the style of glass, it just needs to be large enough to hold ice and 10 oz of liquid.)
[[File:Bartender_Photo.jpg|left|thumb|Bartender with Boston Shaker]]
[[File:Cocktail-strainer.jpg|thumb|Hawthorne Strainer]]
[[File:Jigger.jpg|center|thumb|160x160px|Jigger]]
In addition to these materials, you will also need to ingredients to the recipes we'll prepare. Those will be discussed when we talk about mixing drinks. Many of these materials can bought online or found at secondhand or thrift shops depending on your area. There may also be a specialty store near you catering to restaurants and bars. Having these materials at home are an invaluable way to practice mixing drinks, with the added benefit of allowing you to entertain guests at home! If you cannot acquire these materials, the course is still completable in theory, but you will miss out on the practical aspects of the course. No materials are needed for lesson one.

In addition to these materials, you will also need to ingredients to the recipes we'll prepare. Those will be discussed when we talk about mixing drinks. Many of these materials can bought online or found at secondhand or thrift shops depending on your area. There may also be a specialty store near you catering to restaurants and bars. Having these materials at home are an invaluable way to practice mixing drinks, with the added benefit of allowing you to entertain guests at home! If you cannot acquire these materials, the course is still completable in theory, but you will miss out on the practical aspects of the course. No materials are needed for lesson one.

In addition to these materials, you will also need to ingredients to the recipes we'll prepare. This will be discussed when we talk about mixing drinks.

Start here with Lesson One: Bartending Basics.

= Bartending Basics =
What is a [[wikipedia:Bartender|bartender]]? At its simplest, a bartender serves drinks in a bar. In reality, a bartender selects, mixes, pours, and serves drinks while hosting patrons and creating a hospitable atmosphere at a bar, pub, restaurant, nightclub, living room or [[wikipedia:Parking_lot|parking lot]]. A bartender must have a variety of skills, both technical and interpersonal in order to be successful. A bartender can work in many locations, but this course will focus on bartenders that tend to tend in bars.

This lesson covers: 

#Sources of Alcohol
#Types of Alcohol
# Tools of Bartending
# Safety

== Sources of Alcohol ==
Behind the bar, you are the expert in the room on alcohol. While no one can claim to know everything on alcohol, it's important to have the basics. This and the next lesson focus on learning what alcohol is and what kind of drinks and liquors are made with it.

Every alcoholic drink you've ever had has included the same basic type of alcohol: [[wikipedia:Ethanol|Ethanol]]. Ethanol is one of three types of alcohol and the only alcohol humans can safely drink. All alcoholic beverages whether it be beer, wine or a spirit all contain Ethanol.

There are hundreds of different beverages and liquors that are served at bars around the world.  The method of refining raw ingredients into a final product with Ethanol is what distinguishes each type of beverage. Fortunately they're all related to each other (since they all contain Ethanol) and have some major categories to guide us in understanding what they are. The first step to make any type of alcohol is [[wikipedia:fermentation|fermentation]].

=== Fermentation ===
[[File:40168_2022_1274_Fig6.webp|thumb|People get very scientific with it.]]
Using a process that has existed for thousands of years, we can employ a fungus called [[wikipedia:Yeast#Uses|yeast]] to create alcohol. This process is called fermentation. At it base, fermentation for alcohol production is the process of using yeast to convert sugars to ethanol. 

&quot;Sugars&quot; is intentionally plural. There are many places you can find sugar suitable for fermentation, and you can derive sugar from grains, fruits and vegetables. The source of the sugar is often what defines what type of final alcoholic beverage you get. We'll talk more on how fermenting different materials yields different drinks in a moment. 

Fermentation happens in a couple of steps.

# '''Mashing''': Grains like barley or rice are milled into a coarse flour and fruits are mashed into a pulp. These are sometimes mixed with hot water, where enzymes convert starches to sugars. For beer, this mash is then often boiled.
# '''Fermentation''': The mash is transferred to a fermentation vessel. Yeast is added, and fermentation begins. Over a varying period of time (often 5-10 days), yeast consumes the sugars, producing alcohol and carbon dioxide. This is called primary fermentation.
# '''Conditioning and Packaging''': After primary fermentation, there is often secondary fermentation or conditioning to add or mature flavors and to clarify. It’s then sometimes carbonated and packaged.

Now fermented, there is some amount of alcohol in your drink. But what if ''some'' isn't enough? A secondary process called [[wikipedia:Distillation|distillation]] allowed for fermented drinks and solutions to be concentrated into spirits. 

=== Distillation ===
[[File:Alambins_industrials_per_a_la_destil·lació_de_licors_a_Catalunya.jpg|left|thumb|Industrial stills for liquor production in Catalonia]]
Distillation is a process by which brewers can concentrate the level of alcohol in a drink. Fermentation alone cannot make alcohol concentrations like that of vodka or gin. To reach that level of concentration, [[wiktionary:distiller|distillers]] (referring to both the apparatus that distills and the person that runs it) can boil off and collect the alcohol produced by fermentation. Distillation happens in a few steps:

# The fermented substance is placed into a vessel called a [[wikipedia:Still|still]] (this is where di''still''ation gets its name).
# The ferment is heated slowly from room temperature until it reaches the boiling point of ethanol. The boiling point of ethanol is lower than the boiling point of water, so the ethanol will vaporize, leaving the water behind.
# The ethanol vapor rises through a column until it reaches the condenser. In the condenser, the vapor makes contact with the a cold surface that cools it back into a liquid state.
# The ethanol is collected, and the process is often repeated to further concentrate and purify the result. Sometimes the process is done without having to stop and restart in a process called continuous distillation.

The result of distillation is a liquor with a higher amount of alcohol by volume (ABV) than before. 

=== Alcohol By Volume ===
The amount of alcohol in a drink is calculated based on the percent of ethanol compared to non-ethanol in the total beverage, which is called Alcohol by Volume or ABV for short. For example, if a beverage has an ABV of 50%, half of liquid in the drink will be ethanol and the other half will be everything else.

ABV is important to understand, because it is the basis of drinking and serving drinks safely as well as how many drinks are categorized. While all alcohol is made through fermentation, the process of fermentation alone can only create a concentration of alcohol of about 5% to 20% ABV. Spirits are generally distilled liquids that have a higher ABV (20% or more, as high as 95%). This distinction lower-ABV drinks like wine, beer and cider versus higher-ABV spirits like vodka, gin and whiskey. 

== Types of Alcohol ==
Fermentation and distilling are the core of all alcohol production, but there many more ways that brewers and distillers affect their final product. This incredible flow chart shows many of the processes of fermentation, distillation, carbonation, ageing and processing that make different alcoholic drinks. Take a look at it and compare the different starting ingredients with each other, and then compare the starting ingredients with their final products.
[[File:Alcohol_Flow_Chart.svg|center|frame|Alcohol Flow Chart]]
There is so much to learn about alcohol, and while you should be knowledgeable on the different types, you by no means need to be an expert (see: sommelier) to be a bartender. Below follows an overview of each family of alcohol, but if you want to learn more you can click on the links to Wikipedia pages in each subsection below, or I recommend browsing the [[wikipedia:Portal:Liquor|Liquor Portal]] to view all that Wikipedia has to offer.

Hopefully it's easier to see now how beverages and spirits can both be categorized based on what they're fermented from and how concentrated their alcohol content is. Let's use these categories to talk more about each major type of alcohol.

=== Wine, Beer and Cider (Low-ABV Drinks) ===

==== Wine ====
We call fermented grape juice [[wikipedia:wine|wine]]. Yeast is added to pressed grapes to develop it into one of four types: [[wikipedia:Red_wine|red]], [[wikipedia:white_wine|white]], [[wikipedia:rose_wine|rosé]], and [[wikipedia:Orange_wine|orange]]. The type of grape and the duration of the contact with the skins of the grapes help to define which type of wine is produced. The table below is from the Wikipedia page on wine.
{| class=&quot;wikitable&quot;
|+Colors of wine
!
!Long contact with grape skins
!Short contact with grape skins
|-
!Red grapes
|'''''Red wine''''', made from dark-colored red grape varieties. The actual color of the wine can range from dark pink to almost black. The juice from red grapes is actually pale gray; the color of red wine and some of its flavor (notably tannins) comes from phenolics in the skin, seeds and stem fragments of the grape, extracted by allowing the grapes to soak in the juice.
|'''''Rosé wine''''', which gains color from red grape skins, but not enough to qualify it as a red wine. The color can range from a very pale pink to pale red.
There are two primary ways to produce rosé wine. The preferred technique is allowing a short period of maceration after crushing red grapes, which extracts a certain amount of color. The juice is then fermented like a white wine. An alternative is blending a small amount of finished red wine into finished white wine.
|-
!White grapes
|'''''Orange wine''''', sometimes called amber wine, is made with white grapes but with the skins allowed to macerate during and beyond fermentation, similar to red wine production. This results in their darker color compared to white wines, and produces a deliberately astringent result.
|'''''White wine''''', typically made from white grape varieties (those with yellow or green skins), and range from practically colorless to golden. When skin contact is used, to improve the flavor or to increase the body or aging potential, it is usually limited to between four and 24 hours; any longer leads to bitterness.
|}
''Common names: There are thousands of wine varietals. Red wines like Cabernet Sauvignon, Merlot, Pinot Noir, Malbec, Zinfandel; white wines like Chardonnay, Sauvignon Blanc, Pinot Grigio, Riesling; and rosé wines like Provence, White Zinfandel and Pink Moscato.'' 

In addition to the colors of wine, wine can be [[wikipedia:Sparkling_wine|sparkling]] or still. Wine can also be [[wikipedia:Fortified_wine|fortified]]. Fortifying a wine is the process of adding a distilled spirit to the fermented wine.  

''Common names: Sparkling wine includes Champagne, Prosecco and other Brut wines. Fortified wines include Port, Sherry and Vermouth.''
==== Fruit Wine, Cider and Perry ====
[[wikipedia:Fruit_wine|Fruit wine]] is mostly what it sounds like: wine made from fruits other than grapes. A variety of fruits can be used, some common ones include cherry, plum, dandelion and pineapple. Fruit wine is far less common than traditional grape wines, and is often called by the fruit it's fermented from (ex. &quot;cherry wine&quot; or &quot;dandelion wine&quot;). 

[[wikipedia:Cider|Cider]] is a beverage made specifically from fermenting apples. Despite being a sort of wine fermented from fruit, it isn't considered a fruit wine due to its unique cultural history as a beverage. Its name can cause some confusion particularly in the United States and Canada, where &quot;cider&quot; also refers to unfiltered and sometimes spiced apple juice that has not been fermented. The term &quot;hard cider&quot; is sometimes used to distinguish the alcoholic beverage due to this overlap. Cider can also be carbonated to make a sparkling cider. 

[[wikipedia:Perry|Perry]], or Pear Cider, is another fruit wine exception. Fermented pears create a drink called perry, that though technically is a wine made from fruit, is considered its own thing. 

''Common names: There are thousands of brands of fruit wine, cider and perry. You may have some on tap or bottled; it's mostly important to know the names of what you have.''
[[File:Beer_in_glasses_and_steins_on_a_table_with_bottles_in_the_background_and_a_brick_wall_(15700131777).jpg|thumb|264x264px|There are many varieties of beer.]]

==== Beer ====
[[wikipedia:beer|Beer]] is an ancient drink that has been brewed around the world for hundreds of years. Most beers have four basic ingredients: grain, hops, yeast and water. [[wikipedia:Hops|Hops]] are a type of dried flower that give beer flavor. While there are thousands of brands of beer, there are only a few main types:  

* '''Lagers''' - One of the two main types of beer, with Ales. Lighter and crisp, refreshing, less bitter than Ale.
** Amber Lager - A medium, amber color with a slight bitterness. ''Common Names: Yuengling''
** Pale Lager - Light in color and flavor, not very bitter. Easy drinking and very popular. ''Common Names: Red Stripe, Heineken, Pabst Blue Ribbon (PBR), Hamms''
** Pilsner - Medium amber in color. Hoppy and flavorful. ''Common Names: Modelo Especial, Stella Artois, Pilsner Urquell''

* '''Ales''' - The other main type of beer, with Lager. Richer and flavorful, more bitter than Lager.
** Pale Ale - A diverse group of beer that various depending on its country of origin. Generally balanced taste that pairs with many foods. ''Common Names: Sierra Nevada, Boulevard''
** Indian Pale Ale (IPA) - Often very bitter and fruity. A unique tasting beer with bite. ''Common Names: Voodoo Ranger, many other large brands have an IPA''
** Stouts and Porters - Dark beers that have rich, roasted flavors often imitating chocolate or coffee. These beers can feel like a meal. ''Common Names: Guinness, Imperial Stouts, Breakfast Stouts''
** Wheat Beer - Ales that have some wheat added to them. It can give them a fruity, almost banana like flavor. ''Common Names: Blue Moon''
* Radlers and Shandys - Beers that have been mixed with a fruit juice. Radlers can be any type of citrus while Shandys are always lemon juice. Very refreshing, summary drinks.
* Sour Beer - Beer that's, well, sour. Tangy and punchy, usually served in a smaller &quot;sour beer&quot; glass since they pack a punch.
* Light Beer - Any beer with an especially low ABV is a light beer (less than 4% ABV), though the term is sometimes also used for low-calorie beers. ''Common Names: Bud Light, Coors Light''

==== Sake ====
[[wikipedia:sake|Sake]] is a drink brewed from rice, and is sometimes also called rice wine though it is brewed more similarly to beer. Like beer and wine, sake has a wide range of flavor profiles though it is often fruity and light. It is sometimes served lightly warmed in ceramic cups depending on the variety of sake. 

''Common Names: Dassai, Hakutsuru, Kubota''

=== Spirits (High-ABV Drinks) ===
[[File:Alcoholic beverages.jpg|thumb|High-ABV spirits like vodka and brandy.]]

==== Vodka ====
[[wikipedia:vodka|Vodka]] is a spirit that can be distilled from many sources including grains, potatoes and sugarcane before the pure ethanol is diluted with water. It is generally a neutral spirit, meaning it has little taste and is mostly pure ethanol and water. This isn't entirely true though, as each vodka has its own impurities and subtle flavor profiles. Vodka is also commonly flavored, as its neutral taste allows it to take on flavor easily. It can served up, often freezer chilled, or mixed into many cocktails.

''Common Names: Absolut, Smirnoff, Kettle One''

==== Gin ====
[[wikipedia:Gin|Gin]] is a spirit flavored with [[wikipedia:juniper_berries|juniper berries]] and other botanicals. It has a distinct herbal flavor, and different brands will have their own flavor profile, and perceived dryness.
[[File:Agave tequilana 2.jpg|thumb|Agave is a plant native to Mexico.]]
''Common Names: Beefeater, Tanqueray, Hendrick's, Plymouth''

==== Mezcal (Tequila) ====
[[wikipedia:mezcal|Mezcal]] is a spirit distilled from Agave. The most common type of mezcal is [[wikipedia:tequila|Tequila]], which is distilled only from [[wikipedia:blue_agave|blue agave]]. Many people mistake mezcal for a type of tequila due to tequila's popularity, but it is the other way around. Mezcal and tequila are often aged, which impart a darker, more amber color the longer it sits. Mezcal often has a smokey flavor and can also be sweet, fruity or earthy, and can be enjoyed chilled, up or in many popular cocktails.

''Common Names: Casamigos, Patrón, Jose Cuervo''

==== Rum ====
[[wikipedia:rum|Rum]] is a spirit distilled from [[wikipedia:sugarcane|sugarcane]] then aged in barrels. The less-aged rum has a lighter color, and is called &quot;light rum&quot;. It is most commonly used in cocktails, like the Mojito and the Daiquiri. &quot;Aged&quot; or &quot;Dark&quot; rum has a much deeper flavor is often drank straight or on the rocks. Rum originates from and maintains strong ties to the history of the people of the [[wikipedia:caribbean|Caribbean]] and appears commonly in cultural cuisines as well as drinks. I encourage you to learn about the history of rum, as it can teach a lot about sugarcane as a [[wikipedia:Sugar_plantations_in_the_Caribbean|plantation crop]] and the[[wikipedia:Atlantic_slave_trade|Transatlantic Slave Trade]]. 

''Common Names: Baccardi, Captain Morgan, Malibu'' 

==== Brandy ====
[[wikipedia:Brandy|Brandy]] is made by distilling wine. While most brandy is made from grapes, it can be made from any fruit to yield a [[wikipedia:Fruit_brandy|fruit brandy]]. 

''Common names: Cognac and Armagnac''

==== Whiskey ====
[[wikipedia:whiskey|Whiskey]] is made by fermenting, distilling then aging in barrels various grains. Whiskey drinkers can be particular, as there are many types of whiskey. Some whiskey is &quot;malted&quot;, meaning it uses grains that have been malted. Malting is the process of allowing grains to germinate (begin to sprout), before heating them to halt the germination. This yields &quot;malt whiskey&quot;. The specific grain used also yields different types of whiskey, for example fermenting rye yields &quot;rye whiskey&quot;. &quot;[[wikipedia:Scotch_whisky|Scotch]]&quot; is simply whiskey from Scotland and is also called Scotch Whiskey.

''Common Names: Jameson, Jack Daniel's, Crown Royale''

==== Soju ====
[[wikipedia:soju|Soju]] is a made by distilling rice or other grains. Similarly to vodka, it is a neutral spirit that is most flavorless. It is often flavored with fruits and florals. It can be served up or mixed into cocktails.

''Common Names: Jinro''

==== Liqueurs ====
[[wikipedia:liqueur|Liqueurs]] are spirits that have been heavily flavored and sweetened. They are used very frequently in cocktails to impart their particular flavor. Liqueurs can be used to add a flavor that would otherwise be troublesome, like coffee, liquorish, or almond. Sometimes they're served up as a dessert or a [[wikipedia:digestif|digestif]]. 

''Common Names: Kahlua'' 

== Tools of Bartending ==
You'll be using both common and specialty tools as a bartender.  A collection of such tools are below. Start by trying to name as many of them as you can. Note how many are familiar to you, how many you've seen but cannot name, and how many are unfamiliar. A key to the image follows.
[[File:Bartools2.jpg|border|center]]
Answer key: (1) champagne bottle stopper, (2) kitchen knife, (3) ice tongs, (4) ice scoop, (5) ice bucket, (6) small bar spoon, (7) cocktail-pick, (8) jigger, (9) mesh strainer, (10) boston shaker (metal bottom), (11) bar spoon, (12) lime/lemon squeezer, (13) hawthorn strainer, (14) zester, (15) boston shaker (mixing glass), (16) muddler, (17) citrus reamer, (18) fine grater, (19) Y-peeler, (20) wine key.

How'd you do? Some of these items you'll see more often than others. I want to draw special attention these crucial pieces:

* '''Jigger''' (8) - A small double sided measuring device used to quickly portion spirits and other ingredients. Though the exact measurements may change depending on where you are and the manufacturer, in the US most jiggers measure 1.5 oz on one end and 0.75 oz on the other.
* '''Juicer and Peeler''' (12 &amp; 19) - Many cocktails rely on citrus for flavor and fragrance. The handheld juicer and the peeler are commonly used both at the start of the shift to prep the citrus for the day, collecting fresh citrus juice and peels for adding to cocktails.
* '''The Boston Shaker''' (10 &amp; 15) - This is the industry standard for fast, versatile and accurate cocktail mixing. Coming in two parts, often one side is glass and the other is metal, though often both are metal. The two parts are sealed together allowing for a cocktail to be shaker vigorously inside before being cracked apart without spilling.
* '''The Hawthorn Strainer''' (23) - The final piece to any Boston Shaker. This uniquely shaped strainer allows for cocktails to be poured into a glass while leaving the ice in the shaker.

These are tools you will see every time you step in to bar to make a cocktail.

== Safety ==
A good bartender is the difference between a safe time for your bar patrons and an unsafe environment. It is your responsibility to keep both yourself and your bar patrons safe. It is important first to understand how alcohol affects the body in order to understand how to properly make and serve drinks, so first let's review ABV.

Alcohol by Volume is a percentage of ethanol to all other contents of a drink. However, ABV does not tell you everything about drink. A standard beer has about 5% ABV while a shot of whiskey has about 40% ABV, but because you would only drink about 1.5 oz of whiskey, and would drink 12 oz beer, the total amount of alcohol ingested would actually be about the same. This is the idea behind a &quot;standard drink&quot;. A standard drink contains one &quot;unit&quot; of alcohol regardless of how much liquid it contains in total. The chart below shows some examples of standard drink equivalents. Each of the below drinks contain one unit of alcohol.
[[File:NIH standard drink comparison.jpg|center|frame|National Institute of Health's &quot;standard drink&quot;.]]
Some general best practices when consuming alcohol:

* One unit of alcohol per hour is around what a liver can process. This is generally a safer cadence of drinking.
* Drink a glass of water for every unit of alcohol consumed to stay hydrated.
* Avoid drinking on an empty stomach. Eating while consuming alcohol slows the alcohol's absorption into the blood and curbs its effects.
* Don't order a double. Ordering and serving drinks with more than one standard unit of alcohol in it can cause you or others to become more impaired more quickly than intended. Try to keep it to the cadence of one standard drink per hour.

Standards should be upheld both with your own drinking and the drinking of the patrons at your bar. Some standards to uphold in your bar are:

* '''Never drink and drive'''. Alcohol impairs motor coordination and clouds judgement. Drinking and then driving puts your own life and the lives of everyone around you in immediate danger. Even if you are under the legal limit in your area, you are impaired after one drink. This should be enforced both for yourself and for your bar patrons.
* '''You can say no'''. It's always okay to say no to a drink. Ordering something without alcohol is always an option. Good friends don't pressure you to drink when you don't want to. You can leave a situation when you are uncomfortable.
* '''Alcohol is not medicine'''. Addiction is real and treatable. Help is available.
* '''Abuse isn't acceptable'''. You are not a punching bag. It is not bad hospitality to stand up for yourself. 

Sometimes people drink more than they intended and may become too inebriated to be served. Some signs that this is the case are:

* Slurred speech
* Stumbling
* Glazed over or unfocused eyes
* Speaking too loudly or two softly
* Repeating questions or ordering from multiple bartenders

If you meet a bar patron who has had too much to drink, you can help by offering water, tea, coffee or food, helping them find their group, or asking another member of the staff at your bar for help. It may be that what's needed is to no longer serve the bar patron. This is called &quot;cutting them off&quot;. When a bar patron is cut off, they will no longer be served alcohol for the rest of the service. This can be embarrassing, both for the bar patron and the bartender, so it's best practice to be kind, clear and discrete. If a patron is aggressive however, all bets are off and they should be removed from the bar.

Bartenders have legal obligations depending on where you work. This can include checking the legal age of the patrons at your bar and monitoring the alcohol intake of the patrons at your bar. Sometimes, a bartender can be held individually liable for the failure to meet these obligations. Take the time to find a reputable source to discover what the legal obligation of bartenders in your area are now.

This concludes Lesson 1: Bartending Basics. You can move on now to Lesson 2: Mixing Drinks. 

= Mixing Drinks =
Congratulations! You've made it to the fun part of the course. Mixing and serving drinks are fundamental to bartending, and they're fun to do. In this lesson we'll focus on the most common and most important recipes you'll encounter working at a bar along with the basic techniques and know-how needed to complete them.

In this lesson we'll discuss:

# Glassware
# Typology of cocktails
# Preparation and Skills
# Essential Cocktails
This is the part of the course that really benefits from hands-on practice. When we talk about preparing fruit and mixing cocktails, follow along! It isn't necessary to build an entire bar in your home in order to practice, but having some of the basics and making substitutions when needed will go a long way to making these actions feel natural and easy. If you have a bartending gig in your future, even doing each of these recipes once will put you leagues ahead of your first try being on the clock.

== Glassware ==
Before you can mix a drink, you have to know what glass you're gonna pour it in. Wikipedians have done an excellent job of creating a near comprehensive list of the glasses commonly used to serve alcohol. I've transcluded a navigation box below to all of their pages. Take the time to familiarize yourself with these glasses and their uses by clicking on the pages and digesting the contained information. You don't have to know the history and origin of every glass, just be familiar their shape and general use. The most important ones for this course are:

* Collins
* Highball
* Old fashioned (or Lowball or Rocks)
* Champaign Coupe (or just Coupe)
* Champaign Flute
* Cocktail Glass (or Martini glass)
* Hurricane Glass
* Wine Glass (Red and White)

When we get to cocktails, each recipe will place the drink in a specific type of glass, and you can return here as a reference tool if you need it.

{{Navbox
| name     =  Glassware
| title    = [[:wikipedia:List of glassware|Glassware]]
|listclass = hlist
|state = expanded
|group1 = [[:wikipedia:Tumbler (glass)|Tumblers]]
|list1  =
* {{Nowrap|[[File:Collins glass silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Collins glass|Collins glass]]}}
* {{Nowrap|[[File:Granyonyi Stakan Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Faceted glass|Faceted glass]]}}
* {{Nowrap|[[File:Highball glass silhouette.svg|16x28px|link=|class=skin-invert]] [[:wikipedia:Highball glass|Highball glass]]}}
* {{Nowrap|[[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Old fashioned glass|Old fashioned glass]]}}
* {{Nowrap|[[File:Shot glass.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Shot glass|Shot glass]]}}

|group2 = [[:wikipedia:Beer glassware|Beer glassware]]
|list2  =
* {{Nowrap|[[File:beer stein.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Beer stein|Beer stein]]}}
* {{Nowrap|[[File:Pilsner glass silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Beer glassware#Pilsner glass|Pilsner glass]]}}
* {{Nowrap|[[File:Pint glass.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Pint glass|Pint Glass]]}}
* {{Nowrap|[[File:Pony Glass Silhouette.svg|14x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Pony glass|Pony glass]]}}
* {{Nowrap|[[File:Tankard Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Tankard|Tankard]]}}
* {{Nowrap|[[File:Wheat beer glass silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Wheat beer glass|Wheat beer glass]]}}
* {{Nowrap|[[File:yardglass.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Yard of ale|Yard glass]]}}

|group3 = [[:wikipedia:Stemware|Stemware]]
|list3  =
* {{Nowrap|[[File:Absinthe Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Absinthiana#Absinthe glass|Absinthe glass]]}}
* {{Nowrap|[[File:Chalice Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Chalice|Chalice]]}}
* {{Nowrap|[[File:Coppa Asti sagoma.svg|24x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Champagne glass#Coupe|Champagne coupe]]}}
* {{Nowrap|[[File:flutesil.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Champagne glass#Flute|Champagne flute]]}}
* {{Nowrap|[[File:Coctail glass.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Cocktail glass|Cocktail glass]]}}
* {{Nowrap|[[File:Glencairn Whisky Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Glencairn whisky glass|Glencairn whisky glass]]}}
* {{Nowrap|[[File:Hurricane Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Hurricane glass|Hurricane glass]]}}
* {{Nowrap|[[File:Margarita Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Margarita#Glass|Margarita glass]]}}
* {{Nowrap|[[File:Alcohol glass nick and nora.svg|24x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Nick &amp; Nora (glass)|Nick &amp; Nora]]}}
* [[:wikipedia:Rummer|Rummer]]
* {{Nowrap|[[File:Sherry Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Wine glass#Sherry glass|Sherry glass]]}}
* {{Nowrap|[[File:brandysnifsi.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Snifter|Snifter]]}}
* {{Nowrap|[[File:wineglass.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Wine glass|Wine glass]]}}

}}

== Typology of Cocktails (Cocktail Families) ==
One last thing before we starting preparing garnishes and mixing drinks! I promise! Cocktails and their recipes didn't just appear from nothing in a vacuum. They were developed over time as tastes changed and different ingredients became available or popular. Because of this process, we can group cocktails into broad categories called Cocktail Families. Not all bartenders and mixologists agree on how cocktails should be grouped or how many families there are, but by understanding generally how cocktails relate to each other you can prioritize your learning and remember what goes in which cocktail easier. 

The families we're going to talk about are: 

* Sour Cocktails
* Ancestral/Aromatic Cocktails
* Lengthened Cocktails
* Frozen Cocktails

This isn't by any means comprehensive, but they'll cover most the cocktails we'll explore shortly.

=== Sour Cocktails ===
[[wikipedia:Sour_(cocktail)|Sour cocktails]] are an old family of cocktails who's drinks that are all, well sour. The template for a sour cocktail is simple:

&lt;u&gt;Shake&lt;/u&gt; with ice: Spirit + Sweetener + Something Sour + Egg (sometimes)

The &quot;Something Sour&quot; is most often a citrus juice like lemon or lime. Many sour cocktails will have &quot;sour&quot; in the name, like a Gin Sour, Rum Sour, Whiskey Sour, or Amaretto Sour. Some other notable sour cocktails are the Daiquiri, Cosmopolitan and Margarita. Note that it doesn't matter what base spirit you use, any base spirit can be used in a sour. A sub-family of sour cocktails replaces the traditional sweetener of simple syrup with a liqueur. These cocktails are called a &quot;Daisy&quot;.

=== Ancestral or Aromatic Cocktails ===
Ancestral or Aromatic Cocktails are spirit-forward drinks that use bitters and sometimes liqueur to give an aromatic taste. The template to an aromatic cocktail is:

&lt;u&gt;Stir&lt;/u&gt; with ice: Spirit + Sweetener + Bitters 

These cocktails let the taste of the spirit shine while deepening the flavor with bitters. Some common aromatic cocktails are the Martini, Manhattan, Old Fashioned, Sazerac and Negroni.

=== Lengthened Cocktails ===
Lengthened Cocktails are drinks that have been &quot;lengthened&quot; by adding a carbonated beverage. Two major sub-families of lengthened cocktails are &quot;Highballs&quot; and &quot;Spritz&quot; cocktails. A Highball is just a spirit lengthened with non-alcoholic mixer, think Gin and Tonic or Rum and Coke. A Spritz is a wine lengthened with a mixer, most often soda or sparkling wine. An example you'll find more in Spain is the Kalimotxo (KAL-EE-MO-KO), a drink with Red Wine and Coca Cola. Don't knock it 'till you try it! There are hundreds of combinations to make Lengthened Cocktails, Highballs and Spritz are just the major subtypes.

=== Frozen Cocktails ===
Frozen drinks are frozen! They are just so unlike a non-frozen drink that they get their own family. Frozen drinks are either blended with ice or churned over frozen metal to make a slushy consistency. Many frozen cocktails were adapted from classic cocktails, like a Frozen Daiquiri, but others are always frozen, like a Piña Colada. These are often &quot;batched&quot; or made ahead of time, especially if they're going to come from a slushy machine.

== Preparation and Skills ==
In a bar, preparation is all the steps that are taken before a service starts to get ready in advance. Sometimes this is things that are done at close the night before, and sometimes these are things done in the hours before a service. Some examples of prep are:

* Stocking fridges with beer and wine
* Stocking the bar's backstock of spirits
* Juicing citrus
* Cutting garnishes for drinks
* Making syrups
* Topping up bottles
* Cleaning and preparing tools for the service

All of these make a good bartender. We'll talk more about how good prep is good hospitality, and how it defines a good bartender later. Some of these are self explanatory, but some merit further investigation to make you a more effective and efficient bartender.

While you go through this section, I recommend you actually follow along and prepare the ingredients listed ''especially the lemons and limes''. Immediately following prep, we'll start mixing cocktails that use these ingredients. Store them in an air tight, labeled and dated container if you plan on mixing some drinks within the next day or two. Cut citrus doesn't last that long, so don't overkeep it. You can always get more practice by preparing more garnishes. A lemon wedge can be used for a dozen other things anyway (I like one with my Coke).
[[File:Hellingshoek_antoineren_02.png|left|thumb|241x241px|This isn't me. I don't know who this man is. Thanks to him for uploading this picture though. Curl your fingers away from the knife.]]

=== Before you prep ===
Most preparation can and should be done with a small paring knife. A larger chef knife can be used for large fruits but is harder to maneuver especially when working with small, round fruits and fragile herbs. When cutting something with a knife, curl the fingers of your non-dominant hand away from the blade. This will help protect your fingertips and cut faster with less risk.

You should always have clean hands when you handle food. Before touching food, wash your hands with soap and water for at least 20 seconds, and dry them with a clean towel. Rewash to your hands whenever you step away from your task or touch something dirty like your phone.

=== Juicing Citrus ===
[[File:Citrus × aurantium - fruits cut.jpg|thumb|The lengthwise cut (left) is unsuitable for juicing. The widthwise cut (right) is perfect to juice.]]
Most bars will have one or two types of juicer: a handheld citrus squeezer and/or a countertop citrus reamer. The handheld squeezer is faster and more efficient but cannot fit larger citrus like grapefruit. There are few wrongs ways to juice citrus, but doing it well will save you a lot of time during the prep. There are a few things to keep in mind when juicing citrus at the bar:

# Start with room temperature citrus. Cold citrus will yield less juice. If you have a particularly hard lime or lemon, place it on the counter and put your palm on top. Roll the citrus back and forth while applying firm pressure. This can free up some juice.
# Cut the fruit widthwise down the middle, so that you have two even sections. The cut face of the citrus should look like a wagon wheel with dot of pith in the center.
# Place your fruit cut-side down in the squeezer. Squeeze the citrus through a mesh strainer to remove any seeds or pulp. Squeeze with strength from your arms, not your wrists to avoid injury over time.
# Always label and date fresh juice and store in a sealed container in a refrigerator (Ex. Lime Juice Date: 2/27).

=== Making Simple Syrup ===
Simple syrup is an appropriately named combination of sugar and water made by combining a ratio of 1 parts granulated sugar and 1 parts water. It's a common ingredient used to sweeten cocktails by pre-dissolving the sugar. Here's the recipe:

# Place your sealable storage container on a scale and tare the scale so that it shows zero.
# Add a quantity of white granulated sugar to your container and note the measurement (ex. 200 grams of sugar).
# Add an equal amount of room temperature water (ex. 200 grams of water, making a total measurement of 400 grams).
# Seal the container and shake the mixture until the sugar has dissolved. It may be cloudy, but it with clarify as the mixture settles.
# Label and date your syrup (ex. Simple Syrup Date: 8/29).
If you're preparing this at home and don't have a scale (you're probably American), you can measure by volume, but it will be less accurate and may affect the final result. In a bar, always measure by weight.

=== Preparing Garnishes ===
There are so many things that can be put in drinks to improve their overall look and taste. There are a few, however, that you will find at most bars. If your bar has a specialty garnish, they should train you on how to prepare the house garnishes. For fruit garnishes, they should be used the same day they're cut and discarded at the end of the service. Let's go over some garnishes you'll find just about anywhere.
[[File:Knife_by_Bob_Kramer_(17970647591).jpg|thumb|How beautiful [[File:Face-kiss.svg|20x20px]] ]]

==== Cutting Citrus Half-Moons ====

# Cut the fruit in half lengthwise pole-to-pole. The cut-side of the fruit should have a stripe of pith down the center.
# Place the halves cut-side down, so that the nubs on each end are facing to your left and right.
# Cutting straight ahead, make slices about a quarter inch thick.
# Store in a sealed, labeled and dated container in a refrigerator until needed.

==== Cutting Citrus Wedges ====

# Trim the top and bottom of the fruit to create two flat surfaces. The flat surfaces should look like a wagon wheel with a white dot of pith in the center, or may be entirely pith. This step can be skipped for grapefruit, as they are large enough to rest on a side without additional cuts.
# Rest the fruit on its end, and cut it in half lengthwise. This should reveal a stripe of pith down the center of the cut face.
# Looking at the cut-side, make a shallow cut perpendicular to the pith extending about a half inch either side. Cut through about half of the meat of the fruit, but not all the way through to the peel.
# Placing the fruit cut-side down, cut the half further into quarters, then angling your knife, cut those fourths in half as well. Smaller fruits may only yield three wedges per half fruit.
# Store in a sealed, labeled and dated container in a refrigerator until needed.

==== Cutting Citrus Wheels ====

# Trim the nub off one end of the fruit. The cut should be deep enough to reveal a small wagon wheel shape with a dot of pith in the center.
# With the cut side facing to your left or right depending on your dominant hand, slice straight away from you creating about quarter inch thick circles until you no longer have enough fruit to safely hold while cutting.
# You many way to score a slit in the wheels depending on if you are putting them on the rim of a glass.
# Store in a sealed, labeled and dated container in a refrigerator until needed.

==== Cutting Citrus Peels and Twists ====

# Holding the fruit in your non-dominant hand, drag a y-peeler starting away from you and pulling towards you.
# If peeling an orange or grapefruit, using a paring knife, trim away excess pith from the back of the peel.
# Optionally, using a paring knife, trim the edges of the peel to make straight even sides.

==== Cutting Grapefruit &quot;Horses Neck&quot; ====

# Holding the grapefruit in your non-dominant hand, use a y-peeler to peel around the circumference of the fruit creating a long peel.
# Optionally, trim the ends with a paring knife for a cleaner look.

==== Preparing Pineapple Wedges ====

# Using a large chef knife, cut off the top of the pineapple.
# Resting on its bottom with the cut-side up, cut the pineapple in half from top to bottom.
# Resting the pineapple on its cut-side, angle your knife and cut the fruit into three equal parts creating triangle shapes.
# Take each section and slice into about inch thick wedges.
# You may way to score the point of each wedge to be able to hang it on a glass.
# Store in a sealed, labeled and dated container in a refrigerator until needed.

==== Preparing Mint ====

# Place a damp paper towel in the bottom of a sealable storage container.
# Take a sprig of mint in one hand, and pick the leaves at the bottom of the stem. Leave the ones at the top of the sprig attached to the stem. Place the leaves in the prepared container.
# Bunch the stems of the picked sprigs together and trim them to be only a couple inches long.

=== Essential Skills ===
There are a couple techniques that are used over and over in bartending. The two most essential to mix cocktails are shaking and stirring. Refer back to these steps the first few times you mix a cocktail in the next section. 
[[File:Bartender Photo.jpg|thumb|Shaking a sealed Boston shaker horizontally.]]

==== Using a Boston Shaker ====
A Boston shaker comes in two parts. One will be a large metal mixing cup, and the other will either be a smaller metal mixing cup or a glass mixing cup. Using a shaker comes in a few steps:

# We will start by building a shaken cocktail in the small or glass mixing cup. Add the liquid ingredients (including egg, if being used), then add the ice after. The longer the liquids are in contact with the ice, the more they will be diluted as the ice melts.
# The large mixing cup is placed over top the smaller mixing cup with enough force to create a seal. Feel free to tap the back of the large tin with the palm of your hand to ensure a seal.
# With one hand on either cup holding them together, the assembled cups are then flipped over so that the drink is primarily in the larger mixing cup. 
# The most efficient way to shake is horizontally, so that the drinks sloshes vigorously from front to back. Shaking vertically will achieve the same result, but it is more energy intensive and slower as it throws the drink upwards before slamming it back down. You will feel the metal become cold, and it may frost as the drink is chilled. These are signs that you can stop shaking.
# At this point, the cold from the ice has caused the metal tins to contract and shrink slightly which will bind them together tightly. To unseal the shaker, hold the larger cup or place it on a counter. Identify which was the smaller cup is leaning. With the palm of your hand, strike the smaller cup away from the direction it's leaning. There should be a crack as the seal is broken (this is called &quot;cracking&quot; a shaker open). You can also twist or squeeze the larger tin to release the seal if it's made of metal.
# To strain, place the Hawthorn strainer over the mouth of the larger cup, and pour the drink through the strainer.
You can watch an example of [https://www.youtube.com/watch?v=ai70a6xjSzo someone using a Boston shaker] on YouTube (not my video), if you'd like more instruction.[[File:Stirring with a bar spoon.jpg|thumb|230x230px|Stirring with a Bar Spoon]]
==== Stirring a Cocktail ====
It sounds simple, but there's a right way and a wrong way to stir a cocktail. The goal is to incorporate and chill the ingredients without agitating, decarbonating or introducing air. Here are a few simple steps:

# Insert a bar spoon to the bottom of the mixing glass containing you ingredients and ice. Push the spoon all the way to the bottom edge of the mixing glass.
# Swirl the spoon so that it stays in contact with the edge of the glass, rotating the ice in a circle.

