{
  "id": "1108.2361",
  "version": "v1",
  "published": "2011-08-11T09:37:24.000Z",
  "updated": "2011-08-11T09:37:24.000Z",
  "title": "An elementary proof of a congruence by Skula and Granville",
  "authors": [
    "Romeo Mestrovic"
  ],
  "comment": "pages 7",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $p\\ge 5$ be a prime, and let $q_p(2):=(2^{p-1}-1)/p$ be the Fermat quotient of $p$ to base 2. The following curious congruence was conjectured by L. Skula and proved by A. Granville $$ q_p(2)^2\\equiv -\\sum_{k=1}^{p-1}\\frac{2^k}{k^2}\\pmod{p}. $$ In this note we establish the above congruence by entirely elementary number theory arguments.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-08-11T09:37:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B75",
      "11A07",
      "11B65",
      "05A19",
      "05A19"
    ],
    "keywords": [
      "elementary proof",
      "elementary number theory arguments",
      "fermat quotient",
      "curious congruence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1108.2361M"
    }
  }
}