Talk:Prime number/Archive 9

Improving towards Good Article criteria
Please discuss what changes are necessary for this article to achieve GA status.

My thoughts:
 * 1) Well written
 * 2) Prose, spelling grammar: acceptable
 * 3) Manual of style guidelines
 * 4) More consistency in the markup for mathematical notation is needed
 * 5) Verifiable
 * 6) Reference list: yes
 * 7) Reliable sources: yes
 * 8) No original research: yes
 * 9) No plagiarism or copyright violations: unsure
 * 10) Broad in its coverage
 * 11) Addresses the main aspects: in my opinion, yes
 * 12) Does not go into unnecessary detail: some work could be done
 * 13) It is in my opinion that this article is on the border of going into too much detail
 * 14) The second half to section 5.4 seems out of place. Perhaps the search for primes could be its own section or subsection
 * 15) Neutral: yes
 * 16) Stable: yes
 * 17) Illustrated
 * 18) Contains images: yes, nine
 * 19) Article lead: Demonstration that 7 is prime
 * 20) Section 1: Demonstration that 12 is composite and 11 is prime
 * 21) Section 3: Sieve of Eratosthenes animation
 * 22) Section 6.2: Plot of the prime counting function
 * 23) Section 6.3: Illustration of an arithmetic progression with modulo 9
 * 24) Section 6.4: Ulam spiral illustration
 * 25) Section 6.4: Table illustrating a Ulam spiral
 * 26) Section 7.1: Plot of the Riemann zeta function
 * 27) Section 8.2: animation of the construction of a pentagon
 * 28) Images are tagged with copyright status: yes to all
 * 29) Images are relevant and captioned: some work needed
 * 30) Images 1 and 2 could possibly be re-ordered considering Image 2 demonstrates that 12 is composite, and both prime and composite numbers are defined in the article lead
 * 31) Image 3 could possibly be moved to the Sieves section
 * 32) Caption to Image 4 could be improved (e.g. what conclusion can we make from the comparison of these three functions?)
 * 33) Image 5 has no caption
 * 34) Not sure if both Images 6 and 7 need to be included. At the very least, it would be preferred if Image 7 were an actual image and not a table of numbers.
 * 35) Caption to Image 8 could be improved (e.g. what does this plot tell us about prime numbers?)
 * 36) Caption to Image 9 could be improved (e.g. why is this animation relevant to Fermat primes, or prime numbers in general?)

Derek M (talk) 23:01, 8 January 2018 (UTC)


 * There need to be some typesetting improvements. Formulas that are set apart should generally be done in  mode, while inline formulas can be either way depending on what's needed, but it should be consistent between either plain wiki markup (i.e.,   giving n), or template markup (i.e., either   or   giving $n$).  I'm not crazy about, but it's probably better than the alternative.  There's also some inconsistency between showing multiplication with a dot or with a cross; personally, I think a dot should be used.  That was just a quick glance, I'm sure there's more.  –Deacon Vorbis (carbon &bull; videos) 23:30, 8 January 2018 (UTC)
 * I guess it seems that such considerations don't seem to affect GA status, but I think they're fairly important for a math-heavy article. –Deacon Vorbis (carbon &bull; videos) 23:36, 8 January 2018 (UTC)
 * It's in pretty good shape, but I think some topics that should be mentioned in the lead aren't (notably the entire history section and the Riemann hypothesis subsection). Re the math formatting: I don't have strong opinions between &lt;math&gt; and math, but I prefer either of them over plain wiki markup because I think it's important for variable names to have close to the same appearance in inline formulas as they do in displayed formulas. Additionally, whichever way we end up going on this we should aim for more consistency than we have now. Another area where we should try to be more consistent is reference formatting. Currently we have some references given in their entirety in a footnote, some given by a short harvnb footnote pointing to an entry in the references, and some expanded in inline text (e.g. Euclid's "Book IX, Proposition 20"). Also, in a GA review, we are likely to get dinged for having many sentences (at the ends of their paragraphs) and some entire non-lead paragraphs without sources. —David Eppstein (talk) 00:12, 9 January 2018 (UTC)

I have added the math notation suggestions to my list above.

Can you explain in further detail what you mean by "some topics that should be mentioned in the lead aren't (notably the entire history section and the Riemann hypothesis subsection)"?

I don't have prior experience in assisting improving an article to GA status, so I can only compare this article to other Top Importance, GA mathematics articles like Derivative. I've noticed that the lead to Prime number goes into much more detail than the other article's lead. Additionally I think it is a bit disorganized in comparison. This is what it looks like now:


 * 1) Definition of prime and composite. Example. Role in number theory (fundamental theorem of arithmetic). 1 is excluded.
 * 2) Primality: Trial division, faster algorithms e.g. Miller-Rabin, AKS. Faster algorithms for primes of special forms. Size of largest known prime.
 * 3) Infinitely many primes. No known formula that separates primes from composites. The distribution of primes can be modelled. Prime number theorem.
 * 4) Open questions: Goldbach's and twin prime conjecture. Questions of primes have lead research in other fields. Uses for large prime numbers in technology e.g. cryptography. Uses in algebra.

