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== Joke: There are no Uninteresting Numbers ==

First, one must consider what numbers actually are interesting: 1 is unity, 2 is the smallest even and smallest prime number, 3 is the smallest odd prime number, 4 is the first whole number that is the square of another integer, 5 is significant in Biology (many vertebrates have some version or form of 5 digits), 6 is the first perfect number, 7 is a historically important number in many religions and mythologies, etc, etc....

Assume that one can eventually find and group all of the Uninteresting numbers. By the principle of Well Ordering, they can be arranged from smallest to largest. Let's look carefully at this group:

The first number that one comes across is the smallest of all the Uninteresting numbers known to exist in the universe. Well, that's interesting....


--[[User:Daydreamer302000|Daydreamer302000]] 16:37, 28 February 2007 (UTC)
:We have an article about the [[interesting number paradox]]. [[User:PrimeHunter|PrimeHunter]] ([[User talk:PrimeHunter|talk]]) 10:34, 22 October 2018 (UTC)

::But then the next Uninteresting number is really the smallest, which robs the previous one of all interest.  [[User:BMJ-pdx|BMJ-pdx]] ([[User talk:BMJ-pdx|talk]]) 14:57, 26 July 2021 (UTC)

== Alternate proof is really the same proof ==

It looks to me like the only difference is the wording used and the fact that the "alternate" proof assumes [[Euclid's lemma]] without proving it.

Since the proof of Euclid's lemma can be found in its own main article, I think it makes sense either to simply reference Euclid's lemma here (removing its proof from this page, and condensing the two uniqueness proofs into one since they're really the same), or to build up the entire proof of unique prime factorization from first principles, with separate sections for the division algorithm, [[Bézout's identity]], etc.

[[Special:Contributions/18.189.112.192|18.189.112.192]] ([[User talk:18.189.112.192|talk]]) 06:44, 22 December 2011 (UTC)

:I agree, the alternate proof is the same as the main proof but skips steps (including Euclids lemma) to pull "Therefore,p_1q_1|s-p_1q_1" out of nowhere. Further the proofs concludes by stating nothing more than it's initial assumption (that s is the smallst integer with a non-unique factorization) and thus does not prove anything. I have removed the alternate proof. [[Special:Contributions/213.246.84.162|213.246.84.162]] ([[User talk:213.246.84.162|talk]]) 11:43, 1 April 2012 (UTC) Ali
:: Someone has now added a good alternate proof so this discussion is now mute. [[Special:Contributions/213.246.117.133|213.246.117.133]] ([[User talk:213.246.117.133|talk]]) 08:43, 14 May 2012 (UTC) Ali

:::I've rewritten the article in an atttempt to make it clearer and less redundant.

:::[[User:Virginia-American|Virginia-American]] ([[User talk:Virginia-American|talk]]) 19:27, 1 July 2012 (UTC)

::::The alternate proof is not something one finds in textbooks, it probably violates the [[WP:ORIGINAL|original research policy]].  [[User:MvH|MvH]] ([[User talk:MvH|talk]]) 01:20, 6 April 2014 (UTC)MvH

:::::The Alternate Proof should bee removed. It is "doublicated" material. It seems to be "own work". It is not relevant for an encyclopedia to have multiple proofs.
:::::[[User:LMSchmitt|LMSchmitt]] 08:49, 30 September 2020 (UTC)

::::::The 'alternative proof' needs an authoritative reference.  The one given here is a version, although less elegant and less concise, of the 'abnormal number' proof given in Hardy and Wright, and which they attribute to F. A. Lindemann (1933).  The reference to the Mathworld article on 'abnormal numbers', which has no context in the current article, suggest that the original context has been edited out and perhaps should be replaced. [[Special:Contributions/14.201.149.49|14.201.149.49]] ([[User talk:14.201.149.49|talk]]) 07:24, 24 December 2020 (UTC)

