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Small gaps between primes or almost primes

D. A. Goldston·S. W. Graham·J. Pintz·C. Y. Yilidirm
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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.

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D. A. Goldston, S. W. Graham, J. Pintz, C. Y. Yilidirm. 2005-06-03. Small gaps between primes or almost primes. https://arxiv.org/abs/math/0506067

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