{
  "id": "2202.08116",
  "version": "v1",
  "published": "2022-02-10T13:40:16.000Z",
  "updated": "2022-02-10T13:40:16.000Z",
  "title": "A new approach to odd perfect numbers via GCDs",
  "authors": [
    "Jose Arnaldo Bebita Dris"
  ],
  "comment": "9 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $q^k n^2$ be an odd perfect number with special prime $q$. Define the GCDs $$G = \\gcd\\bigg(\\sigma(q^k),\\sigma(n^2)\\bigg)$$ $$H = \\gcd\\bigg(n^2,\\sigma(n^2)\\bigg)$$ and $$I = \\gcd\\bigg(n,\\sigma(n^2)\\bigg).$$ We prove that $G \\times H = I^2$. (Note that it is trivial to show that $G \\mid I$ and $I \\mid H$ both hold.) We then compute expressions for $G, H,$ and $I$ in terms of $\\sigma(q^k)/2, n,$ and $\\gcd\\bigg(\\sigma(q^k)/2,n\\bigg)$. Afterwards, we prove that if $G = H = I$, then $\\sigma(q^k)/2$ is not squarefree. Other natural and related results are derived further. Lastly, we conjecture that the set $$\\mathscr{A} = \\{m : \\gcd(m,\\sigma(m^2))=\\gcd(m^2,\\sigma(m^2))\\}$$ has asymptotic density zero.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-02-10T13:40:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A05",
      "11A25"
    ],
    "keywords": [
      "odd perfect number",
      "asymptotic density zero",
      "special prime",
      "expressions",
      "squarefree"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}