The third largest prime divisor

of an odd perfect number

exceeds one hundred

Author:
Douglas E. Iannucci

Journal:
Math. Comp. **69** (2000), 867-879

MSC (1991):
Primary 11A25, 11Y70

Published electronically:
May 17, 1999

MathSciNet review:
1651762

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Abstract | References | Similar Articles | Additional Information

Abstract: Let denote the sum of positive divisors of the natural number . Such a number is said to be *perfect* if . It is well known that a number is even and perfect if and only if it has the form where is prime.

It is unknown whether or not odd perfect numbers exist, although many conditions necessary for their existence have been found. For example, Cohen and Hagis have shown that the largest prime divisor of an odd perfect number must exceed , and Iannucci showed that the second largest must exceed . In this paper, we prove that the third largest prime divisor of an odd perfect number must exceed 100.

**1.**Leonard M. Adleman, Carl Pomerance, and Robert S. Rumely,*On distinguishing prime numbers from composite numbers*, Ann. of Math. (2)**117**(1983), no. 1, 173–206. MR**683806**, 10.2307/2006975**2.**Peter Hagis Jr. and Graeme L. Cohen,*Every odd perfect number has a prime factor which exceeds 10⁶*, Math. Comp.**67**(1998), no. 223, 1323–1330. MR**1484897**, 10.1090/S0025-5718-98-00982-X**3.**Peter Hagis Jr.,*On the second largest prime divisor of an odd perfect number*, Analytic number theory (Philadelphia, Pa., 1980) Lecture Notes in Math., vol. 899, Springer, Berlin-New York, 1981, pp. 254–263. MR**654532****4.**Peter Hagis Jr.,*The third largest prime factor of an odd multiperfect number exceeds 100*, Bull. Malaysian Math. Soc. (2)**9**(1986), no. 2, 43–49. MR**896344****5.**D. Iannucci,*The second largest prime divisor of an odd perfect number exceeds ten thousand*, to appear in Math. Comp.**6.**Peter L. Montgomery,*New solutions of 𝑎^{𝑝-1}≡1\pmod{𝑝²}*, Math. Comp.**61**(1993), no. 203, 361–363. MR**1182246**, 10.1090/S0025-5718-1993-1182246-5**7.**Carl Pomerance,*The second largest prime factor of an odd perfect number*, Math. Comput.**29**(1975), 914–921. MR**0371801**, 10.1090/S0025-5718-1975-0371801-7

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Additional Information

**Douglas E. Iannucci**

Affiliation:
University of the Virgin Islands, 2 John Brewers Bay, St. Thomas, VI 00802

Email:
diannuc@uvi.edu

DOI:
https://doi.org/10.1090/S0025-5718-99-01127-8

Keywords:
Perfect numbers,
cyclotomic polynomials

Received by editor(s):
December 12, 1997

Received by editor(s) in revised form:
January 26, 1998, and June 2, 1998

Published electronically:
May 17, 1999

Additional Notes:
This paper presents the main result of the author’s doctoral dissertation completed at Temple University in 1995 under the direction of Peter Hagis, Jr.

Article copyright:
© Copyright 2000
American Mathematical Society