{
  "id": "math/0501070",
  "version": "v2",
  "published": "2005-01-05T18:40:18.000Z",
  "updated": "2006-04-12T19:02:36.000Z",
  "title": "New techniques for bounds on the total number of Prime Factors of an Odd Perfect Number",
  "authors": [
    "Kevin G. Hare"
  ],
  "comment": "9 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\sigma(n)$ denote the sum of the positive divisors of $n$. We say that $n$ is perfect if $\\sigma(n) = 2 n$. Currently there are no known odd perfect numbers. It is known that if an odd perfect number exists, then it must be of the form $N = p^\\alpha \\prod_{j=1}^k q_j^{2 \\beta_j}$, where $p, q_1, ..., q_k$ are distinct primes and $p \\equiv \\alpha\\equiv 1 \\pmod{4}$. Define the total number of prime factors of $N$ as $\\Omega(N) := \\alpha + 2 \\sum_{j=1}^k \\beta_j$. Sayers showed that $\\Omega(N) \\geq 29$. This was later extended by Iannucci and Sorli to show that $\\Omega(N) \\geq 37$. This was extended by the author to show that $\\Omega(N) \\geq 47$. Using an idea of Carl Pomerance this paper extends these results. The current new bound is $\\Omega(N) \\geq 75$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2006-04-12T19:02:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A25",
      "11Y70"
    ],
    "keywords": [
      "odd perfect number",
      "prime factors",
      "total number",
      "techniques",
      "distinct primes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}