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Mathematics of Computation

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More on the total number of prime factors of an odd perfect number

Author: Kevin G. Hare
Journal: Math. Comp. 74 (2005), 1003-1008
MSC (2000): Primary 11A25, 11Y70
Published electronically: June 29, 2004
MathSciNet review: 2114661
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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 $p, q_1, \ldots, q_k$ 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 paper extends these results to show that $\Omega(N) \geq 47$.

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Additional Information

Kevin G. Hare
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1

Keywords: Perfect numbers, divisor function, prime numbers
Received by editor(s): October 24, 2003
Received by editor(s) in revised form: December 2, 2003
Published electronically: June 29, 2004
Additional Notes: This research was supported, in part, by NSERC of Canada.
Article copyright: © Copyright 2004 by the author