Primes in tuples I

  title={Primes in tuples I},
  author={D. A. Goldston and Janos Pintz and C. Y. Yildirim},
  journal={Annals of Mathematics},
We introduce a method for showing that there exist prime numbers which are very close together. The method depends on the level of distribution of primes in arithmetic progressions. Assuming the Elliott-Halberstam conjecture, we prove that there are infinitely often primes differing by 16 or less. Even a much weaker conjecture implies that there are infinitely often primes a bounded distance apart. Unconditionally, we prove that there exist consecutive primes which are closer than any… 

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In a previous paper, the authors proved that in any system of three linear forms satisfying obvious necessary local conditions, there are at least two forms that infinitely often assume E 2 -values;

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In an earlier work it was shown that the Elliott-Halberstam conjecture implies the existence of infinitely many gaps of size at most 16 between consecutive primes. In the present work we show that

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α e−t dt as x→∞, for any two fixed real numbers β > α ≥ 0. Gallagher’s calculation [Ga] shows that this conjecture can be deduced from the Hardy–Littlewood prime k-tuples conjecture (see [S2]). Hence

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is greater than (c2/2)n. A simple calculation now shows that the primes satisfying (4) also satisfy the first inequality of (3) i΀ = e(ci) is chosen small enough. The second inequality of (3) is


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Small gaps between primes exist

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The difference between consecutive prime numbers. II

  • Mathematics
    Mathematical Proceedings of the Cambridge Philosophical Society
  • 1940
In a previous paper, under the same title, I considered the problem of how far apart two consecutive primes can be. The present paper is concerned with the opposite question. How near together can

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In early 2005, Dan Goldston, Janos Pintz, and Cem Yildirim [12] made a spectacular breakthrough in the study of prime numbers. Resolving a long-standing open problem, they proved that there are

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We calculate the triple correlations for the truncated divisor sum λR(n). The λR(n) behave over certain averages just as the prime counting von Mangoldt function Λ(n) does or is conjectured to do. We