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

Author(s): Peter Hagis Jr.; Graeme L. Cohen.
Journal: Math. Comp. 67 (1998), 1323-1330.
MSC (1991): Primary 11A25, 11Y70
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Abstract: It is proved here that every odd perfect number is divisible by a prime greater than $10^{6}$.

References:

1.
A. S. Bang, Taltheoretiske Undersøgelser, Tidsskrift Math. 5, IV (1886), 70-80, 130-137.

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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.

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W. L. McDaniel, On multiple prime divisors of cyclotomic polynomials, Math. Comp. 28 (1974), 847-850. MR 52:8022

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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

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T. Nagell, Introduction to Number Theory, second edition, Chelsea, New York, 1964. MR 30:4714

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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: 10.1090/S0025-5718-98-00982-X
PII: S 0025-5718(98)00982-X
Received by editor(s): October 24, 1995
Received by editor(s) in revised form: July 10, 1996
Copyright of article: Copyright 1998, American Mathematical Society

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