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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2024 MCQ for Mathematics of Computation is 1.78.

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New techniques for bounds on the total number of prime factors of an odd perfect number
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by Kevin G. Hare;
Math. Comp. 76 (2007), 2241-2248
DOI: https://doi.org/10.1090/S0025-5718-07-02033-9
Published electronically: May 30, 2007

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 $p, q_1, \cdots , q_k$ 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 $\Omega (N) \geq 75$.
References
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Bibliographic Information
  • Kevin G. Hare
  • Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1
  • Email: kghare@math.uwaterloo.ca
  • Received by editor(s): July 25, 2005
  • Received by editor(s) in revised form: October 10, 2005
  • Published electronically: May 30, 2007
  • Additional Notes: The research of the author was supported in part by NSERC of Canada.
  • © Copyright 2007 by the author
  • Journal: Math. Comp. 76 (2007), 2241-2248
  • MSC (2000): Primary 11A25, 11Y70
  • DOI: https://doi.org/10.1090/S0025-5718-07-02033-9
  • MathSciNet review: 2336293