SMALL GAPS BETWEEN PRIMES OR ALMOST PRIMES

@article{Goldston2005SMALLGB,
  title={SMALL GAPS BETWEEN PRIMES OR ALMOST PRIMES},
  author={D. Goldston and S. Graham and J. Pintz and C. Y. Yilidirm},
  journal={Transactions of the American Mathematical Society},
  year={2005},
  volume={361},
  pages={5285-5330}
}
Let p n denote the n th prime. Goldston, Pintz, and Yildirim recently proved that li m in f (pn+1 ― p n ) n→∞ log p n = 0. We give an alternative proof of this result. We also prove some corresponding results for numbers with two prime factors. Let q n denote the n th number that is a product of exactly two distinct primes. We prove that lim inf(qn+1―q n ) n→∞ 26. If an appropriate generalization of the Elliott-Halberstam Conjecture is true, then the above bound can be improved to 6. 
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