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Proceedings of the London Mathematical Society Advance Access originally published online on November 5, 2008
Proceedings of the London Mathematical Society 2009 98(3):741-774; doi:10.1112/plms/pdn046
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© 2008 London Mathematical Society

Small gaps between products of two primes

D. A. Goldston

Department of Mathematics
San Jose State University
San Jose, CA 95192
USA
goldston@math.sjsu.edu

S. W. Graham

Department of Mathematics
Central Michigan University
Mount Pleasant, MI 48859
USA

J. Pintz

Rényi Mathematical Institute of the Hungarian Academy of Sciences
P.O.B. 127
H-1364 Budapest
Hungary
pintz@renyi.hu

C. Y. Yildirim

Department of Mathematics
Bogaziçi University,
Istanbul 34342
Turkey
and
Feza Gürsey Enstitüsü
Çengelköy
Istanbul
P.K. 6, 81220
Turkey
yalciny@boun.edu.tr

Received 20 December 2007.

Let qn denote the nth number that is a product of exactly two distinct primes. We prove that qn+1 – qn ≤ 6 infinitely often. This sharpens an earlier result of the authors, which had 26 in place of 6. More generally, we prove that if {nu} is any positive integer, then (qn+{nu} – qn) ≤ {nu} e{nu} – {gamma} (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.

2000 Mathematics Subject Classification 11N25 (primary), 11N36 (secondary).

The first and second authors were supported in part by NSF Grants, the third author by OTKA grants No. 43623, 49693, 67676, and the Balaton program, and the fourth author by TÜBITAK.


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