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The primes contain arbitrarily long arithmetic progressions
 Ann. of Math
"... Abstract. We prove that there are arbitrarily long arithmetic progressions of primes. ..."
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Cited by 151 (26 self)
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Abstract. We prove that there are arbitrarily long arithmetic progressions of primes.
Obstructions to uniformity, and arithmetic patterns in the primes, preprint
"... Abstract. In this expository article, we describe the recent approach, motivated by ergodic theory, towards detecting arithmetic patterns in the primes, and in particular establishing in [26] that the primes contain arbitrarily long arithmetic progressions. One of the driving philosophies is to iden ..."
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Abstract. In this expository article, we describe the recent approach, motivated by ergodic theory, towards detecting arithmetic patterns in the primes, and in particular establishing in [26] that the primes contain arbitrarily long arithmetic progressions. One of the driving philosophies is to identify precisely what the obstructions could be that prevent the primes (or any other set) from behaving “randomly”, and then either show that the obstructions do not actually occur, or else convert the obstructions into usable structural information on the primes. 1.
Generalising the HardyLittlewood method for primes
 In: Proceedings of the international congress of mathematicians
, 2007
"... Abstract. The HardyLittlewood method is a wellknown technique in analytic number theory. Among its spectacular applications are Vinogradov’s 1937 result that every sufficiently large odd number is a sum of three primes, and a related result of Chowla and Van der Corput giving an asymptotic for the ..."
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Abstract. The HardyLittlewood method is a wellknown technique in analytic number theory. Among its spectacular applications are Vinogradov’s 1937 result that every sufficiently large odd number is a sum of three primes, and a related result of Chowla and Van der Corput giving an asymptotic for the number of 3term progressions of primes, all less than N. This article surveys recent developments of the author and T. Tao, in which the HardyLittlewood method has been generalised to obtain, for example, an asymptotic for the number of 4term arithmetic progressions of primes less than N.
Arithmetic progressions and the primes  El Escorial lectures
 Collectanea Mathematica (2006), Vol. Extra., 3788 (Proceedings of the 7th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial
"... Abstract. We describe some of the machinery behind recent progress in establishing infinitely many arithmetic progressions of length k in various sets of integers, in particular in arbitrary dense subsets of the integers, and in the primes. 1. ..."
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Abstract. We describe some of the machinery behind recent progress in establishing infinitely many arithmetic progressions of length k in various sets of integers, in particular in arbitrary dense subsets of the integers, and in the primes. 1.
Equidistribution and Primes by
"... We begin by reviewing various classical problems concerning the existence of primes or numbers with few prime factors as well as some of the key developments towards resolving these long standing questions. Then we put the theory in a natural and general geometric context of actions on affine nspac ..."
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We begin by reviewing various classical problems concerning the existence of primes or numbers with few prime factors as well as some of the key developments towards resolving these long standing questions. Then we put the theory in a natural and general geometric context of actions on affine nspace and indicate what can be established there. The methods used to develop a combinational sieve in this context involve automorphic forms, expander graphs and unexpectedly arithmetic combinatorics. Applications to classical problems such as the divisibility of the areas of Pythagorean triangles and of the curvatures of the circles in an integral Apollonian packing, are given. This is an expanded version of the lecture that I had intended to give at the conference honoring Bourguignon on the occasion of his 60 th birthday. Equidistribution and Primes 2 (1) I have chosen to talk on this topic because I believe it has a wide appeal and also there have been some interesting developments in recent years on some of these classical problems. The questions that we discuss are generalizations of the twin prime conjecture; that there are infinitely many primes p such that p + 2 is also a prime. I am not sure who first asked
THREE TOPICS IN ADDITIVE PRIME NUMBER THEORY
, 710
"... Abstract. We discuss, in varying degrees of detail, three contemporary themes in prime number theory. Topic 1: the work of Goldston, Pintz and Yıldırım on short gaps between primes. Topic 2: the work of Mauduit and Rivat, establishing that 50% of the primes have odd digit sum in base 2. Topic 3: wor ..."
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Abstract. We discuss, in varying degrees of detail, three contemporary themes in prime number theory. Topic 1: the work of Goldston, Pintz and Yıldırım on short gaps between primes. Topic 2: the work of Mauduit and Rivat, establishing that 50% of the primes have odd digit sum in base 2. Topic 3: work of Tao and the author on linear equations in primes. Introduction. These notes are to accompany two lectures I am scheduled to give at the Current Developments in Mathematics conference at Harvard in November 2007. The title of those lectures is ‘A good new millennium for primes’, but I have chosen a rather drier title for these notes for two reasons. Firstly, the title of the lectures was unashamedly stolen (albeit with permission) from Andrew Granville’s entertaining