Small gaps between products of two primes
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 ν is any positive integer, then (qn+ν − qn) ≤ ν eν − γ (1+o(1)) in...
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| Veröffentlicht in: | Proceedings of the London Mathematical Society Jg. 98; H. 3; S. 741 - 774 |
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| Hauptverfasser: | , , , |
| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
Oxford University Press
01.05.2009
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| ISSN: | 0024-6115, 1460-244X |
| Online-Zugang: | Volltext |
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| Zusammenfassung: | 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 ν is any positive integer, then (qn+ν − qn) ≤ ν eν − 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. |
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| Bibliographie: | ark:/67375/HXZ-K24WJJM0-X ArticleID:pdn046 2000 Mathematics Subject Classification 11N25 (primary), 11N36 (secondary). istex:5D881B3214CA89DA97F8495611F149FDEE44B8CD |
| ISSN: | 0024-6115 1460-244X |
| DOI: | 10.1112/plms/pdn046 |