Talk:Euclid's theorem

message: Providing this argument isn't false (because it definitely is not a proof) it may be the most easy-to-understand argument for the infinitude of the primes.
Rather than ordering numbers by quantity, order them in a different way. Order them like this, where all the letters are the prime numbers (which we know not the quantitative value of yet):
A, B, 2A, 2B, C, 3A, 3B, 3C, 2C, D, 4A, 4B, 4C, 4D, 3D, 2D, E, 5A, 5B, 5C, 5D, 5E, 4E, 3E, 2E, F, 6A, 6B, 6C, 6D, 6E, 6F, 5F, 4F, 3F, 2F, G, 7A, 7B, 7C, 7D, 7E, 7F, 7G, 6G, 5G, 4G, 3G, 2G, H,.......
I think the pattern can be seen by now, provided there are no typos. It's kind of organizing numbers in such a way that they are put in order by groupings of primes rather than by quantity. A is the prime of lowest value, B is the prime of second lowest value, etc.
Using this system, every prime and composite number should be able to be generated eventually, even though they'll be generated out of quantitative order, and some composite numbers will be repeated. (If this were a proof I'd probably have to reference the fundamental theorem of arithmetic here.) Provided all primes and composites are generated by the pattern above, it's pretty easy to see that by following the pattern, you'll have to generate a prime (represented by the letter characters) in order to keep counting once you run out of multiples of previous primes. Therefore the primes are infinite.
Theboombody (talk) 15:53, 1 March 2014 (UTC)
replies: text: This argument doesn't work. You've assumed what you are trying to prove -- that at the point in your pattern where you go "...4X, 3X, 2X, hmm, next would come the 25th prime, is there a 25th prime?" there will always be a new prime. 109.246.123.214 (talk) 22:46, 28 April 2019 (UTC)
thread_title:  New"?" Easier Argument For the Infinitude of Primes? 

message: (moved this from my talk page, where it doesn't serve any useful purpose. Sapphorain (talk) 20:13, 6 September 2018 (UTC)
replies: text: « Well-known » might in some cases (in everyday’s life and in the media) not be a weasel word, but certainly not in any specialized scholarly subject. In the present case its use is completely subjective. I am a number theorist, and even though the theorem itself and its classical historic proofs are of course « well-known » to me, some of the proofs presented in the article were not. For a mathematician working in another field I should say that only Euclid’s is «  well-known ». It is doubtful that even this proof is familiar to a non-mathematician scientist. And  the great majority of other people’s knowledge concerning prime numbers comprises nothing more than the definition. Wikipedia is not a mathematical, let alone a number theory, encyclopedia.Sapphorain (talk) 20:13, 6 September 2018 (UTC)
thread_title:  "Weasel word" 

message: I find some statements in this article misleading or out of topic:
(1) Using "Proof by construction" as the title gives the impression, that Saidak's proof is something special, or rare, or unusual etc. However, there are several other direct (induction) proofs of the infinitude of primes. For instance,
(i) consider the infinite sequence e(0), ..., e(i), e(i+1), ... given by e(0) := 2 and e(n+1) := 1 + e(n)*e(n-1)*...*e(0). Prove by induction that all members of the sequence are pairwise co-prime, which entails the truth of the statement;
(ii) consider the infinite sequence f(0), ..., f(i), f(i+1), ... of Fermat numbers defined by f(n) := 1 + 2 ^ (2 ^ n). Prove by induction that all members of the sequence are pairwise co-prime, which entails the truth of the statement;
(iii) consider the infinite sequence a(0), ..., a(i), a(i+1), ... given by a(0) := 3 and a(n+1) := pf(a(n)!-1), where pf(x) is the smallest prime factor of x (if x ≥ 2). Prove by induction that a(m) < a(n) for all m < n, which entails the truth of the statement;
(iv) consider the infinite sequence b(0), ..., b(i), b(i+1), ... given by b(0) := 2 and b(n+1) := pf(b(n)!+1). Prove by induction that b(m) < b(n) for all m < n, which entails the truth of the statement (this is the proof skeched in subsection "Variations" of section "Euclid's proof).
(2) Why is it stressed that Euclid's Lemma is not used in Saidak's proof? This is neither unique nor eminent - none of the proofs under (1) use Euclid's Lemma.
(3) What is the remark that Saidak's proof does not resort to "reductio ad absurdum" good for? Neither the proofs in (1) nor in this article are indirect proofs. In particular it is explicitly mentioned in section "Euclid's proof" that it is not by contradiction. 
(4) Whether the infinitude of primes is proved directly or indirectly is irrelevant for the subject and respective material should be eliminated as it may irritate people. Also using three different notions (proof by contradiction, indirect proof, reductio ad absurdum) for the same subject may confuse people. This should be discussed elsewhere, i.e. in an article about (mathematical) Logic or Proof Techniques or something similar. 
(5) "Proof by construction" is the wrong term for Saidak's proof. It is an induction proof (like the proofs in (1) ).
Jack Rusell (talk) 21:36, 15 November 2019 (UTC)
thread_title:  Misleading and obsolete notions 

message: The section Proof using information theory has a quite wrong contribution. Although proofs in similar directions appear even earlier, it is Chaitin that related the infinitude of primes and information theory in: G. J. Chaitin, Toward a mathematical definition of life, in The Maximum
Entropy Formalism, R. D. Levine and M. Tribus, eds., MIT Press, Cambridge, 1979, 477–498.
:This proof can easily dramatically simplified and does not requires information theory: Given an integer , the beginning of the proof shows that, if n\le N and n = {p_1}^{e_1} {p_2}^{e_2} \cdots {p_k}^{e_k}, then e_i<\lg N for every . So, the tuples (e_1, \ldots, e_k) with this property encode a set of numbers containing all integers less than or equal to . This shows N\le (\lg N)^k for every integer N, a contradiction. There is no need of information theory or incompressibility theorem.
:IMO, a proof that can easily be simplified dramatically does not belong to an encyclopedia. The simplified proof could replace it if it can be sourced. Unfortunately I do not know any source. As it is possible that a source exists, I'll simply tag the section with . D.Lazard (talk) 13:34, 14 September 2021 (UTC)
replies: text: Is that the correct tag to use? The page currently says "This section's factual accuracy is disputed.", which seems strange as no one has actually contested factual accuracy - whether there is a simpler similar proof or not, the facts of the current proof seem correct and are sourced. 2A00:79E0:4A:203:8521:3101:B81A:EB5D (talk) 14:55, 13 October 2021 (UTC)
replies: text: The disputed fact is whether information theory is really used here. D.Lazard (talk) 16:51, 13 October 2021 (UTC)
thread_title:  Wrong contribution regarding information theory