{
  "id": "1311.6803",
  "version": "v2",
  "published": "2013-11-10T17:08:11.000Z",
  "updated": "2014-10-03T15:39:42.000Z",
  "title": "A Sufficient Condition for Disproving Descartes's Conjecture on Odd Perfect Numbers",
  "authors": [
    "Jose Arnaldo B. Dris"
  ],
  "comment": "8 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\sigma(x)$ be the sum of the divisors of $x$. If $N$ is odd and $\\sigma(N) = 2N$, then the odd perfect number $N$ is said to be given in Eulerian form if $N = {q^k}{n^2}$ where $q$ is prime with $q \\equiv k \\equiv 1 \\pmod 4$ and $\\gcd(q,n) = 1$. In this note, we show that $q < n$ implies that Descartes's conjecture (previously Sorli's conjecture), $k = \\nu_{q}(N) = 1$, is not true. This then implies an unconditional proof for the biconditional $$k = \\nu_{q}(N) = 1 \\Longleftrightarrow n < q.$$ Lastly, following a recent result of Cohen and Sorli, we show that if $q < n$, then either $q > 5$ or $k > 5$ is true.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-11-10T17:08:11.000Z",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-10-03T15:39:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A05",
      "11J25",
      "11J99"
    ],
    "keywords": [
      "odd perfect number",
      "disproving descartess conjecture",
      "sufficient condition",
      "sorlis conjecture",
      "unconditional proof"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1311.6803D"
    }
  }
}