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Claim. Theorem 1.1 of Ben Green and Terence Tao, The primes contain arbitrarily long arithmetic progressions, Ann. of Math. (2) 167 (2008), no. 2, 481--547, states: "The prime numbers contain infinitely many arithmetic progressions of length $k$ for all $k$." The reciprocals of the primes have a divergent sum, so this is the conclusion of Problem 3 for $A$ the set of primes. The paper's Theorem 1.2 extends it to every subset of the primes of positive relative upper density. The proof combines Szemerédi's theorem with a transference principle, which carries it to sets of positive relative density with respect to a pseudorandom measure, and the Goldston–Yıldırım sieve estimates, which place the primes inside such a measure; the paper records Erdős's conjecture as its Conjecture 2.2 and notes that it would imply Theorem 1.1. The source card is green_2008_primes_contain_arbitrarily_long_arithmetic_progressions.

Covers. The special case $A$ the set of primes.

Depends on. Nothing in this wiki.

Acceptance. Refereed publication: Ann. of Math. (2) 167 (2008), no. 2, 481--547, doi:10.4007/annals.2008.167.481; the arXiv version was first posted on 8 April 2004, the date of this page. Not reviewed: the site labels Problem 3 OPEN, so its commentary's credit to [GrTa08] is not acceptance of a part of this problem; the curator's PROVED label and credit to the same theorem belong to Problem 219.