==== Serving Sparking Wine ====

# Use the tab to peel away the foil.
# Keep one hand on top of the cage and cork. Lift the handle and twist it open.
# Keep the cage on the cork. Hold the cage and the cork with your dominant hand.
# Holding the bottle in your non-dominant hand, twist the bottle back and forth. The goal is to gradually release the pressure in the bottle, not to make a loud &quot;pop&quot;. Ideally, you should open the bottle almost silently.
# Holding the bottle at the base with one hand (if possible), pour slowly into the glass as the bubbles will climb.

[[File:Pouring wine into a decanter.jpg|left|thumb|Pouring wine|296x296px]]

==== Uncorking Still Wine ====

# Find a wine key with a double-hinge.
# Using the knife, cut the foil around the lip of the bottle. Using the bottle opener end of the key, peel back to foil to expose the cork.
# Insert the corkscrew into the center of the cork by twisting until most of the screw is embedded.
# Place the first hinge on the lip of the bottle, and use it as leverage to lift up on the wine key (lifting the cork partially).
# Place the second hinge of the lip of the bottle, and completely remove the cork.
'''Serving Beer'''

# Whether pouring from a bottle or a tap, tilt the glass 45° and pour the beer down the side of glass to prevent too much foam, or &quot;head&quot;, from developing.
# At the end of the pour, straighten the glass to allow for some head to form.

== Essential Cocktails ==
Here we are, finally mixing some drinks! It can be overwhelming to open a book of cocktail recipes and see the hundreds of drinks with unique names and recipes. Take a deep breath, because there isn't a bartender in the world that knows all of those cocktails by heart, and you aren't expected to either. Over time, you'll pick up on more drinks and their makeup, but that takes time! That's why it's important to focus on the essential cocktails that every bartender knows and that any bar patron might order. What follows are 20 cocktail recipes for classic cocktails that you might make every day at a bar. 

I highly encourage you to make these drinks at home. If you intend to become a bartender, the difference between your first attempt at a recipe being in the privacy of your home and on your first day on the job is ''enormous''. That raises the question, &quot;How am I supposed to supply all the ingredients needed to make all these drinks‽&quot; Start with things you can buy at the grocery store:

* Lemons, Limes and Oranges
* Club Soda
* White Sugar

You may already have these things from trying out the preparation techniques above. If you don't, go out and grab these ingredients, cut some garnishes, and prepare some simple syrup.

The most expensive part of any cocktail is the spirit. I've color coded the following recipes by which spirit they use. The list generally starts with easy recipes and becomes more complicated as you descend. I recommend staring with one type of spirit and trying a few recipes that use that spirit. For example, with only vodka, you can make a Highball, Cosmopolitan, Moscow Mule, Bloody Mary and a (Vodka) Martini. That's one fifth of the list done, with a spirit you likely already have at home.

I don't recommend buying fancy or expensive spirits for this. Buy something cheap or middle-of-the-road to practice with. That's like what your bar patrons are going to be ordering most often anyhow.

All that being said, 20 recipes is a lot to learn. Take your time, maybe make one or two a day. The recipes aren't changing, and some are very simple. You by no means have to drink what you make. You DO have to TASTE what you make. Bartenders, when trying a drink on the job, will smell a drink, swirl it in their mouth and spit it out. Do this at least twice for each drink if you don't intend to drink them. Take note of the taste, the balance of the drink and your personal preference for or against it. Please don't make all 20 in one night and get hammered. A daiquiri will be a daiquiri tomorrow, and it'll be sweeter for the waiting.

Well, get started!

=== Spirit Color Key ===
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Vodka
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Gin
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Rum
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Tequila
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Whiskey
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Other/Mixed
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=== Highball ===
Highballs are a class of cocktail sometimes called &quot;plus one&quot; cocktails because they're a blend of a spirit plus a mixer. Some common highballs are Gin and Tonic, Rum and Coke, Whiskey and Ginger Ale, and Scotch and Soda.

{{RoundBoxTop|theme=11}} 
'''Glass:''' Highball [[File:Highball glass silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Wedges with soda, Lime Wedges with Tonic, Cola and Ginger Beer/Ale

'''Ingredients'''

2 oz of Spirit

Mixer of choice, to top

'''Combine''' Spirit and ice. Stir. Add Mixer to fill. Drag spoon through drink a few times to gently combine. Garnish
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=== Cosmopolitan ===
{{RoundBoxTop|theme=1}} 
'''Glass:''' Coupe or Martini Nowrap|[[File:Coctail glass.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Twist

'''Ingredients'''

1 ½ oz Vodka (Preferably Citron Vodka)

⅓ oz Lime Juice

⅓ oz Cointreau (or other orange liqueur)

⅓ oz Sweet Cranberry Juice

'''Shake''' all ingredients with ice until chilled. Strain into glass. Garnish. 
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=== Margarita ===
{{RoundBoxTop|theme=4}} 
'''Glass:''' Rocks [[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lime Wedge and ask preference for Salt, Sugar, or Tajin rim

'''Ingredients'''

2 oz Tequila

½ oz Agave Nectar

1 oz Lime Juice

½ oz Triple Sec

'''Moisten''' the rim of the glass with a lime wedge, then roll exterior of glass in rimming material careful to avoid getting any inside the glass. Combine all ingredients in shaker with ice. Shake. Add ice to glass. Strain. Garnish.
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=== Paloma ===
{{RoundBoxTop|theme=4}} 
'''Glass:''' Highball [[File:Highball glass silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Grapefruit Wedge or Lime Wheel

'''Ingredients'''

2 oz Tequila

½ oz Lime Juice

Dash of Salt

Grapefruit Soda, to top

'''Combine''' Tequila, Sale and Lime Juice in glass with ice. Stir to chill. Add Grapefruit Soda to fill. Drag spoon through drink a few times to gently combine. Garnish.
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=== Whiskey Sour ===
{{RoundBoxTop|theme=5}} 
'''Glass:''' Sour or Coupe [[File:Coppa Asti sagoma.svg|24x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Wheel and Cherry

'''Ingredients'''

2 oz Whiskey

1 oz Lemon Juice

½ tsp. Sugar or Simple Syrup

'''Shake''' ingredients with ice. Strain. Garnish.
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=== Moscow Mule ===
{{RoundBoxTop|theme=1}} 
'''Glass:''' Mule Mug [[File:Alcohol_glass_copper_mug.svg|24x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lime wheel (optional)

'''Ingredients'''

2 oz Vodka

½ oz Lime Juice

3 oz Ginger Beer

'''Combine''' ingredients in a copper mug filled with ice. Stir. Garnish, optionally.
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=== Spritz ===
Spritz are a class of cocktails that follow the same general 3-2-1 recipe of 3 oz Prosecco, 2 oz liqueur, 1 oz club soda. The most common is the Aperol Spritz.

{{RoundBoxTop|theme=11}} 
'''Glass:''' Highball or Wine Glass [[File:wineglass.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Orange Wheel

'''Ingredients'''

3 oz Prosecco

2 oz Liqueur (Aperol, for an Aperol Spritz)

1 oz Club Soda

'''Combine''' Prosecco and Liqueur with ice in glass. Stir gently. Top with Club Soda. Garnish.
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=== Mojito ===
{{RoundBoxTop|theme=3}} 
'''Glass:''' Highball [[File:Highball glass silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Mint Sprig

'''Ingredients'''

1 ½ oz White Rum

5 to 6 mint leaves

1 oz Lime Juice

2 tsp Sugar

Soda water

'''Muddle''' the Mint, Lime Juice and Sugar until well mixed but not mushy. Add the Rum and fill the glass with ice. Top with Soda Water. Garnish.
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=== Daiquiri ===
{{RoundBoxTop|theme=3}} 
'''Glass:''' Coupe [[File:Coppa Asti sagoma.svg|24x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lime wheel (optional)

'''Ingredients'''

1 ½ oz White Rum

2 tsp Sugar

1 oz Fresh Lime Juice

'''Shake''' ingredients with ice until chilled and frothy. Strain into a coupe glass. Garnish, optionally.
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=== French 75 ===
{{RoundBoxTop|theme=2}} 
'''Glass:''' Champagne Flute [[File:flutesil.svg|24x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Twist

'''Ingredients'''

1 oz Gin

½ oz Lemon Juice

½ oz Simple Syrup

3 oz Champagne (or other sparkling wine)

'''Shake''' Gin, Lemon Juice, and Simple Syrup with ice. Strain into glass. Top with Champagne. Garnish.
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=== Gimlet ===
{{RoundBoxTop|theme=2}} 
'''Glass:''' Coupe [[File:Coppa Asti sagoma.svg|24x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lime Wheel

'''Ingredients'''

2 ½ oz Gin

½ oz Lime Juice

½ oz Simple Syrup

'''Shake''' Gin, Lime Juice, and Simple Syrup with ice. Strain into glass. Garnish.
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=== Manhattan ===
{{RoundBoxTop|theme=5}} 
'''Glass:''' Rocks [[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Twist and/or Cherry

'''Ingredients'''

1 ½ oz Rye Whiskey

½ oz Sweet Vermouth

1 Dash Angostura Bitters

'''Chill''' rocks glass by combining ice and water, or use pre-chilled glass. Combine ingredients in a mixer with ice. Stir. Strain. Garnish.
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=== Mai Tai ===
{{RoundBoxTop|theme=3}} 
'''Glass:''' Rocks [[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Pineapple Wedge, Cherry, and Mint Spring

'''Ingredients'''

1 oz Light Rum

1 oz Gold Rum

½ oz Orange Curaçao

½ oz Orgeat

½ oz Lime Juice

'''Shake''' ingredients with ice. Strain into ice filled glass. Garnish.
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=== Pina Colada ===
{{RoundBoxTop|theme=3}} 
'''Glass:''' Hurricane [[File:Hurricane Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Pineapple Wedge

'''Ingredients'''

1 ½ oz Rum (Preferably Coconut, like Malibu)

1 ½ oz Coconut Cream

1 oz Pineapple Juice

½ oz Lime Juice

'''Combine''' ingredients with ice in a blender. Blend on high for 20-30 seconds or until a flowing slushy consistency. If too stiff add water, if too thin add ice and re-blend for 2-3 seconds. Pour into glass. Garnish.
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=== Bloody Mary ===
{{RoundBoxTop|theme=1}}
'''Glass:''' Pint or Highball [[File:Highball glass silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Wheel or Celery Stalk (or anything savory)

'''Ingredients'''

2 oz Vodka

6 dashes Worcestershire sauce

3 dashes Tobasco Sauce

Pinch of Salt

Pinch of Ground Black Pepper

½ Lemon, juiced

5 oz Tomato Juice

'''Combine''' all ingredients in a glass. Add ice and stir. Garnish.
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=== Negroni ===
{{RoundBoxTop|theme=2}}
'''Glass:''' Rocks [[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Orange Twist or Orange Wheel

'''Ingredients'''

1 oz Dry Gin

1 oz Sweet Vermouth

1 oz Compari

1 oz Sparkling Water

'''Combine''' all ingredients in a glass. Add ice, preferably 2 large rocks. Stir. Garnish.

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=== Old Fashioned ===
{{RoundBoxTop|theme=5}} 
'''Glass:''' Rocks [[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Orange twist

'''Ingredients'''

2 oz Whiskey

1 Sugar Cube

4 dashes Aromatic Bitters

1 dash Soda Water

'''Muddle''' Sugar, Bitters and Soda Water in a glass until a uniform syrup is made. Add Whiskey. Add Ice. Stir. Express Orange Twist over drink then add to glass.
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=== Tom Collins ===
{{RoundBoxTop|theme=2}} 
'''Glass:''' Collins or Highball [[File:Collins glass silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Wheel and Cherry (optional)

'''Ingredients'''

2 oz Gin

1 oz Lemon Juice

½ oz Simple Syrup

Club Soda, to top

'''Combine''' Gin, Lemon Juice, and Simple Syrup in glass. Add ice to top. Fill with Club Soda. Stir gently. Garnish.
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=== Martini ===
Martinis are kind of their own thing, and there's some vocab to go over. There are many drinks that call themselves a martini, and the Classic Martini itself has many variations to suit each bar patron's taste. To make a Martini to order, first you have to have a conversation with your bar patron. 

# First, ask if they prefer Gin or Vodka. Gin is traditional, but it can be made with either.
# Second determine how much Vermouth they would like. The amount of Vermouth will determine how &quot;Dry&quot; or &quot;Wet&quot; the Martini is.

&quot;Extra Dry&quot; or &quot;In and Out&quot; = Vermouth is swirled in the glass then poured out, then the drink is entirely Gin or Vodka

&quot;Dry&quot; = 1 part Vermouth to 5 parts Gin or Vodka

&quot;Wet&quot; = 1 part Vermouth to 2 parts Gin or Vodka

&quot;50/50&quot; = 1 part Vermouth to 1 part Gin or Vodka

&quot;Dirty&quot; = 1 part Olive Brine to 5 parts Gin or Vodka, instead of Vermouth. Garnish with Olives instead of lemon.

The following recipe is for a Dry Martini. You can modify it to make it a Dirty Martini or to suit your taste of Vermouth. Martinis are often stirred, but they can be shaken on request.

{{RoundBoxTop|theme=11}} 
'''Glass:''' Martini [[File:Coctail glass.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon twist

'''Ingredients'''

2 oz Gin or Vodka (patron preference)

½ oz Dry Vermouth

'''Chill''' glassware by pouring water and ice in a glass, or use a pre-chilled glass. Combine ingredients with ice in a mixing glass. Stir until chilled. Strain into chilled glass. Garnish.
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=== Espresso Martini ===
The Espresso Martini is really not much like a Classic Martini at all. The word Martini really gets thrown around. Nonetheless, it's delicious. If your bar has an espresso machine, you'll be trained on how to pull an espresso shot. Otherwise, your bar may use cold brew concentrate made in house or from a can.

{{RoundBoxTop|theme=1}} 
'''Glass:''' Martini [[File:Coctail glass.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' 3 Coffee Beans

'''Ingredients'''

2 oz Vodka

½ oz Coffee Liqueur (often Kahlúa)

1 oz Espresso, or cold brew concentrate

1/4 oz Simple Syrup

'''Chill''' glass with water and ice, or use pre-chilled glass. Shake ingredients with ice. Strain into glass. Garnish by place 3 coffee beans in center.
{{RoundBoxBottom}}
Hey! Congrats! You've made it to the bottom of the list. These 20 recipes are just the beginning of the possibilities when it comes to mixing drinks, but if you can nail these down you can get through just about any shift at any bar. That was an undertaking, and you did a great job.

This concludes Lesson 2: Mixing Drinks. You can move on now to Lesson 3: Hospitality.

= Hospitality =
In this lesson, we'll be talking about some points of service and hospitality that are as essential to bartending as alcohol. The sections in this lesson are:
# What is Hospitality?
# Steps of Service
# Internal Hospitality
# Developing Taste

== What is Hospitality? ==
Mixing drinks is only half the job, maybe even less than half. The real job is [[wiktionary:hospitality|hospitality]].&lt;blockquote&gt;The act or service of welcoming, receiving, hosting, or entertaining guests; an appropriate attitude of openness, respect, and generosity toward guests. &lt;/blockquote&gt;In German, the word for hospitality is ''gastfreundschaft'' or guest-friendship. This encapsulates the charge of all bartenders to make the patrons of our bars feel welcome from the moment the walk through our doors to the moment they leave. Ideally, they still feel welcome long after they've left—they might just come back!

'''If you want to be there, so will they.''' People can immediately tell if you want to be there or not. If your vibe's a drag, people won't want to talk to you. There's the catch: ''people need to talk to you to get drinks''. If people don't want to talk to you, they'll leave, and they won't come back. The entire purpose of a bar is not only pushed forward by you, it's ''gate-kept'' by you. If there's no bartender, there's no bar; if there's a poor bartender, there's a poor bar. If you can find time to smile while behind the bar, you'll find your job is actually easier.

== Steps of Service ==
Service and hospitality are related, but separate ideas. Service is the mechanics of bartending: mixing drinks, closing tabs, cleaning and clearing. Hospitality is how you make your patrons feel. Keeping your service up will go a long ways towards making an environment hospitable. If the drink service is slow and the bar is messy, patrons are not gonna feel good. That's an example of service impacting hospitality.

When a patron comes to a bar, there's steps to how the they should be served that blend hospitality and service techniques. Let's go over the steps of providing good service.

# &lt;u&gt;Greet the Patron&lt;/u&gt; - Make eye contact with each person as they enter or approach your bar. Try to keep a relaxed, or at least welcoming, posture. Greet them warmly, making it clear that you see them and that you're happy they're here. Offer them a menu and pour them a water depending on your bar.
# &lt;u&gt;Take the Order&lt;/u&gt; - Once they've had a moment, or if they look immediately ready, ask what they would like to order or if they have any questions. Repeat what they've order back to them so that they can let you know if you misheard.
# &lt;u&gt;Mix the Drinks&lt;/u&gt; - Maybe they've ordered beer or wine that just need to be poured, or maybe they've ordered several shaken cocktails. Now's the time to make them. While you're making drinks, be sure to look up at the bar and greet people as they arrive.
# &lt;u&gt;Drop the Drinks&lt;/u&gt; - Bring finished drinks to guests as soon as they're finished. Mixed drinks have a short shelf life and should be  delivered promptly.
# &lt;u&gt;Check in&lt;/u&gt; - Give them a few moments to taste their drinks. Check in with an open ended question. &quot;How is everything?&quot; Many people won't tell you if they don't like the drink, so feel free to try to make them comfortable with giving honest feedback. You can always remake a drink!
# &lt;u&gt;Clear and Ask&lt;/u&gt; - When they're done with their drinks, clear any unneeded dishes and give the bar a wipe (avoiding pushing any crumbs onto your patron's lap). Then ask if they'd like another round. It's best to keep this question open and neutral. We don't need to push people to get another drink. A simple, &quot;What's next?&quot; gives patrons the floor to tell you what they need.
# &lt;u&gt;Close Out&lt;/u&gt; - When they've asked for a check, close out and bring the check quickly. The dead time when a patron wants to leave but can't yet is a vibe killer. You should prioritize closing them out and letting them leave quickly. When you drop the check, make eye contact, smile and thank them for coming.

== Internal Hospitality ==
Hospitality isn't something reserved for the patrons of your bar. Hospitality is something that should be shown to your teammates and everyone you work with. Think of it like this: &lt;blockquote&gt;If you're willing to give 100% attention, kindness and respect to complete strangers who walk up to your bar, shouldn't you be willing to give even more to the people you work with and see every day?&lt;/blockquote&gt;The people you work with at your bar will be the most valuable and important resource you have, making them feel good when they're around you is equally if not more important than patron hospitality. They will give you invaluable tips, unlock opportunities you've never imagined, and maybe become lifelong friends, but only if you treat them with the same, or more, hospitality you give to bar patrons.

One step further, hospitality is something you can show yourself. If we return to the definition of hospitality ''as an appropriate attitude of openness, respect, and generosity'', you can absolutely show hospitality to yourself. Respect your needs, be generous with time away from work, and be open to making mistakes and learning. You'll find that bartending will be more pleasurable and that positive feeling will make your job even easier.

== Developing Taste ==
Oftentimes patrons of your bar will ask your opinion on drinks, the menu and even the area around the bar. It's good hospitality to be able to answer their questions to the best of your ability. Here's some tips on each of those topics.

=== Tasting Alcohol ===
Tasting alcohol, including straight spirits, is a unique experience that is often undervalued. Skilled sommeliers can detect subtle notes at different moments when drinking a wine. Something similar can be done with spirits. Just a reminder that you don't have to swallow a drink to taste it. It's common for bartenders to spit a drink out when tasting it, especially on the job. To effectively taste a spirit:

# Smell the spirit. Don't stick your nose straight into it! Hover your nose above the spirit so that you can get a natural whiff of the aroma.
# Take a sip and swirl it around your mouth. Your mouth can hold on to tastes for quite a while, and this first swirl will clear the taste of whatever you've eaten and drank earlier in the day.
# Take another sip and appreciate the flavor. This is the part that takes time, patience, and skill. There are several ways to break up flavors. What did you taste the second it hit your tongue? What about after spitting it out or swallowing? You can try to pick up on the biggest notes first. Is it smokey? Sour? Bitter? Then try honing in on something more subtle. Are there herbal notes? Is it fruity?

=== Giving Recommendations ===
One of the most common responses you'll get when taking someone's order is some variation of &quot;I don't know&quot;, &quot;What you recommend?&quot;, and &quot;What do you like?&quot;. All of these mean basically the same thing: &quot;Talk to me about what's on your menu and what you can make.&quot; 

If you bar has a menu, this should be an easier task as the first thing you'll do after you arrive is memorize the menu. Giving recommendations isn't as easy as knowing the menu though. In these moments, it's important to engage with the patron to find out what kind of drink would suit their taste.

My preferred way of navigating a drink recommendation conversation is treating it like a &quot;choose your own adventure&quot; story. Offer a series of choices that bring both of you closer to a good drink choice. Start with broad strokes of flavor: &quot;Do you like a drink that's sweet, smokey, or sour?&quot; Oftentimes, that is enough to select off a menu. If they say something smokey, you might think about drinks that have a whiskey or mezcal base. Sweet or sour could have all kinds of spirits. You can ask if they have a preferred alcohol. You can also ask if they've had any drinks in the past that they've really liked.

== Conclusion ==
Well, that's the basics.

''If you're reading this, this lesson is still under construction! Feel free to add, edit, or [[User:Sabbier|message me]].''</text>
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      <comment>Replaced content with &quot;Hi ! I'm a librarian that occasionally edits on Wikipedia and Wikiversity. I created the Wikiversity course on [[Bartending]]. Feel free to say hi :)  == Things I'm working on == [[Creating Wikiversity Courses]]  === Bartending Wikiversity Course ===  ==== Notes to Myself ====  # Add an Advanced Bartending lesson with more recipes/mixology/significant bartenders/freepouring?  ==== Course Outline ====  # Bartending Basics ## Sources of alcohol ## Types of alcohol ## T...&quot;</comment>
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      <text bytes="1005" sha1="3cbu8jzvdcd9sd7k3xuzk32fdvnddwd" xml:space="preserve">Hi ! I'm a librarian that occasionally edits on Wikipedia and Wikiversity. I created the Wikiversity course on [[Bartending]]. Feel free to say hi :)

== Things I'm working on ==
[[Creating Wikiversity Courses]]

=== Bartending Wikiversity Course ===

==== Notes to Myself ====

# Add an Advanced Bartending lesson with more recipes/mixology/significant bartenders/freepouring?

==== Course Outline ====

# Bartending Basics
## Sources of alcohol
## Types of alcohol
## Tools of Bartending
## Safety
# Mixing Drinks
## Glassware
## Typology of Cocktails
## Preparation and Skills
## Essential Cocktails
# Hospitality
## What is Hospitality?
## Steps of Service
## Internal Hospitality
## Developing Taste
# Industry 
## History of pubs/bartenders/mixologists
## Wages + tips
## Hours + Working conditions
## Unions
# Advanced Topics
## Free-pouring
## Other recipes to know
## Round building
## Mixology
# Recommended Reading List
## Mr. Boston's Official Bartender's Guide - Available on Internet Archive</text>
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      <text bytes="2765" sha1="rpf5wa1m8hc68a5n94s210nxgetil67" xml:space="preserve">= &lt;big&gt;There is a way to solve Unbalanced Bridge Circuits by using only Ohms Law instead of using Kirchhoff's more difficult method.  [mesh currents]&lt;/big&gt; =

=== &lt;big&gt;This is not about a typical current divider which separates 2 PARALLEL currents.    This is a Nodal Current Splitter.    It separates 2 SERIES currents just like a voltage divider separates 2 series voltages.   It does that by using the difference between the 2 series currents &amp; tells you what that difference is at the same time.&lt;/big&gt; ===

=== &lt;big&gt;To see how nodal current splitters can solve problems using Ohms Law instead of Kirchhoff's more complicated method.    Consider the simple T-FRAME circuit shown below.&lt;/big&gt;  &lt;big&gt;&lt;u&gt;Where&lt;/u&gt;:&lt;/big&gt; ===

=== &lt;big&gt;&lt;math&gt;I_1=\frac{\ V_1-I_3R_3\ }{R_1}\qquad\ I_2=\frac{\ V_2-I_3R_3\ }{R_2}&lt;/math&gt;&lt;/big&gt; ===

=== &lt;big&gt;These 2 equations are much more than just adaptions of Ohms Law. They are actually powerful problem solving formulas that have been hiding right in front of our eyes ever since Kirchhoff's Mesh equations showed up. They are not simple Current Dividers ether. These 2 are ''&lt;u&gt;Common Mode&lt;/u&gt;'' Nodal Current Splitters. These 2 formulas split the '''Common mode Nodal Current''' '''I&lt;sub&gt;3&lt;/sub&gt;''' into 2 '''Discrete Nodal Currents''' '''I&lt;sub&gt;1&lt;/sub&gt; &amp; I&lt;sub&gt;2&lt;/sub&gt;'''. The other type of Nodal Current Splitter could be called ''&lt;u&gt;Discrete Mode&lt;/u&gt;'' Nodal Current Splitters. Wait until you see how Discrete mode Nodal Current Splitters can shred unbalanced Wheatstone Bridge problems. And do it using only 1 Nodal Current instead of 3 Kirchhoff's Mesh equations or the usually required 5 nodal current equations.&lt;/big&gt; ===

= &lt;big&gt;The T-Frame Circuit&lt;/big&gt; =

=== &lt;big&gt;Here you can derive the formula for I&lt;sub&gt;3&lt;/sub&gt; using Ohms Law thus bypassing Kirchhoff.&lt;/big&gt; ===
[[File:Cali Cat.jpg|thumb|609x609px|&lt;big&gt;Cali Cat&lt;/big&gt;]]

=== &lt;big&gt;&lt;math&gt;I_1=\frac{\ V_1-I_3R_3\ }{R_1}\qquad\ I_2=\frac{\ V_2-I_3R_3\ }{R_2}&lt;/math&gt;&lt;/big&gt; ===

=== &lt;big&gt;&lt;math&gt;Therefore:\ I_3=\frac{\ V_1-I_3R_3\ }{R_1}+\frac{\ V_2-I_3R_3\ }{R_2}&lt;/math&gt;&lt;/big&gt; ===

=== &lt;big&gt;&lt;math&gt;I_3=\frac{\ V_1R_2+V_2R_1-I_3R_2R_3-I_3R_1R_3\ }{R_1R_2}&lt;/math&gt;&lt;/big&gt; ===

=== &lt;big&gt;&lt;math&gt;I_3R_1R_2+I_3R_1R_3+I_3R_2R_3=\ V_1R_2+V_2R_1&lt;/math&gt;&lt;/big&gt; ===

=== &lt;big&gt;&lt;math&gt;I_3=\frac{V_1R_2+V_2R_1}{\ R_1R_2+R_3\bigl(R_1+R_2\bigr)\ }&lt;/math&gt;&lt;/big&gt; ===

=== &lt;big&gt;Then calculate currents '''I&lt;sub&gt;1&lt;/sub&gt;''' &amp; '''I&lt;sub&gt;2&lt;/sub&gt;''' us&lt;/big&gt;&lt;big&gt;ing the original 2 ''Common''&lt;/big&gt; ''&lt;big&gt;M&lt;/big&gt;&lt;big&gt;ode&lt;/big&gt;'' &lt;big&gt;Nodal Current Splitters&lt;/big&gt; &lt;big&gt;use&lt;/big&gt;&lt;big&gt;d&lt;/big&gt; &lt;big&gt;to d&lt;/big&gt;&lt;big&gt;erived I&lt;sub&gt;3&lt;/sub&gt; from in the 1st place.&lt;/big&gt; ===

=== &lt;big&gt;&lt;math&gt;I_1=\frac{\ V_1-I_3R_3\ }{R_1}\qquad\ I_2=\frac{\ V_2-I_3R_3\ }{R_2}&lt;/math&gt;&lt;/big&gt; ===

= &lt;big&gt;The H-Frame Circuit&lt;/big&gt; =</text>
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    <revision>
      <id>2819847</id>
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        <username>Tet-Math6</username>
        <id>3103001</id>
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      <text bytes="2760" sha1="2ru58dxteci9pz8zclccn4oz4yxme0f" xml:space="preserve">= &lt;big&gt;There is a way to solve Unbalanced Bridge Circuits by using only Ohms Law instead of using Kirchhoff's more difficult method.  [mesh currents]&lt;/big&gt; =

=== &lt;big&gt;This is not about a typical current divider which separates 2 PARALLEL currents.    This is a Nodal Current Splitter.    It separates 2 SERIES currents just like a voltage divider separates 2 series voltages.   It does that by using the difference between the 2 series currents &amp; tells you what that difference is at the same time.&lt;/big&gt; ===

=== &lt;big&gt;To see how nodal current splitters can solve problems using Ohms Law instead of Kirchhoff's complicated method.    Consider the simple T-FRAME circuit shown below.&lt;/big&gt;  &lt;big&gt;&lt;u&gt;Where&lt;/u&gt;:&lt;/big&gt; ===

=== &lt;big&gt;&lt;math&gt;I_1=\frac{\ V_1-I_3R_3\ }{R_1}\qquad\ I_2=\frac{\ V_2-I_3R_3\ }{R_2}&lt;/math&gt;&lt;/big&gt; ===

=== &lt;big&gt;These 2 equations are much more than just adaptions of Ohms Law. They are actually powerful problem solving formulas that have been hiding right in front of our eyes ever since Kirchhoff's Mesh equations showed up. They are not simple Current Dividers ether. These 2 are ''&lt;u&gt;Common Mode&lt;/u&gt;'' Nodal Current Splitters. These 2 formulas split the '''Common mode Nodal Current''' '''I&lt;sub&gt;3&lt;/sub&gt;''' into 2 '''Discrete Nodal Currents''' '''I&lt;sub&gt;1&lt;/sub&gt; &amp; I&lt;sub&gt;2&lt;/sub&gt;'''. The other type of Nodal Current Splitter could be called ''&lt;u&gt;Discrete Mode&lt;/u&gt;'' Nodal Current Splitters. Wait until you see how Discrete mode Nodal Current Splitters can shred unbalanced Wheatstone Bridge problems. And do it using only 1 Nodal Current instead of 3 Kirchhoff's Mesh equations or the usually required 5 nodal current equations.&lt;/big&gt; ===

= &lt;big&gt;The T-Frame Circuit&lt;/big&gt; =

=== &lt;big&gt;Here you can derive the formula for I&lt;sub&gt;3&lt;/sub&gt; using Ohms Law thus bypassing Kirchhoff.&lt;/big&gt; ===
[[File:Cali Cat.jpg|thumb|609x609px|&lt;big&gt;Cali Cat&lt;/big&gt;]]

=== &lt;big&gt;&lt;math&gt;I_1=\frac{\ V_1-I_3R_3\ }{R_1}\qquad\ I_2=\frac{\ V_2-I_3R_3\ }{R_2}&lt;/math&gt;&lt;/big&gt; ===

=== &lt;big&gt;&lt;math&gt;Therefore:\ I_3=\frac{\ V_1-I_3R_3\ }{R_1}+\frac{\ V_2-I_3R_3\ }{R_2}&lt;/math&gt;&lt;/big&gt; ===

=== &lt;big&gt;&lt;math&gt;I_3=\frac{\ V_1R_2+V_2R_1-I_3R_2R_3-I_3R_1R_3\ }{R_1R_2}&lt;/math&gt;&lt;/big&gt; ===

=== &lt;big&gt;&lt;math&gt;I_3R_1R_2+I_3R_1R_3+I_3R_2R_3=\ V_1R_2+V_2R_1&lt;/math&gt;&lt;/big&gt; ===

=== &lt;big&gt;&lt;math&gt;I_3=\frac{V_1R_2+V_2R_1}{\ R_1R_2+R_3\bigl(R_1+R_2\bigr)\ }&lt;/math&gt;&lt;/big&gt; ===

=== &lt;big&gt;Then calculate currents '''I&lt;sub&gt;1&lt;/sub&gt;''' &amp; '''I&lt;sub&gt;2&lt;/sub&gt;''' us&lt;/big&gt;&lt;big&gt;ing the original 2 ''Common''&lt;/big&gt; ''&lt;big&gt;M&lt;/big&gt;&lt;big&gt;ode&lt;/big&gt;'' &lt;big&gt;Nodal Current Splitters&lt;/big&gt; &lt;big&gt;use&lt;/big&gt;&lt;big&gt;d&lt;/big&gt; &lt;big&gt;to d&lt;/big&gt;&lt;big&gt;erived I&lt;sub&gt;3&lt;/sub&gt; from in the 1st place.&lt;/big&gt; ===

=== &lt;big&gt;&lt;math&gt;I_1=\frac{\ V_1-I_3R_3\ }{R_1}\qquad\ I_2=\frac{\ V_2-I_3R_3\ }{R_2}&lt;/math&gt;&lt;/big&gt; ===

= &lt;big&gt;The H-Frame Circuit&lt;/big&gt; =</text>
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    <revision>
      <id>2819849</id>
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      <timestamp>2026-07-30T09:12:32Z</timestamp>
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        <username>Tet-Math6</username>
        <id>3103001</id>
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      <text bytes="2751" sha1="3uiaqu8e8mrtfc0il28n5sepannumyf" xml:space="preserve">= &lt;big&gt;''There is a way to solve Unbalanced Bridge Circuits by using only Ohms Law instead of using Kirchhoff's more difficult mesh current method.''&lt;/big&gt; =

=== &lt;big&gt;This is not about a typical current divider which separates 2 PARALLEL currents.    This is a Nodal Current Splitter.    It separates 2 SERIES currents just like a voltage divider separates 2 series voltages.   It does that by using the difference between the 2 series currents &amp; tells you what that difference is at the same time.&lt;/big&gt; ===

=== &lt;big&gt;To see how nodal current splitters can solve problems using Ohms Law instead of Kirchhoff's complicated method.    Consider the simple T-FRAME circuit shown below.&lt;/big&gt;  &lt;big&gt;&lt;u&gt;Where&lt;/u&gt;:&lt;/big&gt; ===

=== &lt;big&gt;&lt;math&gt;I_1=\frac{\ V_1-I_3R_3\ }{R_1}\qquad\ I_2=\frac{\ V_2-I_3R_3\ }{R_2}&lt;/math&gt;&lt;/big&gt; ===

=== &lt;big&gt;These 2 equations are much more than just adaptions of Ohms Law. They are actually powerful problem solving formulas that have been hiding right in front of our eyes ever since Kirchhoff's method showed up. They are not simple Current Dividers ether. These 2 are ''&lt;u&gt;Common Mode&lt;/u&gt;'' Nodal Current Splitters. These 2 formulas split the '''Common mode Nodal Current''' '''I&lt;sub&gt;3&lt;/sub&gt;''' into 2 '''Discrete Nodal Currents''' '''I&lt;sub&gt;1&lt;/sub&gt; &amp; I&lt;sub&gt;2&lt;/sub&gt;'''. The other type of Nodal Current Splitter could be called ''&lt;u&gt;Discrete Mode&lt;/u&gt;'' Nodal Current Splitters.  You will see how ''Discrete mode'' Nodal Current Splitters can solve unbalanced Wheatstone Bridge problems. And do it using only 1 Nodal Current instead of 3 Kirchhoff's Mesh equations or the usually required 5 nodal current equations.&lt;/big&gt; ===

= &lt;big&gt;The T-Frame Circuit&lt;/big&gt; =

=== &lt;big&gt;Here you can derive the formula for I&lt;sub&gt;3&lt;/sub&gt; using Ohms Law thus bypassing Kirchhoff.&lt;/big&gt; ===
[[File:Cali Cat.jpg|thumb|609x609px|&lt;big&gt;Cali Cat&lt;/big&gt;]]

=== &lt;big&gt;&lt;math&gt;I_1=\frac{\ V_1-I_3R_3\ }{R_1}\qquad\ I_2=\frac{\ V_2-I_3R_3\ }{R_2}&lt;/math&gt;&lt;/big&gt; ===

=== &lt;big&gt;&lt;math&gt;Therefore:\ I_3=\frac{\ V_1-I_3R_3\ }{R_1}+\frac{\ V_2-I_3R_3\ }{R_2}&lt;/math&gt;&lt;/big&gt; ===

=== &lt;big&gt;&lt;math&gt;I_3=\frac{\ V_1R_2+V_2R_1-I_3R_2R_3-I_3R_1R_3\ }{R_1R_2}&lt;/math&gt;&lt;/big&gt; ===

=== &lt;big&gt;&lt;math&gt;I_3R_1R_2+I_3R_1R_3+I_3R_2R_3=\ V_1R_2+V_2R_1&lt;/math&gt;&lt;/big&gt; ===

=== &lt;big&gt;&lt;math&gt;I_3=\frac{V_1R_2+V_2R_1}{\ R_1R_2+R_3\bigl(R_1+R_2\bigr)\ }&lt;/math&gt;&lt;/big&gt; ===

=== &lt;big&gt;Then calculate currents '''I&lt;sub&gt;1&lt;/sub&gt;''' &amp; '''I&lt;sub&gt;2&lt;/sub&gt;''' us&lt;/big&gt;&lt;big&gt;ing the original 2 ''Common''&lt;/big&gt; ''&lt;big&gt;M&lt;/big&gt;&lt;big&gt;ode&lt;/big&gt;'' &lt;big&gt;Nodal Current Splitters&lt;/big&gt; &lt;big&gt;use&lt;/big&gt;&lt;big&gt;d&lt;/big&gt; &lt;big&gt;to d&lt;/big&gt;&lt;big&gt;erived I&lt;sub&gt;3&lt;/sub&gt; from in the 1st place.&lt;/big&gt; ===

=== &lt;big&gt;&lt;math&gt;I_1=\frac{\ V_1-I_3R_3\ }{R_1}\qquad\ I_2=\frac{\ V_2-I_3R_3\ }{R_2}&lt;/math&gt;&lt;/big&gt; ===