Some things that could improve the lead:


 * The last part of paragraph 2 seems out of place for that paragraph. This should probably be moved to its own paragraph along with the "Uses for large prime numbers in cryptography" part of paragraph 4.
 * Some sort of "why is this important?" conclusion to paragraph 3 would be useful. For example, it is interesting that there are an infinite many primes because you'd expect they'd continue to get rarer and rarer and eventually we'd "run out"

Additionally, my area of interest is in distributed computing and thus in my opinion I think this article would benefit from a section dedicated to the search for large primes. Derek M (talk) 14:58, 9 January 2018 (UTC)
 * This article also has some stale cleanup tags and associated categories: Articles with dead external links from May 2011; Articles with unsourced statements from August 2015; Wikipedia articles needing clarification from June 2011; Articles containing potentially dated statements from February 2011; Articles with specifically marked weasel-worded phrases from June 2014. All these need to be cleared up before it can be nominated for GA (they would be cause for immediate failure under the GA rules). —David Eppstein (talk) 08:02, 11 January 2018 (UTC)


 * Maybe I am fully off track with my suggestion to find a better top-right image for this article. Even the boring "primes within 1 to 100" seems to me more appropriate than the current, dubious commercial for Cuisinaire Rods®, which to my perception are innately unable to demonstrate any prime property. They just demonstrate the PR-abilities of someone pushing C. Gattegno, and his idiosyncratic methods to success in popularity about teaching methods with these rods and his Silent Way, possibly mostly based on mass material in WP. Purgy (talk) 12:20, 18 January 2018 (UTC)
 * I agree. I'm also not entirely happy with the Eratosthenes animation. It colors an initially monochromatic square, but what do all the colors mean? Has anyone checked whether the colors meet our accessibility standards? And what do the two different dimensions mean, when the sequence of integers is only one-dimensional? I think File:Eratosthenes sall.jpg (pixelated as it is) is far clearer. —David Eppstein (talk) 01:11, 19 January 2018 (UTC)
 * I actually like the image we currently have (of course, we should absolutely avoid referring explicitly to commercial material, though). IMO, David's image (which looks nice visually!, what about 1 though?) primarily visualizes composite numbers. Then we would have to say in words (e.g. the image caption) that primes are the numbers which are not composite. The image with the rods (or some other equivalent visualization) has the advantage of directly showing that a certain number is a prime, which is the most immediate task of the article (and its lead). It also gives an opportunity to refer to it later in the text, when discussing elementary primality tests. Jakob.scholbach (talk) 08:08, 19 January 2018 (UTC)
 * What the current image actually shows (I think) is a highly non-optimized version of trial division that goes all the way up to n-1. I don't see how that's a clear demonstration of the definition. —David Eppstein (talk) 08:21, 19 January 2018 (UTC)


 * On a partly independent note, I suggest to avoid the word "divisor" in the article lead, at least in the first sentence. It is unnecessary jargon here. I believe the target audience for the article lead, at least partly, can be a 10 year old kid. So the sentence

For example, 5 is prime because 1 and 5 are its only positive integer divisors, whereas 6 is composite because it has the divisors 2 and 3 in addition to 1 and 6.
 * should be reworded, IMO, to something like:

5 is prime because it cannot be written as a product other than 5 * 1 or 1 * 5, whereas 6 is the product of 2 * 3 (in addition to 1 * 6), and therefore is compiste.
 * Do you agree? If yes, and if our goal with the image is to illustrate the definition, then the image is just fine. And yes, of course it is suboptimal, but this is not the point in the lead.
 * How do you like the image on the right? Jakob.scholbach (talk) 08:31, 19 January 2018 (UTC)
 * I agree with both the intended level and the specific complaints about "divisor" and have edited along the lines you suggest. I think, by the way, that "&times;" is probably more recognizable to a 10-year-old as a symbol of multiplication than either "*" or "&middot;", so I have used that notation — opinions on that? —David Eppstein (talk) 19:12, 19 January 2018 (UTC)

I prefer the current image on Eratosthenes and suggested an explicative caption. David Eppstein's suggestion is certainly more compact, but I am in doubt if the small upper bound suffices to make the mechanism obvious to beginners. Additionally, I am not willing to sacrifice indisputable advantages (e.g.: colors) in favour of some irrational inclusion missions. In principle, I share a certain reluctance to animated diagrams, they should at least be stoppable, but their didactic power is not negligible. I was not bold enough to exchange the top images, but I certainly prefer David Eppstein's suggestions. The reservations regarding 1 and 2 are imho covered by the dimensionality argument: both have simply only one-dimensional arrangements (in arbitrary directions). OTOH, I do not see why obviously commensurable rods (if not in inch, perhaps in mm) should be able to introduce a rigorous notion of primes, not plagued by misleading analogies.

The suggestion by Jakob.scholbach does not show the connection to primes like David Eppstein's does.

I share the opinion of "divisor" being jargon, but "evenly divide" is elementary accessible. Dis-proving any existence is hard, and taking care of non-commutativity is excessive, so my suggestion (introducing the jargon at the end) is 5 is prime because it is not evenly divisible besides by 1 and 5, whereas 6 is composite for being evenly divisible by 2 and 3, additionally to the obvious trivial divisors 1 and 6.

