Small Gaps between Primes or Almost Primes

  title={Small Gaps between Primes or Almost Primes},
  author={D. A. Goldston and Sean W Graham and Janos Pintz and C. Y. Yildirim},
Let pn denote the nth prime. Goldston, Pintz, and Yıldırım recently proved that lim inf n→∞ (pn+1 − pn) log pn = 0. We give an alternative proof of this result. We also prove some corresponding results for numbers with two prime factors. Let qn denote the nth number that is a product of exactly two distinct primes. We prove that lim inf n→∞ (qn+1 − qn) ≤ 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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