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

Full-text PDF Free Access

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 .

**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, 857–868. MR**1094940**, 10.1090/S0025-5718-1991-1094940-3**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, 21–35. Combinatorial algorithms. MR**2016472**, 10.1016/S1570-8667(03)00004-2**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, 1027–1032. MR**572873**, 10.1090/S0025-5718-1980-0572873-9**6.**Peter Hagis Jr.,*Sketch of a proof that an odd perfect number relatively prime to 3 has at least eleven prime factors*, Math. Comp.**40**(1983), no. 161, 399–404. MR**679455**, 10.1090/S0025-5718-1983-0679455-1**7.**Kevin G. Hare,*More on the total number of prime factors of an odd perfect number*, Math. Comp.**74**(2005), no. 250, 1003–1008 (electronic). MR**2114661**, 10.1090/S0025-5718-04-01683-7**8.**D. E. Iannucci and R. M. Sorli,*On the total number of prime factors of an odd perfect number*, Math. Comp.**72**(2003), no. 244, 2077–2084 (electronic). MR**1986824**, 10.1090/S0025-5718-03-01522-9**9.**Masao Kishore,*Odd perfect numbers not divisible by 3. II*, Math. Comp.**40**(1983), no. 161, 405–411. MR**679456**, 10.1090/S0025-5718-1983-0679456-3**10.**Pace P. Nielsen,*Odd perfect numbers have at least nine distinct factors*, Math. Comp. (to appear).**11.**Karl K. Norton,*Remarks on the number of factors of an odd perfect number*, Acta Arith.**6**(1960/1961), 365–374. MR**0147434****12.**Carl Pomerance,*Odd perfect numbers are divisible by at least seven distinct primes*, Acta Arith.**25**(1973/74), 265–300. MR**0340169****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.

Retrieve articles in *Mathematics of Computation*
with MSC (2000):
11A25,
11Y70

Retrieve articles in all journals with MSC (2000): 11A25, 11Y70

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