= &lt;big&gt;The H-Frame Circuit&lt;/big&gt; =</text>
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  <page>
    <title>Communication Law in Ecuador</title>
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      <id>2819749</id>
      <parentid>2819710</parentid>
      <timestamp>2026-07-29T12:19:35Z</timestamp>
      <contributor>
        <username>MathXplore</username>
        <id>2888076</id>
      </contributor>
      <minor />
      <comment>added [[Category:Ecuador]] using [[Help:Gadget-HotCat|HotCat]]</comment>
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      <text bytes="8145" sha1="p8jjxfcud2herpn0ieutjgwg7wkszyi" xml:space="preserve">== '''1. SOURCES OF THE LAW OF ECUADOR''' ==
The Ecuadorian legal framework governing communication is primarily based on the Constitution of the Republic of [[Comparative law and justice/Ecuador|Ecuador]] (2008), which recognizes freedom of expression, the right to communication, access to information, privacy, cultural diversity, and the rights of indigenous peoples. The principal statutory source is the Organic Law on Communication (Ley Orgánica de Comunicación, LOC), originally enacted in 2013 and substantially reformed in 2019. Other important sources include the Organic Law on Personal Data Protection (2021), the Organic Comprehensive Criminal Code (COIP), the Organic Law on Transparency and Access to Public Information (LOTAIP), electoral legislation regulating political advertising, consumer protection law, and telecommunications legislation. Ecuador is also bound by international treaties, particularly the American Convention on Human Rights, the International Covenant on Civil and Political Rights, and relevant decisions of the Inter-American Court of Human Rights, which possess constitutional hierarchy under Articles 11 and 417–424 of the Constitution when they recognize more favorable rights.&lt;ref&gt;{{Cite journal|title=Cómo deducir derechos humanos de la ley natural y de otras disciplinas {{!}} Ius Humani. Revista de Derecho|url=https://www.iushumani.org/index.php/iushumani/article/view/327|language=es-ES|doi=10.31207/ih.v12i2.327}}&lt;/ref&gt;

----

== '''2. PRINCIPLES OF COMMUNICATION LAW &amp; THE MEDIA''' ==
Ecuador recognizes communication as both a '''fundamental right''' and a '''public interest activity'''. Every person has the right to seek, receive, produce, and disseminate truthful, verified, timely, contextualized, plural, and uncensored information. The Constitution guarantees freedom of expression while promoting diversity, interculturality, multilingualism, and equal access to media. The Organic Law on Communication establishes principles including freedom of expression, editorial independence, pluralism, non-discrimination, universal access, transparency of media ownership, protection of children and adolescents, accountability of public media, and promotion of community media. The State regulates the communications sector through independent administrative authorities but may not impose prior authorization based on editorial content.&lt;ref&gt;{{Cite journal|title=El Derecho al Olvido en la Era Digital: Un Análisis Comparativo Entre España y Perú {{!}} Ius Humani. Revista de Derecho|url=https://www.iushumani.org/index.php/iushumani/article/view/466|language=es-ES}}&lt;/ref&gt;
----

== '''3. CENSORSHIP &amp; VIOLENT CONTENT''' ==
The Constitution expressly prohibits '''prior censorship''', except in narrowly defined circumstances involving the protection of children and adolescents in public entertainment. The Organic Law on Communication likewise forbids prior censorship by public authorities or private actors exercising dominant market power. Nevertheless, media organizations remain subject to '''subsequent liability''' for unlawful publications, including defamation, discrimination, or violations of constitutional rights. Broadcasters must classify audiovisual content according to age appropriateness and adopt measures limiting children's exposure to excessive violence, pornography, or material harmful to their development. Incitement to violence, terrorism, genocide, discrimination, or hate speech may be sanctioned under the Criminal Code.
----

== '''4. TRUTH, HONOR &amp; TOLERANCE''' ==
The Constitution protects both '''freedom of expression''' and the '''rights to honor, reputation, dignity, and personal integrity'''. Individuals whose reputation has been harmed by false or inaccurate information possess the constitutional '''right of rectification and reply''', which media organizations must provide promptly under conditions established by law. Journalists are encouraged to verify facts, distinguish news from opinion, and respect professional ethics, although truthfulness is generally enforced through subsequent civil or criminal liability rather than governmental prepublication review. Ecuadorian law also prohibits discriminatory speech based on race, ethnicity, nationality, religion, sex, sexual orientation, disability, age, or other protected characteristics. Public debate receives broader constitutional protection when it concerns public officials or matters of legitimate public interest.
----

== '''5. CULTURAL AND RELIGIOUS EXPRESSIONS''' ==
Ecuador defines itself as an '''intercultural and plurinational State'''. Communication law protects the dissemination of cultural, linguistic, artistic, and religious expressions belonging to all communities, including indigenous peoples, Afro-Ecuadorians, and Montubio populations. Media are encouraged to promote national identity while respecting cultural diversity and religious pluralism. Public authorities may not discriminate among religious beliefs nor suppress lawful religious expression. Community media receive special legal recognition as instruments for preserving local languages, traditions, and cultural identities. Cultural programming and educational content constitute public policy objectives within Ecuador's communication system.
----

== '''6. PRIVACY &amp; DATA PROTECTION''' ==
The Constitution guarantees the rights to '''privacy''', '''personal and family intimacy''', '''confidential communications''', and protection against arbitrary interference. The '''Organic Law on Personal Data Protection (2021)''' establishes comprehensive rules governing the collection, processing, storage, transfer, and deletion of personal data. Individuals possess rights of access, rectification, updating, erasure, portability, and objection to data processing. Data controllers must process information lawfully, transparently, proportionally, and for legitimate purposes. Journalistic activities enjoy certain exceptions when processing personal data serves the public interest, provided that reporting respects constitutional guarantees concerning honor, dignity, and private life.
----

== '''7. RIGHT TO BODILY, SPIRITUAL AND DIGITAL IDENTITY''' ==
Ecuadorian constitutional law protects the '''right to personal identity''', including one's name, image, voice, honor, reputation, nationality, ethnicity, culture, language, beliefs, and personal development. Individuals enjoy autonomy over the commercial use of their image and biometric characteristics except where legitimate public interests justify disclosure. Religious freedom protects spiritual identity and the right to profess, change, or reject any religion without coercion. The Personal Data Protection Law further safeguards digital identity by regulating biometric data, online profiling, automated decision-making, and sensitive personal information. Unauthorized identity theft, digital impersonation, or misuse of biometric information may give rise to civil, administrative, or criminal liability.
----

== '''8. RIGHT TO REJECT INFORMATION, CLOTHING &amp; HUMAN EXHIBITIONS''' ==
Ecuadorian constitutional principles recognize human dignity as the foundation of all rights. Individuals generally possess the freedom to '''accept or reject unwanted communications''', commercial advertising, intrusive publicity, or unauthorized use of their personal image. Personal autonomy also protects decisions regarding clothing, appearance, religious attire, and cultural dress, subject only to reasonable legal limitations based on public safety, health, or institutional requirements. No person may be publicly exhibited, photographed, filmed, or commercially exploited in a manner that violates dignity, privacy, or personal integrity without lawful justification or valid consent. Special protection is afforded to children, victims of crime, persons deprived of liberty, and other vulnerable groups against degrading or sensationalistic media exposure. Digital dissemination of intimate images without consent may result in civil damages and criminal sanctions under Ecuadorian law.

== References ==

[[Category:Ecuador]]</text>
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        <id>2888076</id>
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      <text bytes="8162" sha1="kiiod5obzn19ie1pvixp9hce8haa4s7" xml:space="preserve">== '''1. SOURCES OF THE LAW OF ECUADOR''' ==
The Ecuadorian legal framework governing communication is primarily based on the Constitution of the Republic of [[Comparative law and justice/Ecuador|Ecuador]] (2008), which recognizes freedom of expression, the right to communication, access to information, privacy, cultural diversity, and the rights of indigenous peoples. The principal statutory source is the Organic Law on Communication (Ley Orgánica de Comunicación, LOC), originally enacted in 2013 and substantially reformed in 2019. Other important sources include the Organic Law on Personal Data Protection (2021), the Organic Comprehensive Criminal Code (COIP), the Organic Law on Transparency and Access to Public Information (LOTAIP), electoral legislation regulating political advertising, consumer protection law, and telecommunications legislation. Ecuador is also bound by international treaties, particularly the American Convention on Human Rights, the International Covenant on Civil and Political Rights, and relevant decisions of the Inter-American Court of Human Rights, which possess constitutional hierarchy under Articles 11 and 417–424 of the Constitution when they recognize more favorable rights.&lt;ref&gt;{{Cite journal|title=Cómo deducir derechos humanos de la ley natural y de otras disciplinas {{!}} Ius Humani. Revista de Derecho|url=https://www.iushumani.org/index.php/iushumani/article/view/327|language=es-ES|doi=10.31207/ih.v12i2.327}}&lt;/ref&gt;

----

== '''2. PRINCIPLES OF COMMUNICATION LAW &amp; THE MEDIA''' ==
Ecuador recognizes communication as both a '''fundamental right''' and a '''public interest activity'''. Every person has the right to seek, receive, produce, and disseminate truthful, verified, timely, contextualized, plural, and uncensored information. The Constitution guarantees freedom of expression while promoting diversity, interculturality, multilingualism, and equal access to media. The Organic Law on Communication establishes principles including freedom of expression, editorial independence, pluralism, non-discrimination, universal access, transparency of media ownership, protection of children and adolescents, accountability of public media, and promotion of community media. The State regulates the communications sector through independent administrative authorities but may not impose prior authorization based on editorial content.&lt;ref&gt;{{Cite journal|title=El Derecho al Olvido en la Era Digital: Un Análisis Comparativo Entre España y Perú {{!}} Ius Humani. Revista de Derecho|url=https://www.iushumani.org/index.php/iushumani/article/view/466|language=es-ES}}&lt;/ref&gt;
----

== '''3. CENSORSHIP &amp; VIOLENT CONTENT''' ==
The Constitution expressly prohibits '''prior censorship''', except in narrowly defined circumstances involving the protection of children and adolescents in public entertainment. The Organic Law on Communication likewise forbids prior censorship by public authorities or private actors exercising dominant market power. Nevertheless, media organizations remain subject to '''subsequent liability''' for unlawful publications, including defamation, discrimination, or violations of constitutional rights. Broadcasters must classify audiovisual content according to age appropriateness and adopt measures limiting children's exposure to excessive violence, pornography, or material harmful to their development. Incitement to violence, terrorism, genocide, discrimination, or hate speech may be sanctioned under the Criminal Code.
----

== '''4. TRUTH, HONOR &amp; TOLERANCE''' ==
The Constitution protects both '''freedom of expression''' and the '''rights to honor, reputation, dignity, and personal integrity'''. Individuals whose reputation has been harmed by false or inaccurate information possess the constitutional '''right of rectification and reply''', which media organizations must provide promptly under conditions established by law. Journalists are encouraged to verify facts, distinguish news from opinion, and respect professional ethics, although truthfulness is generally enforced through subsequent civil or criminal liability rather than governmental prepublication review. Ecuadorian law also prohibits discriminatory speech based on race, ethnicity, nationality, religion, sex, sexual orientation, disability, age, or other protected characteristics. Public debate receives broader constitutional protection when it concerns public officials or matters of legitimate public interest.
----

== '''5. CULTURAL AND RELIGIOUS EXPRESSIONS''' ==
Ecuador defines itself as an '''intercultural and plurinational State'''. Communication law protects the dissemination of cultural, linguistic, artistic, and religious expressions belonging to all communities, including indigenous peoples, Afro-Ecuadorians, and Montubio populations. Media are encouraged to promote national identity while respecting cultural diversity and religious pluralism. Public authorities may not discriminate among religious beliefs nor suppress lawful religious expression. Community media receive special legal recognition as instruments for preserving local languages, traditions, and cultural identities. Cultural programming and educational content constitute public policy objectives within Ecuador's communication system.
----

== '''6. PRIVACY &amp; DATA PROTECTION''' ==
The Constitution guarantees the rights to '''privacy''', '''personal and family intimacy''', '''confidential communications''', and protection against arbitrary interference. The '''Organic Law on Personal Data Protection (2021)''' establishes comprehensive rules governing the collection, processing, storage, transfer, and deletion of personal data. Individuals possess rights of access, rectification, updating, erasure, portability, and objection to data processing. Data controllers must process information lawfully, transparently, proportionally, and for legitimate purposes. Journalistic activities enjoy certain exceptions when processing personal data serves the public interest, provided that reporting respects constitutional guarantees concerning honor, dignity, and private life.
----

== '''7. RIGHT TO BODILY, SPIRITUAL AND DIGITAL IDENTITY''' ==
Ecuadorian constitutional law protects the '''right to personal identity''', including one's name, image, voice, honor, reputation, nationality, ethnicity, culture, language, beliefs, and personal development. Individuals enjoy autonomy over the commercial use of their image and biometric characteristics except where legitimate public interests justify disclosure. Religious freedom protects spiritual identity and the right to profess, change, or reject any religion without coercion. The Personal Data Protection Law further safeguards digital identity by regulating biometric data, online profiling, automated decision-making, and sensitive personal information. Unauthorized identity theft, digital impersonation, or misuse of biometric information may give rise to civil, administrative, or criminal liability.
----

== '''8. RIGHT TO REJECT INFORMATION, CLOTHING &amp; HUMAN EXHIBITIONS''' ==
Ecuadorian constitutional principles recognize human dignity as the foundation of all rights. Individuals generally possess the freedom to '''accept or reject unwanted communications''', commercial advertising, intrusive publicity, or unauthorized use of their personal image. Personal autonomy also protects decisions regarding clothing, appearance, religious attire, and cultural dress, subject only to reasonable legal limitations based on public safety, health, or institutional requirements. No person may be publicly exhibited, photographed, filmed, or commercially exploited in a manner that violates dignity, privacy, or personal integrity without lawful justification or valid consent. Special protection is afforded to children, victims of crime, persons deprived of liberty, and other vulnerable groups against degrading or sensationalistic media exposure. Digital dissemination of intimate images without consent may result in civil damages and criminal sanctions under Ecuadorian law.

== References ==

[[Category:Ecuador]]
[[Category:Law]]</text>
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      <text bytes="3504" sha1="dp17nzmr8vmabpy3odpi2b2h9wefjlh" xml:space="preserve">= Bartending for Beginners =

=== Introduction ===
Mixing drinks, talking to people, and making money: that's bartending! If you find that definition to be less-than-enough, this course will guide you through the essential skills of a bartender. This course is intended for adults of legal drinking age where they live who want to start bartending in a professional establishment either as a side gig or a career. Adults wanting to learn about bartending in general or how to mix drinks at home can also benefit from this course. No prior experience is required. By the end of the course, you should know: how to mix common drinks ordered at different types of bars and how they are related to one another; the liquors, spirits, ales, wines and other drinks used at the bar; the tools of the trade; the basics of safety as a bartender; and how to create a hospitable environment for bar patrons

This course makes use of the [[wikipedia:Wikimedia_Foundation#Projects_and_initiatives|Wikimedia ecosystem]] of projects. Throughout the course, there will be links to Wikipedia pages, Wiktionary entries, and images from Wikimedia Commons. When a link is casually included in a lesson, I encourage you to browse its contents. Occasionally, clicking a link and digesting its contents will be formally assigned as part of the course.

=== Structure of the Course ===
This course is comprised of lessons which each focus on a component of bartending. It is recommended to complete them in order, as each lesson will build on the previous. The lessons can be found below:

# Bartending Basics
# Mixing Drinks
# Hospitality

=== Course Objectives ===
By the end of the course, you should be able to:

* Share information about and identify the different types of alcohol
* Identify and use the tools of a bartender to complete common techniques
* Mix the 20 essential cocktails
* Serve bar patrons safely and hospitably

=== Recommended Materials ===
Bartending is a physical practice, and theory alone is not enough to prepare yourself. There will be activities that you complete at home as part of the course. The materials required to complete these activities are:

# Boston shaker (Preferably not a Cobbler shaker or a Parisian shaker, though they may be easier to find.)
# Hawthorn Strainer
# Bar Spoon
# Jigger
# A glass for mixing, and a glass to pour drinks into (Don't worry about the style of glass, it just needs to be large enough to hold ice and 10 oz of liquid.)

[[File:Bartender_Photo.jpg|left|thumb|Bartender with Boston Shaker]]
[[File:Cocktail-strainer.jpg|thumb|Hawthorne Strainer]]
[[File:Jigger.jpg|center|thumb|160x160px|Jigger]]


Many of these materials can bought online or found at secondhand or thrift shops depending on your area. There may also be a specialty store near you catering to restaurants and bars. Having these materials at home are an invaluable way to practice mixing drinks, with the added benefit of allowing you to entertain guests at home! If you cannot acquire these materials, the course is still able to be completed in theory, but you will miss out on the practical aspects of the course. No materials are needed for lesson one.

In addition to these materials, you will also need the ingredients to the recipes we'll prepare. This will be discussed when we talk about mixing drinks.

Start here with Lesson One: Bartending Basics.

&lt;nowiki&gt;{{CourseCat}}&lt;/nowiki&gt;

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      <text bytes="3594" sha1="3rid0ma8e8yul3s6aoqe6y13alxbx3w" xml:space="preserve">= Bartending for Beginners =

=== Introduction ===
Mixing drinks, talking to people, and making money: that's bartending! If you find that definition to be less-than-enough, this course will guide you through the essential skills of a bartender. This course is intended for adults of legal drinking age where they live who want to start bartending in a professional establishment either as a side gig or a career. Adults wanting to learn about bartending in general or how to mix drinks at home can also benefit from this course. No prior experience is required. By the end of the course, you should know: how to mix common drinks ordered at different types of bars and how they are related to one another; the liquors, spirits, ales, wines and other drinks used at the bar; the tools of the trade; the basics of safety as a bartender; and how to create a hospitable environment for bar patrons

This course makes use of the [[wikipedia:Wikimedia_Foundation#Projects_and_initiatives|Wikimedia ecosystem]] of projects. Throughout the course, there will be links to Wikipedia pages, Wiktionary entries, and images from Wikimedia Commons. When a link is casually included in a lesson, I encourage you to browse its contents. Occasionally, clicking a link and digesting its contents will be formally assigned as part of the course.

=== Structure of the Course ===
This course is comprised of lessons which each focus on a component of bartending. It is recommended to complete them in order, as each lesson will build on the previous. The lessons can be found below:

# [[Bartending/Bartending Basics|Bartending Basics]]
# [[Bartending/Mixing Drinks|Mixing Drinks]]
# [[Bartending/Hospitality|Hospitality]]

=== Course Objectives ===
By the end of the course, you should be able to:

* Share information about and identify the different types of alcohol
* Identify and use the tools of a bartender to complete common techniques
* Mix the 20 essential cocktails
* Serve bar patrons safely and hospitably

=== Recommended Materials ===
Bartending is a physical practice, and theory alone is not enough to prepare yourself. There will be activities that you complete at home as part of the course. The materials required to complete these activities are:

# Boston shaker (Preferably not a Cobbler shaker or a Parisian shaker, though they may be easier to find.)
# Hawthorn Strainer
# Bar Spoon
# Jigger
# A glass for mixing, and a glass to pour drinks into (Don't worry about the style of glass, it just needs to be large enough to hold ice and 10 oz of liquid.)

[[File:Bartender_Photo.jpg|left|thumb|Bartender with Boston Shaker]]
[[File:Cocktail-strainer.jpg|thumb|Hawthorne Strainer]]
[[File:Jigger.jpg|center|thumb|160x160px|Jigger]]


Many of these materials can bought online or found at secondhand or thrift shops depending on your area. There may also be a specialty store near you catering to restaurants and bars. Having these materials at home are an invaluable way to practice mixing drinks, with the added benefit of allowing you to entertain guests at home! If you cannot acquire these materials, the course is still able to be completed in theory, but you will miss out on the practical aspects of the course. No materials are needed for lesson one.

In addition to these materials, you will also need the ingredients to the recipes we'll prepare. This will be discussed when we talk about mixing drinks.

Start here with [[Bartending/Bartending Basics|Lesson One: Bartending Basics]].

[[:Category:Professional and vocational studies]] [[:Category:Professions]] [[:Category:Drinks]]</text>
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      <text bytes="3597" sha1="thwsiunq5r4bwaq10mksu6l9yhzc465" xml:space="preserve">= Bartending for Beginners =

=== Introduction ===
Mixing drinks, talking to people, and making money: that's bartending! If you find that definition to be less-than-enough, this course will guide you through the essential skills of a bartender. This course is intended for adults of legal drinking age where they live who want to start bartending in a professional establishment either as a side gig or a career. Adults wanting to learn about bartending in general or how to mix drinks at home can also benefit from this course. No prior experience is required. By the end of the course, you should know: how to mix common drinks ordered at different types of bars and how they are related to one another; the liquors, spirits, ales, wines and other drinks used at the bar; the tools of the trade; the basics of safety as a bartender; and how to create a hospitable environment for bar patrons

This course makes use of the [[wikipedia:Wikimedia_Foundation#Projects_and_initiatives|Wikimedia ecosystem]] of projects. Throughout the course, there will be links to Wikipedia pages, Wiktionary entries, and images from Wikimedia Commons. When a link is casually included in a lesson, I encourage you to browse its contents. Occasionally, clicking a link and digesting its contents will be formally assigned as part of the course.

=== Structure of the Course ===
This course is comprised of lessons which each focus on a component of bartending. It is recommended to complete them in order, as each lesson will build on the previous. The lessons can be found below:

# [[Bartending/Bartending Basics|Bartending Basics]]
# [[Bartending/Mixing Drinks|Mixing Drinks]]
# [[Bartending/Hospitality|Hospitality]]

=== Course Objectives ===
By the end of the course, you should be able to:

* Share information about and identify the different types of alcohol
* Identify and use the tools of a bartender to complete common techniques
* Mix the 20 essential cocktails
* Serve bar patrons safely and hospitably

=== Recommended Materials ===
Bartending is a physical practice, and theory alone is not enough to prepare yourself. There will be activities that you complete at home as part of the course. The materials required to complete these activities are:

# Boston shaker (Preferably not a Cobbler shaker or a Parisian shaker, though they may be easier to find.)
# Hawthorn Strainer
# Bar Spoon
# Jigger
# A glass for mixing, and a glass to pour drinks into (Don't worry about the style of glass, it just needs to be large enough to hold ice and 10 oz of liquid.)

[[File:Bartender_Photo.jpg|left|thumb|Bartender with Boston Shaker]]
[[File:Cocktail-strainer.jpg|thumb|Hawthorne Strainer]]
[[File:Jigger.jpg|center|thumb|160x160px|Jigger]]


Many of these materials can bought online or found at secondhand or thrift shops depending on your area. There may also be a specialty store near you catering to restaurants and bars. Having these materials at home are an invaluable way to practice mixing drinks, with the added benefit of allowing you to entertain guests at home! If you cannot acquire these materials, the course is still able to be completed in theory, but you will miss out on the practical aspects of the course. No materials are needed for lesson one.

In addition to these materials, you will also need the ingredients to the recipes we'll prepare. This will be discussed when we talk about mixing drinks.

Start here with [[Bartending/Bartending Basics|Lesson One: Bartending Basics]].

[[:Category:Professional and vocational studies]] [[:Category:Professions]] [[:Category:Drinks]]</text>
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      <text bytes="3689" sha1="n3a1oziasxarg4yalzklfe15ajkt5dp" xml:space="preserve">= Bartending for Beginners =

=== Introduction ===
Mixing drinks, talking to people, and making money: that's bartending! If you find that definition to be less-than-enough, this course will guide you through the essential skills of a bartender. This course is intended for adults of legal drinking age where they live who want to start bartending in a professional establishment either as a side gig or a career. Adults wanting to learn about bartending in general or how to mix drinks at home can also benefit from this course. No prior experience is required. By the end of the course, you should know: how to mix common drinks ordered at different types of bars and how they are related to one another; the liquors, spirits, ales, wines and other drinks used at the bar; the tools of the trade; the basics of safety as a bartender; and how to create a hospitable environment for bar patrons

This course makes use of the [[wikipedia:Wikimedia_Foundation#Projects_and_initiatives|Wikimedia ecosystem]] of projects. Throughout the course, there will be links to Wikipedia pages, Wiktionary entries, and images from Wikimedia Commons. When a link is casually included in a lesson, I encourage you to browse its contents. Occasionally, clicking a link and digesting its contents will be formally assigned as part of the course.
[[File:Bartender at Sylvarum cocktail bar, Alicante.jpg|thumb|451x451px|This could be you!]]

=== Structure of the Course ===
This course is composed of lessons which each focus on a component of bartending. It is recommended to complete them in order, as each lesson will build on the previous. The lessons can be found below:

# [[Bartending/Bartending Basics|Bartending Basics]]
# [[Bartending/Mixing Drinks|Mixing Drinks]]
# [[Bartending/Hospitality|Hospitality]]

=== Course Objectives ===
By the end of the course, you should be able to:

* Share information about and identify the different types of alcohol
* Identify and use the tools of a bartender to complete common techniques
* Mix the 20 essential cocktails
* Serve bar patrons safely and hospitably

=== Recommended Materials ===
Bartending is a physical practice, and theory alone is not enough to prepare yourself. There will be activities that you complete at home as part of the course. The materials required to complete these activities are:

# Boston shaker (Preferably not a Cobbler shaker or a Parisian shaker, though they may be easier to find.)
# Hawthorn Strainer
# Bar Spoon
# Jigger
# A glass for mixing, and a glass to pour drinks into (Don't worry about the style of glass, it just needs to be large enough to hold ice and 10 oz of liquid.)

[[File:Bartender_Photo.jpg|left|thumb|Bartender with Boston Shaker]]
[[File:Cocktail-strainer.jpg|thumb|Hawthorne Strainer]]
[[File:Jigger.jpg|center|thumb|160x160px|Jigger]]


Many of these materials can bought online or found at secondhand or thrift shops depending on your area. There may also be a specialty store near you catering to restaurants and bars. Having these materials at home are an invaluable way to practice mixing drinks, with the added benefit of allowing you to entertain guests at home! If you cannot acquire these materials, the course is still able to be completed in theory, but you will miss out on the practical aspects of the course. No materials are needed for lesson one.

In addition to these materials, you will also need the ingredients to the recipes we'll prepare. This will be discussed when we talk about mixing drinks.

Start here with [[Bartending/Bartending Basics|Lesson One: Bartending Basics]].

[[:Category:Professional and vocational studies]] [[:Category:Professions]] [[:Category:Drinks]]</text>
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      <text bytes="3686" sha1="gf6chnr618q0s4y3tddhv9hpk3p6weh" xml:space="preserve">= Bartending for Beginners =

=== Introduction ===
Mixing drinks, talking to people, and making money: that's bartending! If you find that definition to be less-than-enough, this course will guide you through the essential skills of a bartender. This course is intended for adults of legal drinking age where they live who want to start bartending in a professional establishment either as a side gig or a career. Adults wanting to learn about bartending in general or how to mix drinks at home can also benefit from this course. No prior experience is required. By the end of the course, you should know: how to mix common drinks ordered at different types of bars and how they are related to one another; the liquors, spirits, ales, wines and other drinks used at the bar; the tools of the trade; the basics of safety as a bartender; and how to create a hospitable environment for bar patrons

This course makes use of the [[wikipedia:Wikimedia_Foundation#Projects_and_initiatives|Wikimedia ecosystem]] of projects. Throughout the course, there will be links to Wikipedia pages, Wiktionary entries, and images from Wikimedia Commons. When a link is casually included in a lesson, I encourage you to browse its contents. Occasionally, clicking a link and digesting its contents will be formally assigned as part of the course.
[[File:Bartender at Sylvarum cocktail bar, Alicante.jpg|thumb|451x451px|This could be you!]]

=== Structure of the Course ===
This course is composed of lessons which each focus on a component of bartending. It is recommended to complete them in order, as each lesson will build on the previous. The lessons can be found below:

# [[Bartending/Bartending Basics|Bartending Basics]]
# [[Bartending/Mixing Drinks|Mixing Drinks]]
# [[Bartending/Hospitality|Hospitality]]

=== Course Objectives ===
By the end of the course, you should be able to:

* Share information about and identify the different types of alcohol
* Identify and use the tools of a bartender to complete common techniques
* Mix the 20 essential cocktails
* Serve bar patrons safely and hospitably

=== Recommended Materials ===
Bartending is a physical practice, and theory alone is not enough to prepare yourself. There will be activities that you complete at home as part of the course. The materials required to complete these activities are:

# Boston shaker (Preferably not a Cobbler shaker or a Parisian shaker, though they may be easier to find.)
# Hawthorn Strainer
# Bar Spoon
# Jigger
# A glass for mixing, and a glass to pour drinks into (Don't worry about the style of glass, it just needs to be large enough to hold ice and 10 oz of liquid.)

[[File:Bartender_Photo.jpg|left|thumb|Bartender with Boston Shaker]]
[[File:Cocktail-strainer.jpg|thumb|Hawthorne Strainer]]
[[File:Jigger.jpg|center|thumb|160x160px|Jigger]]


Many of these materials can bought online or found at secondhand or thrift shops depending on your area. There may also be a specialty store near you catering to restaurants and bars. Having these materials at home are an invaluable way to practice mixing drinks, with the added benefit of allowing you to entertain guests at home! If you cannot acquire these materials, the course is still able to be completed in theory, but you will miss out on the practical aspects of the course. No materials are needed for lesson one.

In addition to these materials, you will also need the ingredients to the recipes we'll prepare. This will be discussed when we talk about mixing drinks.

Start here with [[Bartending/Bartending Basics|Lesson One: Bartending Basics]].

[[Category:Professional and vocational studies]]
[[Category:Professions]]
[[Category:Drinks]]</text>
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      <text bytes="23882" sha1="2ghnt19ixd5xa5fjnnq0uk37bj42hm9" xml:space="preserve">= Bartending Basics =
What is a [[wikipedia:Bartender|bartender]]? At its simplest, a bartender serves drinks at a bar. In reality, a bartender selects, mixes, pours, and serves drinks while hosting patrons and creating a hospitable atmosphere at a bar, pub, restaurant, nightclub, living room or [[wikipedia:Parking_lot|parking lot]]. A bartender must have a variety of skills, both technical and interpersonal in order to be successful. A bartender can work in many locations, but this course will focus on bartenders that tend to tend in bars.

This lesson covers:

# Sources of Alcohol
# Types of Alcohol
# Tools of Bartending
# Safety

== Sources of Alcohol ==
Behind the bar, you are the expert in the room on alcohol. While no one can claim to know everything on alcohol, it's important to have the basics. This and the next lesson focus on learning what alcohol is and what kind of drinks and liquors are made with it.

Every alcoholic drink you've ever had has included the same basic type of alcohol: [[wikipedia:Ethanol|Ethanol]]. Ethanol is one of three types of alcohol and the only alcohol humans can safely drink. All alcoholic beverages whether it be beer, wine or a spirit all contain Ethanol.

There are hundreds of different beverages and liquors that are served at bars around the world.  The method of refining raw ingredients into a final product with Ethanol is what distinguishes each type of beverage. Fortunately they're all related to each other (since they all contain Ethanol) and have some major categories to guide us in understanding what they are. The first step to make any type of alcohol is [[wikipedia:fermentation|fermentation]].

=== Fermentation ===
[[File:40168_2022_1274_Fig6.webp|thumb|People get very scientific with it.]]
Using a process that has existed for thousands of years, we can employ a fungus called [[wikipedia:Yeast#Uses|yeast]] to create alcohol. This process is called fermentation. At it base, fermentation for alcohol production is the process of using yeast to convert sugars to ethanol.

&quot;Sugars&quot; is intentionally plural. There are many places you can find sugar suitable for fermentation, and you can derive sugar from grains, fruits and vegetables. The source of the sugar is often what defines what type of final alcoholic beverage you get. We'll talk more on how fermenting different materials yields different drinks in a moment.

Fermentation happens in a couple of steps.

# '''Mashing''': Grains like barley or rice are milled into a coarse flour and fruits are mashed into a pulp. These are sometimes mixed with hot water, where enzymes convert starches to sugars. For beer, this mash is then often boiled.
# '''Fermentation''': The mash is transferred to a fermentation vessel. Yeast is added, and fermentation begins. Over a varying period of time (often 5-10 days), yeast consumes the sugars, producing alcohol and carbon dioxide. This is called primary fermentation.
# '''Conditioning and Packaging''': After primary fermentation, there is often secondary fermentation or conditioning to add or mature flavors and to clarify. It’s then sometimes carbonated and packaged.

Now fermented, there is some amount of alcohol in your drink. But what if ''some'' isn't enough? A secondary process called [[wikipedia:Distillation|distillation]] allowed for fermented drinks and solutions to be concentrated into spirits.

=== Distillation ===
[[File:Alambins_industrials_per_a_la_destil·lació_de_licors_a_Catalunya.jpg|left|thumb|Industrial stills for liquor production in Catalonia]]
Distillation is a process by which brewers can concentrate the level of alcohol in a drink. Fermentation alone cannot make alcohol concentrations like that of vodka or gin. To reach that level of concentration, [[wiktionary:distiller|distillers]] (referring to both the apparatus that distills and the person that runs it) can boil off and collect the alcohol produced by fermentation. Distillation happens in a few steps:

# The fermented substance is placed into a vessel called a [[wikipedia:Still|still]] (this is where di''still''ation gets its name).
# The ferment is heated slowly from room temperature until it reaches the boiling point of ethanol. The boiling point of ethanol is lower than the boiling point of water, so the ethanol will vaporize, leaving the water behind.
# The ethanol vapor rises through a column until it reaches the condenser. In the condenser, the vapor makes contact with the a cold surface that cools it back into a liquid state.
# The ethanol is collected, and the process is often repeated to further concentrate and purify the result. Sometimes the process is done without having to stop and restart in a process called continuous distillation.

The result of distillation is a liquor with a higher amount of alcohol by volume (ABV) than before.

=== Alcohol By Volume ===
The amount of alcohol in a drink is calculated based on the percent of ethanol compared to non-ethanol in the total beverage, which is called Alcohol by Volume or ABV for short. For example, if a beverage has an ABV of 50%, half of liquid in the drink will be ethanol and the other half will be everything else.

ABV is important to understand, because it is the basis of drinking and serving drinks safely as well as how many drinks are categorized. While all alcohol is made through fermentation, the process of fermentation alone can only create a concentration of alcohol of about 5% to 20% ABV. Spirits are generally distilled liquids that have a higher ABV (20% or more, as high as 95%). This distinction lower-ABV drinks like wine, beer and cider versus higher-ABV spirits like vodka, gin and whiskey.

== Types of Alcohol ==
Fermentation and distilling are the core of all alcohol production, but there many more ways that brewers and distillers affect their final product. This incredible flow chart shows many of the processes of fermentation, distillation, carbonation, ageing and processing that make different alcoholic drinks. Take a look at it and compare the different starting ingredients with each other, and then compare the starting ingredients with their final products.
[[File:Alcohol_Flow_Chart.svg|center|frame|Alcohol Flow Chart]]
There is so much to learn about alcohol, and while you should be knowledgeable on the different types, you by no means need to be an expert (see: sommelier) to be a bartender. Below follows an overview of each family of alcohol, but if you want to learn more you can click on the links to Wikipedia pages in each subsection below, or I recommend browsing the [[wikipedia:Portal:Liquor|Liquor Portal]] to view all that Wikipedia has to offer.

Hopefully it's easier to see now how beverages and spirits can both be categorized based on what they're fermented from and how concentrated their alcohol content is. Let's use these categories to talk more about each major type of alcohol.

=== Wine, Beer and Cider (Low-ABV Drinks) ===

==== Wine ====
We call fermented grape juice [[wikipedia:wine|wine]]. Yeast is added to pressed grapes to develop it into one of four types: [[wikipedia:Red_wine|red]], [[wikipedia:white_wine|white]], [[wikipedia:rose_wine|rosé]], and [[wikipedia:Orange_wine|orange]]. The type of grape and the duration of the contact with the skins of the grapes help to define which type of wine is produced. The table below is from the Wikipedia page on wine.
{| class=&quot;wikitable&quot;
|+Colors of wine
!
!Long contact with grape skins
!Short contact with grape skins
|-
!Red grapes
|'''''Red wine''''', made from dark-colored red grape varieties. The actual color of the wine can range from dark pink to almost black. The juice from red grapes is actually pale gray; the color of red wine and some of its flavor (notably tannins) comes from phenolics in the skin, seeds and stem fragments of the grape, extracted by allowing the grapes to soak in the juice.
|'''''Rosé wine''''', which gains color from red grape skins, but not enough to qualify it as a red wine. The color can range from a very pale pink to pale red.
There are two primary ways to produce rosé wine. The preferred technique is allowing a short period of maceration after crushing red grapes, which extracts a certain amount of color. The juice is then fermented like a white wine. An alternative is blending a small amount of finished red wine into finished white wine.
|-
!White grapes
|'''''Orange wine''''', sometimes called amber wine, is made with white grapes but with the skins allowed to macerate during and beyond fermentation, similar to red wine production. This results in their darker color compared to white wines, and produces a deliberately astringent result.
|'''''White wine''''', typically made from white grape varieties (those with yellow or green skins), and range from practically colorless to golden. When skin contact is used, to improve the flavor or to increase the body or aging potential, it is usually limited to between four and 24 hours; any longer leads to bitterness.
|}
''Common names: There are thousands of wine varietals. Red wines like Cabernet Sauvignon, Merlot, Pinot Noir, Malbec, Zinfandel; white wines like Chardonnay, Sauvignon Blanc, Pinot Grigio, Riesling; and rosé wines like Provence, White Zinfandel and Pink Moscato.''

In addition to the colors of wine, wine can be [[wikipedia:Sparkling_wine|sparkling]] or still. Wine can also be [[wikipedia:Fortified_wine|fortified]]. Fortifying a wine is the process of adding a distilled spirit to the fermented wine.

''Common names: Sparkling wine includes Champagne, Prosecco and other Brut wines. Fortified wines include Port, Sherry and Vermouth.''

==== Fruit Wine, Cider and Perry ====
[[wikipedia:Fruit_wine|Fruit wine]] is mostly what it sounds like: wine made from fruits other than grapes. A variety of fruits can be used, some common ones include cherry, plum, dandelion and pineapple. Fruit wine is far less common than traditional grape wines, and is often called by the fruit it's fermented from (ex. &quot;cherry wine&quot; or &quot;dandelion wine&quot;).

[[wikipedia:Cider|Cider]] is a beverage made specifically from fermenting apples. Despite being a sort of wine fermented from fruit, it isn't considered a fruit wine due to its unique cultural history as a beverage. Its name can cause some confusion particularly in the United States and Canada, where &quot;cider&quot; also refers to unfiltered and sometimes spiced apple juice that has not been fermented. The term &quot;hard cider&quot; is sometimes used to distinguish the alcoholic beverage due to this overlap. Cider can also be carbonated to make a sparkling cider.

[[wikipedia:Perry|Perry]], or Pear Cider, is another fruit wine exception. Fermented pears create a drink called perry, that though technically is a wine made from fruit, is considered its own thing.

''Common names: There are thousands of brands of fruit wine, cider and perry. You may have some on tap or bottled; it's mostly important to know the names of what you have.''
[[File:Beer_in_glasses_and_steins_on_a_table_with_bottles_in_the_background_and_a_brick_wall_(15700131777).jpg|thumb|264x264px|There are many varieties of beer.]]