Addendum 08:53, 19 January 2018 (UTC)Purgy (talk) 08:43, 19 January 2018 (UTC)


 * The colored animation has some irritating features: when a new prime is added, its multiples with the existing primes are not colored, whereas its other multiples for which it is not the smallest factor are colored. For example, when 3 is added, 6 is not colored but 12 and 18 are; when 5 is added, 10 and 15 are not colored but 20 and 30 are.  This seems hard to justify, would be difficult to explain in the caption, and is not currently explained correctly in the caption.  --JBL (talk) 12:59, 19 January 2018 (UTC)
 * As I see it, re-marking numbers only happens for the square and higher "already marked multiples above the square" correction 16:11, 19 January 2018 (UTC)  of the prime in consideration. So for 3 all multiples above and including 3x3=9 are colored in green (not 6), and for 5 all multiples above and including 25 (not 10, 15 and(!) not 20) are made blue. Accordingly, 56, 63, 70, 84, 98 and 105 are recolored in the course of dealing with 7, but not 14, 21, 28, 35, 42. This seems to be quite reasonable, and my suggested caption might not be exhaustive, but I do not think it is incorrect. Purgy (talk) 15:06, 19 January 2018 (UTC)

Thoughts about the new lead image
I suggest to get rid of one of the alternative factorizations of twelve, and just keep either 2x6 or 3x4. This would allow for a narrower pic (keeping 3x4 even more), and the factorizations are not exhaustive, anyway.

Adjusting the colors to the sieve-image (primes in magenta, 3-rows in green, ...) and narrowing the vertical space between "prime" and "composite" are overrun finesses, but maybe easy to do for sages (hint, hint ... :) ) having opened the file in an editor.

For the caption I'd like to read something about the number of dots in one row and the number of dot-rows representing the two "smaller numbers" in the "product", making up the considered number, e.g., something like this:

..., the prime numbers are the ones that cannot be expressed as the product of two smaller numbers, one factor given by the number of dots in a row, the other by the number of dot-rows.

The caption I consider as quite appropriate to change, but the image changes are grain size. Purgy (talk) 11:57, 20 January 2018 (UTC)
 * The image colors are deliberately chosen from a restricted palette that I have been using from User:KSmrq. I have checked them with a color blind friend and found them to be highly distinguishable from each other, even for the color blind. If you want to use the sieve image colors instead, can you provide some evidence that they meet MOS:CONTRAST? —David Eppstein (talk) 18:19, 20 January 2018 (UTC)
 * To follow up on this, I checked the contrast of some of the color pairs of the sieve image against MOS:CONTRAST "Ensure the contrast of the text with its background reaches at least Web Content Accessibility Guidelines (WCAG) 2.0's AA level, and AAA level when feasible" using the tool at https://snook.ca/technical/colour_contrast/colour.html#fg=000000,bg=99C0F8 . The black-on-blue (3238C6) used for #5 is not compliant at any level. The black-on-red (BE3F3B) used for #2 is compliant only at the AA level, and only at font sizes of 18+ (not true for this image). And the black-on-magenta (BD45C6) used for larger primes is compliant for AA, but not for AAA at small font sizes. The same is true for the black-on-lighter-blue (6D73D7) used for multiples of five, and the black-on-lighter-red (CF7571) used for all even numbers. I didn't check whether the pairs of lighter colors that are supposed to contrast with each other are also distinguishable from each other (because the tools is intended to check contrast of text on background, not of contrasting color fields) but I suspect not. If we consider that the text in the image is not really a pure black because of its anti-aliasing, the situation becomes even worse; using the color of darkest pixel I found in the gray-on-black text gives 30322F-on-BBBBBB, also not compliant at AAA for small font sizes. In conclusion, this image does not appear to pass MOS:CONTRAST and should not be used as guidance in creating other images. —David Eppstein (talk) 20:48, 20 January 2018 (UTC)
 * I almost feel like apologizing ... Certainly, I did not want to have the colors of the sieve pic as a guidance. I just thought them to be way harder to change, compared to the (perfectly selected) colors of the lead-pic. I came to mention the colors just because I was dragged myself to consider them in detail. At first sight I did not notice myself the slight difference in coloring the primes and their multiples. Nevertheless, even given all the detrimental color/contrast properties, I would not expunge this image (it was once featured?) without some substitute.
 * The reason for asking for a narrower image was triggered by the high share of whitespace in the whole image, which could be reduced by trimming at the right (after removing the 2x6 dot-block) and, perhaps between prime/composite.
 * What do you think about a more explicative caption? Purgy (talk) 11:25, 21 January 2018 (UTC)
 * Anyway, I've tightened the image. Better? I think there's definitely room for improvement in the caption but the aim should be to work better for the target 10-year-old audience for the first paragraph of the article rather than to be more explicative. —David Eppstein (talk) 19:58, 21 January 2018 (UTC)
 * - Not only the layout of the lead image is now perfect, but also the colors are so, imho. Depicting all composites in one color not only fits to the heading, but also takes out any inappropriate hint to any prime number. I revised my original stance.
 * - Re the caption, I still think that being explicit about the factorization in two smaller numbers within the pic supports intuitive understanding by the targeted readership. E.g., like so?

