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 Elliott-Halberstam Conjecture is true, then the above bound can be improved to 6.
- Publication:
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arXiv Mathematics e-prints
- Pub Date:
- June 2005
- arXiv:
- arXiv:math/0506067
- Bibcode:
- 2005math......6067G
- Keywords:
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- Number Theory;
- 11N25 (primary) 11N05;
- 11N36 (secondary)
- E-Print:
- 49 pages