{
  "id": "1407.7894",
  "version": "v1",
  "published": "2014-07-29T21:33:58.000Z",
  "updated": "2014-07-29T21:33:58.000Z",
  "title": "A primality criterion based on a Lucas' congruence",
  "authors": [
    "Romeo Mestrovic"
  ],
  "comment": "6 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $p$ be a prime. In 1878 \\'{E}. Lucas proved that the congruence $$ {p-1\\choose k}\\equiv (-1)^k\\pmod{p}$$ holds for any nonnegative integer $k\\in\\{0,1,\\ldots,p-1\\}$. The converse statement was given in Problem 1494 of {\\it Mathematics Magazine} proposed in 1997 by E. Deutsch and I.M. Gessel. In this note we generalize this converse assertion by the following result: If $n>1$ and $q>1$ are integers such that $$ {n-1\\choose k}\\equiv (-1)^k \\pmod{q}$$ for every integer $k\\in\\{0,1,\\ldots, n-1\\}$, then $q$ is a prime and $n$ is a power of $q$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-07-29T21:33:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A51",
      "11A07",
      "05A10",
      "11B65"
    ],
    "keywords": [
      "primality criterion",
      "congruence",
      "converse statement",
      "mathematics magazine",
      "converse assertion"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.7894M"
    }
  }
}