... not products of (at least) two smaller numbers, like is shown by the quadrangles for composites.
 * - Re Ishango bone: may I suggest to reconsider engraved vs. included?
 * - Re Rhind papyrus: I came to insert just this pic, because the current one is sort of ubiquitous in references, and I looked for an attractive, mathy looking image. In fact, I inserted both pics to the History section to make some soft science noise (Paleolithikum, Rhind) and create optical wellbeing.
 * Please, do not assign any significance to my opinion beyond your own considerations, it's just my desire to explain myself that makes me look so insistingly. Purgy (talk) 08:21, 22 January 2018 (UTC)
 * I would support changing the current image of the papyrus to one that showed part of its table of 2/n Egyptian fraction expansions, if you want to find and upload one. (I don't think there should be a problem with copyright.) The issue was merely that the other image was from a part of the papyrus that had nothing to do with prime numbers. —David Eppstein (talk) 20:04, 22 January 2018 (UTC)

Removal of unsourced applications
It became a well acknowledged factoid that trivial facts are hard to source soundly. I think that the coprime numbers of teeth on on gears within a gearbox is such a fact within engineering, as is the selection of checkdigits for accounting numbers, according to their properties wrt excluding zero divisors which leads to primes, among bankers. So I think that the removal of the parts referring to these topics in the Applications section is a bit harsh. Maybe these mentionings generate interest within a casual readership, and, imho, they are appropriate within this article for this reason. Not shooing away any good sources! To be honest, I am more bored by the abundance of (non-famous) conjectures about prime gaps. Purgy (talk) 08:37, 24 January 2018 (UTC)
 * Checkdigits rely on the specific properties of the numbers 10 and 11, not on primes more generally. Gear ratios are a possibility, though, if more substantial sourcing can be found. In any case, some thought should go into the overall structure of the article rather than shoving things in where they sort of fit. I'm not at all certain the current structure is right, but with the current structure those topics were part of the top paragraph of a section with subsections, whose content should have been purely a summary of those subsections. —David Eppstein (talk) 08:48, 24 January 2018 (UTC)
 * I admit not to be a (math) purist when stating my above opinions, but I do have some sexyness of articles in mind (imho, "prime gaps" are unattractive, "gearboxes" and "accounts" appeal to certain user groups, as do words like "skin") ;). I also do not want the least to revert the specific edit, or to remove some conjectures, I just humbly dare to invest in some PR and (as said) make unscientific, soft sciency noise. Purgy (talk) 09:21, 24 January 2018 (UTC)

Suggestions on structure
- Placing "History" as first section. - Including "Primality of one" into the History section, since the debate about it seems to be settled. - As a consequence of above, re-title the heading to "Factorization" and include "Integer factorization", including the associated complexity remarks, and perhaps hinting to the Generalizations for non-unique factorizable constructs. This would lead to one more new heading "Testing primality". - Shifting also the clumsy "Number of prime numbers" in to this "Factorization" paragraph, motivated by not only the naturals being infinite, but also their necessary factors. - "In games, arts, and literature" could find a place under "Applications". Just my two cents. Purgy (talk) 14:56, 29 January 2018 (UTC)

Complete?
Ok, I've completed a pass over the entire article, working to make the sourcing complete and avoid unnecessary technicality. Opinions on whether it's now in shape for a GA nomination or whether additional work (e.g. major restructuring to make the section order more logical) would be a good idea first? —David Eppstein (talk) 20:42, 27 January 2018 (UTC)
 * It looks OK for a GA nomination to me. XOR&#39;easter (talk) 23:11, 27 January 2018 (UTC)
 * Whoops! I wiki-linked "place" to the wrong, er, place. But I do think it should be linked somewhere, both to parallel the other terms in that sentence and to provide further information about one of the more opaque terms I've seen in a while. Maybe the target should be Absolute value (algebra)? XOR&#39;easter (talk) 23:57, 27 January 2018 (UTC)
 * I left it unlinked because I thought that complete field immediately following it in the same clause was the correct target, and had a link already. —David Eppstein (talk) 00:30, 28 January 2018 (UTC)
 * I've gone ahead and made a GA nom. We can continue dealing with minor issues like the link discussion above (which I don't want to cut off prematurely) while we wait for a reviewer. —David Eppstein (talk) 04:48, 28 January 2018 (UTC)

Having gone away for a bit and now read over that section again, I think I'm basically fine with it as written. I had a general sense of disquiet with the way the term place was introduced, I think because it is a very ordinary word being used in a technical way for which the everyday sense is not really an indicator. Instances of that often distract me and make me wonder just how that word ended up with that meaning. Is the connection obvious to everyone but me? Is it a literal translation of a nineteenth-century German word applied when the subject was understood less than it is now, kept on for historical reasons? Is this a case like magma, where nobody really knows what Bourbaki were thinking, but we have some sensible lexicographic guesses? XOR&#39;easter (talk) 18:35, 30 January 2018 (UTC)
 * My folk etymology is that a place is where something local happens in the local–global principle. "Locale" and "localization" were already taken for other meanings. I think the term is due to Weil. "Prime" has a common everyday English meaning, too, roughly synonymous with "first". I wonder how it acquired its number-theoretic meaning? —David Eppstein (talk) 19:02, 30 January 2018 (UTC)

How to deal with "primorial"?
Recently I encountered for the first time in my life the term "primorial prime". It is contained as table entry in a table about largest primes of several types, and links to the homonymous article, which itself links to the article about the primorial function. A (the?) notation for this function, a postfixed hashtag, is also employed in this table.

