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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 150 (26 self)
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Abstract. We prove that there are arbitrarily long arithmetic progressions of primes.
Mijajlović: On Kurepa problems in number theory
 Publ. Inst. Math. (N.S
, 1995
"... Dedicated to the memory of Prof.Duro Kurepa ..."
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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Cited by 5 (2 self)
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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.
Long arithmetic progressions of primes
 Mathematics Proceedings
"... Abstract. This is an article for a general mathematical audience on the author’s work, joint with Terence Tao, establishing that there are arbitrarily long arithmetic progressions of primes. 1. introduction and history This is a description of recent work of the author and Terence Tao [11] on primes ..."
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Abstract. This is an article for a general mathematical audience on the author’s work, joint with Terence Tao, establishing that there are arbitrarily long arithmetic progressions of primes. 1. introduction and history This is a description of recent work of the author and Terence Tao [11] on primes in arithmetic progression. It is based on seminars given for a general mathematical
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"... gave occasion for an informal lecture in which I discussed various old and new questions on number theory, geometry and analysis. In the following list, I record these problems, with the addition of references and of a few further questions. A. NUMBER THEORY 1. It is known [35, Vol. 1, Section 581 t ..."
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gave occasion for an informal lecture in which I discussed various old and new questions on number theory, geometry and analysis. In the following list, I record these problems, with the addition of references and of a few further questions. A. NUMBER THEORY 1. It is known [35, Vol. 1, Section 581 that n(2x) < 257(x) for sufficiently large x. Is it true that (1) n(x + y) < B(X) + n(y)? Ungrir has verified the inequality for y 5 41. Hardy and Littlewood [29, p. 691 have proved that (2) P(X + Y) n(x) < cy/log y. In the same paper, they discuss many interesting conjectures. They put h0 sup [R(x+ Y) x00 n(x)1 = p(y), and they conjecture that p(y)> y/log y, and that perhaps n(y) p(y) oo as y) 00. Hardy and Littlewood deduce (2) by Bruds method. A very difficult conjecture, weaker than (1) but much stronger than (2), is that corresponding to each s> 0 there exists a yE such that, for y> yE, dx + Y) dy) < (1 + dy/log yIt has not yet been disproved that p(y) = 1 for all y. If p(y)> 1 for some y, then lim inf (h+l pn) < 00. 2. About seventy years ago, Piltz [38] conjectured that, for each c> 0, pn+l pn = O(n&). Cram & conjectured [7, p. 241 that pn+l pn = O((log n)?. If lim sup (p,+l p,)/(log n) ” = 1, then, for each E> 0, infinitely many of the intervals [n, n + (1 s)(log n)“] contain no primes, but for n> ns, there is a prime between n and n + (1 + E)(log n)“. The Riemann hypothesis implies that prl+1 pn < n&+1/2 [35, Vol. 1, p. 3381. Thus the old conjecture that there is always a prime between two consecutive squares already goes beyond the Riemann hypothesis. 292 PAUL ERDOS The first big achievement in this direction is due to Hoheisel [32], who showed that pn+i Pn < n lv6. Ingham [34, p. 2561 proved that 1 6 can be taken to be 5/8. In the opposite direction, I have proved [9, p. 1241 that, for a certain c> 0 and for infinitely many n, Pnfl