On the total number of prime factors of an odd perfect number

Authors:
D. E. Iannucci and R. M. Sorli

Journal:
Math. Comp. **72** (2003), 2077-2084

MSC (2000):
Primary 11A25, 11Y70

Published electronically:
May 8, 2003

MathSciNet review:
1986824

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We say is *perfect* if , where denotes the sum of the positive divisors of . No odd perfect numbers are known, but it is well known that if such a number exists, it must have prime factorization of the form , where , , ..., are distinct primes and . We prove that if or for all , , then . We also prove as our main result that , where . This improves a result of Sayers given in 1986.

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Additional Information

**D. E. Iannucci**

Affiliation:
University of the Virgin Islands, St. Thomas, Virgin Islands 00802

Email:
diannuc@uvi.edu

**R. M. Sorli**

Affiliation:
Department of Mathematical Sciences, University of Technology, Sydney, Broadway, 2007, Australia

Email:
rons@maths.uts.edu.au

DOI:
https://doi.org/10.1090/S0025-5718-03-01522-9

Keywords:
Odd perfect numbers,
factorization

Received by editor(s):
November 7, 2001

Published electronically:
May 8, 2003

Additional Notes:
The authors are grateful for the advice and assistance given by Graeme Cohen

Article copyright:
© Copyright 2003
American Mathematical Society