:::::::A correct version of this proof, which does not assume Euclid's lemma, appears on p. 45 of Dawson, John W., Bruce S. Babcock, and Steven H. Weintraub. 2015. Why prove it again: alternative proofs in mathematical practice. http://public.ebookcentral.proquest.com/choice/publicfullrecord.aspx?p=3567731. The alternative proof is valuable since the proof of Euclid's Lemma is not, in itself, 'obvious' to most readers. Arguably, the "alternative proof" is the more intuitive one, although it is less commonly seen. (NOTE: in a previous edit I wrote: "As it stands, the 'alternative proof' assumes Euclid's Lemma (" p1 must occur in the factorization of either q1 − p1 or Q.")," thinking that this was an error. However, at that point in the proof, the FTA (and therefore EL) can be assumed true for numbers less than s, so this is not a fault in the proof; it is how the proof is supposed to work! I will simply add the reference. [[User:TheNothingNihilates|TheNothingNihilates]] ([[User talk:TheNothingNihilates|talk]]) 16:39, 10 September 2021 (UTC)
::::::::The alternate proof is still wrong. It assumes that p1 must either divide (q1-p1) or Q because each of them is assumed to have a unique factorization. But assuming that p1 is part of it too much of a jump for a proper proof. [[User:Pavlix|Pavlix]] ([[User talk:Pavlix|talk]]) 11:11, 27 March 2022 (UTC)
:::::::::"Each of them is assumed to have a unique factorization": their product is also assumed to have a unique factorization. All these assumptions are parts of  the induction hypothesis. So, the proof is correct, since p1 appear explicitly in a factorization of their product. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 13:45, 27 March 2022 (UTC)
::::::::::Since about 2005, several years after I retired from teaching at Stanford, I've been maintaining the website [http://boole.stanford.edu/dotsigs.html Sayings of Chairman Pratt].  The third last entry in the section [http://boole.stanford.edu/dotsigs.html#ScienceAndTechnology Science and Technology] is titled "Number theory" and offers "World's shortest rigorous proof of unique factorization".  The proof I gave (for uniqueness) is as follows.
::::::::::''Let the least counterexample be pr = qs with p<q prime and r,s nonempty strings of primes. By leastness no prime occurs on both sides. Moreover p cannot divide q-p. Hence p(r-s) = (q-p)s yields a smaller counterexample.''
::::::::::Although I'd never seen a shorter proof I also didn't expect it to be original.  But now that I read this discussion I'm starting to wonder.  [[User:Vaughan Pratt|Vaughan Pratt]] ([[User talk:Vaughan Pratt|talk]]) 01:48, 14 April 2022 (UTC)
:::::::::::Or you saw it from a colleague and don't remember. See [https://books.google.ca/books?id=XB2nd2ovakIC&lpg=PA574&ots=YS6SH4zuHd&pg=PA574&redir_esc=y#v=onepage&q&f=false]https://books.google.ca/books?id=XB2nd2ovakIC&lpg=PA574&ots=YS6SH4zuHd&pg=PA574&redir_esc=y#v=onepage&q&f=false [[Special:Contributions/199.85.124.43|199.85.124.43]] ([[User talk:199.85.124.43|talk]]) 19:38, 18 July 2023 (UTC)

==Flawed==
Factors are not needed to describe a number's properties and this theorem fails to predict the position of primes. Scrap it <!-- Template:Unsigned --><small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:2601:449:8200:a430:483f:9db0:e740:6d4f|2601:449:8200:a430:483f:9db0:e740:6d4f]] ([[User talk:2601:449:8200:a430:483f:9db0:e740:6d4f#top|talk]] • [[Special:Contributions/2601:449:8200:a430:483f:9db0:e740:6d4f|contribs]]) 09:43, 19 February 2018 (UTC)</small> <br>
<small>... says who? Please, add comments at the bottom and sign them.</small>
:Some, if not most, prefer to keep it, and what is it that one "needs". Factors are useful to many, and what is "predict the position"?. [[User:Purgy Purgatorio|Purgy]] ([[User talk:Purgy Purgatorio|talk]]) 11:20, 19 February 2018 (UTC)

:This theorem is very important to prove that certain sets are countable. For example, that "the set of finite words over a countable alphabet is countable" is easily proven with the help of this theorem. The quoted statement is important in Model Theory. [[User:LMSchmitt|LMSchmitt]] 09:10, 30 September 2020 (UTC)
::who says that you can even factor numbers if not for this theorem though? doesnt any mathematical operation, or action require some theorem or logical argument to state this is valid? [[Special:Contributions/2A02:A46E:D6AB:0:E03E:572F:C70E:EA59|2A02:A46E:D6AB:0:E03E:572F:C70E:EA59]] ([[User talk:2A02:A46E:D6AB:0:E03E:572F:C70E:EA59|talk]]) 17:40, 15 January 2025 (UTC)