==== Beer ====
[[wikipedia:beer|Beer]] is an ancient drink that has been brewed around the world for hundreds of years. Most beers have four basic ingredients: grain, hops, yeast and water. [[wikipedia:Hops|Hops]] are a type of dried flower that give beer flavor. While there are thousands of brands of beer, there are only a few main types:

* '''Lagers''' - One of the two main types of beer, with Ales. Lighter and crisp, refreshing, less bitter than Ale.
** Amber Lager - A medium, amber color with a slight bitterness. ''Common Names: Yuengling''
** Pale Lager - Light in color and flavor, not very bitter. Easy drinking and very popular. ''Common Names: Red Stripe, Heineken, Pabst Blue Ribbon (PBR), Hamms''
** Pilsner - Medium amber in color. Hoppy and flavorful. ''Common Names: Modelo Especial, Stella Artois, Pilsner Urquell''

* '''Ales''' - The other main type of beer, with Lager. Richer and flavorful, more bitter than Lager.
** Pale Ale - A diverse group of beer that various depending on its country of origin. Generally balanced taste that pairs with many foods. ''Common Names: Sierra Nevada, Boulevard''
** Indian Pale Ale (IPA) - Often very bitter and fruity. A unique tasting beer with bite. ''Common Names: Voodoo Ranger, many other large brands have an IPA''
** Stouts and Porters - Dark beers that have rich, roasted flavors often imitating chocolate or coffee. These beers can feel like a meal. ''Common Names: Guinness, Imperial Stouts, Breakfast Stouts''
** Wheat Beer - Ales that have some wheat added to them. It can give them a fruity, almost banana like flavor. ''Common Names: Blue Moon''
* Radlers and Shandys - Beers that have been mixed with a fruit juice. Radlers can be any type of citrus while Shandys are always lemon juice. Very refreshing, summary drinks.
* Sour Beer - Beer that's, well, sour. Tangy and punchy, usually served in a smaller &quot;sour beer&quot; glass since they pack a punch.
* Light Beer - Any beer with an especially low ABV is a light beer (less than 4% ABV), though the term is sometimes also used for low-calorie beers. ''Common Names: Bud Light, Coors Light''

==== Sake ====
[[wikipedia:sake|Sake]] is a drink brewed from rice, and is sometimes also called rice wine though it is brewed more similarly to beer. Like beer and wine, sake has a wide range of flavor profiles though it is often fruity and light. It is sometimes served lightly warmed in ceramic cups depending on the variety of sake.

''Common Names: Dassai, Hakutsuru, Kubota''

=== Spirits (High-ABV Drinks) ===
[[File:Alcoholic_beverages.jpg|thumb|High-ABV spirits like vodka and brandy.]]

==== Vodka ====
[[wikipedia:vodka|Vodka]] is a spirit that can be distilled from many sources including grains, potatoes and sugarcane before the pure ethanol is diluted with water. It is generally a neutral spirit, meaning it has little taste and is mostly pure ethanol and water. This isn't entirely true though, as each vodka has its own impurities and subtle flavor profiles. Vodka is also commonly flavored, as its neutral taste allows it to take on flavor easily. It can served up, often freezer chilled, or mixed into many cocktails.

''Common Names: Absolut, Smirnoff, Kettle One''

==== Gin ====
[[wikipedia:Gin|Gin]] is a spirit flavored with [[wikipedia:juniper_berries|juniper berries]] and other botanicals. It has a distinct herbal flavor, and different brands will have their own flavor profile, and perceived dryness.
[[File:Agave_tequilana_2.jpg|thumb|Agave is a plant native to Mexico.]]
''Common Names: Beefeater, Tanqueray, Hendrick's, Plymouth''

==== Mezcal (Tequila) ====
[[wikipedia:mezcal|Mezcal]] is a spirit distilled from Agave. The most common type of mezcal is [[wikipedia:tequila|Tequila]], which is distilled only from [[wikipedia:blue_agave|blue agave]]. Many people mistake mezcal for a type of tequila due to tequila's popularity, but it is the other way around. Mezcal and tequila are often aged, which impart a darker, more amber color the longer it sits. Mezcal often has a smokey flavor and can also be sweet, fruity or earthy, and can be enjoyed chilled, up or in many popular cocktails.

''Common Names: Casamigos, Patrón, Jose Cuervo''

==== Rum ====
[[wikipedia:rum|Rum]] is a spirit distilled from [[wikipedia:sugarcane|sugarcane]] then aged in barrels. The less-aged rum has a lighter color, and is called &quot;light rum&quot;. It is most commonly used in cocktails, like the Mojito and the Daiquiri. &quot;Aged&quot; or &quot;Dark&quot; rum has a much deeper flavor is often drank straight or on the rocks. Rum originates from and maintains strong ties to the history of the people of the [[wikipedia:caribbean|Caribbean]] and appears commonly in cultural cuisines as well as drinks. I encourage you to learn about the history of rum, as it can teach a lot about sugarcane as a [[wikipedia:Sugar_plantations_in_the_Caribbean|plantation crop]] and the[[wikipedia:Atlantic_slave_trade|Transatlantic Slave Trade]].

''Common Names: Baccardi, Captain Morgan, Malibu''

==== Brandy ====
[[wikipedia:Brandy|Brandy]] is made by distilling wine. While most brandy is made from grapes, it can be made from any fruit to yield a [[wikipedia:Fruit_brandy|fruit brandy]].

''Common names: Cognac and Armagnac''

==== Whiskey ====
[[wikipedia:whiskey|Whiskey]] is made by fermenting, distilling then aging in barrels various grains. Whiskey drinkers can be particular, as there are many types of whiskey. Some whiskey is &quot;malted&quot;, meaning it uses grains that have been malted. Malting is the process of allowing grains to germinate (begin to sprout), before heating them to halt the germination. This yields &quot;malt whiskey&quot;. The specific grain used also yields different types of whiskey, for example fermenting rye yields &quot;rye whiskey&quot;. &quot;[[wikipedia:Scotch_whisky|Scotch]]&quot; is simply whiskey from Scotland and is also called Scotch Whiskey.

''Common Names: Jameson, Jack Daniel's, Crown Royale''

==== Soju ====
[[wikipedia:soju|Soju]] is a made by distilling rice or other grains. Similarly to vodka, it is a neutral spirit that is most flavorless. It is often flavored with fruits and florals. It can be served up or mixed into cocktails.

''Common Names: Jinro''

==== Liqueurs ====
[[wikipedia:liqueur|Liqueurs]] are spirits that have been heavily flavored and sweetened. They are used very frequently in cocktails to impart their particular flavor. Liqueurs can be used to add a flavor that would otherwise be troublesome, like coffee, liquorish, or almond. Sometimes they're served up as a dessert or a [[wikipedia:digestif|digestif]].

''Common Names: Kahlua, Aperol, Campari, Amaretto, Cointreau, and many, many more''

== Tools of Bartending ==
You'll be using both common and specialty tools as a bartender.  A collection of such tools are below. Start by trying to name as many of them as you can. Note how many are familiar to you, how many you've seen but cannot name, and how many are unfamiliar. A key to the image follows.
[[File:Bartools2.jpg|border|center]]
Answer key: (1) champagne bottle stopper, (2) kitchen knife, (3) ice tongs, (4) ice scoop, (5) ice bucket, (6) small bar spoon, (7) cocktail-pick, (8) jigger, (9) mesh strainer, (10) boston shaker (metal bottom), (11) bar spoon, (12) lime/lemon squeezer, (13) hawthorn strainer, (14) zester, (15) boston shaker (mixing glass), (16) muddler, (17) citrus reamer, (18) fine grater, (19) Y-peeler, (20) wine key.

How'd you do? Some of these items you'll see more often than others. I want to draw special attention these crucial pieces:

* '''Jigger''' (8) - A small double sided measuring device used to quickly portion spirits and other ingredients. Though the exact measurements may change depending on where you are and the manufacturer, in the US most jiggers measure 1.5 oz on one end and 0.75 oz on the other.
* '''Juicer and Peeler''' (12 &amp; 19) - Many cocktails rely on citrus for flavor and fragrance. The handheld juicer and the peeler are commonly used both at the start of the shift to prep the citrus for the day, collecting fresh citrus juice and peels for adding to cocktails.
* '''The Boston Shaker''' (10 &amp; 15) - This is the industry standard for fast, versatile and accurate cocktail mixing. Coming in two parts, often one side is glass and the other is metal, though often both are metal. The two parts are sealed together allowing for a cocktail to be shaker vigorously inside before being cracked apart without spilling.
* '''The Hawthorn Strainer''' (23) - The final piece to any Boston Shaker. This uniquely shaped strainer allows for cocktails to be poured into a glass while leaving the ice in the shaker.

These are tools you will see every time you step in to bar to make a cocktail.

== Safety ==
A good bartender is the difference between a safe time for your bar patrons and an unsafe environment. It is your responsibility to keep both yourself and your bar patrons safe. It is important first to understand how alcohol affects the body in order to understand how to properly make and serve drinks, so first let's review ABV.

Alcohol by Volume is a percentage of ethanol to all other contents of a drink. However, ABV does not tell you everything about drink. A standard beer has about 5% ABV while a shot of whiskey has about 40% ABV, but because you would only drink about 1.5 oz of whiskey, and would drink 12 oz beer, the total amount of alcohol ingested would actually be about the same. This is the idea behind a &quot;standard drink&quot;. A standard drink contains one &quot;unit&quot; of alcohol regardless of how much liquid it contains in total. The chart below shows some examples of standard drink equivalents. Each of the below drinks contain one unit of alcohol.
[[File:NIH_standard_drink_comparison.jpg|center|frame|National Institute of Health's &quot;standard drink&quot;.]]
Some general best practices when consuming alcohol:

* One unit of alcohol per hour is around what a liver can process. This is generally a safer cadence of drinking.
* Drink a glass of water for every unit of alcohol consumed to stay hydrated.
* Avoid drinking on an empty stomach. Eating while consuming alcohol slows the alcohol's absorption into the blood and curbs its effects.
* Don't order a double. Ordering and serving drinks with more than one standard unit of alcohol in it can cause you or others to become more impaired more quickly than intended. Try to keep it to the cadence of one standard drink per hour.

Standards should be upheld both with your own drinking and the drinking of the patrons at your bar. Some standards to uphold in your bar are:

* '''Never drink and drive'''. Alcohol impairs motor coordination and clouds judgement. Drinking and then driving puts your own life and the lives of everyone around you in immediate danger. Even if you are under the legal limit in your area, you are impaired after one drink. This should be enforced both for yourself and for your bar patrons.
* '''You can say no'''. It's always okay to say no to a drink. Ordering something without alcohol is always an option. Good friends don't pressure you to drink when you don't want to. You can leave a situation when you are uncomfortable.
* '''Alcohol is not medicine'''. Addiction is real and treatable. Help is available.
* '''Abuse isn't acceptable'''. You are not a punching bag. It is not bad hospitality to stand up for yourself.

Sometimes people drink more than they intended and may become too inebriated to be served. Some signs that this is the case are:

* Slurred speech
* Stumbling
* Glazed over or unfocused eyes
* Speaking too loudly or two softly
* Repeating questions or ordering from multiple bartenders

If you meet a bar patron who has had too much to drink, you can help by offering water, tea, coffee or food, helping them find their group, or asking another member of the staff at your bar for help. It may be that what's needed is to no longer serve the bar patron. This is called &quot;cutting them off&quot;. When a bar patron is cut off, they will no longer be served alcohol for the rest of the service. This can be embarrassing, both for the bar patron and the bartender, so it's best practice to be kind, clear and discrete. If a patron is aggressive however, all bets are off and they should be removed from the bar.

Bartenders have legal obligations depending on where you work. This can include checking the legal age of the patrons at your bar and monitoring the alcohol intake of the patrons at your bar. Sometimes, a bartender can be held individually liable for the failure to meet these obligations. Take the time to find a reputable source to discover what the legal obligation of bartenders in your area are now.

This concludes Lesson 1: Bartending Basics. You can move on now to Lesson 2: Mixing Drinks.</text>
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      <text bytes="24106" sha1="hkwvf3ubidki001hntmtd2jdaf16430" xml:space="preserve">= Bartending Basics =
What is a [[wikipedia:Bartender|bartender]]? At its simplest, a bartender serves drinks at a bar. In reality, a bartender selects, mixes, pours, and serves drinks while hosting patrons and creating a hospitable atmosphere at a bar, pub, restaurant, nightclub, living room or [[wikipedia:Parking_lot|parking lot]]. A bartender must have a variety of skills, both technical and interpersonal in order to be successful. A bartender can work in many locations, but this course will focus on bartenders that tend to tend in bars.

This lesson covers:

# [[Bartending/Bartending Basics#Sources of Alcohol|Sources of Alcohol]]
# [[Bartending/Bartending Basics#Types of Alcohol|Types of Alcohol]]
# [[Bartending/Bartending Basics#Tools of Bartending|Tools of Bartending]]
# [[Bartending/Bartending Basics#Safety|Safety]]

== Sources of Alcohol ==
Behind the bar, you are the expert in the room on alcohol. While no one can claim to know everything on alcohol, it's important to have the basics. This and the next lesson focus on learning what alcohol is and what kind of drinks and liquors are made with it.

Every alcoholic drink you've ever had has included the same basic type of alcohol: [[wikipedia:Ethanol|Ethanol]]. Ethanol is one of three types of alcohol and the only alcohol humans can safely drink. All alcoholic beverages whether it be beer, wine or a spirit all contain Ethanol.

There are hundreds of different beverages and liquors that are served at bars around the world.  The method of refining raw ingredients into a final product with Ethanol is what distinguishes each type of beverage. Fortunately they're all related to each other (since they all contain Ethanol) and have some major categories to guide us in understanding what they are. The first step to make any type of alcohol is [[wikipedia:fermentation|fermentation]].

=== Fermentation ===
[[File:40168_2022_1274_Fig6.webp|thumb|People get very scientific with it.]]
Using a process that has existed for thousands of years, we can employ a fungus called [[wikipedia:Yeast#Uses|yeast]] to create alcohol. This process is called fermentation. At it base, fermentation for alcohol production is the process of using yeast to convert sugars to ethanol.

&quot;Sugars&quot; is intentionally plural. There are many places you can find sugar suitable for fermentation, and you can derive sugar from grains, fruits and vegetables. The source of the sugar is often what defines what type of final alcoholic beverage you get. We'll talk more on how fermenting different materials yields different drinks in a moment.

Fermentation happens in a couple of steps.

# '''Mashing''': Grains like barley or rice are milled into a coarse flour and fruits are mashed into a pulp. These are sometimes mixed with hot water, where enzymes convert starches to sugars. For beer, this mash is then often boiled.
# '''Fermentation''': The mash is transferred to a fermentation vessel. Yeast is added, and fermentation begins. Over a varying period of time (often 5-10 days), yeast consumes the sugars, producing alcohol and carbon dioxide. This is called primary fermentation.
# '''Conditioning and Packaging''': After primary fermentation, there is often secondary fermentation or conditioning to add or mature flavors and to clarify. It’s then sometimes carbonated and packaged.

Now fermented, there is some amount of alcohol in your drink. But what if ''some'' isn't enough? A secondary process called [[wikipedia:Distillation|distillation]] allowed for fermented drinks and solutions to be concentrated into spirits.

=== Distillation ===
[[File:Alambins_industrials_per_a_la_destil·lació_de_licors_a_Catalunya.jpg|left|thumb|Industrial stills for liquor production in Catalonia]]
Distillation is a process by which brewers can concentrate the level of alcohol in a drink. Fermentation alone cannot make alcohol concentrations like that of vodka or gin. To reach that level of concentration, [[wiktionary:distiller|distillers]] (referring to both the apparatus that distills and the person that runs it) can boil off and collect the alcohol produced by fermentation. Distillation happens in a few steps:

# The fermented substance is placed into a vessel called a [[wikipedia:Still|still]] (this is where di''still''ation gets its name).
# The ferment is heated slowly from room temperature until it reaches the boiling point of ethanol. The boiling point of ethanol is lower than the boiling point of water, so the ethanol will vaporize, leaving the water behind.
# The ethanol vapor rises through a column until it reaches the condenser. In the condenser, the vapor makes contact with the a cold surface that cools it back into a liquid state.
# The ethanol is collected, and the process is often repeated to further concentrate and purify the result. Sometimes the process is done without having to stop and restart in a process called continuous distillation.

The result of distillation is a liquor with a higher amount of alcohol by volume (ABV) than before.

=== Alcohol By Volume ===
The amount of alcohol in a drink is calculated based on the percent of ethanol compared to non-ethanol in the total beverage, which is called Alcohol by Volume or ABV for short. For example, if a beverage has an ABV of 50%, half of liquid in the drink will be ethanol and the other half will be everything else.

ABV is important to understand, because it is the basis of drinking and serving drinks safely as well as how many drinks are categorized. While all alcohol is made through fermentation, the process of fermentation alone can only create a concentration of alcohol of about 5% to 20% ABV. Spirits are generally distilled liquids that have a higher ABV (20% or more, as high as 95%). This distinction lower-ABV drinks like wine, beer and cider versus higher-ABV spirits like vodka, gin and whiskey.

== Types of Alcohol ==
Fermentation and distilling are the core of all alcohol production, but there many more ways that brewers and distillers affect their final product. This incredible flow chart shows many of the processes of fermentation, distillation, carbonation, ageing and processing that make different alcoholic drinks. Take a look at it and compare the different starting ingredients with each other, and then compare the starting ingredients with their final products.
[[File:Alcohol_Flow_Chart.svg|center|frame|Alcohol Flow Chart]]
There is so much to learn about alcohol, and while you should be knowledgeable on the different types, you by no means need to be an expert (see: sommelier) to be a bartender. Below follows an overview of each family of alcohol, but if you want to learn more you can click on the links to Wikipedia pages in each subsection below, or I recommend browsing the [[wikipedia:Portal:Liquor|Liquor Portal]] to view all that Wikipedia has to offer.

Hopefully it's easier to see now how beverages and spirits can both be categorized based on what they're fermented from and how concentrated their alcohol content is. Let's use these categories to talk more about each major type of alcohol.

=== Wine, Beer and Cider (Low-ABV Drinks) ===

==== Wine ====
We call fermented grape juice [[wikipedia:wine|wine]]. Yeast is added to pressed grapes to develop it into one of four types: [[wikipedia:Red_wine|red]], [[wikipedia:white_wine|white]], [[wikipedia:rose_wine|rosé]], and [[wikipedia:Orange_wine|orange]]. The type of grape and the duration of the contact with the skins of the grapes help to define which type of wine is produced. The table below is from the Wikipedia page on wine.
{| class=&quot;wikitable&quot;
|+Colors of wine
!
!Long contact with grape skins
!Short contact with grape skins
|-
!Red grapes
|'''''Red wine''''', made from dark-colored red grape varieties. The actual color of the wine can range from dark pink to almost black. The juice from red grapes is actually pale gray; the color of red wine and some of its flavor (notably tannins) comes from phenolics in the skin, seeds and stem fragments of the grape, extracted by allowing the grapes to soak in the juice.
|'''''Rosé wine''''', which gains color from red grape skins, but not enough to qualify it as a red wine. The color can range from a very pale pink to pale red.
There are two primary ways to produce rosé wine. The preferred technique is allowing a short period of maceration after crushing red grapes, which extracts a certain amount of color. The juice is then fermented like a white wine. An alternative is blending a small amount of finished red wine into finished white wine.
|-
!White grapes
|'''''Orange wine''''', sometimes called amber wine, is made with white grapes but with the skins allowed to macerate during and beyond fermentation, similar to red wine production. This results in their darker color compared to white wines, and produces a deliberately astringent result.
|'''''White wine''''', typically made from white grape varieties (those with yellow or green skins), and range from practically colorless to golden. When skin contact is used, to improve the flavor or to increase the body or aging potential, it is usually limited to between four and 24 hours; any longer leads to bitterness.
|}
''Common names: There are thousands of wine varietals. Red wines like Cabernet Sauvignon, Merlot, Pinot Noir, Malbec, Zinfandel; white wines like Chardonnay, Sauvignon Blanc, Pinot Grigio, Riesling; and rosé wines like Provence, White Zinfandel and Pink Moscato.''

In addition to the colors of wine, wine can be [[wikipedia:Sparkling_wine|sparkling]] or still. Wine can also be [[wikipedia:Fortified_wine|fortified]]. Fortifying a wine is the process of adding a distilled spirit to the fermented wine.

''Common names: Sparkling wine includes Champagne, Prosecco and other Brut wines. Fortified wines include Port, Sherry and Vermouth.''

==== Fruit Wine, Cider and Perry ====
[[wikipedia:Fruit_wine|Fruit wine]] is mostly what it sounds like: wine made from fruits other than grapes. A variety of fruits can be used, some common ones include cherry, plum, dandelion and pineapple. Fruit wine is far less common than traditional grape wines, and is often called by the fruit it's fermented from (ex. &quot;cherry wine&quot; or &quot;dandelion wine&quot;).

[[wikipedia:Cider|Cider]] is a beverage made specifically from fermenting apples. Despite being a sort of wine fermented from fruit, it isn't considered a fruit wine due to its unique cultural history as a beverage. Its name can cause some confusion particularly in the United States and Canada, where &quot;cider&quot; also refers to unfiltered and sometimes spiced apple juice that has not been fermented. The term &quot;hard cider&quot; is sometimes used to distinguish the alcoholic beverage due to this overlap. Cider can also be carbonated to make a sparkling cider.

[[wikipedia:Perry|Perry]], or Pear Cider, is another fruit wine exception. Fermented pears create a drink called perry, that though technically is a wine made from fruit, is considered its own thing.

''Common names: There are thousands of brands of fruit wine, cider and perry. You may have some on tap or bottled; it's mostly important to know the names of what you have.''
[[File:Beer_in_glasses_and_steins_on_a_table_with_bottles_in_the_background_and_a_brick_wall_(15700131777).jpg|thumb|264x264px|There are many varieties of beer.]]

==== Beer ====
[[wikipedia:beer|Beer]] is an ancient drink that has been brewed around the world for hundreds of years. Most beers have four basic ingredients: grain, hops, yeast and water. [[wikipedia:Hops|Hops]] are a type of dried flower that give beer flavor. While there are thousands of brands of beer, there are only a few main types:

* '''Lagers''' - One of the two main types of beer, with Ales. Lighter and crisp, refreshing, less bitter than Ale.
** Amber Lager - A medium, amber color with a slight bitterness. ''Common Names: Yuengling''
** Pale Lager - Light in color and flavor, not very bitter. Easy drinking and very popular. ''Common Names: Red Stripe, Heineken, Pabst Blue Ribbon (PBR), Hamms''
** Pilsner - Medium amber in color. Hoppy and flavorful. ''Common Names: Modelo Especial, Stella Artois, Pilsner Urquell''

* '''Ales''' - The other main type of beer, with Lager. Richer and flavorful, more bitter than Lager.
** Pale Ale - A diverse group of beer that various depending on its country of origin. Generally balanced taste that pairs with many foods. ''Common Names: Sierra Nevada, Boulevard''
** Indian Pale Ale (IPA) - Often very bitter and fruity. A unique tasting beer with bite. ''Common Names: Voodoo Ranger, many other large brands have an IPA''
** Stouts and Porters - Dark beers that have rich, roasted flavors often imitating chocolate or coffee. These beers can feel like a meal. ''Common Names: Guinness, Imperial Stouts, Breakfast Stouts''
** Wheat Beer - Ales that have some wheat added to them. It can give them a fruity, almost banana like flavor. ''Common Names: Blue Moon''
* Radlers and Shandys - Beers that have been mixed with a fruit juice. Radlers can be any type of citrus while Shandys are always lemon juice. Very refreshing, summary drinks.
* Sour Beer - Beer that's, well, sour. Tangy and punchy, usually served in a smaller &quot;sour beer&quot; glass since they pack a punch.
* Light Beer - Any beer with an especially low ABV is a light beer (less than 4% ABV), though the term is sometimes also used for low-calorie beers. ''Common Names: Bud Light, Coors Light''

==== Sake ====
[[wikipedia:sake|Sake]] is a drink brewed from rice, and is sometimes also called rice wine though it is brewed more similarly to beer. Like beer and wine, sake has a wide range of flavor profiles though it is often fruity and light. It is sometimes served lightly warmed in ceramic cups depending on the variety of sake.

''Common Names: Dassai, Hakutsuru, Kubota''

=== Spirits (High-ABV Drinks) ===
[[File:Alcoholic_beverages.jpg|thumb|High-ABV spirits like vodka and brandy.]]

==== Vodka ====
[[wikipedia:vodka|Vodka]] is a spirit that can be distilled from many sources including grains, potatoes and sugarcane before the pure ethanol is diluted with water. It is generally a neutral spirit, meaning it has little taste and is mostly pure ethanol and water. This isn't entirely true though, as each vodka has its own impurities and subtle flavor profiles. Vodka is also commonly flavored, as its neutral taste allows it to take on flavor easily. It can served up, often freezer chilled, or mixed into many cocktails.

''Common Names: Absolut, Smirnoff, Kettle One''

==== Gin ====
[[wikipedia:Gin|Gin]] is a spirit flavored with [[wikipedia:juniper_berries|juniper berries]] and other botanicals. It has a distinct herbal flavor, and different brands will have their own flavor profile, and perceived dryness.
[[File:Agave_tequilana_2.jpg|thumb|Agave is a plant native to Mexico.]]
''Common Names: Beefeater, Tanqueray, Hendrick's, Plymouth''

==== Mezcal (Tequila) ====
[[wikipedia:mezcal|Mezcal]] is a spirit distilled from Agave. The most common type of mezcal is [[wikipedia:tequila|Tequila]], which is distilled only from [[wikipedia:blue_agave|blue agave]]. Many people mistake mezcal for a type of tequila due to tequila's popularity, but it is the other way around. Mezcal and tequila are often aged, which impart a darker, more amber color the longer it sits. Mezcal often has a smokey flavor and can also be sweet, fruity or earthy, and can be enjoyed chilled, up or in many popular cocktails.

''Common Names: Casamigos, Patrón, Jose Cuervo''

==== Rum ====
[[wikipedia:rum|Rum]] is a spirit distilled from [[wikipedia:sugarcane|sugarcane]] then aged in barrels. The less-aged rum has a lighter color, and is called &quot;light rum&quot;. It is most commonly used in cocktails, like the Mojito and the Daiquiri. &quot;Aged&quot; or &quot;Dark&quot; rum has a much deeper flavor is often drank straight or on the rocks. Rum originates from and maintains strong ties to the history of the people of the [[wikipedia:caribbean|Caribbean]] and appears commonly in cultural cuisines as well as drinks. I encourage you to learn about the history of rum, as it can teach a lot about sugarcane as a [[wikipedia:Sugar_plantations_in_the_Caribbean|plantation crop]] and the[[wikipedia:Atlantic_slave_trade|Transatlantic Slave Trade]].

''Common Names: Baccardi, Captain Morgan, Malibu''

==== Brandy ====
[[wikipedia:Brandy|Brandy]] is made by distilling wine. While most brandy is made from grapes, it can be made from any fruit to yield a [[wikipedia:Fruit_brandy|fruit brandy]].

''Common names: Cognac and Armagnac''

==== Whiskey ====
[[wikipedia:whiskey|Whiskey]] is made by fermenting, distilling then aging in barrels various grains. Whiskey drinkers can be particular, as there are many types of whiskey. Some whiskey is &quot;malted&quot;, meaning it uses grains that have been malted. Malting is the process of allowing grains to germinate (begin to sprout), before heating them to halt the germination. This yields &quot;malt whiskey&quot;. The specific grain used also yields different types of whiskey, for example fermenting rye yields &quot;rye whiskey&quot;. &quot;[[wikipedia:Scotch_whisky|Scotch]]&quot; is simply whiskey from Scotland and is also called Scotch Whiskey.

''Common Names: Jameson, Jack Daniel's, Crown Royale''

==== Soju ====
[[wikipedia:soju|Soju]] is a made by distilling rice or other grains. Similarly to vodka, it is a neutral spirit that is most flavorless. It is often flavored with fruits and florals. It can be served up or mixed into cocktails.

''Common Names: Jinro''

==== Liqueurs ====
[[wikipedia:liqueur|Liqueurs]] are spirits that have been heavily flavored and sweetened. They are used very frequently in cocktails to impart their particular flavor. Liqueurs can be used to add a flavor that would otherwise be troublesome, like coffee, liquorish, or almond. Sometimes they're served up as a dessert or a [[wikipedia:digestif|digestif]].

''Common Names: Kahlua, Aperol, Campari, Amaretto, Cointreau, and many, many more''

== Tools of Bartending ==
You'll be using both common and specialty tools as a bartender.  A collection of such tools are below. Start by trying to name as many of them as you can. Note how many are familiar to you, how many you've seen but cannot name, and how many are unfamiliar. A key to the image follows.
[[File:Bartools2.jpg|border|center]]
Answer key: (1) champagne bottle stopper, (2) kitchen knife, (3) ice tongs, (4) ice scoop, (5) ice bucket, (6) small bar spoon, (7) cocktail-pick, (8) jigger, (9) mesh strainer, (10) boston shaker (metal bottom), (11) bar spoon, (12) lime/lemon squeezer, (13) hawthorn strainer, (14) zester, (15) boston shaker (mixing glass), (16) muddler, (17) citrus reamer, (18) fine grater, (19) Y-peeler, (20) wine key.

How'd you do? Some of these items you'll see more often than others. I want to draw special attention these crucial pieces:

* '''Jigger''' (8) - A small double sided measuring device used to quickly portion spirits and other ingredients. Though the exact measurements may change depending on where you are and the manufacturer, in the US most jiggers measure 1.5 oz on one end and 0.75 oz on the other.
* '''Juicer and Peeler''' (12 &amp; 19) - Many cocktails rely on citrus for flavor and fragrance. The handheld juicer and the peeler are commonly used both at the start of the shift to prep the citrus for the day, collecting fresh citrus juice and peels for adding to cocktails.
* '''The Boston Shaker''' (10 &amp; 15) - This is the industry standard for fast, versatile and accurate cocktail mixing. Coming in two parts, often one side is glass and the other is metal, though often both are metal. The two parts are sealed together allowing for a cocktail to be shaker vigorously inside before being cracked apart without spilling.
* '''The Hawthorn Strainer''' (23) - The final piece to any Boston Shaker. This uniquely shaped strainer allows for cocktails to be poured into a glass while leaving the ice in the shaker.

These are tools you will see every time you step in to bar to make a cocktail.

== Safety ==
A good bartender is the difference between a safe time for your bar patrons and an unsafe environment. It is your responsibility to keep both yourself and your bar patrons safe. It is important first to understand how alcohol affects the body in order to understand how to properly make and serve drinks, so first let's review ABV.

Alcohol by Volume is a percentage of ethanol to all other contents of a drink. However, ABV does not tell you everything about drink. A standard beer has about 5% ABV while a shot of whiskey has about 40% ABV, but because you would only drink about 1.5 oz of whiskey, and would drink 12 oz beer, the total amount of alcohol ingested would actually be about the same. This is the idea behind a &quot;standard drink&quot;. A standard drink contains one &quot;unit&quot; of alcohol regardless of how much liquid it contains in total. The chart below shows some examples of standard drink equivalents. Each of the below drinks contain one unit of alcohol.
[[File:NIH_standard_drink_comparison.jpg|center|frame|National Institute of Health's &quot;standard drink&quot;.]]
Some general best practices when consuming alcohol:

* One unit of alcohol per hour is around what a liver can process. This is generally a safer cadence of drinking.
* Drink a glass of water for every unit of alcohol consumed to stay hydrated.
* Avoid drinking on an empty stomach. Eating while consuming alcohol slows the alcohol's absorption into the blood and curbs its effects.
* Don't order a double. Ordering and serving drinks with more than one standard unit of alcohol in it can cause you or others to become more impaired more quickly than intended. Try to keep it to the cadence of one standard drink per hour.

Standards should be upheld both with your own drinking and the drinking of the patrons at your bar. Some standards to uphold in your bar are:

* '''Never drink and drive'''. Alcohol impairs motor coordination and clouds judgement. Drinking and then driving puts your own life and the lives of everyone around you in immediate danger. Even if you are under the legal limit in your area, you are impaired after one drink. This should be enforced both for yourself and for your bar patrons.
* '''You can say no'''. It's always okay to say no to a drink. Ordering something without alcohol is always an option. Good friends don't pressure you to drink when you don't want to. You can leave a situation when you are uncomfortable.
* '''Alcohol is not medicine'''. Addiction is real and treatable. Help is available.
* '''Abuse isn't acceptable'''. You are not a punching bag. It is not bad hospitality to stand up for yourself.

Sometimes people drink more than they intended and may become too inebriated to be served. Some signs that this is the case are:

* Slurred speech
* Stumbling
* Glazed over or unfocused eyes
* Speaking too loudly or two softly
* Repeating questions or ordering from multiple bartenders

If you meet a bar patron who has had too much to drink, you can help by offering water, tea, coffee or food, helping them find their group, or asking another member of the staff at your bar for help. It may be that what's needed is to no longer serve the bar patron. This is called &quot;cutting them off&quot;. When a bar patron is cut off, they will no longer be served alcohol for the rest of the service. This can be embarrassing, both for the bar patron and the bartender, so it's best practice to be kind, clear and discrete. If a patron is aggressive however, all bets are off and they should be removed from the bar.

Bartenders have legal obligations depending on where you work. This can include checking the legal age of the patrons at your bar and monitoring the alcohol intake of the patrons at your bar. Sometimes, a bartender can be held individually liable for the failure to meet these obligations. Take the time to find a reputable source to discover what the legal obligation of bartenders in your area are now.

This concludes Lesson 1: Bartending Basics. You can move on now to [[Bartending/Mixing Drinks|Lesson 2: Mixing Drinks]].</text>
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      <text bytes="24246" sha1="9itdyyu0lwkqmnck57j45ahjsxsn85k" xml:space="preserve">= Bartending Basics =
What is a [[wikipedia:Bartender|bartender]]? At its simplest, a bartender serves drinks at a bar. In reality, a bartender selects, mixes, pours, and serves drinks while hosting patrons and creating a hospitable atmosphere at a bar, pub, restaurant, nightclub, living room or [[wikipedia:Parking_lot|parking lot]]. A bartender must have a variety of skills, both technical and interpersonal in order to be successful. A bartender can work in many locations, but this course will focus on bartenders that tend to tend in bars.

This lesson covers:

# [[Bartending/Bartending Basics#Sources of Alcohol|Sources of Alcohol]]
# [[Bartending/Bartending Basics#Types of Alcohol|Types of Alcohol]]
# [[Bartending/Bartending Basics#Tools of Bartending|Tools of Bartending]]
# [[Bartending/Bartending Basics#Safety|Safety]]

== Sources of Alcohol ==
Behind the bar, you are the expert in the room on alcohol. While no one can claim to know everything on alcohol, it's important to have the basics. This and the next lesson focus on learning what alcohol is and what kind of drinks and liquors are made with it.

Every alcoholic drink you've ever had has included the same basic type of alcohol: [[wikipedia:Ethanol|Ethanol]]. Ethanol is one of three types of alcohol and the only alcohol humans can safely drink. All alcoholic beverages whether it be beer, wine or a spirit all contain Ethanol.

There are hundreds of different beverages and liquors that are served at bars around the world.  The method of refining raw ingredients into a final product with Ethanol is what distinguishes each type of beverage. Fortunately they're all related to each other (since they all contain Ethanol) and have some major categories to guide us in understanding what they are. The first step to make any type of alcohol is [[wikipedia:fermentation|fermentation]].

=== Fermentation ===
[[File:40168_2022_1274_Fig6.webp|thumb|People get very scientific with it.]]
Using a process that has existed for thousands of years, we can employ a fungus called [[wikipedia:Yeast#Uses|yeast]] to create alcohol. This process is called fermentation. At it base, fermentation for alcohol production is the process of using yeast to convert sugars to ethanol.

&quot;Sugars&quot; is intentionally plural. There are many places you can find sugar suitable for fermentation, and you can derive sugar from grains, fruits and vegetables. The source of the sugar is often what defines what type of final alcoholic beverage you get. We'll talk more on how fermenting different materials yields different drinks in a moment.

Fermentation happens in a couple of steps.

# '''Mashing''': Grains like barley or rice are milled into a coarse flour and fruits are mashed into a pulp. These are sometimes mixed with hot water, where enzymes convert starches to sugars. For beer, this mash is then often boiled.
# '''Fermentation''': The mash is transferred to a fermentation vessel. Yeast is added, and fermentation begins. Over a varying period of time (often 5-10 days), yeast consumes the sugars, producing alcohol and carbon dioxide. This is called primary fermentation.
# '''Conditioning and Packaging''': After primary fermentation, there is often secondary fermentation or conditioning to add or mature flavors and to clarify. It’s then sometimes carbonated and packaged.

Now fermented, there is some amount of alcohol in your drink. But what if ''some'' isn't enough? A secondary process called [[wikipedia:Distillation|distillation]] allowed for fermented drinks and solutions to be concentrated into spirits.

=== Distillation ===
[[File:Alambins_industrials_per_a_la_destil·lació_de_licors_a_Catalunya.jpg|left|thumb|Industrial stills for liquor production in Catalonia]]
Distillation is a process by which brewers can concentrate the level of alcohol in a drink. Fermentation alone cannot make alcohol concentrations like that of vodka or gin. To reach that level of concentration, [[wiktionary:distiller|distillers]] (referring to both the apparatus that distills and the person that runs it) can boil off and collect the alcohol produced by fermentation. Distillation happens in a few steps:

# The fermented substance is placed into a vessel called a [[wikipedia:Still|still]] (this is where di''still''ation gets its name).
# The ferment is heated slowly from room temperature until it reaches the boiling point of ethanol. The boiling point of ethanol is lower than the boiling point of water, so the ethanol will vaporize, leaving the water behind.
# The ethanol vapor rises through a column until it reaches the condenser. In the condenser, the vapor makes contact with the a cold surface that cools it back into a liquid state.
# The ethanol is collected, and the process is often repeated to further concentrate and purify the result. Sometimes the process is done without having to stop and restart in a process called continuous distillation.

The result of distillation is a liquor with a higher amount of alcohol by volume (ABV) than before.

=== Alcohol By Volume ===
The amount of alcohol in a drink is calculated based on the percent of ethanol compared to non-ethanol in the total beverage, which is called Alcohol by Volume or ABV for short. For example, if a beverage has an ABV of 50%, half of liquid in the drink will be ethanol and the other half will be everything else.

ABV is important to understand, because it is the basis of drinking and serving drinks safely as well as how many drinks are categorized. While all alcohol is made through fermentation, the process of fermentation alone can only create a concentration of alcohol of about 5% to 20% ABV. Spirits are generally distilled liquids that have a higher ABV (20% or more, as high as 95%). This distinction lower-ABV drinks like wine, beer and cider versus higher-ABV spirits like vodka, gin and whiskey.

== Types of Alcohol ==
Fermentation and distilling are the core of all alcohol production, but there many more ways that brewers and distillers affect their final product. This incredible flow chart shows many of the processes of fermentation, distillation, carbonation, ageing and processing that make different alcoholic drinks. Take a look at it and compare the different starting ingredients with each other, and then compare the starting ingredients with their final products.
[[File:Alcohol_Flow_Chart.svg|center|frame|Alcohol Flow Chart]]
There is so much to learn about alcohol, and while you should be knowledgeable on the different types, you by no means need to be an expert (see: sommelier) to be a bartender. Below follows an overview of each family of alcohol, but if you want to learn more you can click on the links to Wikipedia pages in each subsection below, or I recommend browsing the [[wikipedia:Portal:Liquor|Liquor Portal]] to view all that Wikipedia has to offer.

Hopefully it's easier to see now how beverages and spirits can both be categorized based on what they're fermented from and how concentrated their alcohol content is. Let's use these categories to talk more about each major type of alcohol.

