{
  "id": "1112.1034",
  "version": "v11",
  "published": "2011-12-05T19:38:56.000Z",
  "updated": "2013-10-29T15:53:47.000Z",
  "title": "Congruences for Franel numbers",
  "authors": [
    "Zhi-Wei Sun"
  ],
  "comment": "12 pages. Final published version",
  "journal": "Adv. in Appl. Math. 51(2013), no. 4, 524-535",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "The Franel numbers given by $f_n=\\sum_{k=0}^n\\binom{n}{k}^3$ ($n=0,1,2,\\ldots$) play important roles in both combinatorics and number theory. In this paper we initiate the systematic investigation of fundamental congruences for the Franel numbers. We mainly establish for any prime $p>3$ the following congruences: \\begin{align*}\\sum_{k=0}^{p-1}(-1)^kf_k&\\equiv\\left(\\frac p3\\right)\\ \\ (\\mbox{mod}\\ p^2), \\\\ \\sum_{k=0}^{p-1}(-1)^k\\,kf_k&\\equiv-\\frac 23\\left(\\frac p3\\right)\\ \\ (\\mbox{mod}\\ p^2), \\\\ \\sum_{k=1}^{p-1}\\frac{(-1)^k}kf_k &\\equiv0\\ \\ (\\mbox{mod}\\ p^2), \\\\ \\sum_{k=1}^{p-1}\\frac{(-1)^k}{k^2}f_k&\\equiv0\\ \\ (\\mbox{mod}\\ p). \\end{align*}",
  "revisions": [
    {
      "version": "v11",
      "updated": "2013-10-29T15:53:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A07",
      "11B65",
      "05A10",
      "11B37",
      "11B75"
    ],
    "keywords": [
      "franel numbers",
      "play important roles",
      "number theory",
      "systematic investigation",
      "fundamental congruences"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1112.1034S"
    }
  }
}