Assuming a negligible importance of these terms and the denotation, especially compared to the eponymous "factorial", I still feel a certain need, as a service to the expected readership, to explain them locally, perhaps in a footnote to the table entry, e.g., like: primorial prime. Purgy (talk) 11:45, 31 January 2018 (UTC)


 * Actually it's the product of the first n primes, not the primes up to n. But anyway I added the footnote. —David Eppstein (talk) 18:16, 31 January 2018 (UTC)
 * Just for WP's sake, the notation n# (as used in the table) yields the product of all prime numbers up to n, and pn# yields the product of the first n primes. May I ask you to check your source? Purgy (talk) 18:38, 31 January 2018 (UTC)
 * You're right, primorial says there are two conflicting definitions, and the book source I used also uses the same definition you gave, not the one in my comment above. Will fix. —David Eppstein (talk) 18:48, 31 January 2018 (UTC)

On "Computation"
Computation → Localizing primes - Concrete primes - Individual primes - Computing primes Wish I were inspiring. Purgy (talk) 14:32, 3 February 2018 (UTC)
 * Could also be "Algorithms". —David Eppstein (talk) 16:13, 3 February 2018 (UTC)
 * Your button is bigger than mine. ;] Purgy (talk) 09:03, 4 February 2018 (UTC)

Efforts to help with clerical details
Recently I tried to deal with some details, either mentioned in the GAR, or on my own estimation. I certainly do not want to disturb the reviewing process, but I missed the reasons for reverting edits in these examples: Sieves: My effort to describe -independently of the animation- the object the sieve is operating on and its product by "a list of subsequent numbers, starting with 2, and generating all the primes in this list" is completely removed as "unintelligible" and the caption I wrote, qualified as "nicely summarizing" by the reviewer, is replaced with a wrong description. My correcting variant in an other style was also reverted. Now it is hopefully correct. History: I tried to emphasize that the first archeological find spectacularly dates back to the paleolithic &mdash;removed, together with an attempt to construct the timely flow from there via the Rhind papyrus to the ancient Greeks. It may well be that this flow is beside the top mathematical facts, but I would at least spend these few words to make this section more interesting to an encyclopedically minded readership. Trial Division: Triggered by the reviewer, asking for detail on "grows too rapidly", I added from the given source the growth of the number of prime divisors, relevant to the considered algorithm and treated in this article. &mdash;reverted, and later on related -far more vaguely- to exponential growth of numbers with respect to the number of their digits: exponential. Why? BTW, for precision sake I put the word "maximally" next to the upper bound sqrt(n) of divisors. This is certainly not perfect. It was intended to hint that the full range of divisors is not necessarily to be exhausted, one lucky guess on a factual divisor might suffice. Simply removing the word "each" would probably do the job better. I do not mind being reverted for preferring "not greater" to "less or equal".

Since, as I repeatedly said already, my prime target is not to interfere with making this article a good one, I suspend any activities of mine until some distinct clarifications. Purgy (talk) 12:17, 5 February 2018 (UTC)

Resource check script

 * Script being tested/under development.
 * Inconsistent use of Location parameter (23 with; 62 without);
 * Adler, Irving (1960).  Missing Identifier/control number, e.g. OCLC;
 * Beiler, Albert H. (1966).  Pub. too early for ISBN, perhaps needs orig-year; Missing Identifier/control number, e.g. OCLC;
 * Deutsch, P.  Missing Year/Date;
 * Euclid's Elements, Book IX, Proposition 20. See David Joyce's English translation of Euclid's proof or Williamson, James (1782). The Elements of Euclid, With Dissertations. Oxford: Clarendon Press. p. 63. Missing Identifier/control number, e.g. OCLC;
 * Hua, L. K. (1965). Pub. too early for ISBN, perhaps needs orig-year;
 * Hardy, Godfrey Harold (1940). Pub. too early for ISBN, perhaps needs orig-year; Missing Identifier/control number, e.g. OCLC;
 * Goodrich, Michael T.; Tamassia, Roberto (2006).  Missing ISBN;
 * Knuth, Donald E. (1998). Missing ISBN; Lingzhi ♦ (talk) 00:04, 9 February 2018 (UTC)
 * OK, I think I've gotten everything except the location parameters. XOR&#39;easter (talk) 19:36, 10 February 2018 (UTC)

