The primes contain arbitrarily long arithmetic progressions

Number theory 1 Minute

You need to know: Prime numbers.

Background: A (non-trivial) arithmetic progression of length n is a sequence a_1, a_2, \dots, a_n of real numbers such that a_i = a_1 + (i-1)d, \, i=1,2,\dots,n for some d\neq 0.

The Theorem: On 8th April 2004, Ben Green and Terence Tao submitted to arxiv a paper in which they proved that for every positive integer n there exist an arithmetic progression of length n consisting of only prime numbers.

Short context: In 1939, Van der Corput proved that the set of primes contains infinitely many arithmetic progressions of length n=3. The Theorem proves this for all n. Before 2004, this was open even for n=4. Moreover, Green and Tao proved that even any positive proportion of primes contains infinitely many arithmetic progressions of length n for all n. This generalises an earlier theorem of Green, which proves this for n=3.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 8.2 of this book for an accessible description of the Theorem.

Go to the list of all theorems

Unknown's avatar

Published by Bogdan Grechuk

Published