== Name for a number's set of prime factors? ==

Isn't there a name for the set comprising a number's prime factors? I see nothing in the various prime/composite-related articles.  [[User:BMJ-pdx|BMJ-pdx]] ([[User talk:BMJ-pdx|talk]]) 15:02, 26 July 2021 (UTC)
:{{u|BMJ-pdx}}, see an answer to the same question in [[Talk:Greatest common divisor]] [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 15:17, 26 July 2021 (UTC)

== Incorrect statement in generalization to gaussian integers ==

It is written
:''the non-zero, non-unit numbers fall into two classes, primes and composites, and that (except for order), the composites have unique factorization as a product of primes''
This is wrong. For example,
:(1+i)(2+i) = (1-i)(2i-1).
It is true only if one chooses one representative for each equivalence class of primes modulo the units. Or, one should add "(except for order '''and factors of units''')", or similar. &mdash; [[User:MFH|MFH]]:[[User talk:MFH|Talk]] 13:04, 7 April 2023 (UTC)

:{{fixed}}. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 15:55, 7 April 2023 (UTC)

== Contradicting/incomplete sources in History section ==

The History section states 

"While [[Euclid]] took the first step on the way to the existence of prime factorization, [[Kamāl al-Dīn al-Fārisī]] took the final step and stated for the first time the fundamental theorem of arithmetic."

In this sentence, two sources are cited:

8. A. Goksel Agargun and E. Mehmet Özkan.[https://core.ac.uk/download/pdf/82721726.pdf "A Historical survey of the Fundamental Theorem of Arithmetic"] (PDF). Historia Mathematica: 209. "One could say that Euclid takes the first step on the way to the existence of prime factorization, and al-Farisi takes the final step by actually proving the existence of a finite prime factorization in his first proposition."

9. Rashed, Roshdi (2002-09-11). [https://books.google.com/books?id=7veIAgAAQBAJ&q=fundamental+theorem+of+arithmetic+discovered+al-farisi&pg=PA385 Encyclopedia of the History of Arabic Science]?. Routledge. p. 385. <nowiki>ISBN 9781134977246</nowiki>. "The famous physicist and mathematician Kamal al-Din al-Farisi compiled a paper in which he set out deliberately to prove the theorem of Ibn Qurra in an algebraic way. This forced him to an understanding of the first arithmetical functions and to a full preparation which led him to state for the first time the fundamental theorem of arithmetic."

The first source however links only to pages 162-173 of "A Historical survey of the Fundamental Theorem of Arithmetic" and on page 172 the authors disagree that al-Farisi stated or proved the uniqueness of prime factorisation:

"In our estimation, al-Farisi neither stated nor proved uniqueness, and it was not his intention to do so."

So the first source disagrees with the second that al-Farisi stated and proved (the entirety of) the FTA. In fact, the authors directly cite and criticise a different work by Rashed, who provided the claim that al-Farisi stated and proved the FTA in the second source. [[User:Muisnotforyou|Muisnotforyou]] ([[User talk:Muisnotforyou|talk]]) 16:10, 1 February 2024 (UTC)

== Base case of the existence proof ==

{{Reply to|Jasper Deng}} There's a disagreement on whether highlighting that 2 is prime is necessary for induction proof to work. My point is that since we're basically doing well-founded induction over natural numbers with the divisibility relation, the base case is not 2 specifically, it is every prime. 2 would serve as the base case for every multiple of 2, but for n = 15 for example, we need 3 and 5 as base cases.

If we remove the "2 is prime" base case and try to run the proof on n = 2, we get that "2 is prime, there is nothing more to prove". So n = 2 is a special case of the base case already established, and there is no need to draw special attention to it.

