The primes contain arbitrarily long arithmetic progressions

2004/04/08 by Ben Green, Terence Tao, Green, Ben +1 · 14 citations
Mathematics · #Analytic Number Theory Research #Dynamical Systems (math.DS) #FOS: Mathematics #Finite Group Theory Research #Limits and Structures in Graph Theory #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.math/0404188

openalex publication_date 2004/04/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Higher correlations of divisor sums related to primes III: k-correlations
• On Some Sequences of Integers
• On Certain Sets of Integers

Cited by

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• On the Realizability of Prime Conjectures in Heyting Arithmetic
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• The trigonometric polynomial on sums of two squares, an additive problem and generalisation
• Finding arithmetic progressions in dense sets of integers
• Supersimulators
• Small Composite Numbers in Orbits of Linear Maps
• Arithmetic progressions of primes in short intervals beyond the 17/30 barrier
• Quantum Higher Order Fourier Analysis and the Clifford Hierarchy
• On the density version of quadratic Waring's problem and the quadratic Waring--Goldbach problem
• Finite field models in additive combinatorics

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