{
  "id": "1108.4178",
  "version": "v1",
  "published": "2011-08-21T10:45:00.000Z",
  "updated": "2011-08-21T10:45:00.000Z",
  "title": "Congruences for Wolstenholme primes",
  "authors": [
    "Romeo Mestrovic"
  ],
  "comment": "pages 16",
  "categories": [
    "math.NT"
  ],
  "abstract": "A prime number $p$ is said to be a Wolstenholme prime if it satisfies the congruence ${2p-1\\choose p-1} \\equiv 1 \\,\\,(\\bmod{\\,\\,p^4})$. For such a prime $p$, we establish the expression for ${2p-1\\choose p-1}\\,\\,(\\bmod{\\,\\,p^8})$ given in terms of the sums $R_i:=\\sum_{k=1}^{p-1}1/k^i$ ($i=1,2,3,4,5,6)$. Further, the expression in this congruence is reduced in terms of the sums $R_i$ ($i=1,3,4,5$). Using this congruence, we prove that for any Wolstenholme prime, $$ {2p-1\\choose p-1}\\equiv 1 -2p \\sum_{k=1}^{p-1}\\frac{1}{k} -2p^2\\sum_{k=1}^{p-1}\\frac{1}{k^2}\\pmod{p^7}. $$ Moreover, using a recent result of the author \\cite{Me}, we prove that the above congruence implies that a prime $p$ necessarily must be a Wolstenholme prime. Applying a technique of Helou and Terjanian \\cite{HT}, the above congruence is given as the expression involving the Bernoulli numbers.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-08-21T10:45:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B75",
      "11A07",
      "11B65",
      "11B68",
      "05A10"
    ],
    "keywords": [
      "wolstenholme prime",
      "expression",
      "congruence implies",
      "prime number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1108.4178M"
    }
  }
}