For references, see page 41 of Elementary Number Theory by Rosen, or page 2 of Introduction to the Theory of Numbers by Hardy and Wright. [[User:Sheddow|Sheddow]] ([[User talk:Sheddow|talk]]) 19:12, 23 August 2024 (UTC)

:In strong induction, the induction step simply proves the statement for ''n'' whenever it is true for all ''k'' with 1 < ''k'' < ''n''. Without the base case, the induction step proves nothing.—[[User:Anita5192|Anita5192]] ([[User talk:Anita5192|talk]]) 19:24, 23 August 2024 (UTC)
::We're technically doing well-founded induction. [https://math.stackexchange.com/questions/2792061/understanding-the-definition-of-well-founded-induction]https://math.stackexchange.com/questions/2792061/understanding-the-definition-of-well-founded-induction The base cases of well-founded induction are the minimal elements, which under divisibility are the primes. If we were doing induction over the naturals under the normal ordering (<), then the base case would be 1 (or 0).
::To reiterate, I'm not saying there's ''no'' base case. I'm saying the base cases are every prime, so there's no need to draw special attention to 2. If 2 were the only base case, how would you prove that 15 is a product of primes?
::And feel free to check out the references I mentioned, they contain a proof of the fundamental theorem of arithmetic without mentioning that 2 is prime. [[User:Sheddow|Sheddow]] ([[User talk:Sheddow|talk]]) 19:45, 23 August 2024 (UTC)

:::For ''n'' = 15, we would already have proved that 3 and 5 are prime; again beginning with the base case 2.—[[User:Anita5192|Anita5192]] ([[User talk:Anita5192|talk]]) 20:24, 23 August 2024 (UTC)
::::Well-founded induction says that if you want to prove that <math>\forall a\in A, P(a)</math> for some proposition <math>P(a)</math> it is sufficient to prove that for all <math>a \in A</math>
::::<math>(\forall b \in A, b \prec a \implies P(b)) \implies P(a)</math>
::::Let me give a quick proof. Assume for sake of contradiction that the set <math>S = \{s \in A, \neg P(s)\}</math> is non-empty. Then it has a minimal element <math>a</math>, ie. if <math>k \prec a</math> then <math>k \notin S</math>. But this is the same as saying that <math>k \prec a \implies P(k)</math>, so we can use the inductive hypothesis to conclude that <math>P(a)</math>, which contradicts the fact that <math>a \in S</math>, so <math>S</math> must be empty.
::::{{break}}
::::Whether strong induction requires a base case depends on the exact version you use. If you use the inductive hypothesis
::::<math>(\forall b \in \Bbb N, b < a \implies P(b)) \implies P(a)</math>
::::then you don't need a base case, since it is just well-founded induction with <math><</math>. However, if you use
::::<math>(\forall b \in \Bbb N, b \leq a \implies P(b)) \implies P(a+1)</math>
::::you need a base case, since <math>\leq</math> is not a well-ordering. 
::::The idea is basically that when <math>\prec</math> is a well-ordering and <math>a</math> is a minimal element, then there are no hypotheses in the premise, so you're forced to prove the "base case" <math>P(a)</math>. If we use <math>\leq</math> there are no minimal elements, so the premise is never empty.
::::{{break}}
::::I realized by the way that we don't need to do well-founded induction on the divisibility relation in this case, we can just use the strong induction described above. I'll just give a somewhat formal proof for completeness.
::::Let <math>P(n)</math> be the proposition that <math>n</math> can be written as a product of primes.
::::Let <math>n \in \Bbb N_{\geq 2}</math>. Assume <math>\forall k \in \Bbb N_{\geq 2}, k < n \implies P(k)</math>. If <math>n</math> is prime then we have <math>P(n)</math> trivially. Otherwise there exist <math>a, b</math> such that <math>n = ab</math> and <math>a, b \in \Bbb N_{\geq 2}</math> and <math>a, b < n</math>. Then we can use the inductive hypothesis to conclude <math>P(a)</math> and <math>P(b)</math>, and since <math>n = ab</math> we have <math>P(n)</math>. Thus, <math>\forall n \in \Bbb N_{\geq 2}, P(n)</math>.
::::{{break}}
::::Note that we haven't bypassed the need for a base case; the base cases are just the points where we don't use the inductive hypothesis. You can only use the inductive hypothesis on <math>n</math> when there exists an element strictly less than <math>n</math>.
::::{{break}}
::::Interestingly, this same proof appears here: https://en.wikipedia.org/wiki/Mathematical_induction#Example:_prime_factorization
::::To be fair, this is a minor issue, the proof works either way. It just peeved me a bit to see what seems like an unnecessary repetition. [[User:Sheddow|Sheddow]] ([[User talk:Sheddow|talk]]) 16:21, 24 August 2024 (UTC)

