@article{Goldston2009PrimesIT,
title={Primes in tuples I},
author={D. Goldston and J. Pintz and C. Yildirim},
journal={Annals of Mathematics},
year={2009},
volume={170},
pages={819-862}
}

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… CONTINUE READING