{
  "id": "2109.13652",
  "version": "v2",
  "published": "2021-09-20T04:48:08.000Z",
  "updated": "2022-03-02T08:27:46.000Z",
  "title": "On the quantity $m^2-p^k$ where $p^k m^2$ is an odd perfect number -- Part II",
  "authors": [
    "Jose Arnaldo Bebita Dris",
    "Immanuel Tobias San Diego"
  ],
  "comment": "11 pages, incorporated referee comments and suggestions, added Section 5.1, in press at NNTDM <https://nntdm.net/volume-28-2022/number-1/>",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $p^k m^2$ be an odd perfect number with special prime $p$. Extending previous work of the authors, we prove that the inequality $m < p^k$ follows from $m^2 - p^k = 2^r t$, where $r \\geq 2$ and $\\gcd(2,t)=1$, under the following hypotheses: (a) $m > t > 2^r$, or (b) $m > 2^r > t$. We also prove that the estimate $m^2 - p^k > 2m$ holds. We can also improve this unconditional estimate to $m^2 - p^k > {m^2}/3$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-03-02T08:27:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A05",
      "11A25"
    ],
    "keywords": [
      "odd perfect number",
      "unconditional estimate",
      "special prime",
      "inequality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}