== fundamental theorem of arithmetic and the number "2" ==

2 is a positive whole number

2 can not be expressed as the product of any prime numbers, since a product is a*b*c..etc, 2 is however a trivial factorization of itself.

the fundamental theorem of arithmetic CAN NOT state that 2 is a product of any such numbers, therefore it is logically incorrect.

this is basically a proof, given "my" definition of a product. (this point is raised in the wiki definition of product )

the fundamental theorem of arithmetic must state that you have unique factorizations to represent any whole positive integer, not a product, else there is a logical mistake in the maths, unless maybe a product is defined as a factorization itself, or as a sequence https://en.wikipedia.org/wiki/Product_(mathematics)

the wiki page on products actually recognizes my issue with the definitions in some capacity.

[[Special:Contributions/2A02:A46E:D6AB:0:65E6:8701:1AA0:782A|2A02:A46E:D6AB:0:65E6:8701:1AA0:782A]] ([[User talk:2A02:A46E:D6AB:0:65E6:8701:1AA0:782A|talk]]) 16:33, 15 January 2025 (UTC)

:See [[Product (mathematics)#Product of a sequence]]: "The product of a sequence consisting of only one number is just that number itself; the product of no factors at all is known as the [[empty product]], and is equal to 1."—[[User:Anita5192|Anita5192]] ([[User talk:Anita5192|talk]]) 18:11, 15 January 2025 (UTC)
::that seems valid, thank you for your patience [[Special:Contributions/2A02:A46E:D6AB:0:E03E:572F:C70E:EA59|2A02:A46E:D6AB:0:E03E:572F:C70E:EA59]] ([[User talk:2A02:A46E:D6AB:0:E03E:572F:C70E:EA59|talk]]) 18:31, 15 January 2025 (UTC)

== "up to the order of the factors"? ==

In the opining statement:

"In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the order of the factors."

What is "up to the order of the factors" intended to convey? A unique factorisation consists of a set of factors. Order is not significant, surely?  If this is true, the "up to" clause is superfluous, irrelevant and messy. If not, a cited rationalisation would be welcome. [[Special:Contributions/86.143.2.4|86.143.2.4]] ([[User talk:86.143.2.4|talk]]) 10:15, 17 March 2025 (UTC)

:A factorization is not a set of factors, but, formally it is a [[multiset]]. Clearly, the order of the factors is not significant, and the "[[up to]]" clause is useful for recalling that. Moreover, the word "represented" is fundamental: {{tmath|2\times 3}} and {{tmath|3\times 2}} are different as representations of 6 as a product, even if they are the same as factorizations (commutativity of integer multiplication is a theorem that is not so easy to prove). If you can propose a better formulation, without unneeded technicalities or pedantry, it would be welcome. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 10:51, 17 March 2025 (UTC)
::You're correct about multiset, of course.
::However, it is still superfluous to reference order. You will not find usage of set (or multiset) in serious mathematical texts that accompany it by a reminder that order is irrelevant. To do so in Wikipedia conveys an impression of poor quality and inadequate editorial oversight. It is not 'useful', it is clumsy and poorly written. A better formulation is simply to omit the superfluous wordage.
::Of course, some Wikipedia editors might prefer the latter, which would be a reflection of Wikipedia's modus operandi. [[Special:Contributions/86.143.2.4|86.143.2.4]] ([[User talk:86.143.2.4|talk]]) 09:55, 19 March 2025 (UTC)
:::I agree that "superflous wordage" must be omitted. This includes the words "set" and "multiset". Also, "order" does not refer here to any mathematical order. It refers to the order in which the factors are written. As the fundamental theorem of arithmetic predates the concepts of sets, multisets and [[order (mathematics)]] of more that 2,000 years, using them here would be both pedantry and an historical mistake. There is no "editorial oversight". [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 10:28, 19 March 2025 (UTC)
::::"As the fundamental theorem of arithmetic predates the concepts of sets, multisets and order... using them here would be both pedantry and an historical mistake."
::::Not pedantic but precise use of terminology. Historical order is irrelevant if a more recent term is precise and appropriate for the article. Of course such terminology wouldn't have been used with its current (mathematical) meaning when the original theorem was formulated, but neither would English. That is neither here nor there. The article is about the theorem, not about historical linguistics.
::::(Btw, "a historical" is correct English usage. The indefinite article is chosen according to  pronunciation, not spelling. "An istorical" would be correct, were there such a word).
::::'There is no "editorial oversight"'.
::::Exactly my point: a major cause of the highly variable quality of Wikipedia article content, selection and length, unlike  actual encyclopaedias which benefit from the consistency resulting from editorial oversight. There would not seem to be a way to address this long-standing problem that is compatible with free-for-all, often-amateur editing. [[Special:Contributions/86.143.2.4|86.143.2.4]] ([[User talk:86.143.2.4|talk]]) 20:35, 19 March 2025 (UTC)

