# The primes contain arbitrarily long arithmetic progressions

@article{Green2004ThePC,
title={The primes contain arbitrarily long arithmetic progressions},
author={Ben Green and Terence Tao},
journal={Annals of Mathematics},
year={2004},
volume={167},
pages={481-547}
}
• Published 8 April 2004
• Mathematics
• 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…
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## References

SHOWING 1-10 OF 65 REFERENCES
On sets of integers containing k elements in arithmetic progression
In 1926 van der Waerden [13] proved the following startling theorem : If the set of integers is arbitrarily partitioned into two classes then at least one class contains arbitrarily long arithmetic
Additive properties of dense subsets of sifted sequences
• Mathematics
• 2001
In this paper we show that if A is a subset of the primes with positive lower relative density then A +A must have positive lower density at least C1= log log(1= ) in the natural numbers. Our
Arithmetic progressions of length three in subsets of a random set
• Mathematics
• 1996
0. Introduction. In 1936 Erdős and Turan [ET 36] asked whether for every natural number k and every positive constant α, every subset A of [n] = {0, 1, . . . , n − 1} with at least αn elements
Roth's theorem in the primes
We show that any set containing a positive proportion of the primes contains a 3-term arithmetic progression. An important ingredient is a proof that the primes enjoy the so-called Hardy-Littlewood
XXIV.—Sets of Integers Containing not more than a Given Number of Terms in Arithmetical Progression*
Sets of integers are constructed having the property that n members are in arithmetical progression only if they are all equal; here n is any integer greater than or equal to 3. Previous results have
Six Primes and an Almost Prime in Four Linear Equations
• A. Balog
• Mathematics
• 1998
Abstract There are infinitely many triplets of primes $p,\,q,\,r$ such that the arithmetic means of any two of them, $\frac{p+q}{2},\,\frac{p+r}{2},\,\frac{q+r}{2}$ are also primes. We give an
On Sets of Integers Containing No Four Elements in Arithmetic Progression
• Mathematics
In what follows we use capital letters to denote sequences of integers, A + B to denote the sum of two sets of integers formed elementwise, and A-q B to denote the complement of the set B with
A new proof of Szemerédi's theorem
In 1927 van der Waerden published his celebrated theorem on arithmetic progressions, which states that if the positive integers are partitioned into finitely many classes, then at least one of these
Number Theory and Algebraic Geometry: Linear relations amongst sums of two squares
subject to the natural condition that A + B + C should be even. Balog [?] has made important progress on the question of linear relations involving more than 3 primes, but none the less it remains an
A Quantitative Ergodic Theory Proof of Szemerédi's Theorem
• T. Tao
• Mathematics
Electron. J. Comb.
• 2006
A quantitative, self-contained version of this ergodic theory proof is presented, which is elementary'' in the sense that it does not require the axiom of choice, the use of infinite sets or measures, or theUse of the Fourier transform or inverse theorems from additive combinatorics.