Small gaps between products of two primes

Goldston, D.A. and Graham, S.W. and Pintz, János and Yildrim, C.Y. (2009) Small gaps between products of two primes. PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY, 98 (3). pp. 741-774. ISSN 0024-6115

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Abstract

Let qn denote the nth number that is a product of exactly two distinct primes. We prove that qn+1 - qnle; 6 infinitely often. This sharpens an earlier result of the authors, which had 26 in place of 6. More generally, we prove that if ? is any positive integer, then (qn+1 - qn) ≤ e-γ(1 + o(1)) infinitely often. We also prove several other related results on the representation of numbers with exactly two prime factors by linear forms. © 2008 London Mathematical Society.

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