Wikipedia:Peer review/Prime number/archive1

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The article has been nominated as a good article per the review in Talk:Prime number/GA1 and promoted eight years ago. I would like to request some improvements through the wholehearted users on commentary, for the sake of a featured article. Thanks. Dedhert.Jr (talk) 07:41, 19 May 2026 (UTC)

Some comments on the intro:

Is there a way to "future-proof" the phrasing "As of October 2024 the largest known prime number..."? As written, this looks like it could be something that has gone out of date. Does it mean that the largest known prime number was found in October 2024, or that the sentence was written in October 2024 and no one has updated it yet?

I am a little uncomfortable with this sentence: "Such questions spurred the development of various branches of number theory, focusing on analytic or algebraic aspects of numbers." The terms analytic and algebraic are wiki-linked, but that doesn't make them familiar, really, and the sentence is structured in such a way that it implies the reader knows what they mean. Contrast this with the last sentence of the paragraph: "In abstract algebra, objects that behave in a generalized way like prime numbers include prime elements and prime ideals." Here, unfamiliar things are introduced as objects that behave like primes. If you the random reader don't know what they are, and you probably don't, you at least come away with the idea that they're like prime numbers somehow. I feel like the "Such questions spurred..." line could be tweaked slightly to make it a bit less intimidating in this regard. Stepwise Continuous Dysfunction (talk) 16:41, 22 May 2026 (UTC)

In "Definition and examples", take the last line of the second paragraph: "Yet another way to express the same thing...". How is this "another way" to give the definition, rather than just the same definition again? Is this line necessary? Stepwise Continuous Dysfunction (talk) 16:44, 22 May 2026 (UTC)

Next section: "The Rhind Mathematical Papyrus, from around 1550 BC, has Egyptian fraction expansions of different forms for prime and composite numbers." I don't know what this means. An integer doesn't have an Egyptian fraction expansion; fractions do. Perhaps the article should say something like, "The Rhind Mathematical Papyrus, from around 1550 BC, contains Egyptian fraction expansions of numbers of the form ${\displaystyle 2/n}$. These were chosen in a way that suggests the author may have recognized a distinction between prime and composite." I don't think the given source could support that statement by itself (it barely mentions primes), but if my memories and the Rhind Mathematical Papyrus and Egyptian fraction articles are reasonably accurate, a statement like that could be justifiable. Stepwise Continuous Dysfunction (talk) 01:34, 23 May 2026 (UTC)

From what I have gathered, the Egyptian fraction is written as the fraction ${\displaystyle 2/n}$, where in some cases, prime and composite numbers are the denominators. See Egyptian fraction § Calculation methods and Rhind Mathematical Papyrus 2/n table § Explanations Dedhert.Jr (talk) 03:45, 23 May 2026 (UTC)

No. An Egyptian fraction is a sum of distinct unit fractions. 2/n is not a unit fraction. The Rhind papyrus contains expansions as sums of distinct unit fractions for the rationals that in modern notation would be written 2/n, for n ≤ 101; for instance, they expanded 2/37 as 1/24 + 1/111 + 1/296. There are often multiple choices for how to expand a given rational and the expansions used on the Rhind papyrus appear to use different forms for prime and composite numbers. For instance, for a composite n = ab you can expand 2/a and then multiply all the denominators by b, and the Egyptians did this for most but not all of the composite n in this range. See Egyptian fraction#Calculation methods for details. It would be kind of off-topic to expand all this in our article on prime numbers, but it does hint that maybe the Egyptians had some idea about distinguishing prime from composite numbers, and that's all we should need to say. The details are available for those who care to follow the Wikilinks.

As for sourcing, what the source (the MathSciNet review, not the paper reviewed by it) says of relevance is "This procedure explains why for prime numbers in the Rhind Papyrus apart form the first part all the others are multiples of p, p a prime." Which doesn't say much, but I think is enough to justify what we say, that the prime numbers had expansions of a different form (without making any stronger claim that the Egyptians knew this was because they were prime). If you want a more-developed source, Wilbur Knorr (writing about how the expansions in the Rhind papyrus might have been calculated) writes "There are two methods, depending on whether n has proper divisors or not.", p. 136 of Knorr, Wilbur (1982), "Techniques of fractions in ancient Egypt and Greece", Historia Mathematica, 9 (2): 133–171, doi:10.1016/0315-0860(82)90001-5, MR 0662138. —David Eppstein (talk) 06:30, 23 May 2026 (UTC)

I added an efn saying some of this, to avoid putting it into the main text. Also, in the main text, I changed "for prime and composite numbers" to "for fractions with prime and composite denominators". —David Eppstein (talk) 07:03, 23 May 2026 (UTC)

Looks good. Thanks. Stepwise Continuous Dysfunction (talk) 02:14, 24 May 2026 (UTC)

Prime number § Infinitude mentions "Kummer's elegant proof". This is a POV statement (claims of elegance in mathematics are widely but certainly not universally agreed upon), and neither the Ernst Kummer article nor the Euclid's theorem article, which discusses various proofs of infinitude, follow up on it. Stepwise Continuous Dysfunction (talk) 01:50, 23 May 2026 (UTC)

Prime number § Analytic properties says, "Although conjectures have been formulated about the proportions of primes in higher-degree polynomials, they remain unproven". I don't think the article follows up on this later. Should we insert some specifics about these conjectures, perhaps mentioning the Bunyakovsky conjecture or Schinzel's hypothesis H after the Ulam spiral? Stepwise Continuous Dysfunction (talk) 02:12, 23 May 2026 (UTC)

Some replies:

I have updated the year on the largest prime number. I have removed "elegant" for Kummer's proof; unless there's a reason on why many mathematicians regarded Kummer's proof as elegant by showing the description. Dedhert.Jr (talk) 03:07, 23 May 2026 (UTC)

