{
  "id": "1108.1174",
  "version": "v1",
  "published": "2011-08-04T19:25:00.000Z",
  "updated": "2011-08-04T19:25:00.000Z",
  "title": "On the mod $p^7$ determination of ${2p-1\\choose p-1}$",
  "authors": [
    "Romeo Mestrovic"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "In this paper we prove that for any prime $p\\ge 11$ holds $$ {2p-1\\choose p-1}\\equiv 1 -2p \\sum_{k=1}^{p-1}\\frac{1}{k} +4p^2\\sum_{1\\le i<j\\le p-1}\\frac{1}{ij}\\pmod{p^7}. $$ This is a generalization of the famous Wolstenholme's theorem which asserts that ${2p-1\\choose p-1} \\equiv 1 \\,\\,(\\bmod\\,\\,p^3)$ for all primes $p\\ge 5$. Our proof is elementary and it does not use a standard technique involving the classic formula for the power sums in terms of the Bernoulli numbers. Notice that the above congruence reduced modulo $p^6$, $p^5$ and $p^4$ yields related congruences obtained by R. Tauraso, J. Zhao and J.W.L. Glaisher, respectively.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-08-04T19:25:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B75",
      "11A07",
      "11B65",
      "11B68",
      "05A10"
    ],
    "keywords": [
      "determination",
      "famous wolstenholmes theorem",
      "standard technique",
      "classic formula",
      "power sums"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1108.1174M"
    }
  }
}