=== Wine, Beer and Cider (Low-ABV Drinks) ===

==== Wine ====
We call fermented grape juice [[wikipedia:wine|wine]]. Yeast is added to pressed grapes to develop it into one of four types: [[wikipedia:Red_wine|red]], [[wikipedia:white_wine|white]], [[wikipedia:rose_wine|rosé]], and [[wikipedia:Orange_wine|orange]]. The type of grape and the duration of the contact with the skins of the grapes help to define which type of wine is produced. The table below is from the Wikipedia page on wine.
{| class=&quot;wikitable&quot;
|+Colors of wine
!
!Long contact with grape skins
!Short contact with grape skins
|-
!Red grapes
|'''''Red wine''''', made from dark-colored red grape varieties. The actual color of the wine can range from dark pink to almost black. The juice from red grapes is actually pale gray; the color of red wine and some of its flavor (notably tannins) comes from phenolics in the skin, seeds and stem fragments of the grape, extracted by allowing the grapes to soak in the juice.
|'''''Rosé wine''''', which gains color from red grape skins, but not enough to qualify it as a red wine. The color can range from a very pale pink to pale red.
There are two primary ways to produce rosé wine. The preferred technique is allowing a short period of maceration after crushing red grapes, which extracts a certain amount of color. The juice is then fermented like a white wine. An alternative is blending a small amount of finished red wine into finished white wine.
|-
!White grapes
|'''''Orange wine''''', sometimes called amber wine, is made with white grapes but with the skins allowed to macerate during and beyond fermentation, similar to red wine production. This results in their darker color compared to white wines, and produces a deliberately astringent result.
|'''''White wine''''', typically made from white grape varieties (those with yellow or green skins), and range from practically colorless to golden. When skin contact is used, to improve the flavor or to increase the body or aging potential, it is usually limited to between four and 24 hours; any longer leads to bitterness.
|}
''Common names: There are thousands of wine varietals. Red wines like Cabernet Sauvignon, Merlot, Pinot Noir, Malbec, Zinfandel; white wines like Chardonnay, Sauvignon Blanc, Pinot Grigio, Riesling; and rosé wines like Provence, White Zinfandel and Pink Moscato.''

In addition to the colors of wine, wine can be [[wikipedia:Sparkling_wine|sparkling]] or still. Wine can also be [[wikipedia:Fortified_wine|fortified]]. Fortifying a wine is the process of adding a distilled spirit to the fermented wine.

''Common names: Sparkling wine includes Champagne, Prosecco and other Brut wines. Fortified wines include Port, Sherry and Vermouth.''

==== Fruit Wine, Cider and Perry ====
[[wikipedia:Fruit_wine|Fruit wine]] is mostly what it sounds like: wine made from fruits other than grapes. A variety of fruits can be used, some common ones include cherry, plum, dandelion and pineapple. Fruit wine is far less common than traditional grape wines, and is often called by the fruit it's fermented from (ex. &quot;cherry wine&quot; or &quot;dandelion wine&quot;).

[[wikipedia:Cider|Cider]] is a beverage made specifically from fermenting apples. Despite being a sort of wine fermented from fruit, it isn't considered a fruit wine due to its unique cultural history as a beverage. Its name can cause some confusion particularly in the United States and Canada, where &quot;cider&quot; also refers to unfiltered and sometimes spiced apple juice that has not been fermented. The term &quot;hard cider&quot; is sometimes used to distinguish the alcoholic beverage due to this overlap. Cider can also be carbonated to make a sparkling cider.

[[wikipedia:Perry|Perry]], or Pear Cider, is another fruit wine exception. Fermented pears create a drink called perry, that though technically is a wine made from fruit, is considered its own thing.

''Common names: There are thousands of brands of fruit wine, cider and perry. You may have some on tap or bottled; it's mostly important to know the names of what you have.''
[[File:Beer_in_glasses_and_steins_on_a_table_with_bottles_in_the_background_and_a_brick_wall_(15700131777).jpg|thumb|264x264px|There are many varieties of beer.]]

==== Beer ====
[[wikipedia:beer|Beer]] is an ancient drink that has been brewed around the world for hundreds of years. Most beers have four basic ingredients: grain, hops, yeast and water. [[wikipedia:Hops|Hops]] are a type of dried flower that give beer flavor. While there are thousands of brands of beer, there are only a few main types:

* '''Lagers''' - One of the two main types of beer, with Ales. Lighter and crisp, refreshing, less bitter than Ale.
** Amber Lager - A medium, amber color with a slight bitterness. ''Common Names: Yuengling''
** Pale Lager - Light in color and flavor, not very bitter. Easy drinking and very popular. ''Common Names: Red Stripe, Heineken, Pabst Blue Ribbon (PBR), Hamms''
** Pilsner - Medium amber in color. Hoppy and flavorful. ''Common Names: Modelo Especial, Stella Artois, Pilsner Urquell''

* '''Ales''' - The other main type of beer, with Lager. Richer and flavorful, more bitter than Lager.
** Pale Ale - A diverse group of beer that various depending on its country of origin. Generally balanced taste that pairs with many foods. ''Common Names: Sierra Nevada, Boulevard''
** Indian Pale Ale (IPA) - Often very bitter and fruity. A unique tasting beer with bite. ''Common Names: Voodoo Ranger, many other large brands have an IPA''
** Stouts and Porters - Dark beers that have rich, roasted flavors often imitating chocolate or coffee. These beers can feel like a meal. ''Common Names: Guinness, Imperial Stouts, Breakfast Stouts''
** Wheat Beer - Ales that have some wheat added to them. It can give them a fruity, almost banana like flavor. ''Common Names: Blue Moon''
* Radlers and Shandys - Beers that have been mixed with a fruit juice. Radlers can be any type of citrus while Shandys are always lemon juice. Very refreshing, summary drinks.
* Sour Beer - Beer that's, well, sour. Tangy and punchy, usually served in a smaller &quot;sour beer&quot; glass since they pack a punch.
* Light Beer - Any beer with an especially low ABV is a light beer (less than 4% ABV), though the term is sometimes also used for low-calorie beers. ''Common Names: Bud Light, Coors Light''

==== Sake ====
[[wikipedia:sake|Sake]] is a drink brewed from rice, and is sometimes also called rice wine though it is brewed more similarly to beer. Like beer and wine, sake has a wide range of flavor profiles though it is often fruity and light. It is sometimes served lightly warmed in ceramic cups depending on the variety of sake.

''Common Names: Dassai, Hakutsuru, Kubota''

=== Spirits (High-ABV Drinks) ===
[[File:Alcoholic_beverages.jpg|thumb|High-ABV spirits like vodka and brandy.]]

==== Vodka ====
[[wikipedia:vodka|Vodka]] is a spirit that can be distilled from many sources including grains, potatoes and sugarcane before the pure ethanol is diluted with water. It is generally a neutral spirit, meaning it has little taste and is mostly pure ethanol and water. This isn't entirely true though, as each vodka has its own impurities and subtle flavor profiles. Vodka is also commonly flavored, as its neutral taste allows it to take on flavor easily. It can served up, often freezer chilled, or mixed into many cocktails.

''Common Names: Absolut, Smirnoff, Kettle One''

==== Gin ====
[[wikipedia:Gin|Gin]] is a spirit flavored with [[wikipedia:juniper_berries|juniper berries]] and other botanicals. It has a distinct herbal flavor, and different brands will have their own flavor profile, and perceived dryness.
[[File:Agave_tequilana_2.jpg|thumb|Agave is a plant native to Mexico.]]
''Common Names: Beefeater, Tanqueray, Hendrick's, Plymouth''

==== Mezcal (Tequila) ====
[[wikipedia:mezcal|Mezcal]] is a spirit distilled from Agave. The most common type of mezcal is [[wikipedia:tequila|Tequila]], which is distilled only from [[wikipedia:blue_agave|blue agave]]. Many people mistake mezcal for a type of tequila due to tequila's popularity, but it is the other way around. Mezcal and tequila are often aged, which impart a darker, more amber color the longer it sits. Mezcal often has a smokey flavor and can also be sweet, fruity or earthy, and can be enjoyed chilled, up or in many popular cocktails.

''Common Names: Casamigos, Patrón, Jose Cuervo''

==== Rum ====
[[wikipedia:rum|Rum]] is a spirit distilled from [[wikipedia:sugarcane|sugarcane]] then aged in barrels. The less-aged rum has a lighter color, and is called &quot;light rum&quot;. It is most commonly used in cocktails, like the Mojito and the Daiquiri. &quot;Aged&quot; or &quot;Dark&quot; rum has a much deeper flavor is often drank straight or on the rocks. Rum originates from and maintains strong ties to the history of the people of the [[wikipedia:caribbean|Caribbean]] and appears commonly in cultural cuisines as well as drinks. I encourage you to learn about the history of rum, as it can teach a lot about sugarcane as a [[wikipedia:Sugar_plantations_in_the_Caribbean|plantation crop]] and the[[wikipedia:Atlantic_slave_trade|Transatlantic Slave Trade]].

''Common Names: Baccardi, Captain Morgan, Malibu''

==== Brandy ====
[[wikipedia:Brandy|Brandy]] is made by distilling wine. While most brandy is made from grapes, it can be made from any fruit to yield a [[wikipedia:Fruit_brandy|fruit brandy]].

''Common names: Cognac and Armagnac''

==== Whiskey ====
[[wikipedia:whiskey|Whiskey]] is made by fermenting, distilling then aging in barrels various grains. Whiskey drinkers can be particular, as there are many types of whiskey. Some whiskey is &quot;malted&quot;, meaning it uses grains that have been malted. Malting is the process of allowing grains to germinate (begin to sprout), before heating them to halt the germination. This yields &quot;malt whiskey&quot;. The specific grain used also yields different types of whiskey, for example fermenting rye yields &quot;rye whiskey&quot;. &quot;[[wikipedia:Scotch_whisky|Scotch]]&quot; is simply whiskey from Scotland and is also called Scotch Whiskey.

''Common Names: Jameson, Jack Daniel's, Crown Royale''

==== Soju ====
[[wikipedia:soju|Soju]] is a made by distilling rice or other grains. Similarly to vodka, it is a neutral spirit that is most flavorless. It is often flavored with fruits and florals. It can be served up or mixed into cocktails.

''Common Names: Jinro''

==== Liqueurs ====
[[wikipedia:liqueur|Liqueurs]] are spirits that have been heavily flavored and sweetened. They are used very frequently in cocktails to impart their particular flavor. Liqueurs can be used to add a flavor that would otherwise be troublesome, like coffee, liquorish, or almond. Sometimes they're served up as a dessert or a [[wikipedia:digestif|digestif]].

''Common Names: Kahlua, Aperol, Campari, Amaretto, Cointreau, and many, many more''

== Tools of Bartending ==
You'll be using both common and specialty tools as a bartender.  A collection of such tools are below. Start by trying to name as many of them as you can. Note how many are familiar to you, how many you've seen but cannot name, and how many are unfamiliar. A key to the image follows.
[[File:Bartools2.jpg|border|center]]
Answer key: (1) champagne bottle stopper, (2) kitchen knife, (3) ice tongs, (4) ice scoop, (5) ice bucket, (6) small bar spoon, (7) cocktail-pick, (8) jigger, (9) mesh strainer, (10) boston shaker (metal bottom), (11) bar spoon, (12) lime/lemon squeezer, (13) hawthorn strainer, (14) zester, (15) boston shaker (mixing glass), (16) muddler, (17) citrus reamer, (18) fine grater, (19) Y-peeler, (20) wine key.

How'd you do? Some of these items you'll see more often than others. I want to draw special attention these crucial pieces:

* '''Jigger''' (8) - A small double sided measuring device used to quickly portion spirits and other ingredients. Though the exact measurements may change depending on where you are and the manufacturer, in the US most jiggers measure 1.5 oz on one end and 0.75 oz on the other.
* '''Juicer and Peeler''' (12 &amp; 19) - Many cocktails rely on citrus for flavor and fragrance. The handheld juicer and the peeler are commonly used both at the start of the shift to prep the citrus for the day, collecting fresh citrus juice and peels for adding to cocktails.
* '''The Boston Shaker''' (10 &amp; 15) - This is the industry standard for fast, versatile and accurate cocktail mixing. Coming in two parts, often one side is glass and the other is metal, though often both are metal. The two parts are sealed together allowing for a cocktail to be shaker vigorously inside before being cracked apart without spilling.
* '''The Hawthorn Strainer''' (23) - The final piece to any Boston Shaker. This uniquely shaped strainer allows for cocktails to be poured into a glass while leaving the ice in the shaker.

These are tools you will see every time you step in to bar to make a cocktail.

== Safety ==
A good bartender is the difference between a safe time for your bar patrons and an unsafe environment. It is your responsibility to keep both yourself and your bar patrons safe. It is important first to understand how alcohol affects the body in order to understand how to properly make and serve drinks, so first let's review ABV.

Alcohol by Volume is a percentage of ethanol to all other contents of a drink. However, ABV does not tell you everything about drink. A standard beer has about 5% ABV while a shot of whiskey has about 40% ABV, but because you would only drink about 1.5 oz of whiskey, and would drink 12 oz beer, the total amount of alcohol ingested would actually be about the same. This is the idea behind a &quot;standard drink&quot;. A standard drink contains one &quot;unit&quot; of alcohol regardless of how much liquid it contains in total. The chart below shows some examples of standard drink equivalents. Each of the below drinks contain one unit of alcohol.
[[File:NIH_standard_drink_comparison.jpg|center|frame|National Institute of Health's &quot;standard drink&quot;.]]
Some general best practices when consuming alcohol:

* One unit of alcohol per hour is around what a liver can process. This is generally a safer cadence of drinking.
* Drink a glass of water for every unit of alcohol consumed to stay hydrated.
* Avoid drinking on an empty stomach. Eating while consuming alcohol slows the alcohol's absorption into the blood and curbs its effects.
* Don't order a double. Ordering and serving drinks with more than one standard unit of alcohol in it can cause you or others to become more impaired more quickly than intended. Try to keep it to the cadence of one standard drink per hour.

Standards should be upheld both with your own drinking and the drinking of the patrons at your bar. Some standards to uphold in your bar are:

* '''Never drink and drive'''. Alcohol impairs motor coordination and clouds judgement. Drinking and then driving puts your own life and the lives of everyone around you in immediate danger. Even if you are under the legal limit in your area, you are impaired after one drink. This should be enforced both for yourself and for your bar patrons.
* '''You can say no'''. It's always okay to say no to a drink. Ordering something without alcohol is always an option. Good friends don't pressure you to drink when you don't want to. You can leave a situation when you are uncomfortable.
* '''Alcohol is not medicine'''. Addiction is real and treatable. Help is available.
* '''Abuse isn't acceptable'''. You are not a punching bag. It is not bad hospitality to stand up for yourself.

Sometimes people drink more than they intended and may become too inebriated to be served. Some signs that this is the case are:

* Slurred speech
* Stumbling
* Glazed over or unfocused eyes
* Speaking too loudly or two softly
* Repeating questions or ordering from multiple bartenders

If you meet a bar patron who has had too much to drink, you can help by offering water, tea, coffee or food, helping them find their group, or asking another member of the staff at your bar for help. It may be that what's needed is to no longer serve the bar patron. This is called &quot;cutting them off&quot;. When a bar patron is cut off, they will no longer be served alcohol for the rest of the service. This can be embarrassing, both for the bar patron and the bartender, so it's best practice to be kind, clear and discrete. If a patron is aggressive however, all bets are off and they should be removed from the bar.

Bartenders have legal obligations depending on where you work. This can include checking the legal age of the patrons at your bar and monitoring the alcohol intake of the patrons at your bar. Sometimes, a bartender can be held individually liable for the failure to meet these obligations. Take the time to find a reputable source to discover what the legal obligation of bartenders in your area are now.

This concludes Lesson 1: Bartending Basics. You can move on now to [[Bartending/Mixing Drinks|Lesson 2: Mixing Drinks]].

# [[Bartending/Bartending Basics|Bartending Basics]]
# [[Bartending/Mixing Drinks|Mixing Drinks]]
# [[Bartending/Hospitality|Hospitality]]</text>
      <sha1>9itdyyu0lwkqmnck57j45ahjsxsn85k</sha1>
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    <revision>
      <id>2819822</id>
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      <contributor>
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      <text bytes="24201" sha1="gtdqpiqgariuta0chkru10sg293zoak" xml:space="preserve">= Bartending Basics =
What is a [[wikipedia:Bartender|bartender]]? At its simplest, a bartender serves drinks at a bar. In reality, a bartender selects, mixes, pours, and serves drinks while hosting patrons and creating a hospitable atmosphere at a bar, pub, restaurant, nightclub, living room or [[wikipedia:Parking_lot|parking lot]]. A bartender must have a variety of skills, both technical and interpersonal in order to be successful. A bartender can work in many locations, but this course will focus on bartenders that tend to tend in bars.

This lesson covers:

# [[Bartending/Bartending Basics#Sources of Alcohol|Sources of Alcohol]]
# [[Bartending/Bartending Basics#Types of Alcohol|Types of Alcohol]]
# [[Bartending/Bartending Basics#Tools of Bartending|Tools of Bartending]]
# [[Bartending/Bartending Basics#Safety|Safety]]

== Sources of Alcohol ==
Behind the bar, you are the expert in the room on alcohol. While no one can claim to know everything on alcohol, it's important to have the basics. This and the next lesson focus on learning what alcohol is and what kind of drinks and liquors are made with it.

Every alcoholic drink you've ever had has included the same basic type of alcohol: [[wikipedia:Ethanol|Ethanol]]. Ethanol is one of three types of alcohol and the only alcohol humans can safely drink. All alcoholic beverages whether it be beer, wine or a spirit all contain Ethanol.

There are hundreds of different beverages and liquors that are served at bars around the world.  The method of refining raw ingredients into a final product with Ethanol is what distinguishes each type of beverage. Fortunately they're all related to each other (since they all contain Ethanol) and have some major categories to guide us in understanding what they are. The first step to make any type of alcohol is [[wikipedia:fermentation|fermentation]].

=== Fermentation ===
[[File:40168_2022_1274_Fig6.webp|thumb|People get very scientific with it.]]
Using a process that has existed for thousands of years, we can employ a fungus called [[wikipedia:Yeast#Uses|yeast]] to create alcohol. This process is called fermentation. At it base, fermentation for alcohol production is the process of using yeast to convert sugars to ethanol.

&quot;Sugars&quot; is intentionally plural. There are many places you can find sugar suitable for fermentation, and you can derive sugar from grains, fruits and vegetables. The source of the sugar is often what defines what type of final alcoholic beverage you get. We'll talk more on how fermenting different materials yields different drinks in a moment.

Fermentation happens in a couple of steps.

# '''Mashing''': Grains like barley or rice are milled into a coarse flour and fruits are mashed into a pulp. These are sometimes mixed with hot water, where enzymes convert starches to sugars. For beer, this mash is then often boiled.
# '''Fermentation''': The mash is transferred to a fermentation vessel. Yeast is added, and fermentation begins. Over a varying period of time (often 5-10 days), yeast consumes the sugars, producing alcohol and carbon dioxide. This is called primary fermentation.
# '''Conditioning and Packaging''': After primary fermentation, there is often secondary fermentation or conditioning to add or mature flavors and to clarify. It’s then sometimes carbonated and packaged.

Now fermented, there is some amount of alcohol in your drink. But what if ''some'' isn't enough? A secondary process called [[wikipedia:Distillation|distillation]] allowed for fermented drinks and solutions to be concentrated into spirits.

=== Distillation ===
[[File:Alambins_industrials_per_a_la_destil·lació_de_licors_a_Catalunya.jpg|left|thumb|Industrial stills for liquor production in Catalonia]]
Distillation is a process by which brewers can concentrate the level of alcohol in a drink. Fermentation alone cannot make alcohol concentrations like that of vodka or gin. To reach that level of concentration, [[wiktionary:distiller|distillers]] (referring to both the apparatus that distills and the person that runs it) can boil off and collect the alcohol produced by fermentation. Distillation happens in a few steps:

# The fermented substance is placed into a vessel called a [[wikipedia:Still|still]] (this is where di''still''ation gets its name).
# The ferment is heated slowly from room temperature until it reaches the boiling point of ethanol. The boiling point of ethanol is lower than the boiling point of water, so the ethanol will vaporize, leaving the water behind.
# The ethanol vapor rises through a column until it reaches the condenser. In the condenser, the vapor makes contact with the a cold surface that cools it back into a liquid state.
# The ethanol is collected, and the process is often repeated to further concentrate and purify the result. Sometimes the process is done without having to stop and restart in a process called continuous distillation.

The result of distillation is a liquor with a higher amount of alcohol by volume (ABV) than before.

=== Alcohol By Volume ===
The amount of alcohol in a drink is calculated based on the percent of ethanol compared to non-ethanol in the total beverage, which is called Alcohol by Volume or ABV for short. For example, if a beverage has an ABV of 50%, half of liquid in the drink will be ethanol and the other half will be everything else.

ABV is important to understand, because it is the basis of drinking and serving drinks safely as well as how many drinks are categorized. While all alcohol is made through fermentation, the process of fermentation alone can only create a concentration of alcohol of about 5% to 20% ABV. Spirits are generally distilled liquids that have a higher ABV (20% or more, as high as 95%). This distinction lower-ABV drinks like wine, beer and cider versus higher-ABV spirits like vodka, gin and whiskey.

== Types of Alcohol ==
Fermentation and distilling are the core of all alcohol production, but there many more ways that brewers and distillers affect their final product. This incredible flow chart shows many of the processes of fermentation, distillation, carbonation, ageing and processing that make different alcoholic drinks. Take a look at it and compare the different starting ingredients with each other, and then compare the starting ingredients with their final products.
[[File:Alcohol_Flow_Chart.svg|center|frame|Alcohol Flow Chart]]
There is so much to learn about alcohol, and while you should be knowledgeable on the different types, you by no means need to be an expert (see: sommelier) to be a bartender. Below follows an overview of each family of alcohol, but if you want to learn more you can click on the links to Wikipedia pages in each subsection below, or I recommend browsing the [[wikipedia:Portal:Liquor|Liquor Portal]] to view all that Wikipedia has to offer.

Hopefully it's easier to see now how beverages and spirits can both be categorized based on what they're fermented from and how concentrated their alcohol content is. Let's use these categories to talk more about each major type of alcohol.

=== Wine, Beer and Cider (Low-ABV Drinks) ===

==== Wine ====
We call fermented grape juice [[wikipedia:wine|wine]]. Yeast is added to pressed grapes to develop it into one of four types: [[wikipedia:Red_wine|red]], [[wikipedia:white_wine|white]], [[wikipedia:rose_wine|rosé]], and [[wikipedia:Orange_wine|orange]]. The type of grape and the duration of the contact with the skins of the grapes help to define which type of wine is produced. The table below is from the Wikipedia page on wine.
{| class=&quot;wikitable&quot;
|+Colors of wine
!
!Long contact with grape skins
!Short contact with grape skins
|-
!Red grapes
|'''''Red wine''''', made from dark-colored red grape varieties. The actual color of the wine can range from dark pink to almost black. The juice from red grapes is actually pale gray; the color of red wine and some of its flavor (notably tannins) comes from phenolics in the skin, seeds and stem fragments of the grape, extracted by allowing the grapes to soak in the juice.
|'''''Rosé wine''''', which gains color from red grape skins, but not enough to qualify it as a red wine. The color can range from a very pale pink to pale red.
There are two primary ways to produce rosé wine. The preferred technique is allowing a short period of maceration after crushing red grapes, which extracts a certain amount of color. The juice is then fermented like a white wine. An alternative is blending a small amount of finished red wine into finished white wine.
|-
!White grapes
|'''''Orange wine''''', sometimes called amber wine, is made with white grapes but with the skins allowed to macerate during and beyond fermentation, similar to red wine production. This results in their darker color compared to white wines, and produces a deliberately astringent result.
|'''''White wine''''', typically made from white grape varieties (those with yellow or green skins), and range from practically colorless to golden. When skin contact is used, to improve the flavor or to increase the body or aging potential, it is usually limited to between four and 24 hours; any longer leads to bitterness.
|}
''Common names: There are thousands of wine varietals. Red wines like Cabernet Sauvignon, Merlot, Pinot Noir, Malbec, Zinfandel; white wines like Chardonnay, Sauvignon Blanc, Pinot Grigio, Riesling; and rosé wines like Provence, White Zinfandel and Pink Moscato.''

In addition to the colors of wine, wine can be [[wikipedia:Sparkling_wine|sparkling]] or still. Wine can also be [[wikipedia:Fortified_wine|fortified]]. Fortifying a wine is the process of adding a distilled spirit to the fermented wine.

''Common names: Sparkling wine includes Champagne, Prosecco and other Brut wines. Fortified wines include Port, Sherry and Vermouth.''

==== Fruit Wine, Cider and Perry ====
[[wikipedia:Fruit_wine|Fruit wine]] is mostly what it sounds like: wine made from fruits other than grapes. A variety of fruits can be used, some common ones include cherry, plum, dandelion and pineapple. Fruit wine is far less common than traditional grape wines, and is often called by the fruit it's fermented from (ex. &quot;cherry wine&quot; or &quot;dandelion wine&quot;).

[[wikipedia:Cider|Cider]] is a beverage made specifically from fermenting apples. Despite being a sort of wine fermented from fruit, it isn't considered a fruit wine due to its unique cultural history as a beverage. Its name can cause some confusion particularly in the United States and Canada, where &quot;cider&quot; also refers to unfiltered and sometimes spiced apple juice that has not been fermented. The term &quot;hard cider&quot; is sometimes used to distinguish the alcoholic beverage due to this overlap. Cider can also be carbonated to make a sparkling cider.

[[wikipedia:Perry|Perry]], or Pear Cider, is another fruit wine exception. Fermented pears create a drink called perry, that though technically is a wine made from fruit, is considered its own thing.

''Common names: There are thousands of brands of fruit wine, cider and perry. You may have some on tap or bottled; it's mostly important to know the names of what you have.''
[[File:Beer_in_glasses_and_steins_on_a_table_with_bottles_in_the_background_and_a_brick_wall_(15700131777).jpg|thumb|264x264px|There are many varieties of beer.]]

==== Beer ====
[[wikipedia:beer|Beer]] is an ancient drink that has been brewed around the world for hundreds of years. Most beers have four basic ingredients: grain, hops, yeast and water. [[wikipedia:Hops|Hops]] are a type of dried flower that give beer flavor. While there are thousands of brands of beer, there are only a few main types:

* '''Lagers''' - One of the two main types of beer, with Ales. Lighter and crisp, refreshing, less bitter than Ale.
** Amber Lager - A medium, amber color with a slight bitterness. ''Common Names: Yuengling''
** Pale Lager - Light in color and flavor, not very bitter. Easy drinking and very popular. ''Common Names: Red Stripe, Heineken, Pabst Blue Ribbon (PBR), Hamms''
** Pilsner - Medium amber in color. Hoppy and flavorful. ''Common Names: Modelo Especial, Stella Artois, Pilsner Urquell''

* '''Ales''' - The other main type of beer, with Lager. Richer and flavorful, more bitter than Lager.
** Pale Ale - A diverse group of beer that various depending on its country of origin. Generally balanced taste that pairs with many foods. ''Common Names: Sierra Nevada, Boulevard''
** Indian Pale Ale (IPA) - Often very bitter and fruity. A unique tasting beer with bite. ''Common Names: Voodoo Ranger, many other large brands have an IPA''
** Stouts and Porters - Dark beers that have rich, roasted flavors often imitating chocolate or coffee. These beers can feel like a meal. ''Common Names: Guinness, Imperial Stouts, Breakfast Stouts''
** Wheat Beer - Ales that have some wheat added to them. It can give them a fruity, almost banana like flavor. ''Common Names: Blue Moon''
* Radlers and Shandys - Beers that have been mixed with a fruit juice. Radlers can be any type of citrus while Shandys are always lemon juice. Very refreshing, summary drinks.
* Sour Beer - Beer that's, well, sour. Tangy and punchy, usually served in a smaller &quot;sour beer&quot; glass since they pack a punch.
* Light Beer - Any beer with an especially low ABV is a light beer (less than 4% ABV), though the term is sometimes also used for low-calorie beers. ''Common Names: Bud Light, Coors Light''

==== Sake ====
[[wikipedia:sake|Sake]] is a drink brewed from rice, and is sometimes also called rice wine though it is brewed more similarly to beer. Like beer and wine, sake has a wide range of flavor profiles though it is often fruity and light. It is sometimes served lightly warmed in ceramic cups depending on the variety of sake.

''Common Names: Dassai, Hakutsuru, Kubota''

=== Spirits (High-ABV Drinks) ===
[[File:Alcoholic_beverages.jpg|thumb|High-ABV spirits like vodka and brandy.]]

==== Vodka ====
[[wikipedia:vodka|Vodka]] is a spirit that can be distilled from many sources including grains, potatoes and sugarcane before the pure ethanol is diluted with water. It is generally a neutral spirit, meaning it has little taste and is mostly pure ethanol and water. This isn't entirely true though, as each vodka has its own impurities and subtle flavor profiles. Vodka is also commonly flavored, as its neutral taste allows it to take on flavor easily. It can served up, often freezer chilled, or mixed into many cocktails.

''Common Names: Absolut, Smirnoff, Kettle One''

==== Gin ====
[[wikipedia:Gin|Gin]] is a spirit flavored with [[wikipedia:juniper_berries|juniper berries]] and other botanicals. It has a distinct herbal flavor, and different brands will have their own flavor profile, and perceived dryness.
[[File:Agave_tequilana_2.jpg|thumb|Agave is a plant native to Mexico.]]
''Common Names: Beefeater, Tanqueray, Hendrick's, Plymouth''

==== Mezcal (Tequila) ====
[[wikipedia:mezcal|Mezcal]] is a spirit distilled from Agave. The most common type of mezcal is [[wikipedia:tequila|Tequila]], which is distilled only from [[wikipedia:blue_agave|blue agave]]. Many people mistake mezcal for a type of tequila due to tequila's popularity, but it is the other way around. Mezcal and tequila are often aged, which impart a darker, more amber color the longer it sits. Mezcal often has a smokey flavor and can also be sweet, fruity or earthy, and can be enjoyed chilled, up or in many popular cocktails.

''Common Names: Casamigos, Patrón, Jose Cuervo''

==== Rum ====
[[wikipedia:rum|Rum]] is a spirit distilled from [[wikipedia:sugarcane|sugarcane]] then aged in barrels. The less-aged rum has a lighter color, and is called &quot;light rum&quot;. It is most commonly used in cocktails, like the Mojito and the Daiquiri. &quot;Aged&quot; or &quot;Dark&quot; rum has a much deeper flavor is often drank straight or on the rocks. Rum originates from and maintains strong ties to the history of the people of the [[wikipedia:caribbean|Caribbean]] and appears commonly in cultural cuisines as well as drinks. I encourage you to learn about the history of rum, as it can teach a lot about sugarcane as a [[wikipedia:Sugar_plantations_in_the_Caribbean|plantation crop]] and the[[wikipedia:Atlantic_slave_trade|Transatlantic Slave Trade]].

''Common Names: Baccardi, Captain Morgan, Malibu''

==== Brandy ====
[[wikipedia:Brandy|Brandy]] is made by distilling wine. While most brandy is made from grapes, it can be made from any fruit to yield a [[wikipedia:Fruit_brandy|fruit brandy]].

''Common names: Cognac and Armagnac''

==== Whiskey ====
[[wikipedia:whiskey|Whiskey]] is made by fermenting, distilling then aging in barrels various grains. Whiskey drinkers can be particular, as there are many types of whiskey. Some whiskey is &quot;malted&quot;, meaning it uses grains that have been malted. Malting is the process of allowing grains to germinate (begin to sprout), before heating them to halt the germination. This yields &quot;malt whiskey&quot;. The specific grain used also yields different types of whiskey, for example fermenting rye yields &quot;rye whiskey&quot;. &quot;[[wikipedia:Scotch_whisky|Scotch]]&quot; is simply whiskey from Scotland and is also called Scotch Whiskey.

''Common Names: Jameson, Jack Daniel's, Crown Royale''

==== Soju ====
[[wikipedia:soju|Soju]] is a made by distilling rice or other grains. Similarly to vodka, it is a neutral spirit that is most flavorless. It is often flavored with fruits and florals. It can be served up or mixed into cocktails.

''Common Names: Jinro''

==== Liqueurs ====
[[wikipedia:liqueur|Liqueurs]] are spirits that have been heavily flavored and sweetened. They are used very frequently in cocktails to impart their particular flavor. Liqueurs can be used to add a flavor that would otherwise be troublesome, like coffee, liquorish, or almond. Sometimes they're served up as a dessert or a [[wikipedia:digestif|digestif]].

''Common Names: Kahlua, Aperol, Campari, Amaretto, Cointreau, and many, many more''

== Tools of Bartending ==
You'll be using both common and specialty tools as a bartender.  A collection of such tools are below. Start by trying to name as many of them as you can. Note how many are familiar to you, how many you've seen but cannot name, and how many are unfamiliar. A key to the image follows.
[[File:Bartools2.jpg|border|center]]
Answer key: (1) champagne bottle stopper, (2) kitchen knife, (3) ice tongs, (4) ice scoop, (5) ice bucket, (6) small bar spoon, (7) cocktail-pick, (8) jigger, (9) mesh strainer, (10) boston shaker (metal bottom), (11) bar spoon, (12) lime/lemon squeezer, (13) hawthorn strainer, (14) zester, (15) boston shaker (mixing glass), (16) muddler, (17) citrus reamer, (18) fine grater, (19) Y-peeler, (20) wine key.

How'd you do? Some of these items you'll see more often than others. I want to draw special attention these crucial pieces:

* '''Jigger''' (8) - A small double sided measuring device used to quickly portion spirits and other ingredients. Though the exact measurements may change depending on where you are and the manufacturer, in the US most jiggers measure 1.5 oz on one end and 0.75 oz on the other.
* '''Juicer and Peeler''' (12 &amp; 19) - Many cocktails rely on citrus for flavor and fragrance. The handheld juicer and the peeler are commonly used both at the start of the shift to prep the citrus for the day, collecting fresh citrus juice and peels for adding to cocktails.
* '''The Boston Shaker''' (10 &amp; 15) - This is the industry standard for fast, versatile and accurate cocktail mixing. Coming in two parts, often one side is glass and the other is metal, though often both are metal. The two parts are sealed together allowing for a cocktail to be shaker vigorously inside before being cracked apart without spilling.
* '''The Hawthorn Strainer''' (23) - The final piece to any Boston Shaker. This uniquely shaped strainer allows for cocktails to be poured into a glass while leaving the ice in the shaker.

These are tools you will see every time you step in to bar to make a cocktail.

== Safety ==
A good bartender is the difference between a safe time for your bar patrons and an unsafe environment. It is your responsibility to keep both yourself and your bar patrons safe. It is important first to understand how alcohol affects the body in order to understand how to properly make and serve drinks, so first let's review ABV.

Alcohol by Volume is a percentage of ethanol to all other contents of a drink. However, ABV does not tell you everything about drink. A standard beer has about 5% ABV while a shot of whiskey has about 40% ABV, but because you would only drink about 1.5 oz of whiskey, and would drink 12 oz beer, the total amount of alcohol ingested would actually be about the same. This is the idea behind a &quot;standard drink&quot;. A standard drink contains one &quot;unit&quot; of alcohol regardless of how much liquid it contains in total. The chart below shows some examples of standard drink equivalents. Each of the below drinks contain one unit of alcohol.
[[File:NIH_standard_drink_comparison.jpg|center|frame|National Institute of Health's &quot;standard drink&quot;.]]
Some general best practices when consuming alcohol:

* One unit of alcohol per hour is around what a liver can process. This is generally a safer cadence of drinking.
* Drink a glass of water for every unit of alcohol consumed to stay hydrated.
* Avoid drinking on an empty stomach. Eating while consuming alcohol slows the alcohol's absorption into the blood and curbs its effects.
* Don't order a double. Ordering and serving drinks with more than one standard unit of alcohol in it can cause you or others to become more impaired more quickly than intended. Try to keep it to the cadence of one standard drink per hour.

Standards should be upheld both with your own drinking and the drinking of the patrons at your bar. Some standards to uphold in your bar are:

* '''Never drink and drive'''. Alcohol impairs motor coordination and clouds judgement. Drinking and then driving puts your own life and the lives of everyone around you in immediate danger. Even if you are under the legal limit in your area, you are impaired after one drink. This should be enforced both for yourself and for your bar patrons.
* '''You can say no'''. It's always okay to say no to a drink. Ordering something without alcohol is always an option. Good friends don't pressure you to drink when you don't want to. You can leave a situation when you are uncomfortable.
* '''Alcohol is not medicine'''. Addiction is real and treatable. Help is available.
* '''Abuse isn't acceptable'''. You are not a punching bag. It is not bad hospitality to stand up for yourself.

Sometimes people drink more than they intended and may become too inebriated to be served. Some signs that this is the case are:

* Slurred speech
* Stumbling
* Glazed over or unfocused eyes
* Speaking too loudly or two softly
* Repeating questions or ordering from multiple bartenders

If you meet a bar patron who has had too much to drink, you can help by offering water, tea, coffee or food, helping them find their group, or asking another member of the staff at your bar for help. It may be that what's needed is to no longer serve the bar patron. This is called &quot;cutting them off&quot;. When a bar patron is cut off, they will no longer be served alcohol for the rest of the service. This can be embarrassing, both for the bar patron and the bartender, so it's best practice to be kind, clear and discrete. If a patron is aggressive however, all bets are off and they should be removed from the bar.

Bartenders have legal obligations depending on where you work. This can include checking the legal age of the patrons at your bar and monitoring the alcohol intake of the patrons at your bar. Sometimes, a bartender can be held individually liable for the failure to meet these obligations. Take the time to find a reputable source to discover what the legal obligation of bartenders in your area are now.

This concludes Lesson 1: Bartending Basics. You can move on now to [[Bartending/Mixing Drinks|Lesson 2: Mixing Drinks]].

[[Category:Professional and vocational studies]]
[[Category:Professions]]
[[Category:Drinks]]</text>
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      <text bytes="34922" sha1="qj5wb4e2bsawpovunli53qpld27xwqa" xml:space="preserve">= Mixing Drinks =
Congratulations! You've made it to the fun part of the course. Mixing and serving drinks are fundamental to bartending, and they're fun to do. In this lesson we'll focus on the most common and most important recipes you'll encounter working at a bar along with the basic techniques and know-how needed to complete them.

In this lesson we'll discuss:

# Glassware
# Typology of cocktails
# Preparation and Skills
# Essential Cocktails

This is the part of the course that really benefits from hands-on practice. When we talk about preparing fruit and mixing cocktails, follow along! It isn't necessary to build an entire bar in your home in order to practice, but having some of the basics and making substitutions when needed will go a long way to making these actions feel natural and easy. If you have a bartending gig in your future, even doing each of these recipes once will put you leagues ahead of your first try being on the clock.