== What about omnific integers ==

What about omnific integers? [[User:2007GabrielT|2007GabrielT]] ([[User talk:2007GabrielT|talk]]) 12:16, 1 May 2025 (UTC)

:What is it? Never heard of such a concept. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 13:39, 1 May 2025 (UTC)
::the integers of surreal numbers [[User:2007GabrielT|2007GabrielT]] ([[User talk:2007GabrielT|talk]]) 15:33, 1 May 2025 (UTC)
:::"Omnific" is not used in [[Surreal number]]. Do you have a reliable source for this terminology. In any case, this fundamental theorem is about natural numbers, not surreal numbers. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 16:00, 1 May 2025 (UTC)
::::Yea I find it strange thats its not on that page (or any page for that matter), but here are some soruces: [https://mathworld.wolfram.com/OmnificInteger.html] [https://planetmath.org/surrealnumber] [https://arxiv.org/abs/1710.07304] [http://wolfgang.bertram.perso.math.cnrs.fr/CoN.pdf] [[User:2007GabrielT|2007GabrielT]] ([[User talk:2007GabrielT|talk]]) 16:06, 1 May 2025 (UTC)
:::::In your sources, only Bertram's is really a [[WP:reliable source]]. PlaneteMath and MathWorld are not, being wiki's. ArXiV is not a reliable sourc, unless if the paper has been reliably published. From Beltram, it results that omnific integers are a subtopic of [[surreal number]]s. So, I would agree to adding a section on omnific integers to [[Surreal number]]s. If you have the competence for that, you may do this yourself. Otherwise, you could ask to the main contributors of this article, {{u|Michael K. Edwards}}, {{u|Lhmathies}}, and {{u|Quantling}}. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 13:07, 2 May 2025 (UTC)
::::According to a comment [https://mathoverflow.net/questions/72691/can-we-axiomatize-omnific-integers-without-the-surreal-number-system here], Conway came up with the name and the definition of Omnific integers. That should be enough for WP. (Page 45 of On Numbers and Games, it says. I don't have the book myself, and just because it's in Conway it doesn't have to be in the WP article). But according to a comment on the first answer, '''Oz''' is not the only way of defining an integer subRing of '''No''', and if adding a section to the article on Surreal Numbers, I'd like to see that mentioned.
::::I'm out of my depth, though, if you have to understand transfinitely recursive axiom systems to deal with this. Also it's ten years ago since I was stuck into the subject, even Conway's definition does not have an intuitive meaning for me. Looks like fun, but I doubt I'll have the time to commit to understanding it. [[User:Lhmathies|Lhmathies]] ([[User talk:Lhmathies|talk]]) 06:07, 3 May 2025 (UTC)
:Yes, what about them? Do they have unique prime factorizations? What is even a prime number in the Omnifics once you get beyond infinity? (A product of finite primes is obviously finite, so you'd need infinite primes). [[User:Lhmathies|Lhmathies]] ([[User talk:Lhmathies|talk]]) 06:12, 3 May 2025 (UTC)
::These are not facetious questions, I simply don't know if there is anything interesting to say about infinite integers in this context. Personally, I'd even find it interesting to know if primeness can be defined for Conway's Omnifics, but it probably shouldn't be in this article, or even in the one on Surreal Numbers. [[User:Lhmathies|Lhmathies]] ([[User talk:Lhmathies|talk]]) 12:07, 3 May 2025 (UTC)
::I mean a finite product. Things like 2<sup>ω</sup> are defined in '''No''', but I can't tell off hand if it's an integer. [[User:Lhmathies|Lhmathies]] ([[User talk:Lhmathies|talk]]) 12:13, 3 May 2025 (UTC)