Group theory
In the account of the Burnside theorem, it is not said if the two primes are required to be distinct or not. — Preceding unsigned comment added by 2A00:23C0:7C80:8401:41F9:319F:21B2:8E74 (talk) 16:00, 14 February 2018 (UTC)
 * Luckily, the words "Burnside theorem" in the article are a link, and anyone who is confused by this can click the link and see the extremely clear statement in the first paragraph there. --JBL (talk) 16:37, 14 February 2018 (UTC)
 * There is some ambiguity in whether "only two primes" in this article means at most two primes or exactly two primes. But fortunately, it doesn't matter much because the one-prime case gives cyclic groups, the zero-prime case gives the trivial group, and all are solvable. I think the current wording doesn't need to be made more technical and difficult to read by trying for more precision in a case where that precision is irrelevant. —David Eppstein (talk) 19:01, 14 February 2018 (UTC)

Fundamental theorem of arithmetic
Since you and me discussued the inclusion of al-Farisi's work in the article and the correction of Euclid's contribution, i would welcome your opinion about my edits on the mainspace of this article. , your opinion is also welcome. Since i'm a new user, your insight is important for me. Thanks for your valuable time. Best regards.---Wikaviani (talk) 16:01, 4 May 2018 (UTC)
 * My opinion is that the fundamental theorem clearly appears in the work of Euclid. He formulated it in an awkward way that makes the unique factorization part apply only to squarefree numbers, but it's there too. Modern authors have rewritten the theorem in modern terminology without that flaw. I don't know the history of who first corrected it and wrote out the theorem in what we would recognize as its modern form, but your sources (because they don't count Euclid at all) aren't informative with respect to that question. —David Eppstein (talk) 16:06, 4 May 2018 (UTC)
 * Thanks for your answer. the problem is that i've looked on the internet but i was not able to find a reliable source stating that Euclid CLEARLY stated the fundamental theorem, but all sources state that this theorem is easy to deduce from his work. More, R. Rashed who is a prominent expert on that matter states that al-Farisi stated the theorem for the first time ( page 385). If we consider this theorem with modern expression, Gauss was the first to state it. I think that the article is misleading by making no mention of al-Farisi's contribution while reliable sources support his contribution. Here another source stating that the theorem is easy to deduce from Euclid's work (but the theorem is not in his work) : page 4. Best regards.---Wikaviani (talk) 16:18, 4 May 2018 (UTC)
 * The sources provided by do not state which version of the fundamental theorem of arithmetic has been proved by al-Farisi (existence and/or uniqueness?). A fortiori, they do not give any indication of which result of al-Farisi is not in Euclid's elements. Thus the assertion that al-Farisi was the first to prove the theorem is not verifiable from the given sources. Therefore these sources are not reliable for supporting the assertions added by . al-Farisi's results may be cited in the article only if some sources are provided that describe accurately these results. This is not yet the case. D.Lazard (talk) 16:46, 4 May 2018 (UTC)
 * Hi D. Lazrad, thanks for your comment. The article on prime number and the article on the fundamental theorem of arithmetic do not contain any sources supporting the Euclid claim, but this does not seem to bother you, could you please explain me why ? I would appreciate a mathematician's opinion on this matter. I'm not a specialist of number theory, but maybe , who nicely worked with me and others on the Sharaf al-Din al-Tusi's article, can help us here. Please note that my concerns are not driven by any nationalistic view, i just try to improve the article and make it the most complete possible. But if the community does not want to recocgnise al-Farisi's contribution on this topic, that's fine, let's keep Euclid, but we need reliable sources for any claim included in the articles, i think that our personal opinions are not requested in them, right ?---Wikaviani (talk) 17:23, 4 May 2018 (UTC)
 * We have Euclid's Elements to look at ourselves. It is not difficult to read in translation. —David Eppstein (talk) 17:30, 4 May 2018 (UTC)


 * Here is a source for existence only (the authors wrote an earlier paper dedicated to al-Farsi, see References,2). "The first known proof of existence is due to al-Farisi (died c. 1320), but he did not go on to prove uniqueness, mainly because his interest was in the divisors of a number rather than the factorisation itself." ("The fundamental theorem of arithmetic dissected", A Agargün and CR Fletcher, Mathematical Gazette, 1997) Wiqi(55) 17:32, 4 May 2018 (UTC)
 * If just reading the Elements was enough, there would not be so much confusing history about this theorem. Check the sources i posted above, Andrew Granville says the history of this theorem is confused page 4. He also states that  "The oldest surviving text WITH A CLEAR STATEMENT that every positive integer can be written as a finite product of prime numbers is given by al-Farisi", page 4. This is a very clear statement and contradicts the fact that Euclid CLEARLY stated the theorem.---Wikaviani (talk) 17:45, 4 May 2018 (UTC)
 * Hi Wiqi55, nice to see you again ! in fact al-Farisi "gave all that is needed to prove the uniqueness", see, introduction. This reference also states that the theorem dies not appear in its modern form in Euclid's work and that the history of the FTA is strangely obscure. So this is not that simple to read the Elements and draw conclusions.---Wikaviani (talk) 17:52, 4 May 2018 (UTC)

, you appear to have removed one of my comments from this discussion. Do not do this. Here it is again.
 * Ok, so (e.g. ): Euclid (IX.14) proved uniqueness, but only for squarefree numbers. Many sources agree that existence is a corollary of Euclid VII.30-31, but is not explicitly stated by Euclid. According to this source, al-Farisi took the additional step of stating existence and deriving it as a corollary. This still doesn't tell us who first stated it in the full existence-and-uniqueness-for-all-numbers form. (Certainly at least Gauss did so but that seems very late for this.) —David Eppstein (talk) 17:38, 4 May 2018 (UTC)

