Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

   
 

 

More on the total number of prime factors of an odd perfect number


Author: Kevin G. Hare
Journal: Math. Comp. 74 (2005), 1003-1008
MSC (2000): Primary 11A25, 11Y70
Published electronically: June 29, 2004
MathSciNet review: 2114661
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 $p, q_1, \ldots, q_k$ 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 paper extends these results to show that $\Omega(N) \geq 47$.


References [Enhancements On Off] (What's this?)

  • 1. E. Z. Chein, An odd perfect number has at least 8 prime factors, Ph.D. thesis, Pennsylvania State University, 1979.
  • 2. Graeme L. Cohen, Generalised quasiperfect numbers Ph.D. thesis, University of New South Wales, 1982.
  • 3. -, On the largest component of an odd perfect number, J. Austral. Math. Soc. Ser. A 42 (1987), no. 2, 280-286. MR 87m:11005
  • 4. 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 81k:10004
  • 5. -, 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 85b:11004
  • 6. K. G. Hare, Home page, http://www.math.berkeley.edu/$\sim$kghare, 2002.
  • 7. D. E. Iannucci and M. Sorli, On the total number of prime factors of an odd perfect number, Math. Comp. 72 (2003), no. 244, 2077-2084.MR 2004b:11008
  • 8. Masao Kishore, Odd perfect numbers not divisible by $3$. II, Math. Comp. 40 (1983), no. 161, 405-411. MR 84d:10009
  • 9. Wayne L. McDaniel, On the divisibility of an odd perfect number by the sixth power of a prime, Math. Comp. 25 (1971), 383-385. MR 45:5074
  • 10. Karl K. Norton, Remarks on the number of factors of an odd perfect number, Acta Arith. 6 (1960/1961), 365-374. MR 26:4950
  • 11. 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.
  • 12. R.M. Sorli, Factorization tables, http://www-staff.maths.uts.edu.au/$\sim$rons/fact/fact. htm, 1999.
  • 13. Paul Zimmermann, The ECMNET project, http://www.loria.fr/$\sim$zimmerma/records/ ecmnet.html, 2003.

Similar Articles

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: http://dx.doi.org/10.1090/S0025-5718-04-01683-7
Keywords: Perfect numbers, divisor function, prime numbers
Received by editor(s): October 24, 2003
Received by editor(s) in revised form: December 2, 2003
Published electronically: June 29, 2004
Additional Notes: This research was supported, in part, by NSERC of Canada.
Article copyright: © Copyright 2004 by the author