{
  "id": "math/0609615",
  "version": "v1",
  "published": "2006-09-21T19:05:55.000Z",
  "updated": "2006-09-21T19:05:55.000Z",
  "title": "Small gaps between products of two primes",
  "authors": [
    "D. A. Goldston",
    "S. W. Graham",
    "J. Pintz",
    "C. Y. Yildirim"
  ],
  "comment": "11N25 (primary) 11N36 (secondary)",
  "doi": "10.1112/plms/pdn046",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $q_n$ denote the $n^{th}$ number that is a product of exactly two distinct primes. We prove that $$\\liminf_{n\\to \\infty} (q_{n+1}-q_n) \\le 6.$$ 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 $$ \\liminf_{n\\to \\infty} (q_{n+\\nu}-q_n) \\le C(\\nu) = \\nu e^{\\nu-\\gamma} (1+o(1)).$$ We also prove several other results on the representation of numbers with exactly two prime factors by linear forms.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-09-21T19:05:55.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "small gaps",
      "linear forms",
      "distinct primes",
      "prime factors",
      "earlier result"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......9615G"
    }
  }
}