Discrete mathematics

Mathematics

Arithmetic progression

Arithmetic

Annals of Mathematics

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 suficiently 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.

The primes contain arbitrarily long arithmetic progressions