== Glassware ==
Before you can mix a drink, you have to know what glass you're gonna pour it in. Wikipedians have done an excellent job of creating a near comprehensive list of the glasses commonly used to serve alcohol. I've transcluded a navigation box below to all of their pages. Take the time to familiarize yourself with these glasses and their uses by clicking on the pages and digesting the contained information. You don't have to know the history and origin of every glass, just be familiar their shape and general use. The most important ones for this course are:

* Collins
* Highball
* Old fashioned (or Lowball or Rocks)
* Champaign Coupe (or just Coupe)
* Champaign Flute
* Cocktail Glass (or Martini glass)
* Hurricane Glass
* Wine Glass (Red and White)

When we get to cocktails, each recipe will place the drink in a specific type of glass, and you can return here as a reference tool if you need it.{{Navbox|name=Glassware|title=[[:wikipedia:List of glassware|Glassware]]|listclass=hlist|state=expanded|group1=[[:wikipedia:Tumbler (glass)|Tumblers]]|list1=* {{Nowrap|[[File:Collins glass silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Collins glass|Collins glass]]}}
* {{Nowrap|[[File:Granyonyi Stakan Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Faceted glass|Faceted glass]]}}
* {{Nowrap|[[File:Highball glass silhouette.svg|16x28px|link=|class=skin-invert]] [[:wikipedia:Highball glass|Highball glass]]}}
* {{Nowrap|[[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Old fashioned glass|Old fashioned glass]]}}
* {{Nowrap|[[File:Shot glass.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Shot glass|Shot glass]]}}|group2=[[:wikipedia:Beer glassware|Beer glassware]]|list2=* {{Nowrap|[[File:beer stein.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Beer stein|Beer stein]]}}
* {{Nowrap|[[File:Pilsner glass silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Beer glassware#Pilsner glass|Pilsner glass]]}}
* {{Nowrap|[[File:Pint glass.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Pint glass|Pint Glass]]}}
* {{Nowrap|[[File:Pony Glass Silhouette.svg|14x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Pony glass|Pony glass]]}}
* {{Nowrap|[[File:Tankard Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Tankard|Tankard]]}}
* {{Nowrap|[[File:Wheat beer glass silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Wheat beer glass|Wheat beer glass]]}}
* {{Nowrap|[[File:yardglass.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Yard of ale|Yard glass]]}}|group3=[[:wikipedia:Stemware|Stemware]]|list3=* {{Nowrap|[[File:Absinthe Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Absinthiana#Absinthe glass|Absinthe glass]]}}
* {{Nowrap|[[File:Chalice Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Chalice|Chalice]]}}
* {{Nowrap|[[File:Coppa Asti sagoma.svg|24x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Champagne glass#Coupe|Champagne coupe]]}}
* {{Nowrap|[[File:flutesil.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Champagne glass#Flute|Champagne flute]]}}
* {{Nowrap|[[File:Coctail glass.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Cocktail glass|Cocktail glass]]}}
* {{Nowrap|[[File:Glencairn Whisky Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Glencairn whisky glass|Glencairn whisky glass]]}}
* {{Nowrap|[[File:Hurricane Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Hurricane glass|Hurricane glass]]}}
* {{Nowrap|[[File:Margarita Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Margarita#Glass|Margarita glass]]}}
* {{Nowrap|[[File:Alcohol glass nick and nora.svg|24x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Nick &amp; Nora (glass)|Nick &amp; Nora]]}}
* [[:wikipedia:Rummer|Rummer]]
* {{Nowrap|[[File:Sherry Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Wine glass#Sherry glass|Sherry glass]]}}
* {{Nowrap|[[File:brandysnifsi.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Snifter|Snifter]]}}
* {{Nowrap|[[File:wineglass.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Wine glass|Wine glass]]}}}}

== Typology of Cocktails (Cocktail Families) ==
One last thing before we starting preparing garnishes and mixing drinks! I promise! Cocktails and their recipes didn't just appear from nothing in a vacuum. They were developed over time as tastes changed and different ingredients became available or popular. Because of this process, we can group cocktails into broad categories called Cocktail Families. Not all bartenders and mixologists agree on how cocktails should be grouped or how many families there are, but by understanding generally how cocktails relate to each other you can prioritize your learning and remember what goes in which cocktail easier.

The families we're going to talk about are:

* Sour Cocktails
* Ancestral/Aromatic Cocktails
* Lengthened Cocktails
* Frozen Cocktails

This isn't by any means comprehensive, but they'll cover most the cocktails we'll explore shortly.

=== Sour Cocktails ===
[[wikipedia:Sour_(cocktail)|Sour cocktails]] are an old family of cocktails who's drinks that are all, well sour. The template for a sour cocktail is simple:

&lt;u&gt;Shake&lt;/u&gt; with ice: Spirit + Sweetener + Something Sour + Egg (sometimes)

The &quot;Something Sour&quot; is most often a citrus juice like lemon or lime. Many sour cocktails will have &quot;sour&quot; in the name, like a Gin Sour, Rum Sour, Whiskey Sour, or Amaretto Sour. Some other notable sour cocktails are the Daiquiri, Cosmopolitan and Margarita. Note that it doesn't matter what base spirit you use, any base spirit can be used in a sour. A sub-family of sour cocktails replaces the traditional sweetener of simple syrup with a liqueur. These cocktails are called a &quot;Daisy&quot;.

=== Ancestral or Aromatic Cocktails ===
Ancestral or Aromatic Cocktails are spirit-forward drinks that use bitters and sometimes liqueur to give an aromatic taste. The template to an aromatic cocktail is:

&lt;u&gt;Stir&lt;/u&gt; with ice: Spirit + Sweetener + Bitters

These cocktails let the taste of the spirit shine while deepening the flavor with bitters. Some common aromatic cocktails are the Martini, Manhattan, Old Fashioned, Sazerac and Negroni.

=== Lengthened Cocktails ===
Lengthened Cocktails are drinks that have been &quot;lengthened&quot; by adding a carbonated beverage. Two major sub-families of lengthened cocktails are &quot;Highballs&quot; and &quot;Spritz&quot; cocktails. A Highball is just a spirit lengthened with non-alcoholic mixer, think Gin and Tonic or Rum and Coke. A Spritz is a wine lengthened with a mixer, most often soda or sparkling wine. An example you'll find more in Spain is the Kalimotxo (KAL-EE-MO-KO), a drink with Red Wine and Coca Cola. Don't knock it 'till you try it! There are hundreds of combinations to make Lengthened Cocktails, Highballs and Spritz are just the major subtypes.

=== Frozen Cocktails ===
Frozen drinks are frozen! They are just so unlike a non-frozen drink that they get their own family. Frozen drinks are either blended with ice or churned over frozen metal to make a slushy consistency. Many frozen cocktails were adapted from classic cocktails, like a Frozen Daiquiri, but others are always frozen, like a Piña Colada. These are often &quot;batched&quot; or made ahead of time, especially if they're going to come from a slushy machine.

== Preparation and Skills ==
In a bar, preparation is all the steps that are taken before a service starts to get ready in advance. Sometimes this is things that are done at close the night before, and sometimes these are things done in the hours before a service. Some examples of prep are:

* Stocking fridges with beer and wine
* Stocking the bar's backstock of spirits
* Juicing citrus
* Cutting garnishes for drinks
* Making syrups
* Topping up bottles
* Cleaning and preparing tools for the service

All of these make a good bartender. We'll talk more about how good prep is good hospitality, and how it defines a good bartender later. Some of these are self explanatory, but some merit further investigation to make you a more effective and efficient bartender.

While you go through this section, I recommend you actually follow along and prepare the ingredients listed ''especially the lemons and limes''. Immediately following prep, we'll start mixing cocktails that use these ingredients. Store them in an air tight, labeled and dated container if you plan on mixing some drinks within the next day or two. Cut citrus doesn't last that long, so don't overkeep it. You can always get more practice by preparing more garnishes. A lemon wedge can be used for a dozen other things anyway (I like one with my Coke).
[[File:Hellingshoek_antoineren_02.png|left|thumb|241x241px|This isn't me. I don't know who this man is. Thanks to him for uploading this picture though. Curl your fingers away from the knife.]]

=== Before you prep ===
Most preparation can and should be done with a small paring knife. A larger chef knife can be used for large fruits but is harder to maneuver especially when working with small, round fruits and fragile herbs. When cutting something with a knife, curl the fingers of your non-dominant hand away from the blade. This will help protect your fingertips and cut faster with less risk.

You should always have clean hands when you handle food. Before touching food, wash your hands with soap and water for at least 20 seconds, and dry them with a clean towel. Rewash to your hands whenever you step away from your task or touch something dirty like your phone.

=== Juicing Citrus ===
[[File:Citrus_×_aurantium_-_fruits_cut.jpg|thumb|The lengthwise cut (left) is unsuitable for juicing. The widthwise cut (right) is perfect to juice.]]
Most bars will have one or two types of juicer: a handheld citrus squeezer and/or a countertop citrus reamer. The handheld squeezer is faster and more efficient but cannot fit larger citrus like grapefruit. There are few wrongs ways to juice citrus, but doing it well will save you a lot of time during the prep. There are a few things to keep in mind when juicing citrus at the bar:

# Start with room temperature citrus. Cold citrus will yield less juice. If you have a particularly hard lime or lemon, place it on the counter and put your palm on top. Roll the citrus back and forth while applying firm pressure. This can free up some juice.
# Cut the fruit widthwise down the middle, so that you have two even sections. The cut face of the citrus should look like a wagon wheel with dot of pith in the center.
# Place your fruit cut-side down in the squeezer. Squeeze the citrus through a mesh strainer to remove any seeds or pulp. Squeeze with strength from your arms, not your wrists to avoid injury over time.
# Always label and date fresh juice and store in a sealed container in a refrigerator (Ex. Lime Juice Date: 2/27).

=== Making Simple Syrup ===
Simple syrup is an appropriately named combination of sugar and water made by combining a ratio of 1 parts granulated sugar and 1 parts water. It's a common ingredient used to sweeten cocktails by pre-dissolving the sugar. Here's the recipe:

# Place your sealable storage container on a scale and tare the scale so that it shows zero.
# Add a quantity of white granulated sugar to your container and note the measurement (ex. 200 grams of sugar).
# Add an equal amount of room temperature water (ex. 200 grams of water, making a total measurement of 400 grams).
# Seal the container and shake the mixture until the sugar has dissolved. It may be cloudy, but it with clarify as the mixture settles.
# Label and date your syrup (ex. Simple Syrup Date: 8/29).

If you're preparing this at home and don't have a scale (you're probably American), you can measure by volume, but it will be less accurate and may affect the final result. In a bar, always measure by weight.

=== Preparing Garnishes ===
There are so many things that can be put in drinks to improve their overall look and taste. There are a few, however, that you will find at most bars. If your bar has a specialty garnish, they should train you on how to prepare the house garnishes. For fruit garnishes, they should be used the same day they're cut and discarded at the end of the service. Let's go over some garnishes you'll find just about anywhere.
[[File:Knife_by_Bob_Kramer_(17970647591).jpg|thumb|How beautiful [[File:Face-kiss.svg|20x20px]]]]

==== Cutting Citrus Half-Moons ====

# Cut the fruit in half lengthwise pole-to-pole. The cut-side of the fruit should have a stripe of pith down the center.
# Place the halves cut-side down, so that the nubs on each end are facing to your left and right.
# Cutting straight ahead, make slices about a quarter inch thick.
# Store in a sealed, labeled and dated container in a refrigerator until needed.

==== Cutting Citrus Wedges ====

# Trim the top and bottom of the fruit to create two flat surfaces. The flat surfaces should look like a wagon wheel with a white dot of pith in the center, or may be entirely pith. This step can be skipped for grapefruit, as they are large enough to rest on a side without additional cuts.
# Rest the fruit on its end, and cut it in half lengthwise. This should reveal a stripe of pith down the center of the cut face.
# Looking at the cut-side, make a shallow cut perpendicular to the pith extending about a half inch either side. Cut through about half of the meat of the fruit, but not all the way through to the peel.
# Placing the fruit cut-side down, cut the half further into quarters, then angling your knife, cut those fourths in half as well. Smaller fruits may only yield three wedges per half fruit.
# Store in a sealed, labeled and dated container in a refrigerator until needed.

==== Cutting Citrus Wheels ====

# Trim the nub off one end of the fruit. The cut should be deep enough to reveal a small wagon wheel shape with a dot of pith in the center.
# With the cut side facing to your left or right depending on your dominant hand, slice straight away from you creating about quarter inch thick circles until you no longer have enough fruit to safely hold while cutting.
# You many way to score a slit in the wheels depending on if you are putting them on the rim of a glass.
# Store in a sealed, labeled and dated container in a refrigerator until needed.

==== Cutting Citrus Peels and Twists ====

# Holding the fruit in your non-dominant hand, drag a y-peeler starting away from you and pulling towards you.
# If peeling an orange or grapefruit, using a paring knife, trim away excess pith from the back of the peel.
# Optionally, using a paring knife, trim the edges of the peel to make straight even sides.

==== Cutting Grapefruit &quot;Horses Neck&quot; ====

# Holding the grapefruit in your non-dominant hand, use a y-peeler to peel around the circumference of the fruit creating a long peel.
# Optionally, trim the ends with a paring knife for a cleaner look.

==== Preparing Pineapple Wedges ====

# Using a large chef knife, cut off the top of the pineapple.
# Resting on its bottom with the cut-side up, cut the pineapple in half from top to bottom.
# Resting the pineapple on its cut-side, angle your knife and cut the fruit into three equal parts creating triangle shapes.
# Take each section and slice into about inch thick wedges.
# You may way to score the point of each wedge to be able to hang it on a glass.
# Store in a sealed, labeled and dated container in a refrigerator until needed.

==== Preparing Mint ====

# Place a damp paper towel in the bottom of a sealable storage container.
# Take a sprig of mint in one hand, and pick the leaves at the bottom of the stem. Leave the ones at the top of the sprig attached to the stem. Place the leaves in the prepared container.
# Bunch the stems of the picked sprigs together and trim them to be only a couple inches long.

=== Essential Skills ===
There are a couple techniques that are used over and over in bartending. The two most essential to mix cocktails are shaking and stirring. Refer back to these steps the first few times you mix a cocktail in the next section.
[[File:Bartender_Photo.jpg|thumb|Shaking a sealed Boston shaker horizontally.]]

==== Using a Boston Shaker ====
A Boston shaker comes in two parts. One will be a large metal mixing cup, and the other will either be a smaller metal mixing cup or a glass mixing cup. Using a shaker comes in a few steps:

# We will start by building a shaken cocktail in the small or glass mixing cup. Add the liquid ingredients (including egg, if being used), then add the ice after. The longer the liquids are in contact with the ice, the more they will be diluted as the ice melts.
# The large mixing cup is placed over top the smaller mixing cup with enough force to create a seal. Feel free to tap the back of the large tin with the palm of your hand to ensure a seal.
# With one hand on either cup holding them together, the assembled cups are then flipped over so that the drink is primarily in the larger mixing cup.
# The most efficient way to shake is horizontally, so that the drinks sloshes vigorously from front to back. Shaking vertically will achieve the same result, but it is more energy intensive and slower as it throws the drink upwards before slamming it back down. You will feel the metal become cold, and it may frost as the drink is chilled. These are signs that you can stop shaking.
# At this point, the cold from the ice has caused the metal tins to contract and shrink slightly which will bind them together tightly. To unseal the shaker, hold the larger cup or place it on a counter. Identify which was the smaller cup is leaning. With the palm of your hand, strike the smaller cup away from the direction it's leaning. There should be a crack as the seal is broken (this is called &quot;cracking&quot; a shaker open). You can also twist or squeeze the larger tin to release the seal if it's made of metal.
# To strain, place the Hawthorn strainer over the mouth of the larger cup, and pour the drink through the strainer.

You can watch an example of [https://www.youtube.com/watch?v=ai70a6xjSzo someone using a Boston shaker] on YouTube (not my video), if you'd like more instruction.
[[File:Stirring_with_a_bar_spoon.jpg|thumb|230x230px|Stirring with a Bar Spoon]]

==== Stirring a Cocktail ====
It sounds simple, but there's a right way and a wrong way to stir a cocktail. The goal is to incorporate and chill the ingredients without agitating, decarbonating or introducing air. Here are a few simple steps:

# Insert a bar spoon to the bottom of the mixing glass containing you ingredients and ice. Push the spoon all the way to the bottom edge of the mixing glass.
# Swirl the spoon so that it stays in contact with the edge of the glass, rotating the ice in a circle.

==== Serving Sparking Wine ====

# Use the tab to peel away the foil.
# Keep one hand on top of the cage and cork. Lift the handle and twist it open.
# Keep the cage on the cork. Hold the cage and the cork with your dominant hand.
# Holding the bottle in your non-dominant hand, twist the bottle back and forth. The goal is to gradually release the pressure in the bottle, not to make a loud &quot;pop&quot;. Ideally, you should open the bottle almost silently.
# Holding the bottle at the base with one hand (if possible), pour slowly into the glass as the bubbles will climb.

[[File:Pouring_wine_into_a_decanter.jpg|left|thumb|296x296px|Pouring wine]]

==== Uncorking Still Wine ====

# Find a wine key with a double-hinge.
# Using the knife, cut the foil around the lip of the bottle. Using the bottle opener end of the key, peel back to foil to expose the cork.
# Insert the corkscrew into the center of the cork by twisting until most of the screw is embedded.
# Place the first hinge on the lip of the bottle, and use it as leverage to lift up on the wine key (lifting the cork partially).
# Place the second hinge of the lip of the bottle, and completely remove the cork.

'''Serving Beer'''

# Whether pouring from a bottle or a tap, tilt the glass 45° and pour the beer down the side of glass to prevent too much foam, or &quot;head&quot;, from developing.
# At the end of the pour, straighten the glass to allow for some head to form.

== Essential Cocktails ==
Here we are, finally mixing some drinks! It can be overwhelming to open a book of cocktail recipes and see the hundreds of drinks with unique names and recipes. Take a deep breath, because there isn't a bartender in the world that knows all of those cocktails by heart, and you aren't expected to either. Over time, you'll pick up on more drinks and their makeup, but that takes time! That's why it's important to focus on the essential cocktails that every bartender knows and that any bar patron might order. What follows are 20 cocktail recipes for classic cocktails that you might make every day at a bar.

I highly encourage you to make these drinks at home. If you intend to become a bartender, the difference between your first attempt at a recipe being in the privacy of your home and on your first day on the job is ''enormous''. That raises the question, &quot;How am I supposed to supply all the ingredients needed to make all these drinks‽&quot; Start with things you can buy at the grocery store:

* Lemons, Limes and Oranges
* Club Soda
* White Sugar

You may already have these things from trying out the preparation techniques above. If you don't, go out and grab these ingredients, cut some garnishes, and prepare some simple syrup.

The most expensive part of any cocktail is the spirit. I've color coded the following recipes by which spirit they use. The list generally starts with easy recipes and becomes more complicated as you descend. I recommend staring with one type of spirit and trying a few recipes that use that spirit. For example, with only vodka, you can make a Highball, Cosmopolitan, Moscow Mule, Bloody Mary and a (Vodka) Martini. That's one fifth of the list done, with a spirit you likely already have at home.

I don't recommend buying fancy or expensive spirits for this. Buy something cheap or middle-of-the-road to practice with. That's like what your bar patrons are going to be ordering most often anyhow.

All that being said, 20 recipes is a lot to learn. Take your time, maybe make one or two a day. The recipes aren't changing, and some are very simple. You by no means have to drink what you make. You DO have to TASTE what you make. Bartenders, when trying a drink on the job, will smell a drink, swirl it in their mouth and spit it out. Do this at least twice for each drink if you don't intend to drink them. Take note of the taste, the balance of the drink and your personal preference for or against it. Please don't make all 20 in one night and get hammered. A daiquiri will be a daiquiri tomorrow, and it'll be sweeter for the waiting.

Well, get started!

=== Spirit Color Key ===
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Vodka
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Gin
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Rum
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Tequila
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Whiskey
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Other/Mixed
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=== Highball ===
Highballs are a class of cocktail sometimes called &quot;plus one&quot; cocktails because they're a blend of a spirit plus a mixer. Some common highballs are Gin and Tonic, Rum and Coke, Whiskey and Ginger Ale, and Scotch and Soda.{{RoundBoxTop|theme=11}} 
'''Glass:''' Highball [[File:Highball glass silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Wedges with soda, Lime Wedges with Tonic, Cola and Ginger Beer/Ale

'''Ingredients'''

2 oz of Spirit

Mixer of choice, to top

'''Combine''' Spirit and ice. Stir. Add Mixer to fill. Drag spoon through drink a few times to gently combine. Garnish
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=== Cosmopolitan ===
{{RoundBoxTop|theme=1}} 
'''Glass:''' Coupe or Martini Nowrap|[[File:Coctail glass.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Twist

'''Ingredients'''

1 ½ oz Vodka (Preferably Citron Vodka)

⅓ oz Lime Juice

⅓ oz Cointreau (or other orange liqueur)

⅓ oz Sweet Cranberry Juice

'''Shake''' all ingredients with ice until chilled. Strain into glass. Garnish. 
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=== Margarita ===
{{RoundBoxTop|theme=4}} 
'''Glass:''' Rocks [[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lime Wedge and ask preference for Salt, Sugar, or Tajin rim

'''Ingredients'''

2 oz Tequila

½ oz Agave Nectar

1 oz Lime Juice

½ oz Triple Sec

'''Moisten''' the rim of the glass with a lime wedge, then roll exterior of glass in rimming material careful to avoid getting any inside the glass. Combine all ingredients in shaker with ice. Shake. Add ice to glass. Strain. Garnish.
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=== Paloma ===
{{RoundBoxTop|theme=4}} 
'''Glass:''' Highball [[File:Highball glass silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Grapefruit Wedge or Lime Wheel

'''Ingredients'''

2 oz Tequila

½ oz Lime Juice

Dash of Salt

Grapefruit Soda, to top

'''Combine''' Tequila, Sale and Lime Juice in glass with ice. Stir to chill. Add Grapefruit Soda to fill. Drag spoon through drink a few times to gently combine. Garnish.
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=== Whiskey Sour ===
{{RoundBoxTop|theme=5}} 
'''Glass:''' Sour or Coupe [[File:Coppa Asti sagoma.svg|24x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Wheel and Cherry

'''Ingredients'''

2 oz Whiskey

1 oz Lemon Juice

½ tsp. Sugar or Simple Syrup

'''Shake''' ingredients with ice. Strain. Garnish.
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=== Moscow Mule ===
{{RoundBoxTop|theme=1}} 
'''Glass:''' Mule Mug [[File:Alcohol_glass_copper_mug.svg|24x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lime wheel (optional)

'''Ingredients'''

2 oz Vodka

½ oz Lime Juice

3 oz Ginger Beer

'''Combine''' ingredients in a copper mug filled with ice. Stir. Garnish, optionally.
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=== Spritz ===
Spritz are a class of cocktails that follow the same general 3-2-1 recipe of 3 oz Prosecco, 2 oz liqueur, 1 oz club soda. The most common is the Aperol Spritz.{{RoundBoxTop|theme=11}} 
'''Glass:''' Highball or Wine Glass [[File:wineglass.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Orange Wheel

'''Ingredients'''

3 oz Prosecco

2 oz Liqueur (Aperol, for an Aperol Spritz)

1 oz Club Soda

'''Combine''' Prosecco and Liqueur with ice in glass. Stir gently. Top with Club Soda. Garnish.
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=== Mojito ===
{{RoundBoxTop|theme=3}} 
'''Glass:''' Highball [[File:Highball glass silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Mint Sprig

'''Ingredients'''

1 ½ oz White Rum

5 to 6 mint leaves

1 oz Lime Juice

2 tsp Sugar

Soda water

'''Muddle''' the Mint, Lime Juice and Sugar until well mixed but not mushy. Add the Rum and fill the glass with ice. Top with Soda Water. Garnish.
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=== Daiquiri ===
{{RoundBoxTop|theme=3}} 
'''Glass:''' Coupe [[File:Coppa Asti sagoma.svg|24x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lime wheel (optional)

'''Ingredients'''

1 ½ oz White Rum

2 tsp Sugar

1 oz Fresh Lime Juice

'''Shake''' ingredients with ice until chilled and frothy. Strain into a coupe glass. Garnish, optionally.
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=== French 75 ===
{{RoundBoxTop|theme=2}} 
'''Glass:''' Champagne Flute [[File:flutesil.svg|24x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Twist

'''Ingredients'''

1 oz Gin

½ oz Lemon Juice

½ oz Simple Syrup

3 oz Champagne (or other sparkling wine)

'''Shake''' Gin, Lemon Juice, and Simple Syrup with ice. Strain into glass. Top with Champagne. Garnish.
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=== Gimlet ===
{{RoundBoxTop|theme=2}} 
'''Glass:''' Coupe [[File:Coppa Asti sagoma.svg|24x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lime Wheel

'''Ingredients'''

2 ½ oz Gin

½ oz Lime Juice

½ oz Simple Syrup

'''Shake''' Gin, Lime Juice, and Simple Syrup with ice. Strain into glass. Garnish.
{{RoundBoxBottom}}

=== Manhattan ===
{{RoundBoxTop|theme=5}} 
'''Glass:''' Rocks [[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Twist and/or Cherry

'''Ingredients'''

1 ½ oz Rye Whiskey

½ oz Sweet Vermouth

1 Dash Angostura Bitters

'''Chill''' rocks glass by combining ice and water, or use pre-chilled glass. Combine ingredients in a mixer with ice. Stir. Strain. Garnish.
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=== Mai Tai ===
{{RoundBoxTop|theme=3}} 
'''Glass:''' Rocks [[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Pineapple Wedge, Cherry, and Mint Spring

'''Ingredients'''

1 oz Light Rum

1 oz Gold Rum

½ oz Orange Curaçao

½ oz Orgeat

½ oz Lime Juice

'''Shake''' ingredients with ice. Strain into ice filled glass. Garnish.
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=== Pina Colada ===
{{RoundBoxTop|theme=3}} 
'''Glass:''' Hurricane [[File:Hurricane Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Pineapple Wedge

'''Ingredients'''

1 ½ oz Rum (Preferably Coconut, like Malibu)

1 ½ oz Coconut Cream

1 oz Pineapple Juice

½ oz Lime Juice

'''Combine''' ingredients with ice in a blender. Blend on high for 20-30 seconds or until a flowing slushy consistency. If too stiff add water, if too thin add ice and re-blend for 2-3 seconds. Pour into glass. Garnish.
{{RoundBoxBottom}}

=== Bloody Mary ===
{{RoundBoxTop|theme=1}}
'''Glass:''' Pint or Highball [[File:Highball glass silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Wheel or Celery Stalk (or anything savory)

'''Ingredients'''

2 oz Vodka

6 dashes Worcestershire sauce

3 dashes Tobasco Sauce

Pinch of Salt

Pinch of Ground Black Pepper

½ Lemon, juiced

5 oz Tomato Juice

'''Combine''' all ingredients in a glass. Add ice and stir. Garnish.
{{RoundBoxBottom}}

=== Negroni ===
{{RoundBoxTop|theme=2}}
'''Glass:''' Rocks [[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Orange Twist or Orange Wheel

'''Ingredients'''

1 oz Dry Gin

1 oz Sweet Vermouth

1 oz Compari

1 oz Sparkling Water

'''Combine''' all ingredients in a glass. Add ice, preferably 2 large rocks. Stir. Garnish.

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=== Old Fashioned ===
{{RoundBoxTop|theme=5}} 
'''Glass:''' Rocks [[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Orange twist

'''Ingredients'''

2 oz Whiskey

1 Sugar Cube

4 dashes Aromatic Bitters

1 dash Soda Water

'''Muddle''' Sugar, Bitters and Soda Water in a glass until a uniform syrup is made. Add Whiskey. Add Ice. Stir. Express Orange Twist over drink then add to glass.
{{RoundBoxBottom}}

=== Tom Collins ===
{{RoundBoxTop|theme=2}} 
'''Glass:''' Collins or Highball [[File:Collins glass silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Wheel and Cherry (optional)

'''Ingredients'''

2 oz Gin

1 oz Lemon Juice

½ oz Simple Syrup

Club Soda, to top

'''Combine''' Gin, Lemon Juice, and Simple Syrup in glass. Add ice to top. Fill with Club Soda. Stir gently. Garnish.
{{RoundBoxBottom}}

=== Martini ===
Martinis are kind of their own thing, and there's some vocab to go over. There are many drinks that call themselves a martini, and the Classic Martini itself has many variations to suit each bar patron's taste. To make a Martini to order, first you have to have a conversation with your bar patron.

# First, ask if they prefer Gin or Vodka. Gin is traditional, but it can be made with either.
# Second determine how much Vermouth they would like. The amount of Vermouth will determine how &quot;Dry&quot; or &quot;Wet&quot; the Martini is.

&quot;Extra Dry&quot; or &quot;In and Out&quot; = Vermouth is swirled in the glass then poured out, then the drink is entirely Gin or Vodka

&quot;Dry&quot; = 1 part Vermouth to 5 parts Gin or Vodka

&quot;Wet&quot; = 1 part Vermouth to 2 parts Gin or Vodka

&quot;50/50&quot; = 1 part Vermouth to 1 part Gin or Vodka

&quot;Dirty&quot; = 1 part Olive Brine to 5 parts Gin or Vodka, instead of Vermouth. Garnish with Olives instead of lemon.

The following recipe is for a Dry Martini. You can modify it to make it a Dirty Martini or to suit your taste of Vermouth. Martinis are often stirred, but they can be shaken on request.
{{RoundBoxTop|theme=11}} 
'''Glass:''' Martini [[File:Coctail glass.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon twist

'''Ingredients'''

2 oz Gin or Vodka (patron preference)

½ oz Dry Vermouth

'''Chill''' glassware by pouring water and ice in a glass, or use a pre-chilled glass. Combine ingredients with ice in a mixing glass. Stir until chilled. Strain into chilled glass. Garnish.
{{RoundBoxBottom}}

=== Espresso Martini ===
The Espresso Martini is really not much like a Classic Martini at all. The word Martini really gets thrown around. Nonetheless, it's delicious. If your bar has an espresso machine, you'll be trained on how to pull an espresso shot. Otherwise, your bar may use cold brew concentrate made in house or from a can.{{RoundBoxTop|theme=1}} 
'''Glass:''' Martini [[File:Coctail glass.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' 3 Coffee Beans

'''Ingredients'''

2 oz Vodka

½ oz Coffee Liqueur (often Kahlúa)

1 oz Espresso, or cold brew concentrate

1/4 oz Simple Syrup

'''Chill''' glass with water and ice, or use pre-chilled glass. Shake ingredients with ice. Strain into glass. Garnish by place 3 coffee beans in center.
{{RoundBoxBottom}}

Hey! Congrats! You've made it to the bottom of the list. These 20 recipes are just the beginning of the possibilities when it comes to mixing drinks, but if you can nail these down you can get through just about any shift at any bar. That was an undertaking, and you did a great job.

This concludes Lesson 2: Mixing Drinks. You can move on now to Lesson 3: Hospitality.</text>
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      <text bytes="35160" sha1="mirokfmbmx36dn0hacrcebx400vz6p2" xml:space="preserve">= Mixing Drinks =
Congratulations! You've made it to the fun part of the course. Mixing and serving drinks are fundamental to bartending, and they're fun to do. In this lesson we'll focus on the most common and most important recipes you'll encounter working at a bar along with the basic techniques and know-how needed to complete them.

In this lesson we'll discuss:

# [[Bartending/Mixing Drinks#Glassware|Glassware]]
# [[Bartending/Mixing Drinks#Typology of Cocktails (Cocktail Families)|Typology of cocktails]]
# [[Bartending/Mixing Drinks#Preparation and Skills|Preparation and Skills]]
# [[Bartending/Mixing Drinks#Essential Cocktails|Essential Cocktails]]

This is the part of the course that really benefits from hands-on practice. When we talk about preparing fruit and mixing cocktails, follow along! It isn't necessary to build an entire bar in your home in order to practice, but having some of the basics and making substitutions when needed will go a long way to making these actions feel natural and easy. If you have a bartending gig in your future, even doing each of these recipes once will put you leagues ahead of your first try being on the clock.

== Glassware ==
Before you can mix a drink, you have to know what glass you're gonna pour it in. Wikipedians have done an excellent job of creating a near comprehensive list of the glasses commonly used to serve alcohol. I've transcluded a navigation box below to all of their pages. Take the time to familiarize yourself with these glasses and their uses by clicking on the pages and digesting the contained information. You don't have to know the history and origin of every glass, just be familiar their shape and general use. The most important ones for this course are:

* Collins
* Highball
* Old fashioned (or Lowball or Rocks)
* Champaign Coupe (or just Coupe)
* Champaign Flute
* Cocktail Glass (or Martini glass)
* Hurricane Glass
* Wine Glass (Red and White)

When we get to cocktails, each recipe will place the drink in a specific type of glass, and you can return here as a reference tool if you need it.{{Navbox|name=Glassware|title=[[:wikipedia:List of glassware|Glassware]]|listclass=hlist|state=expanded|group1=[[:wikipedia:Tumbler (glass)|Tumblers]]|list1=* {{Nowrap|[[File:Collins glass silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Collins glass|Collins glass]]}}
* {{Nowrap|[[File:Granyonyi Stakan Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Faceted glass|Faceted glass]]}}
* {{Nowrap|[[File:Highball glass silhouette.svg|16x28px|link=|class=skin-invert]] [[:wikipedia:Highball glass|Highball glass]]}}
* {{Nowrap|[[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Old fashioned glass|Old fashioned glass]]}}
* {{Nowrap|[[File:Shot glass.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Shot glass|Shot glass]]}}|group2=[[:wikipedia:Beer glassware|Beer glassware]]|list2=* {{Nowrap|[[File:beer stein.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Beer stein|Beer stein]]}}
* {{Nowrap|[[File:Pilsner glass silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Beer glassware#Pilsner glass|Pilsner glass]]}}
* {{Nowrap|[[File:Pint glass.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Pint glass|Pint Glass]]}}
* {{Nowrap|[[File:Pony Glass Silhouette.svg|14x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Pony glass|Pony glass]]}}
* {{Nowrap|[[File:Tankard Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Tankard|Tankard]]}}
* {{Nowrap|[[File:Wheat beer glass silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Wheat beer glass|Wheat beer glass]]}}
* {{Nowrap|[[File:yardglass.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Yard of ale|Yard glass]]}}|group3=[[:wikipedia:Stemware|Stemware]]|list3=* {{Nowrap|[[File:Absinthe Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Absinthiana#Absinthe glass|Absinthe glass]]}}
* {{Nowrap|[[File:Chalice Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Chalice|Chalice]]}}
* {{Nowrap|[[File:Coppa Asti sagoma.svg|24x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Champagne glass#Coupe|Champagne coupe]]}}
* {{Nowrap|[[File:flutesil.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Champagne glass#Flute|Champagne flute]]}}
* {{Nowrap|[[File:Coctail glass.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Cocktail glass|Cocktail glass]]}}
* {{Nowrap|[[File:Glencairn Whisky Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Glencairn whisky glass|Glencairn whisky glass]]}}
* {{Nowrap|[[File:Hurricane Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Hurricane glass|Hurricane glass]]}}
* {{Nowrap|[[File:Margarita Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Margarita#Glass|Margarita glass]]}}
* {{Nowrap|[[File:Alcohol glass nick and nora.svg|24x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Nick &amp; Nora (glass)|Nick &amp; Nora]]}}
* [[:wikipedia:Rummer|Rummer]]
* {{Nowrap|[[File:Sherry Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Wine glass#Sherry glass|Sherry glass]]}}
* {{Nowrap|[[File:brandysnifsi.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Snifter|Snifter]]}}
* {{Nowrap|[[File:wineglass.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Wine glass|Wine glass]]}}}}

== Typology of Cocktails (Cocktail Families) ==
One last thing before we starting preparing garnishes and mixing drinks! I promise! Cocktails and their recipes didn't just appear from nothing in a vacuum. They were developed over time as tastes changed and different ingredients became available or popular. Because of this process, we can group cocktails into broad categories called Cocktail Families. Not all bartenders and mixologists agree on how cocktails should be grouped or how many families there are, but by understanding generally how cocktails relate to each other you can prioritize your learning and remember what goes in which cocktail easier.

The families we're going to talk about are:

* Sour Cocktails
* Ancestral/Aromatic Cocktails
* Lengthened Cocktails
* Frozen Cocktails

This isn't by any means comprehensive, but they'll cover most the cocktails we'll explore shortly.

=== Sour Cocktails ===
[[wikipedia:Sour_(cocktail)|Sour cocktails]] are an old family of cocktails who's drinks that are all, well sour. The template for a sour cocktail is simple:

&lt;u&gt;Shake&lt;/u&gt; with ice: Spirit + Sweetener + Something Sour + Egg (sometimes)

The &quot;Something Sour&quot; is most often a citrus juice like lemon or lime. Many sour cocktails will have &quot;sour&quot; in the name, like a Gin Sour, Rum Sour, Whiskey Sour, or Amaretto Sour. Some other notable sour cocktails are the Daiquiri, Cosmopolitan and Margarita. Note that it doesn't matter what base spirit you use, any base spirit can be used in a sour. A sub-family of sour cocktails replaces the traditional sweetener of simple syrup with a liqueur. These cocktails are called a &quot;Daisy&quot;.

=== Ancestral or Aromatic Cocktails ===
Ancestral or Aromatic Cocktails are spirit-forward drinks that use bitters and sometimes liqueur to give an aromatic taste. The template to an aromatic cocktail is:

&lt;u&gt;Stir&lt;/u&gt; with ice: Spirit + Sweetener + Bitters

These cocktails let the taste of the spirit shine while deepening the flavor with bitters. Some common aromatic cocktails are the Martini, Manhattan, Old Fashioned, Sazerac and Negroni.

=== Lengthened Cocktails ===
Lengthened Cocktails are drinks that have been &quot;lengthened&quot; by adding a carbonated beverage. Two major sub-families of lengthened cocktails are &quot;Highballs&quot; and &quot;Spritz&quot; cocktails. A Highball is just a spirit lengthened with non-alcoholic mixer, think Gin and Tonic or Rum and Coke. A Spritz is a wine lengthened with a mixer, most often soda or sparkling wine. An example you'll find more in Spain is the Kalimotxo (KAL-EE-MO-KO), a drink with Red Wine and Coca Cola. Don't knock it 'till you try it! There are hundreds of combinations to make Lengthened Cocktails, Highballs and Spritz are just the major subtypes.

=== Frozen Cocktails ===
Frozen drinks are frozen! They are just so unlike a non-frozen drink that they get their own family. Frozen drinks are either blended with ice or churned over frozen metal to make a slushy consistency. Many frozen cocktails were adapted from classic cocktails, like a Frozen Daiquiri, but others are always frozen, like a Piña Colada. These are often &quot;batched&quot; or made ahead of time, especially if they're going to come from a slushy machine.

== Preparation and Skills ==
In a bar, preparation is all the steps that are taken before a service starts to get ready in advance. Sometimes this is things that are done at close the night before, and sometimes these are things done in the hours before a service. Some examples of prep are:

* Stocking fridges with beer and wine
* Stocking the bar's backstock of spirits
* Juicing citrus
* Cutting garnishes for drinks
* Making syrups
* Topping up bottles
* Cleaning and preparing tools for the service

All of these make a good bartender. We'll talk more about how good prep is good hospitality, and how it defines a good bartender later. Some of these are self explanatory, but some merit further investigation to make you a more effective and efficient bartender.

While you go through this section, I recommend you actually follow along and prepare the ingredients listed ''especially the lemons and limes''. Immediately following prep, we'll start mixing cocktails that use these ingredients. Store them in an air tight, labeled and dated container if you plan on mixing some drinks within the next day or two. Cut citrus doesn't last that long, so don't overkeep it. You can always get more practice by preparing more garnishes. A lemon wedge can be used for a dozen other things anyway (I like one with my Coke).
[[File:Hellingshoek_antoineren_02.png|left|thumb|241x241px|This isn't me. I don't know who this man is. Thanks to him for uploading this picture though. Curl your fingers away from the knife.]]

=== Before you prep ===
Most preparation can and should be done with a small paring knife. A larger chef knife can be used for large fruits but is harder to maneuver especially when working with small, round fruits and fragile herbs. When cutting something with a knife, curl the fingers of your non-dominant hand away from the blade. This will help protect your fingertips and cut faster with less risk.

