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Every odd perfect number has a prime factor which exceeds $\mathrm{10^{6}}$


Authors: Peter Hagis Jr. and Graeme L. Cohen
Journal: Math. Comp. 67 (1998), 1323-1330
MSC (1991): Primary 11A25, 11Y70
DOI: https://doi.org/10.1090/S0025-5718-98-00982-X
MathSciNet review: 1484897
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Abstract: It is proved here that every odd perfect number is divisible by a prime greater than $10^{6}$.


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

  • 1. A. S. Bang, Taltheoretiske Undersøgelser, Tidsskrift Math. 5, IV (1886), 70-80, 130-137.
  • 2. M. S. Brandstein, New lower bound for a factor of an odd perfect number, Abstracts of the Amer. Math. Soc. 3 (1982), 257.
  • 3. J. T. Condict, On an Odd Perfect Number's Largest Prime Divisor, Senior Thesis (Middlebury College), May 1978.
  • 4. P. Hagis, Jr. and W. L. McDaniel, On the largest prime divisor of an odd perfect number. II, Math. Comp. 29 (1975), 922-924. MR 51:8021
  • 5. P. Hagis, Jr. and G. L. Cohen, Every odd perfect number has a prime factor which exceeds $10^{6}$ (with appendix), Research Report No.93-5 (School of Mathematical Sciences, University of Technology, Sydney), July 1993.
  • 6. W. L. McDaniel, On multiple prime divisors of cyclotomic polynomials, Math. Comp. 28 (1974), 847-850. MR 52:8022
  • 7. P. L. Montgomery, New solutions of $a^{p-1}\equiv 1 \text{ \rm (mod $p^{2}$)}$, Math. Comp. 61 (1993), 361-363. MR 94d:11003
  • 8. T. Nagell, Introduction to Number Theory, second edition, Chelsea, New York, 1964. MR 30:4714
  • 9. I. Niven, Irrational Numbers, Wiley, New York, 1956. MR 18:195c

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Additional Information

Peter Hagis Jr.
Affiliation: Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19122

Graeme L. Cohen
Affiliation: Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19122; School of Mathematical Sciences, University of Technology, Sydney, Broadway, NSW 2007, Australia
Email: g.cohen@maths.uts.edu.au

DOI: https://doi.org/10.1090/S0025-5718-98-00982-X
Received by editor(s): October 24, 1995
Received by editor(s) in revised form: July 10, 1996
Article copyright: © Copyright 1998 American Mathematical Society

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