To this I would add: if you are going to fault Euclid for providing all the ingredients for existence but not explicitly stating it, it seems very unfair to credit al-Farisi for providing the ingredients for uniqueness but not explicitly stating it. Especially since a form of uniqueness is already in Euclid. —David Eppstein (talk) 18:07, 4 May 2018 (UTC)
 * The quotation does not give any indication on what is in al-Farisi's work and is not in Euclid's elements. And I have not access to the source without paying. Thus, without further information, the only thing that can be added to the article is Some authors assert that the first known proof of existence is due to al-Farisi (died c. 1320). Another acceptable formulation, would be The theorem has been essentially proved by Euclid, but the first known explicit statement that every natural number greater than one is either a prime or a product of primes is due to al-Farisi. Also, all proofs of uniqueness that I know are corollaries of Euclid's lemma (Elements Book VII, Proposition 30). So, what is "needed to prove the uniqueness" and is not in Elements? Without answer to this question, the sentence is nothing else than an opinion. D.Lazard (talk) 18:10, 4 May 2018 (UTC)

Sorry, which comment have i removed ? if so, it's because the computer i use for this thread is short of RAM and often freeze, that makes my typing slow and makes me do some mistakes. Sorry again. I understand your point of view about Euclid and al-Farisi, but the current version is unfair with al-Farisi because there is not a single mention of him in the article while he seems to have made significant contributions to this topic. Euclid is presented as the one who discovered the theorem. I think D. Lazard's compromise is quite well balanced. , which source do you need to access ? i can help you for that if you want. Also, you're right about the proofs of uniqueness, they are corollaries of Euclid's lemma.---Wikaviani (talk) 18:41, 4 May 2018 (UTC)
 * It's important to note that the place where this material would go (the history section) is really telegraphic; there isn't really room in there to go into a lot of details. The separate fundamental theorem article is where details should go. So what would be needed here is, first, how to clearly and concisely state Euclid's contribution to this theorem (currently "Euclid's Elements proves ... the fundamental theorem of arithmetic") and second, again concisely, state al-Farisi's contribution (maybe something like "In YYYY, al-Farisi provided the first explicit statement of the existence of prime factorizations", but that only makes sense if the earlier sentence about Euclid makes clear that Euclid provided the main ideas of the proof but never stated the fundamental theorem explicitly. —David Eppstein (talk) 20:55, 4 May 2018 (UTC)
 * First, i would like to thank all you guys for the valuable time you have devoted to this discussion. Second i completely agree with you now, David, when you say that this article is not the right place for details about the history of the theorem. Third, i need some time (probably several days) to make accurate researchs before editing the Fundamental theorem of arithmetic's article. With your permission, i would like to ping you before on the talk page to show you what i would like to write in the article before editing the mainspace article (perhaps something close to D. Lazard's proposal), in order to avoid any disruption of Wikipedia. Would this be ok for you ? Thanks again.---Wikaviani (talk) 21:55, 4 May 2018 (UTC)
 * Thinking about it some more, I think Euclid VII.30 and VII.31 really are the uniqueness and existence conditions (respectively) of the fundamental theorem; just written in the language of individual prime factors instead of the whole factorization at once. 31 says there is a prime factor and 30 says that whenever you divide by anything else it will still be there. What's missing from them, compared to the modern formulation, is looking at the whole factorization at once as a single mathematical object. The whole factorization at once is present in IX.14, but only for squarefree numbers and (because it doesn't work for all numbers) only the uniqueness part and not existence. And why is it in book IX instead of next to the other two propositions? It's almost like Euclid took parts of two number theory texts and glommed them together in a not-completely-consistent way (not the only instance of this kind of inconsistency). —David Eppstein (talk) 07:51, 5 May 2018 (UTC)
 * So do you think it's worth including some other mathematicians contributions (i think of Gauss and al-Farisi) or Euclid's work is enough ? As far as i know al-Farisi proved the uniqueness in the general case and Gauss can be credited of stating the theorem in its most modern form and with modern algebraic notations, but maybe i'm wrong, i need more time to make some researchs. But if you think that mentioning Euclid alone is enough, then it's ok for me and i will self revert on the al-Farisi's article to keep some logic between the different articles about this topic. By the way i found the whole paragraph from you that i removed yesterday and again, sorry for that, i was trying to cut/paste a part of the comment i was typing then the keyboard freezed and i made that mistake, stupid machine. Best regards.---Wikaviani (talk) 15:56, 5 May 2018 (UTC)

Semi-protected edit request on 18 June 2018
the math is done wrong 21 is not a prime number as 7x3= 21 Milatarymarshmallow (talk) 06:05, 18 June 2018 (UTC)
 * Pictogram voting question.svg Question: And where does it say that 21 is a prime number? ChamithN   (talk)  06:11, 18 June 2018 (UTC)
 * Possibly this is referring to " the largest number that has been factored by a quantum computer running Shor's algorithm is 21." Also the gears diagram says that 21 is relatively prime to 13.-- ♦Ian Ma c M♦  (talk to me) 07:09, 18 June 2018 (UTC)
 * In which case, there's nothing to fix, as both statements are correct. Regarding Shor's algorithm, it says "the largest number", not "the largest prime number". Also, 21 and 13 are relatively prime since they only have 1 as a common factor. -- ChamithN   (talk)  08:07, 18 June 2018 (UTC)

