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Small gaps between products of two primes
Author(s) -
Goldston D. A.,
Graham S. W.,
Pintz J.,
Yıldırım C. Y.
Publication year - 2009
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/pdn046
Subject(s) - mathematics , integer (computer science) , product (mathematics) , prime (order theory) , combinatorics , representation (politics) , prime number , discrete mathematics , computer science , law , geometry , politics , political science , programming language
Let q n denote the n th number that is a product of exactly two distinct primes. We prove that q n +1 − q n ⩽ 6 infinitely often. This sharpens an earlier result of the authors, which had 26 in place of 6. More generally, we prove that if ν is any positive integer, then ( q n +ν − q n ) ⩽ ν e ν − 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.