{
  "id": "math/0506067",
  "version": "v1",
  "published": "2005-06-03T15:02:45.000Z",
  "updated": "2005-06-03T15:02:45.000Z",
  "title": "Small gaps between primes or almost primes",
  "authors": [
    "D. A. Goldston",
    "S. W. Graham",
    "J. Pintz",
    "C. Y. Yilidirm"
  ],
  "comment": "49 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $p_n$ denote the $n^{th}$ prime. Goldston, Pintz, and Yildirim recently proved that $ \\liminf_{n\\to \\infty} \\frac{(p_{n+1}-p_n)}{\\log p_n} =0.$ We give an alternative proof of this result. We also prove some corresponding results for numbers with two prime factors. Let $q_n$ denote the $n^{th}$ number that is a product of exactly two distinct primes. We prove that $\\liminf_{n\\to \\infty} (q_{n+1}-q_n) \\le 26.$ If an appropriate generalization of the Elliott-Halberstam Conjecture is true, then the above bound can be improved to 6.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-06-03T15:02:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N25",
      "11N05",
      "11N36"
    ],
    "keywords": [
      "small gaps",
      "elliott-halberstam conjecture",
      "distinct primes",
      "appropriate generalization",
      "prime factors"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 49,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......6067G"
    }
  }
}