You should always have clean hands when you handle food. Before touching food, wash your hands with soap and water for at least 20 seconds, and dry them with a clean towel. Rewash to your hands whenever you step away from your task or touch something dirty like your phone.

=== Juicing Citrus ===
[[File:Citrus_×_aurantium_-_fruits_cut.jpg|thumb|The lengthwise cut (left) is unsuitable for juicing. The widthwise cut (right) is perfect to juice.]]
Most bars will have one or two types of juicer: a handheld citrus squeezer and/or a countertop citrus reamer. The handheld squeezer is faster and more efficient but cannot fit larger citrus like grapefruit. There are few wrongs ways to juice citrus, but doing it well will save you a lot of time during the prep. There are a few things to keep in mind when juicing citrus at the bar:

# Start with room temperature citrus. Cold citrus will yield less juice. If you have a particularly hard lime or lemon, place it on the counter and put your palm on top. Roll the citrus back and forth while applying firm pressure. This can free up some juice.
# Cut the fruit widthwise down the middle, so that you have two even sections. The cut face of the citrus should look like a wagon wheel with dot of pith in the center.
# Place your fruit cut-side down in the squeezer. Squeeze the citrus through a mesh strainer to remove any seeds or pulp. Squeeze with strength from your arms, not your wrists to avoid injury over time.
# Always label and date fresh juice and store in a sealed container in a refrigerator (Ex. Lime Juice Date: 2/27).

=== Making Simple Syrup ===
Simple syrup is an appropriately named combination of sugar and water made by combining a ratio of 1 parts granulated sugar and 1 parts water. It's a common ingredient used to sweeten cocktails by pre-dissolving the sugar. Here's the recipe:

# Place your sealable storage container on a scale and tare the scale so that it shows zero.
# Add a quantity of white granulated sugar to your container and note the measurement (ex. 200 grams of sugar).
# Add an equal amount of room temperature water (ex. 200 grams of water, making a total measurement of 400 grams).
# Seal the container and shake the mixture until the sugar has dissolved. It may be cloudy, but it with clarify as the mixture settles.
# Label and date your syrup (ex. Simple Syrup Date: 8/29).

If you're preparing this at home and don't have a scale (you're probably American), you can measure by volume, but it will be less accurate and may affect the final result. In a bar, always measure by weight.

=== Preparing Garnishes ===
There are so many things that can be put in drinks to improve their overall look and taste. There are a few, however, that you will find at most bars. If your bar has a specialty garnish, they should train you on how to prepare the house garnishes. For fruit garnishes, they should be used the same day they're cut and discarded at the end of the service. Let's go over some garnishes you'll find just about anywhere.
[[File:Knife_by_Bob_Kramer_(17970647591).jpg|thumb|How beautiful [[File:Face-kiss.svg|20x20px]]]]

==== Cutting Citrus Half-Moons ====

# Cut the fruit in half lengthwise pole-to-pole. The cut-side of the fruit should have a stripe of pith down the center.
# Place the halves cut-side down, so that the nubs on each end are facing to your left and right.
# Cutting straight ahead, make slices about a quarter inch thick.
# Store in a sealed, labeled and dated container in a refrigerator until needed.

==== Cutting Citrus Wedges ====

# Trim the top and bottom of the fruit to create two flat surfaces. The flat surfaces should look like a wagon wheel with a white dot of pith in the center, or may be entirely pith. This step can be skipped for grapefruit, as they are large enough to rest on a side without additional cuts.
# Rest the fruit on its end, and cut it in half lengthwise. This should reveal a stripe of pith down the center of the cut face.
# Looking at the cut-side, make a shallow cut perpendicular to the pith extending about a half inch either side. Cut through about half of the meat of the fruit, but not all the way through to the peel.
# Placing the fruit cut-side down, cut the half further into quarters, then angling your knife, cut those fourths in half as well. Smaller fruits may only yield three wedges per half fruit.
# Store in a sealed, labeled and dated container in a refrigerator until needed.

==== Cutting Citrus Wheels ====

# Trim the nub off one end of the fruit. The cut should be deep enough to reveal a small wagon wheel shape with a dot of pith in the center.
# With the cut side facing to your left or right depending on your dominant hand, slice straight away from you creating about quarter inch thick circles until you no longer have enough fruit to safely hold while cutting.
# You many way to score a slit in the wheels depending on if you are putting them on the rim of a glass.
# Store in a sealed, labeled and dated container in a refrigerator until needed.

==== Cutting Citrus Peels and Twists ====

# Holding the fruit in your non-dominant hand, drag a y-peeler starting away from you and pulling towards you.
# If peeling an orange or grapefruit, using a paring knife, trim away excess pith from the back of the peel.
# Optionally, using a paring knife, trim the edges of the peel to make straight even sides.

==== Cutting Grapefruit &quot;Horses Neck&quot; ====

# Holding the grapefruit in your non-dominant hand, use a y-peeler to peel around the circumference of the fruit creating a long peel.
# Optionally, trim the ends with a paring knife for a cleaner look.

==== Preparing Pineapple Wedges ====

# Using a large chef knife, cut off the top of the pineapple.
# Resting on its bottom with the cut-side up, cut the pineapple in half from top to bottom.
# Resting the pineapple on its cut-side, angle your knife and cut the fruit into three equal parts creating triangle shapes.
# Take each section and slice into about inch thick wedges.
# You may way to score the point of each wedge to be able to hang it on a glass.
# Store in a sealed, labeled and dated container in a refrigerator until needed.

==== Preparing Mint ====

# Place a damp paper towel in the bottom of a sealable storage container.
# Take a sprig of mint in one hand, and pick the leaves at the bottom of the stem. Leave the ones at the top of the sprig attached to the stem. Place the leaves in the prepared container.
# Bunch the stems of the picked sprigs together and trim them to be only a couple inches long.

=== Essential Skills ===
There are a couple techniques that are used over and over in bartending. The two most essential to mix cocktails are shaking and stirring. Refer back to these steps the first few times you mix a cocktail in the next section.
[[File:Bartender_Photo.jpg|thumb|Shaking a sealed Boston shaker horizontally.]]

==== Using a Boston Shaker ====
A Boston shaker comes in two parts. One will be a large metal mixing cup, and the other will either be a smaller metal mixing cup or a glass mixing cup. Using a shaker comes in a few steps:

# We will start by building a shaken cocktail in the small or glass mixing cup. Add the liquid ingredients (including egg, if being used), then add the ice after. The longer the liquids are in contact with the ice, the more they will be diluted as the ice melts.
# The large mixing cup is placed over top the smaller mixing cup with enough force to create a seal. Feel free to tap the back of the large tin with the palm of your hand to ensure a seal.
# With one hand on either cup holding them together, the assembled cups are then flipped over so that the drink is primarily in the larger mixing cup.
# The most efficient way to shake is horizontally, so that the drinks sloshes vigorously from front to back. Shaking vertically will achieve the same result, but it is more energy intensive and slower as it throws the drink upwards before slamming it back down. You will feel the metal become cold, and it may frost as the drink is chilled. These are signs that you can stop shaking.
# At this point, the cold from the ice has caused the metal tins to contract and shrink slightly which will bind them together tightly. To unseal the shaker, hold the larger cup or place it on a counter. Identify which was the smaller cup is leaning. With the palm of your hand, strike the smaller cup away from the direction it's leaning. There should be a crack as the seal is broken (this is called &quot;cracking&quot; a shaker open). You can also twist or squeeze the larger tin to release the seal if it's made of metal.
# To strain, place the Hawthorn strainer over the mouth of the larger cup, and pour the drink through the strainer.

You can watch an example of [https://www.youtube.com/watch?v=ai70a6xjSzo someone using a Boston shaker] on YouTube (not my video), if you'd like more instruction.
[[File:Stirring_with_a_bar_spoon.jpg|thumb|230x230px|Stirring with a Bar Spoon]]

==== Stirring a Cocktail ====
It sounds simple, but there's a right way and a wrong way to stir a cocktail. The goal is to incorporate and chill the ingredients without agitating, decarbonating or introducing air. Here are a few simple steps:

# Insert a bar spoon to the bottom of the mixing glass containing you ingredients and ice. Push the spoon all the way to the bottom edge of the mixing glass.
# Swirl the spoon so that it stays in contact with the edge of the glass, rotating the ice in a circle.

==== Serving Sparking Wine ====

# Use the tab to peel away the foil.
# Keep one hand on top of the cage and cork. Lift the handle and twist it open.
# Keep the cage on the cork. Hold the cage and the cork with your dominant hand.
# Holding the bottle in your non-dominant hand, twist the bottle back and forth. The goal is to gradually release the pressure in the bottle, not to make a loud &quot;pop&quot;. Ideally, you should open the bottle almost silently.
# Holding the bottle at the base with one hand (if possible), pour slowly into the glass as the bubbles will climb.

[[File:Pouring_wine_into_a_decanter.jpg|left|thumb|296x296px|Pouring wine]]

==== Uncorking Still Wine ====

# Find a wine key with a double-hinge.
# Using the knife, cut the foil around the lip of the bottle. Using the bottle opener end of the key, peel back to foil to expose the cork.
# Insert the corkscrew into the center of the cork by twisting until most of the screw is embedded.
# Place the first hinge on the lip of the bottle, and use it as leverage to lift up on the wine key (lifting the cork partially).
# Place the second hinge of the lip of the bottle, and completely remove the cork.

'''Serving Beer'''

# Whether pouring from a bottle or a tap, tilt the glass 45° and pour the beer down the side of glass to prevent too much foam, or &quot;head&quot;, from developing.
# At the end of the pour, straighten the glass to allow for some head to form.

== Essential Cocktails ==
Here we are, finally mixing some drinks! It can be overwhelming to open a book of cocktail recipes and see the hundreds of drinks with unique names and recipes. Take a deep breath, because there isn't a bartender in the world that knows all of those cocktails by heart, and you aren't expected to either. Over time, you'll pick up on more drinks and their makeup, but that takes time! That's why it's important to focus on the essential cocktails that every bartender knows and that any bar patron might order. What follows are 20 cocktail recipes for classic cocktails that you might make every day at a bar.

I highly encourage you to make these drinks at home. If you intend to become a bartender, the difference between your first attempt at a recipe being in the privacy of your home and on your first day on the job is ''enormous''. That raises the question, &quot;How am I supposed to supply all the ingredients needed to make all these drinks‽&quot; Start with things you can buy at the grocery store:

* Lemons, Limes and Oranges
* Club Soda
* White Sugar

You may already have these things from trying out the preparation techniques above. If you don't, go out and grab these ingredients, cut some garnishes, and prepare some simple syrup.

The most expensive part of any cocktail is the spirit. I've color coded the following recipes by which spirit they use. The list generally starts with easy recipes and becomes more complicated as you descend. I recommend staring with one type of spirit and trying a few recipes that use that spirit. For example, with only vodka, you can make a Highball, Cosmopolitan, Moscow Mule, Bloody Mary and a (Vodka) Martini. That's one fifth of the list done, with a spirit you likely already have at home.

I don't recommend buying fancy or expensive spirits for this. Buy something cheap or middle-of-the-road to practice with. That's like what your bar patrons are going to be ordering most often anyhow.

All that being said, 20 recipes is a lot to learn. Take your time, maybe make one or two a day. The recipes aren't changing, and some are very simple. You by no means have to drink what you make. You DO have to TASTE what you make. Bartenders, when trying a drink on the job, will smell a drink, swirl it in their mouth and spit it out. Do this at least twice for each drink if you don't intend to drink them. Take note of the taste, the balance of the drink and your personal preference for or against it. Please don't make all 20 in one night and get hammered. A daiquiri will be a daiquiri tomorrow, and it'll be sweeter for the waiting.

Well, get started!

=== Spirit Color Key ===
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Vodka
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Gin
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Rum
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Tequila
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Whiskey
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Other/Mixed
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=== Highball ===
Highballs are a class of cocktail sometimes called &quot;plus one&quot; cocktails because they're a blend of a spirit plus a mixer. Some common highballs are Gin and Tonic, Rum and Coke, Whiskey and Ginger Ale, and Scotch and Soda.{{RoundBoxTop|theme=11}} 
'''Glass:''' Highball [[File:Highball glass silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Wedges with soda, Lime Wedges with Tonic, Cola and Ginger Beer/Ale

'''Ingredients'''

2 oz of Spirit

Mixer of choice, to top

'''Combine''' Spirit and ice. Stir. Add Mixer to fill. Drag spoon through drink a few times to gently combine. Garnish
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=== Cosmopolitan ===
{{RoundBoxTop|theme=1}} 
'''Glass:''' Coupe or Martini Nowrap|[[File:Coctail glass.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Twist

'''Ingredients'''

1 ½ oz Vodka (Preferably Citron Vodka)

⅓ oz Lime Juice

⅓ oz Cointreau (or other orange liqueur)

⅓ oz Sweet Cranberry Juice

'''Shake''' all ingredients with ice until chilled. Strain into glass. Garnish. 
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=== Margarita ===
{{RoundBoxTop|theme=4}} 
'''Glass:''' Rocks [[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lime Wedge and ask preference for Salt, Sugar, or Tajin rim

'''Ingredients'''

2 oz Tequila

½ oz Agave Nectar

1 oz Lime Juice

½ oz Triple Sec

'''Moisten''' the rim of the glass with a lime wedge, then roll exterior of glass in rimming material careful to avoid getting any inside the glass. Combine all ingredients in shaker with ice. Shake. Add ice to glass. Strain. Garnish.
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=== Paloma ===
{{RoundBoxTop|theme=4}} 
'''Glass:''' Highball [[File:Highball glass silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Grapefruit Wedge or Lime Wheel

'''Ingredients'''

2 oz Tequila

½ oz Lime Juice

Dash of Salt

Grapefruit Soda, to top

'''Combine''' Tequila, Sale and Lime Juice in glass with ice. Stir to chill. Add Grapefruit Soda to fill. Drag spoon through drink a few times to gently combine. Garnish.
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=== Whiskey Sour ===
{{RoundBoxTop|theme=5}} 
'''Glass:''' Sour or Coupe [[File:Coppa Asti sagoma.svg|24x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Wheel and Cherry

'''Ingredients'''

2 oz Whiskey

1 oz Lemon Juice

½ tsp. Sugar or Simple Syrup

'''Shake''' ingredients with ice. Strain. Garnish.
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=== Moscow Mule ===
{{RoundBoxTop|theme=1}} 
'''Glass:''' Mule Mug [[File:Alcohol_glass_copper_mug.svg|24x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lime wheel (optional)

'''Ingredients'''

2 oz Vodka

½ oz Lime Juice

3 oz Ginger Beer

'''Combine''' ingredients in a copper mug filled with ice. Stir. Garnish, optionally.
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=== Spritz ===
Spritz are a class of cocktails that follow the same general 3-2-1 recipe of 3 oz Prosecco, 2 oz liqueur, 1 oz club soda. The most common is the Aperol Spritz.{{RoundBoxTop|theme=11}} 
'''Glass:''' Highball or Wine Glass [[File:wineglass.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Orange Wheel

'''Ingredients'''

3 oz Prosecco

2 oz Liqueur (Aperol, for an Aperol Spritz)

1 oz Club Soda

'''Combine''' Prosecco and Liqueur with ice in glass. Stir gently. Top with Club Soda. Garnish.
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=== Mojito ===
{{RoundBoxTop|theme=3}} 
'''Glass:''' Highball [[File:Highball glass silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Mint Sprig

'''Ingredients'''

1 ½ oz White Rum

5 to 6 mint leaves

1 oz Lime Juice

2 tsp Sugar

Soda water

'''Muddle''' the Mint, Lime Juice and Sugar until well mixed but not mushy. Add the Rum and fill the glass with ice. Top with Soda Water. Garnish.
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=== Daiquiri ===
{{RoundBoxTop|theme=3}} 
'''Glass:''' Coupe [[File:Coppa Asti sagoma.svg|24x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lime wheel (optional)

'''Ingredients'''

1 ½ oz White Rum

2 tsp Sugar

1 oz Fresh Lime Juice

'''Shake''' ingredients with ice until chilled and frothy. Strain into a coupe glass. Garnish, optionally.
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=== French 75 ===
{{RoundBoxTop|theme=2}} 
'''Glass:''' Champagne Flute [[File:flutesil.svg|24x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Twist

'''Ingredients'''

1 oz Gin

½ oz Lemon Juice

½ oz Simple Syrup

3 oz Champagne (or other sparkling wine)

'''Shake''' Gin, Lemon Juice, and Simple Syrup with ice. Strain into glass. Top with Champagne. Garnish.
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=== Gimlet ===
{{RoundBoxTop|theme=2}} 
'''Glass:''' Coupe [[File:Coppa Asti sagoma.svg|24x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lime Wheel

'''Ingredients'''

2 ½ oz Gin

½ oz Lime Juice

½ oz Simple Syrup

'''Shake''' Gin, Lime Juice, and Simple Syrup with ice. Strain into glass. Garnish.
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=== Manhattan ===
{{RoundBoxTop|theme=5}} 
'''Glass:''' Rocks [[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Twist and/or Cherry

'''Ingredients'''

1 ½ oz Rye Whiskey

½ oz Sweet Vermouth

1 Dash Angostura Bitters

'''Chill''' rocks glass by combining ice and water, or use pre-chilled glass. Combine ingredients in a mixer with ice. Stir. Strain. Garnish.
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=== Mai Tai ===
{{RoundBoxTop|theme=3}} 
'''Glass:''' Rocks [[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Pineapple Wedge, Cherry, and Mint Spring

'''Ingredients'''

1 oz Light Rum

1 oz Gold Rum

½ oz Orange Curaçao

½ oz Orgeat

½ oz Lime Juice

'''Shake''' ingredients with ice. Strain into ice filled glass. Garnish.
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=== Pina Colada ===
{{RoundBoxTop|theme=3}} 
'''Glass:''' Hurricane [[File:Hurricane Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Pineapple Wedge

'''Ingredients'''

1 ½ oz Rum (Preferably Coconut, like Malibu)

1 ½ oz Coconut Cream

1 oz Pineapple Juice

½ oz Lime Juice

'''Combine''' ingredients with ice in a blender. Blend on high for 20-30 seconds or until a flowing slushy consistency. If too stiff add water, if too thin add ice and re-blend for 2-3 seconds. Pour into glass. Garnish.
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=== Bloody Mary ===
{{RoundBoxTop|theme=1}}
'''Glass:''' Pint or Highball [[File:Highball glass silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Wheel or Celery Stalk (or anything savory)

'''Ingredients'''

2 oz Vodka

6 dashes Worcestershire sauce

3 dashes Tobasco Sauce

Pinch of Salt

Pinch of Ground Black Pepper

½ Lemon, juiced

5 oz Tomato Juice

'''Combine''' all ingredients in a glass. Add ice and stir. Garnish.
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=== Negroni ===
{{RoundBoxTop|theme=2}}
'''Glass:''' Rocks [[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Orange Twist or Orange Wheel

'''Ingredients'''

1 oz Dry Gin

1 oz Sweet Vermouth

1 oz Compari

1 oz Sparkling Water

'''Combine''' all ingredients in a glass. Add ice, preferably 2 large rocks. Stir. Garnish.

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=== Old Fashioned ===
{{RoundBoxTop|theme=5}} 
'''Glass:''' Rocks [[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Orange twist

'''Ingredients'''

2 oz Whiskey

1 Sugar Cube

4 dashes Aromatic Bitters

1 dash Soda Water

'''Muddle''' Sugar, Bitters and Soda Water in a glass until a uniform syrup is made. Add Whiskey. Add Ice. Stir. Express Orange Twist over drink then add to glass.
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=== Tom Collins ===
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'''Glass:''' Collins or Highball [[File:Collins glass silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Wheel and Cherry (optional)

'''Ingredients'''

2 oz Gin

1 oz Lemon Juice

½ oz Simple Syrup

Club Soda, to top

'''Combine''' Gin, Lemon Juice, and Simple Syrup in glass. Add ice to top. Fill with Club Soda. Stir gently. Garnish.
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=== Martini ===
Martinis are kind of their own thing, and there's some vocab to go over. There are many drinks that call themselves a martini, and the Classic Martini itself has many variations to suit each bar patron's taste. To make a Martini to order, first you have to have a conversation with your bar patron.

# First, ask if they prefer Gin or Vodka. Gin is traditional, but it can be made with either.
# Second determine how much Vermouth they would like. The amount of Vermouth will determine how &quot;Dry&quot; or &quot;Wet&quot; the Martini is.

&quot;Extra Dry&quot; or &quot;In and Out&quot; = Vermouth is swirled in the glass then poured out, then the drink is entirely Gin or Vodka

&quot;Dry&quot; = 1 part Vermouth to 5 parts Gin or Vodka

&quot;Wet&quot; = 1 part Vermouth to 2 parts Gin or Vodka

&quot;50/50&quot; = 1 part Vermouth to 1 part Gin or Vodka

&quot;Dirty&quot; = 1 part Olive Brine to 5 parts Gin or Vodka, instead of Vermouth. Garnish with Olives instead of lemon.

The following recipe is for a Dry Martini. You can modify it to make it a Dirty Martini or to suit your taste of Vermouth. Martinis are often stirred, but they can be shaken on request.
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'''Glass:''' Martini [[File:Coctail glass.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon twist

'''Ingredients'''

2 oz Gin or Vodka (patron preference)

½ oz Dry Vermouth

'''Chill''' glassware by pouring water and ice in a glass, or use a pre-chilled glass. Combine ingredients with ice in a mixing glass. Stir until chilled. Strain into chilled glass. Garnish.
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=== Espresso Martini ===
The Espresso Martini is really not much like a Classic Martini at all. The word Martini really gets thrown around. Nonetheless, it's delicious. If your bar has an espresso machine, you'll be trained on how to pull an espresso shot. Otherwise, your bar may use cold brew concentrate made in house or from a can.{{RoundBoxTop|theme=1}} 
'''Glass:''' Martini [[File:Coctail glass.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' 3 Coffee Beans

'''Ingredients'''

2 oz Vodka

½ oz Coffee Liqueur (often Kahlúa)

1 oz Espresso, or cold brew concentrate

1/4 oz Simple Syrup

'''Chill''' glass with water and ice, or use pre-chilled glass. Shake ingredients with ice. Strain into glass. Garnish by place 3 coffee beans in center.
{{RoundBoxBottom}}

Hey! Congrats! You've made it to the bottom of the list. These 20 recipes are just the beginning of the possibilities when it comes to mixing drinks, but if you can nail these down you can get through just about any shift at any bar. That was an undertaking, and you did a great job.

This concludes Lesson 2: Mixing Drinks. You can move on now to [[Bartending/Hospitality|Lesson 3: Hospitality]].</text>
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      <text bytes="35255" sha1="g1vwmjrn6fzktk9d4i3wm9q3lkne1d9" xml:space="preserve">= Mixing Drinks =
Congratulations! You've made it to the fun part of the course. Mixing and serving drinks are fundamental to bartending, and they're fun to do. In this lesson we'll focus on the most common and most important recipes you'll encounter working at a bar along with the basic techniques and know-how needed to complete them.

In this lesson we'll discuss:

# [[Bartending/Mixing Drinks#Glassware|Glassware]]
# [[Bartending/Mixing Drinks#Typology of Cocktails (Cocktail Families)|Typology of cocktails]]
# [[Bartending/Mixing Drinks#Preparation and Skills|Preparation and Skills]]
# [[Bartending/Mixing Drinks#Essential Cocktails|Essential Cocktails]]

This is the part of the course that really benefits from hands-on practice. When we talk about preparing fruit and mixing cocktails, follow along! It isn't necessary to build an entire bar in your home in order to practice, but having some of the basics and making substitutions when needed will go a long way to making these actions feel natural and easy. If you have a bartending gig in your future, even doing each of these recipes once will put you leagues ahead of your first try being on the clock.

== Glassware ==
Before you can mix a drink, you have to know what glass you're gonna pour it in. Wikipedians have done an excellent job of creating a near comprehensive list of the glasses commonly used to serve alcohol. I've transcluded a navigation box below to all of their pages. Take the time to familiarize yourself with these glasses and their uses by clicking on the pages and digesting the contained information. You don't have to know the history and origin of every glass, just be familiar their shape and general use. The most important ones for this course are:

* Collins
* Highball
* Old fashioned (or Lowball or Rocks)
* Champaign Coupe (or just Coupe)
* Champaign Flute
* Cocktail Glass (or Martini glass)
* Hurricane Glass
* Wine Glass (Red and White)

When we get to cocktails, each recipe will place the drink in a specific type of glass, and you can return here as a reference tool if you need it.{{Navbox|name=Glassware|title=[[:wikipedia:List of glassware|Glassware]]|listclass=hlist|state=expanded|group1=[[:wikipedia:Tumbler (glass)|Tumblers]]|list1=* {{Nowrap|[[File:Collins glass silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Collins glass|Collins glass]]}}
* {{Nowrap|[[File:Granyonyi Stakan Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Faceted glass|Faceted glass]]}}
* {{Nowrap|[[File:Highball glass silhouette.svg|16x28px|link=|class=skin-invert]] [[:wikipedia:Highball glass|Highball glass]]}}
* {{Nowrap|[[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Old fashioned glass|Old fashioned glass]]}}
* {{Nowrap|[[File:Shot glass.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Shot glass|Shot glass]]}}|group2=[[:wikipedia:Beer glassware|Beer glassware]]|list2=* {{Nowrap|[[File:beer stein.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Beer stein|Beer stein]]}}
* {{Nowrap|[[File:Pilsner glass silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Beer glassware#Pilsner glass|Pilsner glass]]}}
* {{Nowrap|[[File:Pint glass.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Pint glass|Pint Glass]]}}
* {{Nowrap|[[File:Pony Glass Silhouette.svg|14x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Pony glass|Pony glass]]}}
* {{Nowrap|[[File:Tankard Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Tankard|Tankard]]}}
* {{Nowrap|[[File:Wheat beer glass silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Wheat beer glass|Wheat beer glass]]}}
* {{Nowrap|[[File:yardglass.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Yard of ale|Yard glass]]}}|group3=[[:wikipedia:Stemware|Stemware]]|list3=* {{Nowrap|[[File:Absinthe Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Absinthiana#Absinthe glass|Absinthe glass]]}}
* {{Nowrap|[[File:Chalice Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Chalice|Chalice]]}}
* {{Nowrap|[[File:Coppa Asti sagoma.svg|24x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Champagne glass#Coupe|Champagne coupe]]}}
* {{Nowrap|[[File:flutesil.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Champagne glass#Flute|Champagne flute]]}}
* {{Nowrap|[[File:Coctail glass.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Cocktail glass|Cocktail glass]]}}
* {{Nowrap|[[File:Glencairn Whisky Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Glencairn whisky glass|Glencairn whisky glass]]}}
* {{Nowrap|[[File:Hurricane Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Hurricane glass|Hurricane glass]]}}
* {{Nowrap|[[File:Margarita Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Margarita#Glass|Margarita glass]]}}
* {{Nowrap|[[File:Alcohol glass nick and nora.svg|24x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Nick &amp; Nora (glass)|Nick &amp; Nora]]}}
* [[:wikipedia:Rummer|Rummer]]
* {{Nowrap|[[File:Sherry Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Wine glass#Sherry glass|Sherry glass]]}}
* {{Nowrap|[[File:brandysnifsi.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Snifter|Snifter]]}}
* {{Nowrap|[[File:wineglass.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Wine glass|Wine glass]]}}}}

== Typology of Cocktails (Cocktail Families) ==
One last thing before we starting preparing garnishes and mixing drinks! I promise! Cocktails and their recipes didn't just appear from nothing in a vacuum. They were developed over time as tastes changed and different ingredients became available or popular. Because of this process, we can group cocktails into broad categories called Cocktail Families. Not all bartenders and mixologists agree on how cocktails should be grouped or how many families there are, but by understanding generally how cocktails relate to each other you can prioritize your learning and remember what goes in which cocktail easier.

The families we're going to talk about are:

* Sour Cocktails
* Ancestral/Aromatic Cocktails
* Lengthened Cocktails
* Frozen Cocktails

This isn't by any means comprehensive, but they'll cover most the cocktails we'll explore shortly.

=== Sour Cocktails ===
[[wikipedia:Sour_(cocktail)|Sour cocktails]] are an old family of cocktails who's drinks that are all, well sour. The template for a sour cocktail is simple:

&lt;u&gt;Shake&lt;/u&gt; with ice: Spirit + Sweetener + Something Sour + Egg (sometimes)

The &quot;Something Sour&quot; is most often a citrus juice like lemon or lime. Many sour cocktails will have &quot;sour&quot; in the name, like a Gin Sour, Rum Sour, Whiskey Sour, or Amaretto Sour. Some other notable sour cocktails are the Daiquiri, Cosmopolitan and Margarita. Note that it doesn't matter what base spirit you use, any base spirit can be used in a sour. A sub-family of sour cocktails replaces the traditional sweetener of simple syrup with a liqueur. These cocktails are called a &quot;Daisy&quot;.

=== Ancestral or Aromatic Cocktails ===
Ancestral or Aromatic Cocktails are spirit-forward drinks that use bitters and sometimes liqueur to give an aromatic taste. The template to an aromatic cocktail is:

&lt;u&gt;Stir&lt;/u&gt; with ice: Spirit + Sweetener + Bitters

These cocktails let the taste of the spirit shine while deepening the flavor with bitters. Some common aromatic cocktails are the Martini, Manhattan, Old Fashioned, Sazerac and Negroni.

=== Lengthened Cocktails ===
Lengthened Cocktails are drinks that have been &quot;lengthened&quot; by adding a carbonated beverage. Two major sub-families of lengthened cocktails are &quot;Highballs&quot; and &quot;Spritz&quot; cocktails. A Highball is just a spirit lengthened with non-alcoholic mixer, think Gin and Tonic or Rum and Coke. A Spritz is a wine lengthened with a mixer, most often soda or sparkling wine. An example you'll find more in Spain is the Kalimotxo (KAL-EE-MO-KO), a drink with Red Wine and Coca Cola. Don't knock it 'till you try it! There are hundreds of combinations to make Lengthened Cocktails, Highballs and Spritz are just the major subtypes.

=== Frozen Cocktails ===
Frozen drinks are frozen! They are just so unlike a non-frozen drink that they get their own family. Frozen drinks are either blended with ice or churned over frozen metal to make a slushy consistency. Many frozen cocktails were adapted from classic cocktails, like a Frozen Daiquiri, but others are always frozen, like a Piña Colada. These are often &quot;batched&quot; or made ahead of time, especially if they're going to come from a slushy machine.

== Preparation and Skills ==
In a bar, preparation is all the steps that are taken before a service starts to get ready in advance. Sometimes this is things that are done at close the night before, and sometimes these are things done in the hours before a service. Some examples of prep are:

* Stocking fridges with beer and wine
* Stocking the bar's backstock of spirits
* Juicing citrus
* Cutting garnishes for drinks
* Making syrups
* Topping up bottles
* Cleaning and preparing tools for the service

All of these make a good bartender. We'll talk more about how good prep is good hospitality, and how it defines a good bartender later. Some of these are self explanatory, but some merit further investigation to make you a more effective and efficient bartender.

While you go through this section, I recommend you actually follow along and prepare the ingredients listed ''especially the lemons and limes''. Immediately following prep, we'll start mixing cocktails that use these ingredients. Store them in an air tight, labeled and dated container if you plan on mixing some drinks within the next day or two. Cut citrus doesn't last that long, so don't overkeep it. You can always get more practice by preparing more garnishes. A lemon wedge can be used for a dozen other things anyway (I like one with my Coke).
[[File:Hellingshoek_antoineren_02.png|left|thumb|241x241px|This isn't me. I don't know who this man is. Thanks to him for uploading this picture though. Curl your fingers away from the knife.]]

=== Before you prep ===
Most preparation can and should be done with a small paring knife. A larger chef knife can be used for large fruits but is harder to maneuver especially when working with small, round fruits and fragile herbs. When cutting something with a knife, curl the fingers of your non-dominant hand away from the blade. This will help protect your fingertips and cut faster with less risk.

You should always have clean hands when you handle food. Before touching food, wash your hands with soap and water for at least 20 seconds, and dry them with a clean towel. Rewash to your hands whenever you step away from your task or touch something dirty like your phone.

=== Juicing Citrus ===
[[File:Citrus_×_aurantium_-_fruits_cut.jpg|thumb|The lengthwise cut (left) is unsuitable for juicing. The widthwise cut (right) is perfect to juice.]]
Most bars will have one or two types of juicer: a handheld citrus squeezer and/or a countertop citrus reamer. The handheld squeezer is faster and more efficient but cannot fit larger citrus like grapefruit. There are few wrongs ways to juice citrus, but doing it well will save you a lot of time during the prep. There are a few things to keep in mind when juicing citrus at the bar:

# Start with room temperature citrus. Cold citrus will yield less juice. If you have a particularly hard lime or lemon, place it on the counter and put your palm on top. Roll the citrus back and forth while applying firm pressure. This can free up some juice.
# Cut the fruit widthwise down the middle, so that you have two even sections. The cut face of the citrus should look like a wagon wheel with dot of pith in the center.
# Place your fruit cut-side down in the squeezer. Squeeze the citrus through a mesh strainer to remove any seeds or pulp. Squeeze with strength from your arms, not your wrists to avoid injury over time.
# Always label and date fresh juice and store in a sealed container in a refrigerator (Ex. Lime Juice Date: 2/27).

=== Making Simple Syrup ===
Simple syrup is an appropriately named combination of sugar and water made by combining a ratio of 1 parts granulated sugar and 1 parts water. It's a common ingredient used to sweeten cocktails by pre-dissolving the sugar. Here's the recipe:

# Place your sealable storage container on a scale and tare the scale so that it shows zero.
# Add a quantity of white granulated sugar to your container and note the measurement (ex. 200 grams of sugar).
# Add an equal amount of room temperature water (ex. 200 grams of water, making a total measurement of 400 grams).
# Seal the container and shake the mixture until the sugar has dissolved. It may be cloudy, but it with clarify as the mixture settles.
# Label and date your syrup (ex. Simple Syrup Date: 8/29).

If you're preparing this at home and don't have a scale (you're probably American), you can measure by volume, but it will be less accurate and may affect the final result. In a bar, always measure by weight.

=== Preparing Garnishes ===
There are so many things that can be put in drinks to improve their overall look and taste. There are a few, however, that you will find at most bars. If your bar has a specialty garnish, they should train you on how to prepare the house garnishes. For fruit garnishes, they should be used the same day they're cut and discarded at the end of the service. Let's go over some garnishes you'll find just about anywhere.
[[File:Knife_by_Bob_Kramer_(17970647591).jpg|thumb|How beautiful [[File:Face-kiss.svg|20x20px]]]]

==== Cutting Citrus Half-Moons ====

# Cut the fruit in half lengthwise pole-to-pole. The cut-side of the fruit should have a stripe of pith down the center.
# Place the halves cut-side down, so that the nubs on each end are facing to your left and right.
# Cutting straight ahead, make slices about a quarter inch thick.
# Store in a sealed, labeled and dated container in a refrigerator until needed.

==== Cutting Citrus Wedges ====

# Trim the top and bottom of the fruit to create two flat surfaces. The flat surfaces should look like a wagon wheel with a white dot of pith in the center, or may be entirely pith. This step can be skipped for grapefruit, as they are large enough to rest on a side without additional cuts.
# Rest the fruit on its end, and cut it in half lengthwise. This should reveal a stripe of pith down the center of the cut face.
# Looking at the cut-side, make a shallow cut perpendicular to the pith extending about a half inch either side. Cut through about half of the meat of the fruit, but not all the way through to the peel.
# Placing the fruit cut-side down, cut the half further into quarters, then angling your knife, cut those fourths in half as well. Smaller fruits may only yield three wedges per half fruit.
# Store in a sealed, labeled and dated container in a refrigerator until needed.

==== Cutting Citrus Wheels ====

# Trim the nub off one end of the fruit. The cut should be deep enough to reveal a small wagon wheel shape with a dot of pith in the center.
# With the cut side facing to your left or right depending on your dominant hand, slice straight away from you creating about quarter inch thick circles until you no longer have enough fruit to safely hold while cutting.
# You many way to score a slit in the wheels depending on if you are putting them on the rim of a glass.
# Store in a sealed, labeled and dated container in a refrigerator until needed.

==== Cutting Citrus Peels and Twists ====

# Holding the fruit in your non-dominant hand, drag a y-peeler starting away from you and pulling towards you.
# If peeling an orange or grapefruit, using a paring knife, trim away excess pith from the back of the peel.
# Optionally, using a paring knife, trim the edges of the peel to make straight even sides.

==== Cutting Grapefruit &quot;Horses Neck&quot; ====

# Holding the grapefruit in your non-dominant hand, use a y-peeler to peel around the circumference of the fruit creating a long peel.
# Optionally, trim the ends with a paring knife for a cleaner look.

==== Preparing Pineapple Wedges ====

# Using a large chef knife, cut off the top of the pineapple.
# Resting on its bottom with the cut-side up, cut the pineapple in half from top to bottom.
# Resting the pineapple on its cut-side, angle your knife and cut the fruit into three equal parts creating triangle shapes.
# Take each section and slice into about inch thick wedges.
# You may way to score the point of each wedge to be able to hang it on a glass.
# Store in a sealed, labeled and dated container in a refrigerator until needed.

==== Preparing Mint ====

# Place a damp paper towel in the bottom of a sealable storage container.
# Take a sprig of mint in one hand, and pick the leaves at the bottom of the stem. Leave the ones at the top of the sprig attached to the stem. Place the leaves in the prepared container.
# Bunch the stems of the picked sprigs together and trim them to be only a couple inches long.

=== Essential Skills ===
There are a couple techniques that are used over and over in bartending. The two most essential to mix cocktails are shaking and stirring. Refer back to these steps the first few times you mix a cocktail in the next section.
[[File:Bartender_Photo.jpg|thumb|Shaking a sealed Boston shaker horizontally.]]

==== Using a Boston Shaker ====
A Boston shaker comes in two parts. One will be a large metal mixing cup, and the other will either be a smaller metal mixing cup or a glass mixing cup. Using a shaker comes in a few steps:

# We will start by building a shaken cocktail in the small or glass mixing cup. Add the liquid ingredients (including egg, if being used), then add the ice after. The longer the liquids are in contact with the ice, the more they will be diluted as the ice melts.
# The large mixing cup is placed over top the smaller mixing cup with enough force to create a seal. Feel free to tap the back of the large tin with the palm of your hand to ensure a seal.
# With one hand on either cup holding them together, the assembled cups are then flipped over so that the drink is primarily in the larger mixing cup.
# The most efficient way to shake is horizontally, so that the drinks sloshes vigorously from front to back. Shaking vertically will achieve the same result, but it is more energy intensive and slower as it throws the drink upwards before slamming it back down. You will feel the metal become cold, and it may frost as the drink is chilled. These are signs that you can stop shaking.
# At this point, the cold from the ice has caused the metal tins to contract and shrink slightly which will bind them together tightly. To unseal the shaker, hold the larger cup or place it on a counter. Identify which was the smaller cup is leaning. With the palm of your hand, strike the smaller cup away from the direction it's leaning. There should be a crack as the seal is broken (this is called &quot;cracking&quot; a shaker open). You can also twist or squeeze the larger tin to release the seal if it's made of metal.
# To strain, place the Hawthorn strainer over the mouth of the larger cup, and pour the drink through the strainer.

