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), 22412248
MSC (2000):
Primary 11A25, 11Y70
Published electronically:
May 30, 2007
MathSciNet review:
2336293
Fulltext PDF Free Access
Abstract 
References 
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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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 1.
 R. P. Brent, G. L. Cohen, and H. J. J. te Riele, Improved techniques for lower bounds for odd perfect numbers, Math. Comp. 57 (1991), no. 196, 857868. MR 1094940 (92c:11004)
 2.
 E. Z. Chein, An odd perfect number has at least 8 prime factors, Ph.D. thesis, Pennsylvania State University, 1979.
 3.
 Graeme L. Cohen, Generalised quasiperfect numbers, Ph.D. thesis, University of New South Wales, 1982.
 4.
 Graeme L. Cohen and Ronald M. Sorli, On the number of distinct prime factors of an odd perfect number, J. Discrete Algorithms 1 (2003), no. 1, 2135, Combinatorial algorithms. MR 2016472 (2004h:11003)
 5.
 Peter Hagis, Jr., Outline of a proof that every odd perfect number has at least eight prime factors, Math. Comp. 35 (1980), no. 151, 10271032. MR 572873 (81k:10004)
 6.
 , Sketch of a proof that an odd perfect number relatively prime to has at least eleven prime factors, Math. Comp. 40 (1983), no. 161, 399404. MR 679455 (85b:11004)
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 Kevin G. Hare, More on the total number of prime factors of an odd perfect number, Math. Comp. 74 (2005), no. 250, 10031008 (electronic). MR 2114661 (2005h:11010)
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 D. E. Iannucci and M. Sorli, On the total number of prime factors of an odd perfect number, Math. Comp. 72 (2003), no. 244, 20772084. MR 1986824 (2004b:11008)
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 Masao Kishore, Odd perfect numbers not divisible by . II, Math. Comp. 40 (1983), no. 161, 405411. MR 679456 (84d:10009)
 10.
 Pace P. Nielsen, Odd perfect numbers have at least nine distinct factors, Math. Comp. (to appear).
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 Karl K. Norton, Remarks on the number of factors of an odd perfect number, Acta Arith. 6 (1960/1961), 365374. MR 0147434 (26:4950)
 12.
 Carl Pomerance, Odd perfect numbers are divisible by at least seven distinct primes, Acta Arith. 25 (1973/74), 265300. MR 0340169 (49:4925)
 13.
 M. Sayers, An improved lower bound for the total number of prime factors of an odd perfect number, Master's thesis, New South Wales Institute of Technology, 1986.
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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:
http://dx.doi.org/10.1090/S0025571807020339
PII:
S 00255718(07)020339
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
