The primes contain arbitrarily long arithmetic progressions

TL;DR

This paper proves that primes contain arbitrarily long arithmetic progressions by combining Szemeredi's theorem, a new transference principle, and a recent result on almost primes, advancing understanding of prime number structure.

Contribution

Introduces a transference principle that extends Szemeredi's theorem to primes, enabling proof of arbitrarily long prime arithmetic progressions.

Findings

Primes contain arbitrarily long arithmetic progressions.

The transference principle bridges pseudorandom sets and primes.

Utilizes recent results on almost primes for the proof.

Abstract

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 of positive relative density contains progressions of arbitrary length. The third ingredient is a recent result of Goldston and Yildirim. Using this, one may place the primes inside a pseudorandom set of ``almost primes'' with positive relative density.