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Mathematics of Computation

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New techniques for bounds on the total number of prime factors of an odd perfect number

Author: Kevin G. Hare
Journal: Math. Comp. 76 (2007), 2241-2248
MSC (2000): Primary 11A25, 11Y70
Published electronically: May 30, 2007
MathSciNet review: 2336293
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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, \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$.

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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): 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.
Article copyright: © Copyright 2007 by the author