Pokémon
Would it be worth adding, under the Arts and Literature section, that the category of Pokémon known as "Ultra Beasts" has prime numbers as a motif? In the Pokémon video games, Ultra Beasts can only learn moves at levels equal to prime numbers, all of their base stats are prime numbers (with the exception of Naganadel's 121 base Speed), and the quantity of confirmed Ultra Beast species is always a prime number (7 in Pokémon Sun and Moon and 11 in Pokémon UltraSun and UltraMoon). Ron Stoppable (talk) 01:36, 2 August 2018 (UTC)
 * No. –Deacon Vorbis (carbon &bull; videos) 01:41, 2 August 2018 (UTC)
 * WP:POPCULTURE is the guideline here. A person can live without knowing this. It is more on topic at List of Pokémon characters. There is also a problem with sourcing, because I couldn't find much beyond fan sites which mention this.-- ♦Ian Ma c M♦  (talk to me) 05:22, 2 August 2018 (UTC)
 * Absolutely not. Someone interested in Pokémon may be interested in learning that, but someone interested in the mathematics of primes is not likely to, so it doesn't belong in this article. The editor who uses the pseudonym "JamesBWatson" (talk) 11:01, 2 August 2018 (UTC)

set of potential primes > 7
Here is the set of all potential primes greater than 7 (30k + r), r = {1,7,11,13,17,19,23,29}, k is any integer 0 to infinity. the only ones that are not prime are divisible by (30x + r). (30(0) + 1)(30k + r) = 30k + r, so the first prime with r = 1 will be at k = 1. This also means it is not worth checking (30(0) + 1) as a factor of a composite.

If you take all r, (r1 x r2) mod 30 it will give you the valid pairs of composites. Example (11 x 11) mod 30 = 1. So if you take a number n/30 remainder 1 it has potential divisors of (30x + 11)(30y + 11).

(11,11) is a valid pair for remainder 1. 1 and 19 have 6 valid pairs all other r have 4. This is because we have 8 numbers and a • b = b • a. 1 and 19 have 2 more pairs because or the squares of r’s exist there.

So if you take any number n/30 if it does not have a remainder in the set r it is not prime. If it does have a remainder in set r it is prime if it cannot be divided by a number from one of the found valid pairs. 4087 = (30(1) + 7)(30(1) + 1) LandonL (talk) 12:04, 8 June 2024 (UTC)


 * This is essentially the sieve of Erastothenes stopped after having marked the multiples of 2, 3 and 5. D.Lazard (talk) 13:47, 8 June 2024 (UTC)
 * You can use the extend form of composites to tell you all non-primes in order. N - r3 = 900xy + 30(x)(r2) + 30(y)(r1) + (r1)(r2)
 * All potential primes > 5 are 30(z) + r3. You can now print out all primes in order in O(1) time. LandonL (talk) 13:57, 8 June 2024 (UTC)
 * I want to also note the set of all potential primes could be written as 15o + w where w = {-14,-8,-4,-2,2,4,8,14} and o is any positive odd integer.
 * Separate note:
 * Since N = 900xy + 30(x)(r2) + 30(y)(r1) + (r1)(r2), the range, to find a factor on N, for x is 0 to sqrt(n/900) or sqrt(n)/30.
 * So long as r1 <= r2 LandonL (talk) 16:53, 9 June 2024 (UTC)

This is not the place to write original research. Unless you can find a reliably published reference for this material it is off-topic here. —David Eppstein (talk) 17:04, 9 June 2024 (UTC)


 * This isn't original, and this article already has a relevant section describing the idea at a high level and wikilinking to more detailed articles covering the details. –jacobolus (t) 19:26, 9 June 2024 (UTC)
 * I have just corrected the sieve because 2,3,5 should not be in the sieve if you want to see the set of potential primes larger than 5. Which if you subtract the set created by 900xy + 30(x)(r2) + 30(y)(r1) + (r1)(r2) you will have all your prime numbers greater than 5. It is easier to see, if in the composite set you change 1 in r to 31. I know I have never seen the set of potential primes, composites, or primes written out like this. But if it helps others explain things related to prime numbers that’s fine. If I am just an idiot wasting time, that seems fine as well. LandonL (talk) 20:27, 9 June 2024 (UTC)
 * This is essentially just saying that if you are looking for prime numbers you don't have to consider any numbers which are multiples of the first few primes. Not a particularly significant observation. JBW (talk) 21:24, 9 June 2024 (UTC)
 * What I am saying is you can print out all potential primes in order without missing any, and at the same time you could print out all the numbers that are not prime in the set of potential primes in order. You can do a loop within a loop to print out prime numbers in 0(1) time, which seems much faster than what everyone is doing now. LandonL (talk) 21:35, 9 June 2024 (UTC)