{
  "id": "1201.0617",
  "version": "v3",
  "published": "2012-01-03T12:40:50.000Z",
  "updated": "2012-07-21T12:29:37.000Z",
  "title": "Proof of two conjectures of Z.-W. Sun on congruences for Franel numbers",
  "authors": [
    "Victor J. W. Guo"
  ],
  "comment": "8 pages, minor changes, to appear in Integral Transforms Spec. Funct",
  "doi": "10.1080/10652469.2012.715160",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "For all nonnegative integers n, the Franel numbers are defined as $$ f_n=\\sum_{k=0}^n {n\\choose k}^3.$$ We confirm two conjectures of Z.-W. Sun on congruences for Franel numbers: \\sum_{k=0}^{n-1}(3k+2)(-1)^k f_k &\\equiv 0 \\pmod{2n^2}, \\sum_{k=0}^{p-1}(3k+2)(-1)^k f_k &\\equiv 2p^2 (2^p-1)^2 \\pmod{p^5}, where n is a positive integer and p>3 is a prime.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2012-07-21T12:29:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A07",
      "11B65",
      "05A10",
      "05A19"
    ],
    "keywords": [
      "franel numbers",
      "congruences",
      "conjectures"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1201.0617G"
    }
  }
}