You can watch an example of [https://www.youtube.com/watch?v=ai70a6xjSzo someone using a Boston shaker] on YouTube (not my video), if you'd like more instruction.
[[File:Stirring_with_a_bar_spoon.jpg|thumb|230x230px|Stirring with a Bar Spoon]]

==== Stirring a Cocktail ====
It sounds simple, but there's a right way and a wrong way to stir a cocktail. The goal is to incorporate and chill the ingredients without agitating, decarbonating or introducing air. Here are a few simple steps:

# Insert a bar spoon to the bottom of the mixing glass containing you ingredients and ice. Push the spoon all the way to the bottom edge of the mixing glass.
# Swirl the spoon so that it stays in contact with the edge of the glass, rotating the ice in a circle.

==== Serving Sparking Wine ====

# Use the tab to peel away the foil.
# Keep one hand on top of the cage and cork. Lift the handle and twist it open.
# Keep the cage on the cork. Hold the cage and the cork with your dominant hand.
# Holding the bottle in your non-dominant hand, twist the bottle back and forth. The goal is to gradually release the pressure in the bottle, not to make a loud &quot;pop&quot;. Ideally, you should open the bottle almost silently.
# Holding the bottle at the base with one hand (if possible), pour slowly into the glass as the bubbles will climb.

[[File:Pouring_wine_into_a_decanter.jpg|left|thumb|296x296px|Pouring wine]]

==== Uncorking Still Wine ====

# Find a wine key with a double-hinge.
# Using the knife, cut the foil around the lip of the bottle. Using the bottle opener end of the key, peel back to foil to expose the cork.
# Insert the corkscrew into the center of the cork by twisting until most of the screw is embedded.
# Place the first hinge on the lip of the bottle, and use it as leverage to lift up on the wine key (lifting the cork partially).
# Place the second hinge of the lip of the bottle, and completely remove the cork.

'''Serving Beer'''

# Whether pouring from a bottle or a tap, tilt the glass 45° and pour the beer down the side of glass to prevent too much foam, or &quot;head&quot;, from developing.
# At the end of the pour, straighten the glass to allow for some head to form.

== Essential Cocktails ==
Here we are, finally mixing some drinks! It can be overwhelming to open a book of cocktail recipes and see the hundreds of drinks with unique names and recipes. Take a deep breath, because there isn't a bartender in the world that knows all of those cocktails by heart, and you aren't expected to either. Over time, you'll pick up on more drinks and their makeup, but that takes time! That's why it's important to focus on the essential cocktails that every bartender knows and that any bar patron might order. What follows are 20 cocktail recipes for classic cocktails that you might make every day at a bar.

I highly encourage you to make these drinks at home. If you intend to become a bartender, the difference between your first attempt at a recipe being in the privacy of your home and on your first day on the job is ''enormous''. That raises the question, &quot;How am I supposed to supply all the ingredients needed to make all these drinks‽&quot; Start with things you can buy at the grocery store:

* Lemons, Limes and Oranges
* Club Soda
* White Sugar

You may already have these things from trying out the preparation techniques above. If you don't, go out and grab these ingredients, cut some garnishes, and prepare some simple syrup.

The most expensive part of any cocktail is the spirit. I've color coded the following recipes by which spirit they use. The list generally starts with easy recipes and becomes more complicated as you descend. I recommend staring with one type of spirit and trying a few recipes that use that spirit. For example, with only vodka, you can make a Highball, Cosmopolitan, Moscow Mule, Bloody Mary and a (Vodka) Martini. That's one fifth of the list done, with a spirit you likely already have at home.

I don't recommend buying fancy or expensive spirits for this. Buy something cheap or middle-of-the-road to practice with. That's like what your bar patrons are going to be ordering most often anyhow.

All that being said, 20 recipes is a lot to learn. Take your time, maybe make one or two a day. The recipes aren't changing, and some are very simple. You by no means have to drink what you make. You DO have to TASTE what you make. Bartenders, when trying a drink on the job, will smell a drink, swirl it in their mouth and spit it out. Do this at least twice for each drink if you don't intend to drink them. Take note of the taste, the balance of the drink and your personal preference for or against it. Please don't make all 20 in one night and get hammered. A daiquiri will be a daiquiri tomorrow, and it'll be sweeter for the waiting.

Well, get started!

=== Spirit Color Key ===
{{RoundBoxLeft|theme=1|width=50}}
Vodka
{{RoundBoxRight|theme=2|width=50}}
Gin
{{LeftRightBoxClose}}
{{RoundBoxLeft|theme=3|width=50}}
Rum
{{RoundBoxRight|theme=4|width=50}}
Tequila
{{LeftRightBoxClose}}
{{RoundBoxLeft|theme=5|width=50}}
Whiskey
{{RoundBoxRight|theme=11|width=50}}
Other/Mixed
{{LeftRightBoxClose}}
=== Highball ===
Highballs are a class of cocktail sometimes called &quot;plus one&quot; cocktails because they're a blend of a spirit plus a mixer. Some common highballs are Gin and Tonic, Rum and Coke, Whiskey and Ginger Ale, and Scotch and Soda.{{RoundBoxTop|theme=11}} 
'''Glass:''' Highball [[File:Highball glass silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Wedges with soda, Lime Wedges with Tonic, Cola and Ginger Beer/Ale

'''Ingredients'''

2 oz of Spirit

Mixer of choice, to top

'''Combine''' Spirit and ice. Stir. Add Mixer to fill. Drag spoon through drink a few times to gently combine. Garnish
{{RoundBoxBottom}}

=== Cosmopolitan ===
{{RoundBoxTop|theme=1}} 
'''Glass:''' Coupe or Martini Nowrap|[[File:Coctail glass.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Twist

'''Ingredients'''

1 ½ oz Vodka (Preferably Citron Vodka)

⅓ oz Lime Juice

⅓ oz Cointreau (or other orange liqueur)

⅓ oz Sweet Cranberry Juice

'''Shake''' all ingredients with ice until chilled. Strain into glass. Garnish. 
{{RoundBoxBottom}}

=== Margarita ===
{{RoundBoxTop|theme=4}} 
'''Glass:''' Rocks [[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lime Wedge and ask preference for Salt, Sugar, or Tajin rim

'''Ingredients'''

2 oz Tequila

½ oz Agave Nectar

1 oz Lime Juice

½ oz Triple Sec

'''Moisten''' the rim of the glass with a lime wedge, then roll exterior of glass in rimming material careful to avoid getting any inside the glass. Combine all ingredients in shaker with ice. Shake. Add ice to glass. Strain. Garnish.
{{RoundBoxBottom}}

=== Paloma ===
{{RoundBoxTop|theme=4}} 
'''Glass:''' Highball [[File:Highball glass silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Grapefruit Wedge or Lime Wheel

'''Ingredients'''

2 oz Tequila

½ oz Lime Juice

Dash of Salt

Grapefruit Soda, to top

'''Combine''' Tequila, Sale and Lime Juice in glass with ice. Stir to chill. Add Grapefruit Soda to fill. Drag spoon through drink a few times to gently combine. Garnish.
{{RoundBoxBottom}}

=== Whiskey Sour ===
{{RoundBoxTop|theme=5}} 
'''Glass:''' Sour or Coupe [[File:Coppa Asti sagoma.svg|24x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Wheel and Cherry

'''Ingredients'''

2 oz Whiskey

1 oz Lemon Juice

½ tsp. Sugar or Simple Syrup

'''Shake''' ingredients with ice. Strain. Garnish.
{{RoundBoxBottom}}

=== Moscow Mule ===
{{RoundBoxTop|theme=1}} 
'''Glass:''' Mule Mug [[File:Alcohol_glass_copper_mug.svg|24x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lime wheel (optional)

'''Ingredients'''

2 oz Vodka

½ oz Lime Juice

3 oz Ginger Beer

'''Combine''' ingredients in a copper mug filled with ice. Stir. Garnish, optionally.
{{RoundBoxBottom}}

=== Spritz ===
Spritz are a class of cocktails that follow the same general 3-2-1 recipe of 3 oz Prosecco, 2 oz liqueur, 1 oz club soda. The most common is the Aperol Spritz.{{RoundBoxTop|theme=11}} 
'''Glass:''' Highball or Wine Glass [[File:wineglass.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Orange Wheel

'''Ingredients'''

3 oz Prosecco

2 oz Liqueur (Aperol, for an Aperol Spritz)

1 oz Club Soda

'''Combine''' Prosecco and Liqueur with ice in glass. Stir gently. Top with Club Soda. Garnish.
{{RoundBoxBottom}}

=== Mojito ===
{{RoundBoxTop|theme=3}} 
'''Glass:''' Highball [[File:Highball glass silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Mint Sprig

'''Ingredients'''

1 ½ oz White Rum

5 to 6 mint leaves

1 oz Lime Juice

2 tsp Sugar

Soda water

'''Muddle''' the Mint, Lime Juice and Sugar until well mixed but not mushy. Add the Rum and fill the glass with ice. Top with Soda Water. Garnish.
{{RoundBoxBottom}}

=== Daiquiri ===
{{RoundBoxTop|theme=3}} 
'''Glass:''' Coupe [[File:Coppa Asti sagoma.svg|24x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lime wheel (optional)

'''Ingredients'''

1 ½ oz White Rum

2 tsp Sugar

1 oz Fresh Lime Juice

'''Shake''' ingredients with ice until chilled and frothy. Strain into a coupe glass. Garnish, optionally.
{{RoundBoxBottom}}

=== French 75 ===
{{RoundBoxTop|theme=2}} 
'''Glass:''' Champagne Flute [[File:flutesil.svg|24x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Twist

'''Ingredients'''

1 oz Gin

½ oz Lemon Juice

½ oz Simple Syrup

3 oz Champagne (or other sparkling wine)

'''Shake''' Gin, Lemon Juice, and Simple Syrup with ice. Strain into glass. Top with Champagne. Garnish.
{{RoundBoxBottom}}

=== Gimlet ===
{{RoundBoxTop|theme=2}} 
'''Glass:''' Coupe [[File:Coppa Asti sagoma.svg|24x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lime Wheel

'''Ingredients'''

2 ½ oz Gin

½ oz Lime Juice

½ oz Simple Syrup

'''Shake''' Gin, Lime Juice, and Simple Syrup with ice. Strain into glass. Garnish.
{{RoundBoxBottom}}

=== Manhattan ===
{{RoundBoxTop|theme=5}} 
'''Glass:''' Rocks [[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Twist and/or Cherry

'''Ingredients'''

1 ½ oz Rye Whiskey

½ oz Sweet Vermouth

1 Dash Angostura Bitters

'''Chill''' rocks glass by combining ice and water, or use pre-chilled glass. Combine ingredients in a mixer with ice. Stir. Strain. Garnish.
{{RoundBoxBottom}}

=== Mai Tai ===
{{RoundBoxTop|theme=3}} 
'''Glass:''' Rocks [[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Pineapple Wedge, Cherry, and Mint Spring

'''Ingredients'''

1 oz Light Rum

1 oz Gold Rum

½ oz Orange Curaçao

½ oz Orgeat

½ oz Lime Juice

'''Shake''' ingredients with ice. Strain into ice filled glass. Garnish.
{{RoundBoxBottom}}

=== Pina Colada ===
{{RoundBoxTop|theme=3}} 
'''Glass:''' Hurricane [[File:Hurricane Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Pineapple Wedge

'''Ingredients'''

1 ½ oz Rum (Preferably Coconut, like Malibu)

1 ½ oz Coconut Cream

1 oz Pineapple Juice

½ oz Lime Juice

'''Combine''' ingredients with ice in a blender. Blend on high for 20-30 seconds or until a flowing slushy consistency. If too stiff add water, if too thin add ice and re-blend for 2-3 seconds. Pour into glass. Garnish.
{{RoundBoxBottom}}

=== Bloody Mary ===
{{RoundBoxTop|theme=1}}
'''Glass:''' Pint or Highball [[File:Highball glass silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Wheel or Celery Stalk (or anything savory)

'''Ingredients'''

2 oz Vodka

6 dashes Worcestershire sauce

3 dashes Tobasco Sauce

Pinch of Salt

Pinch of Ground Black Pepper

½ Lemon, juiced

5 oz Tomato Juice

'''Combine''' all ingredients in a glass. Add ice and stir. Garnish.
{{RoundBoxBottom}}

=== Negroni ===
{{RoundBoxTop|theme=2}}
'''Glass:''' Rocks [[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Orange Twist or Orange Wheel

'''Ingredients'''

1 oz Dry Gin

1 oz Sweet Vermouth

1 oz Compari

1 oz Sparkling Water

'''Combine''' all ingredients in a glass. Add ice, preferably 2 large rocks. Stir. Garnish.

{{RoundBoxBottom}}

=== Old Fashioned ===
{{RoundBoxTop|theme=5}} 
'''Glass:''' Rocks [[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Orange twist

'''Ingredients'''

2 oz Whiskey

1 Sugar Cube

4 dashes Aromatic Bitters

1 dash Soda Water

'''Muddle''' Sugar, Bitters and Soda Water in a glass until a uniform syrup is made. Add Whiskey. Add Ice. Stir. Express Orange Twist over drink then add to glass.
{{RoundBoxBottom}}

=== Tom Collins ===
{{RoundBoxTop|theme=2}} 
'''Glass:''' Collins or Highball [[File:Collins glass silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Wheel and Cherry (optional)

'''Ingredients'''

2 oz Gin

1 oz Lemon Juice

½ oz Simple Syrup

Club Soda, to top

'''Combine''' Gin, Lemon Juice, and Simple Syrup in glass. Add ice to top. Fill with Club Soda. Stir gently. Garnish.
{{RoundBoxBottom}}

=== Martini ===
Martinis are kind of their own thing, and there's some vocab to go over. There are many drinks that call themselves a martini, and the Classic Martini itself has many variations to suit each bar patron's taste. To make a Martini to order, first you have to have a conversation with your bar patron.

# First, ask if they prefer Gin or Vodka. Gin is traditional, but it can be made with either.
# Second determine how much Vermouth they would like. The amount of Vermouth will determine how &quot;Dry&quot; or &quot;Wet&quot; the Martini is.

&quot;Extra Dry&quot; or &quot;In and Out&quot; = Vermouth is swirled in the glass then poured out, then the drink is entirely Gin or Vodka

&quot;Dry&quot; = 1 part Vermouth to 5 parts Gin or Vodka

&quot;Wet&quot; = 1 part Vermouth to 2 parts Gin or Vodka

&quot;50/50&quot; = 1 part Vermouth to 1 part Gin or Vodka

&quot;Dirty&quot; = 1 part Olive Brine to 5 parts Gin or Vodka, instead of Vermouth. Garnish with Olives instead of lemon.

The following recipe is for a Dry Martini. You can modify it to make it a Dirty Martini or to suit your taste of Vermouth. Martinis are often stirred, but they can be shaken on request.
{{RoundBoxTop|theme=11}} 
'''Glass:''' Martini [[File:Coctail glass.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon twist

'''Ingredients'''

2 oz Gin or Vodka (patron preference)

½ oz Dry Vermouth

'''Chill''' glassware by pouring water and ice in a glass, or use a pre-chilled glass. Combine ingredients with ice in a mixing glass. Stir until chilled. Strain into chilled glass. Garnish.
{{RoundBoxBottom}}

=== Espresso Martini ===
The Espresso Martini is really not much like a Classic Martini at all. The word Martini really gets thrown around. Nonetheless, it's delicious. If your bar has an espresso machine, you'll be trained on how to pull an espresso shot. Otherwise, your bar may use cold brew concentrate made in house or from a can.{{RoundBoxTop|theme=1}} 
'''Glass:''' Martini [[File:Coctail glass.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' 3 Coffee Beans

'''Ingredients'''

2 oz Vodka

½ oz Coffee Liqueur (often Kahlúa)

1 oz Espresso, or cold brew concentrate

1/4 oz Simple Syrup

'''Chill''' glass with water and ice, or use pre-chilled glass. Shake ingredients with ice. Strain into glass. Garnish by place 3 coffee beans in center.
{{RoundBoxBottom}}

Hey! Congrats! You've made it to the bottom of the list. These 20 recipes are just the beginning of the possibilities when it comes to mixing drinks, but if you can nail these down you can get through just about any shift at any bar. That was an undertaking, and you did a great job.

This concludes Lesson 2: Mixing Drinks. You can move on now to [[Bartending/Hospitality|Lesson 3: Hospitality]].

[[Category:Professional and vocational studies]]
[[Category:Professions]]
[[Category:Drinks]]</text>
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      <text bytes="8738" sha1="abs5zufboa2tbt6k1ntvytnm60wsrbg" xml:space="preserve">= Hospitality =
In this lesson, we'll be talking about some points of service and hospitality that are as essential to bartending as alcohol. The sections in this lesson are:

# What is Hospitality?
# Steps of Service
# Internal Hospitality
# Developing Taste

== What is Hospitality? ==
Mixing drinks is only half the job, maybe even less than half. The real job is [[wiktionary:hospitality|hospitality]].&lt;blockquote&gt;The act or service of welcoming, receiving, hosting, or entertaining guests; an appropriate attitude of openness, respect, and generosity toward guests.&lt;/blockquote&gt;In German, the word for hospitality is ''gastfreundschaft'' or guest-friendship. This encapsulates the charge of all bartenders to make the patrons of our bars feel welcome from the moment the walk through our doors to the moment they leave. Ideally, they still feel welcome long after they've left—they might just come back!

'''If you want to be there, so will they.''' People can immediately tell if you want to be there or not. If your vibe's a drag, people won't want to talk to you. There's the catch: ''people need to talk to you to get drinks''. If people don't want to talk to you, they'll leave, and they won't come back. The entire purpose of a bar is not only pushed forward by you, it's ''gate-kept'' by you. If there's no bartender, there's no bar; if there's a poor bartender, there's a poor bar. If you can find time to smile while behind the bar, you'll find your job is actually easier.

== Steps of Service ==
Service and hospitality are related, but separate ideas. Service is the mechanics of bartending: mixing drinks, closing tabs, cleaning and clearing. Hospitality is how you make your patrons feel. Keeping your service up will go a long ways towards making an environment hospitable. If the drink service is slow and the bar is messy, patrons are not gonna feel good. That's an example of service impacting hospitality.

When a patron comes to a bar, there's steps to how the they should be served that blend hospitality and service techniques. Let's go over the steps of providing good service.

# &lt;u&gt;Greet the Patron&lt;/u&gt; - Make eye contact with each person as they enter or approach your bar. Try to keep a relaxed, or at least welcoming, posture. Greet them warmly, making it clear that you see them and that you're happy they're here. Offer them a menu and pour them a water depending on your bar.
# &lt;u&gt;Take the Order&lt;/u&gt; - Once they've had a moment, or if they look immediately ready, ask what they would like to order or if they have any questions. Repeat what they've order back to them so that they can let you know if you misheard.
# &lt;u&gt;Mix the Drinks&lt;/u&gt; - Maybe they've ordered beer or wine that just need to be poured, or maybe they've ordered several shaken cocktails. Now's the time to make them. While you're making drinks, be sure to look up at the bar and greet people as they arrive.
# &lt;u&gt;Drop the Drinks&lt;/u&gt; - Bring finished drinks to guests as soon as they're finished. Mixed drinks have a short shelf life and should be  delivered promptly.
# &lt;u&gt;Check in&lt;/u&gt; - Give them a few moments to taste their drinks. Check in with an open ended question. &quot;How is everything?&quot; Many people won't tell you if they don't like the drink, so feel free to try to make them comfortable with giving honest feedback. You can always remake a drink!
# &lt;u&gt;Clear and Ask&lt;/u&gt; - When they're done with their drinks, clear any unneeded dishes and give the bar a wipe (avoiding pushing any crumbs onto your patron's lap). Then ask if they'd like another round. It's best to keep this question open and neutral. We don't need to push people to get another drink. A simple, &quot;What's next?&quot; gives patrons the floor to tell you what they need.
# &lt;u&gt;Close Out&lt;/u&gt; - When they've asked for a check, close out and bring the check quickly. The dead time when a patron wants to leave but can't yet is a vibe killer. You should prioritize closing them out and letting them leave quickly. When you drop the check, make eye contact, smile and thank them for coming.

== Internal Hospitality ==
Hospitality isn't something reserved for the patrons of your bar. Hospitality is something that should be shown to your teammates and everyone you work with. Think of it like this:&lt;blockquote&gt;If you're willing to give 100% attention, kindness and respect to complete strangers who walk up to your bar, shouldn't you be willing to give even more to the people you work with and see every day?&lt;/blockquote&gt;The people you work with at your bar will be the most valuable and important resource you have, making them feel good when they're around you is equally if not more important than patron hospitality. They will give you invaluable tips, unlock opportunities you've never imagined, and maybe become lifelong friends, but only if you treat them with the same, or more, hospitality you give to bar patrons.

One step further, hospitality is something you can show yourself. If we return to the definition of hospitality ''as an appropriate attitude of openness, respect, and generosity'', you can absolutely show hospitality to yourself. Respect your needs, be generous with time away from work, and be open to making mistakes and learning. You'll find that bartending will be more pleasurable and that positive feeling will make your job even easier.

== Developing Taste ==
Oftentimes patrons of your bar will ask your opinion on drinks, the menu and even the area around the bar. It's good hospitality to be able to answer their questions to the best of your ability. Here's some tips on each of those topics.

=== Tasting Alcohol ===
Tasting alcohol, including straight spirits, is a unique experience that is often undervalued. Skilled sommeliers can detect subtle notes at different moments when drinking a wine. Something similar can be done with spirits. Just a reminder that you don't have to swallow a drink to taste it. It's common for bartenders to spit a drink out when tasting it, especially on the job. To effectively taste a spirit:

# Smell the spirit. Don't stick your nose straight into it! Hover your nose above the spirit so that you can get a natural whiff of the aroma.
# Take a sip and swirl it around your mouth. Your mouth can hold on to tastes for quite a while, and this first swirl will clear the taste of whatever you've eaten and drank earlier in the day.
# Take another sip and appreciate the flavor. This is the part that takes time, patience, and skill. There are several ways to break up flavors. What did you taste the second it hit your tongue? What about after spitting it out or swallowing? You can try to pick up on the biggest notes first. Is it smokey? Sour? Bitter? Then try honing in on something more subtle. Are there herbal notes? Is it fruity?

=== Giving Recommendations ===
One of the most common responses you'll get when taking someone's order is some variation of &quot;I don't know&quot;, &quot;What you recommend?&quot;, and &quot;What do you like?&quot;. All of these mean basically the same thing: &quot;Talk to me about what's on your menu and what you can make.&quot;

If you bar has a menu, this should be an easier task as the first thing you'll do after you arrive is memorize the menu. Giving recommendations isn't as easy as knowing the menu though. In these moments, it's important to engage with the patron to find out what kind of drink would suit their taste.

My preferred way of navigating a drink recommendation conversation is treating it like a &quot;choose your own adventure&quot; story. Offer a series of choices that bring both of you closer to a good drink choice. Start with broad strokes of flavor: &quot;Do you like a drink that's sweet, smokey, or sour?&quot; Oftentimes, that is enough to select off a menu. If they say something smokey, you might think about drinks that have a whiskey or mezcal base. Sweet or sour could have all kinds of spirits. You can ask if they have a preferred alcohol. You can also ask if they've had any drinks in the past that they've really liked.

== Conclusion ==
Well, that's the basics. There's so much to learn about bartending, most of which you won't learn until you start. I hope that this course have given you the confidence to mix drinks and put yourself out there as a bartender.

I have more things that I think could benefit this course, and I intend to eventually add them in subsequent lessons. Those lessons I would consider to be &quot;advanced&quot;, and I'll work on them as I have time. Thank you for taking the time to go through this course. If you have questions, comments, or would like to help with the continuation of this course, feel free to [[User:Sabbier|message me]]''.''

Happy Bartending!
[[File:Hacker-Pschorr Oktoberfest Girl Remix.jpg|center|thumb|Picture of me smiling]]</text>
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      <text bytes="8983" sha1="ku2vingu7vmdxan8azm0s41cvf5t4cw" xml:space="preserve">= Hospitality =
In this lesson, we'll be talking about some points of service and hospitality that are as essential to bartending as alcohol. The sections in this lesson are:

# [[Bartending/Hospitality#What is Hospitality?|What is Hospitality?]]
# [[Bartending/Hospitality#Steps of Service|Steps of Service]]
# [[Bartending/Hospitality#Internal Hospitality|Internal Hospitality]]
# [[Bartending/Hospitality#Developing Taste|Developing Taste]]

== What is Hospitality? ==
Mixing drinks is only half the job, maybe even less than half. The real job is [[wiktionary:hospitality|hospitality]].&lt;blockquote&gt;The act or service of welcoming, receiving, hosting, or entertaining guests; an appropriate attitude of openness, respect, and generosity toward guests.&lt;/blockquote&gt;In German, the word for hospitality is ''gastfreundschaft'' or guest-friendship. This encapsulates the charge of all bartenders to make the patrons of our bars feel welcome from the moment the walk through our doors to the moment they leave. Ideally, they still feel welcome long after they've left—they might just come back!

'''If you want to be there, so will they.''' People can immediately tell if you want to be there or not. If your vibe's a drag, people won't want to talk to you. There's the catch: ''people need to talk to you to get drinks''. If people don't want to talk to you, they'll leave, and they won't come back. The entire purpose of a bar is not only pushed forward by you, it's ''gate-kept'' by you. If there's no bartender, there's no bar; if there's a poor bartender, there's a poor bar. If you can find time to smile while behind the bar, you'll find your job is actually easier.

== Steps of Service ==
Service and hospitality are related, but separate ideas. Service is the mechanics of bartending: mixing drinks, closing tabs, cleaning and clearing. Hospitality is how you make your patrons feel. Keeping your service up will go a long ways towards making an environment hospitable. If the drink service is slow and the bar is messy, patrons are not gonna feel good. That's an example of service impacting hospitality.

When a patron comes to a bar, there's steps to how the they should be served that blend hospitality and service techniques. Let's go over the steps of providing good service.

# &lt;u&gt;Greet the Patron&lt;/u&gt; - Make eye contact with each person as they enter or approach your bar. Try to keep a relaxed, or at least welcoming, posture. Greet them warmly, making it clear that you see them and that you're happy they're here. Offer them a menu and pour them a water depending on your bar.
# &lt;u&gt;Take the Order&lt;/u&gt; - Once they've had a moment, or if they look immediately ready, ask what they would like to order or if they have any questions. Repeat what they've order back to them so that they can let you know if you misheard.
# &lt;u&gt;Mix the Drinks&lt;/u&gt; - Maybe they've ordered beer or wine that just need to be poured, or maybe they've ordered several shaken cocktails. Now's the time to make them. While you're making drinks, be sure to look up at the bar and greet people as they arrive.
# &lt;u&gt;Drop the Drinks&lt;/u&gt; - Bring finished drinks to guests as soon as they're finished. Mixed drinks have a short shelf life and should be  delivered promptly.
# &lt;u&gt;Check in&lt;/u&gt; - Give them a few moments to taste their drinks. Check in with an open ended question. &quot;How is everything?&quot; Many people won't tell you if they don't like the drink, so feel free to try to make them comfortable with giving honest feedback. You can always remake a drink!
# &lt;u&gt;Clear and Ask&lt;/u&gt; - When they're done with their drinks, clear any unneeded dishes and give the bar a wipe (avoiding pushing any crumbs onto your patron's lap). Then ask if they'd like another round. It's best to keep this question open and neutral. We don't need to push people to get another drink. A simple, &quot;What's next?&quot; gives patrons the floor to tell you what they need.
# &lt;u&gt;Close Out&lt;/u&gt; - When they've asked for a check, close out and bring the check quickly. The dead time when a patron wants to leave but can't yet is a vibe killer. You should prioritize closing them out and letting them leave quickly. When you drop the check, make eye contact, smile and thank them for coming.

== Internal Hospitality ==
Hospitality isn't something reserved for the patrons of your bar. Hospitality is something that should be shown to your teammates and everyone you work with. Think of it like this:&lt;blockquote&gt;If you're willing to give 100% attention, kindness and respect to complete strangers who walk up to your bar, shouldn't you be willing to give even more to the people you work with and see every day?&lt;/blockquote&gt;The people you work with at your bar will be the most valuable and important resource you have, making them feel good when they're around you is equally if not more important than patron hospitality. They will give you invaluable tips, unlock opportunities you've never imagined, and maybe become lifelong friends, but only if you treat them with the same, or more, hospitality you give to bar patrons.

One step further, hospitality is something you can show yourself. If we return to the definition of hospitality ''as an appropriate attitude of openness, respect, and generosity'', you can absolutely show hospitality to yourself. Respect your needs, be generous with time away from work, and be open to making mistakes and learning. You'll find that bartending will be more pleasurable and that positive feeling will make your job even easier.

== Developing Taste ==
Oftentimes patrons of your bar will ask your opinion on drinks, the menu and even the area around the bar. It's good hospitality to be able to answer their questions to the best of your ability. Here's some tips on each of those topics.

=== Tasting Alcohol ===
Tasting alcohol, including straight spirits, is a unique experience that is often undervalued. Skilled sommeliers can detect subtle notes at different moments when drinking a wine. Something similar can be done with spirits. Just a reminder that you don't have to swallow a drink to taste it. It's common for bartenders to spit a drink out when tasting it, especially on the job. To effectively taste a spirit:

# Smell the spirit. Don't stick your nose straight into it! Hover your nose above the spirit so that you can get a natural whiff of the aroma.
# Take a sip and swirl it around your mouth. Your mouth can hold on to tastes for quite a while, and this first swirl will clear the taste of whatever you've eaten and drank earlier in the day.
# Take another sip and appreciate the flavor. This is the part that takes time, patience, and skill. There are several ways to break up flavors. What did you taste the second it hit your tongue? What about after spitting it out or swallowing? You can try to pick up on the biggest notes first. Is it smokey? Sour? Bitter? Then try honing in on something more subtle. Are there herbal notes? Is it fruity?

=== Giving Recommendations ===
One of the most common responses you'll get when taking someone's order is some variation of &quot;I don't know&quot;, &quot;What you recommend?&quot;, and &quot;What do you like?&quot;. All of these mean basically the same thing: &quot;Talk to me about what's on your menu and what you can make.&quot;

If you bar has a menu, this should be an easier task as the first thing you'll do after you arrive is memorize the menu. Giving recommendations isn't as easy as knowing the menu though. In these moments, it's important to engage with the patron to find out what kind of drink would suit their taste.

My preferred way of navigating a drink recommendation conversation is treating it like a &quot;choose your own adventure&quot; story. Offer a series of choices that bring both of you closer to a good drink choice. Start with broad strokes of flavor: &quot;Do you like a drink that's sweet, smokey, or sour?&quot; Oftentimes, that is enough to select off a menu. If they say something smokey, you might think about drinks that have a whiskey or mezcal base. Sweet or sour could have all kinds of spirits. You can ask if they have a preferred alcohol. You can also ask if they've had any drinks in the past that they've really liked.

== Conclusion ==
Well, that's the basics. There's so much to learn about bartending, most of which you can't learn until you start doing it. I hope that this course have given you the confidence to mix drinks and put yourself out there as a bartender.

I have more things that I think could benefit this course, and I intend to eventually add them in subsequent lessons. Those lessons I would consider to be &quot;advanced&quot;, and I'll work on them as I have time. Thank you for taking the time to go through this course. If you have questions, comments, or would like to help with the continuation of this course, feel free to [[User:Sabbier|message me]]. I would love to hear from you!

Happy Bartending!
[[File:Hacker-Pschorr Oktoberfest Girl Remix.jpg|center|thumb|Picture of me smiling if I were a German woman]]</text>
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      <text bytes="9078" sha1="60ct8ljy4aogiblbi1zdnfrbn1gjfme" xml:space="preserve">= Hospitality =
In this lesson, we'll be talking about some points of service and hospitality that are as essential to bartending as alcohol. The sections in this lesson are:

# [[Bartending/Hospitality#What is Hospitality?|What is Hospitality?]]
# [[Bartending/Hospitality#Steps of Service|Steps of Service]]
# [[Bartending/Hospitality#Internal Hospitality|Internal Hospitality]]
# [[Bartending/Hospitality#Developing Taste|Developing Taste]]

== What is Hospitality? ==
Mixing drinks is only half the job, maybe even less than half. The real job is [[wiktionary:hospitality|hospitality]].&lt;blockquote&gt;The act or service of welcoming, receiving, hosting, or entertaining guests; an appropriate attitude of openness, respect, and generosity toward guests.&lt;/blockquote&gt;In German, the word for hospitality is ''gastfreundschaft'' or guest-friendship. This encapsulates the charge of all bartenders to make the patrons of our bars feel welcome from the moment the walk through our doors to the moment they leave. Ideally, they still feel welcome long after they've left—they might just come back!

'''If you want to be there, so will they.''' People can immediately tell if you want to be there or not. If your vibe's a drag, people won't want to talk to you. There's the catch: ''people need to talk to you to get drinks''. If people don't want to talk to you, they'll leave, and they won't come back. The entire purpose of a bar is not only pushed forward by you, it's ''gate-kept'' by you. If there's no bartender, there's no bar; if there's a poor bartender, there's a poor bar. If you can find time to smile while behind the bar, you'll find your job is actually easier.

== Steps of Service ==
Service and hospitality are related, but separate ideas. Service is the mechanics of bartending: mixing drinks, closing tabs, cleaning and clearing. Hospitality is how you make your patrons feel. Keeping your service up will go a long ways towards making an environment hospitable. If the drink service is slow and the bar is messy, patrons are not gonna feel good. That's an example of service impacting hospitality.

When a patron comes to a bar, there's steps to how the they should be served that blend hospitality and service techniques. Let's go over the steps of providing good service.

# &lt;u&gt;Greet the Patron&lt;/u&gt; - Make eye contact with each person as they enter or approach your bar. Try to keep a relaxed, or at least welcoming, posture. Greet them warmly, making it clear that you see them and that you're happy they're here. Offer them a menu and pour them a water depending on your bar.
# &lt;u&gt;Take the Order&lt;/u&gt; - Once they've had a moment, or if they look immediately ready, ask what they would like to order or if they have any questions. Repeat what they've order back to them so that they can let you know if you misheard.
# &lt;u&gt;Mix the Drinks&lt;/u&gt; - Maybe they've ordered beer or wine that just need to be poured, or maybe they've ordered several shaken cocktails. Now's the time to make them. While you're making drinks, be sure to look up at the bar and greet people as they arrive.
# &lt;u&gt;Drop the Drinks&lt;/u&gt; - Bring finished drinks to guests as soon as they're finished. Mixed drinks have a short shelf life and should be  delivered promptly.
# &lt;u&gt;Check in&lt;/u&gt; - Give them a few moments to taste their drinks. Check in with an open ended question. &quot;How is everything?&quot; Many people won't tell you if they don't like the drink, so feel free to try to make them comfortable with giving honest feedback. You can always remake a drink!
# &lt;u&gt;Clear and Ask&lt;/u&gt; - When they're done with their drinks, clear any unneeded dishes and give the bar a wipe (avoiding pushing any crumbs onto your patron's lap). Then ask if they'd like another round. It's best to keep this question open and neutral. We don't need to push people to get another drink. A simple, &quot;What's next?&quot; gives patrons the floor to tell you what they need.
# &lt;u&gt;Close Out&lt;/u&gt; - When they've asked for a check, close out and bring the check quickly. The dead time when a patron wants to leave but can't yet is a vibe killer. You should prioritize closing them out and letting them leave quickly. When you drop the check, make eye contact, smile and thank them for coming.

== Internal Hospitality ==
Hospitality isn't something reserved for the patrons of your bar. Hospitality is something that should be shown to your teammates and everyone you work with. Think of it like this:&lt;blockquote&gt;If you're willing to give 100% attention, kindness and respect to complete strangers who walk up to your bar, shouldn't you be willing to give even more to the people you work with and see every day?&lt;/blockquote&gt;The people you work with at your bar will be the most valuable and important resource you have, making them feel good when they're around you is equally if not more important than patron hospitality. They will give you invaluable tips, unlock opportunities you've never imagined, and maybe become lifelong friends, but only if you treat them with the same, or more, hospitality you give to bar patrons.

One step further, hospitality is something you can show yourself. If we return to the definition of hospitality ''as an appropriate attitude of openness, respect, and generosity'', you can absolutely show hospitality to yourself. Respect your needs, be generous with time away from work, and be open to making mistakes and learning. You'll find that bartending will be more pleasurable and that positive feeling will make your job even easier.

== Developing Taste ==
Oftentimes patrons of your bar will ask your opinion on drinks, the menu and even the area around the bar. It's good hospitality to be able to answer their questions to the best of your ability. Here's some tips on each of those topics.

=== Tasting Alcohol ===
Tasting alcohol, including straight spirits, is a unique experience that is often undervalued. Skilled sommeliers can detect subtle notes at different moments when drinking a wine. Something similar can be done with spirits. Just a reminder that you don't have to swallow a drink to taste it. It's common for bartenders to spit a drink out when tasting it, especially on the job. To effectively taste a spirit:

# Smell the spirit. Don't stick your nose straight into it! Hover your nose above the spirit so that you can get a natural whiff of the aroma.
# Take a sip and swirl it around your mouth. Your mouth can hold on to tastes for quite a while, and this first swirl will clear the taste of whatever you've eaten and drank earlier in the day.
# Take another sip and appreciate the flavor. This is the part that takes time, patience, and skill. There are several ways to break up flavors. What did you taste the second it hit your tongue? What about after spitting it out or swallowing? You can try to pick up on the biggest notes first. Is it smokey? Sour? Bitter? Then try honing in on something more subtle. Are there herbal notes? Is it fruity?

=== Giving Recommendations ===
One of the most common responses you'll get when taking someone's order is some variation of &quot;I don't know&quot;, &quot;What you recommend?&quot;, and &quot;What do you like?&quot;. All of these mean basically the same thing: &quot;Talk to me about what's on your menu and what you can make.&quot;

If you bar has a menu, this should be an easier task as the first thing you'll do after you arrive is memorize the menu. Giving recommendations isn't as easy as knowing the menu though. In these moments, it's important to engage with the patron to find out what kind of drink would suit their taste.

My preferred way of navigating a drink recommendation conversation is treating it like a &quot;choose your own adventure&quot; story. Offer a series of choices that bring both of you closer to a good drink choice. Start with broad strokes of flavor: &quot;Do you like a drink that's sweet, smokey, or sour?&quot; Oftentimes, that is enough to select off a menu. If they say something smokey, you might think about drinks that have a whiskey or mezcal base. Sweet or sour could have all kinds of spirits. You can ask if they have a preferred alcohol. You can also ask if they've had any drinks in the past that they've really liked.

== Conclusion ==
Well, that's the basics. There's so much to learn about bartending, most of which you can't learn until you start doing it. I hope that this course have given you the confidence to mix drinks and put yourself out there as a bartender.

I have more things that I think could benefit this course, and I intend to eventually add them in subsequent lessons. Those lessons I would consider to be &quot;advanced&quot;, and I'll work on them as I have time. Thank you for taking the time to go through this course. If you have questions, comments, or would like to help with the continuation of this course, feel free to [[User:Sabbier|message me]]. I would love to hear from you!

Happy Bartending!
[[File:Hacker-Pschorr Oktoberfest Girl Remix.jpg|center|thumb|Picture of me smiling if I were a German woman]]

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Why is the transfer of energy is an ecosystem refferd to as energy flow , not energy cycling [[Special:Contributions/&amp;#126;2026-42320-09|&amp;#126;2026-42320-09]] ([[User talk:&amp;#126;2026-42320-09|talk]]) 09:06, 30 July 2026 (UTC)</text>
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