Small gaps between primes or almost primes

Abstract

Let $p_n$ denote the $n^{\textrm {th}}$ prime. Goldston, Pintz, and Yıldırım recently proved that \begin{equation*} \liminf _{n\to \infty } \frac {(p_{n+1}-p_n)}{\log p_n} =0. \end{equation*} 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 $n^{\textrm {th}}$ number that is a product of exactly two distinct primes. We prove that \begin{equation*} \liminf _{n\to \infty } (q_{n+1}-q_n) \le 26. \end{equation*} If an appropriate generalization of the Elliott-Halberstam Conjecture is true, then the above bound can be improved to $6$.

Locations

• Repository of the Academy's Library (Library of the Hungarian Academy of Sciences) - View - PDF
• arXiv (Cornell University) - View - PDF
• Transactions of the American Mathematical Society - View - PDF