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

DOI:
https://doi.org/10.1090/S0025-5718-07-02033-9

Published electronically:
May 30, 2007

MathSciNet review:
2336293

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Abstract | References | Similar Articles | Additional Information

Abstract: Let denote the sum of the positive divisors of . We say that is perfect if . 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 , where are distinct primes and . Define the total number of prime factors of as . Sayers showed that . This was later extended by Iannucci and Sorli to show that . This was extended by the author to show that . Using an idea of Carl Pomerance this paper extends these results. The current new bound is .

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Additional Information

**Kevin G. Hare**

Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1

Email:
kghare@math.uwaterloo.ca

DOI:
https://doi.org/10.1090/S0025-5718-07-02033-9

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