## Primes in Tuples I

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Citations: | 6 - 1 self |

### BibTeX

@MISC{Goldston_primesin,

author = {D. A. Goldston and J. Pintz and C. Y. Yıldırım},

title = { Primes in Tuples I},

year = {}

}

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### Abstract

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 arbitrarily small multiple of the average spacing, that is, pn+1 − pn lim inf =0. n→ ∞ log pn We will quantify this result further in a later paper (see (1.9) below).