Small gaps between primes or almost primes
Abstract
Let $p_n$ denote the $n^{th}$ prime. Goldston, Pintz, and Yildirim recently proved that $ \liminf_{n\to \infty} \frac{(p_{n+1}p_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 $\liminf_{n\to \infty} (q_{n+1}q_n) \le 26.$ If an appropriate generalization of the ElliottHalberstam Conjecture is true, then the above bound can be improved to 6.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 June 2005
 arXiv:
 arXiv:math/0506067
 Bibcode:
 2005math......6067G
 Keywords:

 Number Theory;
 11N25 (primary) 11N05;
 11N36 (secondary)
 EPrint:
 49 pages