Talk:Green–Tao theorem

Infinitely many arithmetic progressions

I think that the theorem is even stronger than what is said in the article: for any natural number k, there exist infinitely many k-term arithmetic progressions of primes. This is, for any natural number k, there exists infinitely many pairs (p,q) such that

p,p + q,p + 2q,...,p + (k - 1)q

are all prime numbers (cf. http://www.dms.umontreal.ca/~andrew/PDF/PrimePatterns.pdf).

Yes, there are infinitely many for all k. This is a trivial consequence of the theorem, since for any n there is an arithmetic progression of k*n primes, and that can be split into n progressions of k primes (or more when overlapping progressions are allowed). The title of the paper is "The primes contain arbitrarily long arithmetic progressions", and the abstract starts "We prove that there are arbitrarily long arithmetic progressions of primes." This is the formulation the authors and most sources use, so I think we should do that too. PrimeHunter 23:20, 16 August 2007 (UTC)[reply]

Yes, you are right. Thanks for the reply. (The authors also use the "infinitely many" formulation in the theorem 1.1.) Jayme 09:22, 17 August 2007 (UTC)[reply]

proof

I have seen the prooof and it was the simpliest proof i have ever seen. Just obs n! and that n!+1, n!+2, ... are all composite —Preceding unsigned comment added by 128.226.195.85 (talk) 21:00, 14 October 2007 (UTC)[reply]

That's an easy proof that there are arbitrarily long consecutive sequences of composite numbers. The Green–Tao theorem is that there are arbitrarily long arithmetic sequences of prime numbers. —David Eppstein 22:38, 14 October 2007 (UTC)[reply]

To be precise, n!+1 is sometimes a factorial prime, but n!+2 to n!+n is composite for n > 1. PrimeHunter 23:44, 14 October 2007 (UTC)[reply]

Written to be overcomplicated...

Why is the 2nd statement of the theorem, regarding the arbitrary set $A$ , the "main" theorem, although it is in the paper the 2nd statement. The article is written so that people do not understand. --Tensorproduct (talk) 02:51, 17 June 2021 (UTC)[reply]