Talk:Twin prime

Chris De Corte proved that the number of twin primes depend on the number of primes below 2n

This can be found on page 21 of his paper "Derivation of a Prime verification formula to prove the related open problems". — Preceding unsigned comment added by Chrisdecorte (talk • contribs) 04:21, 18 July 2013 (UTC)

Self-published. I can't tell if it's drivel, but I've never seen a serious result first published as a slide show. arXiv is more credible than that. — Arthur Rubin (talk) 04:37, 18 July 2013 (UTC)

It's absolute nonsense. Among other things, it claims to prove the Goldbach conjecture by asserting that if p is an odd prime and 2n is composite, then 2n-p must be prime (page 14), which is immediately contradicted by infinitely many examples like 28-3=25 not being prime. Joshuazucker (talk) 17:34, 15 September 2020 (UTC)

Goldbach requires p+q=2n but then for n not divisible by 3; p,q,n are congruent mod 3 or 2n-3 is prime... Roderick MacPhee (talk) 19:18, 2 September 2024 (UTC)

A paper by Daniel Goldston

http://www.math.sjsu.edu/~goldston/twinprimes.pdf

I want to at least read some of it and skim the rest. This might be a worthy addition to our article. FriendlyRiverOtter (talk) 02:41, 6 February 2018 (UTC)

the link appears to be broken/obsolete.Billymac00 (talk) 12:58, 19 June 2022 (UTC)

5 only prime in 2 distinct pairs

I wouldn't say that (3,5) and (5,7) are distinct, since they overlap. Is there a mathematical reason for the word 'distinct'?--2607:FEA8:D5DF:F945:8135:8E64:C05B:63FC (talk) 16:52, 3 January 2020 (UTC)

"Distinct" means "not the same" or "different". (In some mathematical contexts one can have many copies of equal things -- for example, the collection (1, 1, 2, 3, 3, 3) contains six entries but only three distinct entries.) I agree that the word does not seem necessary where it is used in the article, and have reworded to a less jargon-y phrase. --JBL (talk) 17:02, 3 January 2020 (UTC)

Result to watch: Czelakowski's claim to have proved the conjecture in standard PA

https://link.springer.com/article/10.1007/s11225-022-10017-2

The author claims to have proven that the twin Prime conjecture holds in the standard model of Peano arithmetic. The paper appears in a serious, peer-reviewed mathematical journal. If I understand its main claim correctly, it is weird that it has received so little attention. Hence, I may be misunderstanding the paper. If someone more knowledgeable than me could check the paper, we would understand better whether the result is worth including in this page or not. Many thanks. Mbtnt (talk) 17:27, 26 October 2022 (UTC)

Just a quick fly-by remark: Proving that the result holds in the standard model of PA (that is, the natural numbers) is not the same as actually proving it in PA. The claim that TPC holds in the standard model of PA is exactly the same as the claim that TPC is true. If TPC is provable in PA, then it holds in every model of PA, not just the standard model. --Trovatore (talk) 17:34, 26 October 2022 (UTC)

Yes, in fact I did not said that the author claimed to have proved that PA implies TPC. I always said that the author claims that TPC holds in the standard model of PA. Mbtnt (talk) 18:01, 26 October 2022 (UTC)

Right, I was mainly responding to the heading you chose ("proved...in standard PA"). --Trovatore (talk) 18:02, 26 October 2022 (UTC)

(As an aside, it's a funny way to put it, even in the paper you linked. For a statement σ in the language of arithmetic, proving that σ holds in the standard model of PA is exactly equivalent to proving σ. Why wouldn't Czelakowski just announce that he had proved TPC, full stop?) --Trovatore (talk) 18:10, 26 October 2022 (UTC)

The first mathematical statement in the paper, Theorem 1.1, with the proof "Immediate", is obviously wrong. It claims that the smallest number that is not a multiple of any of the first n primes p1,...,pn is "the least number greater than the numbers p1,...,pn". I think this is just a typo and that it really means "the least prime number greater than...", but this does not produce confidence in the care with which the paper was written and peer-reviewed. —David Eppstein (talk) 21:10, 26 October 2022 (UTC)

It's also immediately suspicious when anyone claims to prove an arithmetical result using forcing. Oh, in some sense it's not impossible; relative consistency results are arithmetical results, after you arithmetize logic, and they're often proved via forcing, but that seems different somehow. Forcing can't change the natural numbers, so to settle naturally-arising questions about the naturals via forcing seems ... unlikely. Of course it would be of great interest if someone did manage it. --Trovatore (talk) 21:47, 26 October 2022 (UTC)

The same author, in another paper in the same journal, claims to use the same methods to prove that there are infinitely many Mersenne primes: doi:10.1007/s11225-022-10015-4 — Preceding unsigned comment added by David Eppstein (talk • contribs)

That one looks short enough, and clear enough at a "local" level, that it might actually be readable. It would be interesting to try; not sure I'll really get around to it though. --Trovatore (talk) 00:11, 27 October 2022 (UTC)

I removed mention in the article.[1] This definitely falls under Wikipedia:Verifiability#Exceptional claims require exceptional sources. If the mathematical community ignores this claim then so should we. PrimeHunter (talk) 23:27, 26 October 2022 (UTC)

I guess it can't be too surprising that he is also an editor of the journal. --JBL (talk) 00:14, 27 October 2022 (UTC)

There is a discussion about this paper on MathOverflow: https://mathoverflow.net/questions/433278/czelakowskis-claimed-proof-of-the-twin-prime-conjecture . It started yesterday. Mbtnt (talk) 15:56, 27 October 2022 (UTC)

The question on MathOverflow has been closed with an accepted answer: the paper is wrong. Mbtnt (talk) 10:53, 29 October 2022 (UTC)

Citation worthy of adding in ?

https://math.stackexchange.com/questions/2324324/proof-of-minor-claim-related-to-the-twin-primes-conjecture Roderick MacPhee (talk) 20:29, 19 April 2023 (UTC)

Restatement

There's a restatement of twin prime conjecture. Namely, no natural number N exist such that all n>N are of forms 6ab+a+b, 6ab-a-b, 6ab-a+b, or 6ab+a-b ... Roderick MacPhee (talk) 18:37, 2 September 2024 (UTC)

With a,b>0 doh Roderick MacPhee (talk) 00:02, 3 September 2024 (UTC)