Regarding to extend this, I have found arXiv (see for the journal). Other than Bunyakovsky conjecture and its extended version Schinzel's hypothesis H, Dickson's conjecture and Bateman–Horn conjecture can be added. Dedhert.Jr (talk) 04:41, 4 June 2026 (UTC)

• Is it possible to remove two citations a lead per MOS:LEADCITE? 🍕BP!🍕 (🔔) 15:30, 24 May 2026 (UTC)

  You ask this and simultaneously request citations in the summary material at the end of a section (diff), after those citation requests were already recently rejected for the same reason that they are summaries of later sourced material? Please engage with the article content more substantively rather than merely looking at the positions where footnotes appear and do not appear. We are not required to remove citations from leads of articles or the starts of sections when those citations are appropriate; neither are we required to add citations to leads of articles or the starts of sections when those parts of the article merely summarize later sourced material. It requires human thought about whether the citations are appropriate for the content, not superficial checks for where footnotes appear. So please, think.

  Now in the case of the two footnotes currently in the lead, it appears they are used only to support the "largest known prime number" claims detailed and sourced in a later table. So I am not convinced they are necessary. There is an "as of 2026" in the lead that does not appear in the article body, but I am not a big fan of this sort of phrasing as it needs constant updates, the sources do not actually support that this is true in 2026 (they are dated 2024), and it puts the article into unnecessary maintenance categories.

  I am also not convinced that the largest prime number factoid is necessary in the lead, and I think that if it is to go in the lead it would be better placed in the paragraph on distribution of primes rather than the paragraph about calculation methods. —David Eppstein (talk) 19:44, 24 May 2026 (UTC)

  Now improved by Stepwise Continuous Dysfunction: diff with content that avoids date-specific wording and fits more naturally in the calculation paragraph. —David Eppstein (talk) 20:41, 24 May 2026 (UTC)

I just noticed this line: The AKS primality test has mathematically proven time complexity, but is slower than elliptic curve primality proving in practice. This is probably completely confusing for anyone who isn't already familiar with computational complexity classes. Everyone else will wonder what "mathematically proven time complexity" is supposed to mean. The point here is, I guess, that "polynomial time" does not always equate to "fast" in practice, but we need a better way to say that. Stepwise Continuous Dysfunction (talk) 19:22, 25 May 2026 (UTC)

It's not really about polynomial time versus real world efficiency. They're all polynomial, and some polynomial algorithms being faster than others doesn't raise any interesting philosophical points. The real point is actually that some primality tests base their validity on unproven mathematical hypotheses (such as generalizations of the Riemann hypothesis), some use random numbers and only give you a very small chance of a false answer rather than an answer that is correct with certainty, and some have neither of these two flaws. But you pay for that certainty: the unconditionally-proven non-random algorithms (AKS) are much slower than the unproven or randomized methods. —David Eppstein (talk) 19:28, 25 May 2026 (UTC)

Hmm, OK. I don't think the current phrasing conveys that adequately. Perhaps the article should say something like, "The AKS primality test, though mathematically proven to [...], is slower than elliptic curve primality proving in practice." Stepwise Continuous Dysfunction (talk) 20:15, 26 May 2026 (UTC)

I get the feeling this discussion is happening out of context. Standing alone, that sentence may be confusing, but in context with its preceding sentence stating that "[ECP]'s runtime analysis is based on heuristic arguments rather than rigorous proofs", I think that what this sentence is saying about proven vs unproven behavior is unambiguous. —David Eppstein (talk) 23:41, 26 May 2026 (UTC)

Maybe it's just the "time complexity" part? Even in the context of the surrounding sentences, I still find that a confusing turn of phrase. It's saying that something is proven, but as written I can't tell what. Stepwise Continuous Dysfunction (talk) 04:51, 27 May 2026 (UTC)

It is proven that AKS runs in polynomial time. The polynomial time bound that we believe ECP to take is unproven, resting on heuristic arguments, but faster. There is a problem with your new wording: one can of course analyze ECP completely rigorously too. Your wording implies inaccurately that this is somehow impossible. It is just that the time we think it takes and that it takes in practice is different from the much larger time that can be proved for it. —David Eppstein (talk) 06:57, 27 May 2026 (UTC)

Feel free to change what I wrote, of course; it was just my attempt to squeeze your previous paragraph into a single sentence. It seems to me that the implication you mention was also present in the previous version. I left unchanged the first line: The elliptic curve primality test is the fastest in practice of the guaranteed-correct primality tests, but its runtime analysis is based on heuristic arguments rather than rigorous proofs. To me, this already suggests that there is somehow no way to do a rigorous analysis, rather than a rigorous analysis only being able to give a loose bound. Stepwise Continuous Dysfunction (talk) 15:26, 27 May 2026 (UTC)

Resolved. Thanks. Stepwise Continuous Dysfunction (talk) 02:40, 28 May 2026 (UTC)

Endnote [d] does not contain a reference itself. Can someone with access to Koblitz (1987) check if it verifies the endnote text? Stepwise Continuous Dysfunction (talk) 03:34, 4 June 2026 (UTC)

Well, strictly speaking Koblitz (p.116) describes the Solovay–Strassen test as checking whether ${\displaystyle a^{(n-1)/2}}$ is congruent to the Jacobi symbol ${\displaystyle \left({\tfrac {a}{n}}\right)}$ modulo ${\displaystyle n}$, rather than whether their difference is divisible by ${\displaystyle n}$, but it's the same thing. —David Eppstein (talk) 05:59, 4 June 2026 (UTC)

Thanks. Stepwise Continuous Dysfunction (talk) 16:39, 5 June 2026 (UTC)