The primes contain arbitrarily long arithmetic progressions

• 成果类型: Article
• 署名作者: Green, Ben; Tao, Terence
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2008.167.481
• 发表日期: 2008
• 页码: 481-547
• 关键词: szemeredi theorem; linear-equations; ergodic averages; PROOF; CONVERGENCE; REGULARITY; density; number; sets

摘要

We prove that there are arbitrarily long arithmetic progressions of primes. There are three major ingredients. The first is Szemeredi's theorem, which asserts that any subset of the integers of positive density contains progressions of arbitrary length. The second, which is the main new ingredient, of this paper, is a, certain transference principle. This allows us to deduce from Szemeredi's theorem that any subset of a sufficiently pseudorandom set (or measure) of positive relative density contains progressions of arbitrary length. The third ingredient is a recent result of Goldston and Yildirim, which we reproduce here. Using this, one may place (a large fraction of) the primes inside a pseudorandom set of almost primes (or more precisely, a pseudorandom measure concentrated on almost primes) with positive relative density.