Small gaps between products of two primes

Abstract

Let $q_n$ denote the $n^{th}$ number that is a product of exactly two distinct primes. We prove that $$\underset{n\to \infty }{\mathrm{liminf}}(q_{n+1}-q_n)\le 6.$$ This sharpens an earlier result of the authors (arXivMath NT/0506067), which had 26 in place of 6. More generally, we prove that if $\nu$ is any positive integer, then $$\underset{n\to \infty }{\mathrm{liminf}}(q_{n+\nu }-q_n)\le C(\nu )=\nu e^{\nu -\gamma }(1+o(1)).$$ We also prove several other results on the representation of numbers with exactly two prime factors by linear forms.

11N25 (primary) 11N36 (secondary)

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• Prime and almost prime integral points on principal homogeneous spaces
• Positive density for consecutive runs of sums of two squares
• Small gaps between the set of products of at most two primes
• The existence of small prime gaps in subsets of the integers
• Almost-prime $k$-tuples