{
  "id": "0911.2415",
  "version": "v16",
  "published": "2009-11-12T19:15:58.000Z",
  "updated": "2011-08-02T16:39:05.000Z",
  "title": "On congruences related to central binomial coefficients",
  "authors": [
    "Zhi-Wei Sun"
  ],
  "journal": "J. Number Theory 131(2011), no.11, 2219-2238",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "It is known that $\\sum_{k=0}^\\infty\\binom{2k}{k}/((2k+1)4^k)=\\pi/2$ and $\\sum_{k=0}^\\infty\\binom{2k}{k}/((2k+1)16^k)=\\pi/3$. In this paper we obtain their p-adic analogues such as $$\\sum_{p/2<k<p}\\binom{2k}{k}/((2k+1)4^k)=3\\sum_{p/2<k<p}\\binom{2k}{k}/((2k+1)16^k)= pE_{p-3} (mod p^2),$$ where p>3 is a prime and E_0,E_1,E_2,... are Euler numbers. Besides these, we also deduce some other congruences related to central binomial coefficients. In addition, we pose some conjectures one of which states that for any odd prime p we have $$\\sum_{k=0}^{p-1}\\binom{2k}{k}^3=4x^2-2p (mod p^2)$$ if (p/7)=1 and p=x^2+7y^2 with x,y integers, and $$\\sum_{k=0}^{p-1}\\binom{2k}{k}^3=0 (mod p^2)$$ if (p/7)=-1, i.e., p=3,5,6 (mod 7).",
  "revisions": [
    {
      "version": "v16",
      "updated": "2011-08-02T16:39:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B65",
      "11A07",
      "05A10",
      "11B68",
      "11E25"
    ],
    "keywords": [
      "central binomial coefficients",
      "congruences",
      "odd prime",
      "euler numbers",
      "p-adic analogues"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0911.2415S"
    }
  }
}