Odd perfect numbers have a prime factor exceeding

Authors:
Takeshi Goto and Yasuo Ohno

Journal:
Math. Comp. **77** (2008), 1859-1868

MSC (2000):
Primary 11A25, 11Y70

Published electronically:
February 12, 2008

Previous version:
Original version posted with incorrect PII checkdigit on first page.

MathSciNet review:
2398799

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Abstract | References | Similar Articles | Additional Information

Abstract: Jenkins in 2003 showed that every odd perfect number is divisible by a prime exceeding . Using the properties of cyclotomic polynomials, we improve this result to show that every perfect number is divisible by a prime exceeding .

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Additional Information

**Takeshi Goto**

Affiliation:
Department of Mathematics, Faculty of Science and Technology, Tokyo University of Science, Noda, Chiba, 278-8510, Japan

Email:
goto_takeshi@ma.noda.tus.ac.jp

**Yasuo Ohno**

Affiliation:
Department of Mathematics, Kinki University Higashi-Osaka, Osaka 577-8502, Japan

Email:
ohno@math.kindai.ac.jp

DOI:
https://doi.org/10.1090/S0025-5718-08-02050-4

Keywords:
Odd perfect numbers,
cyclotomic numbers

Received by editor(s):
December 13, 2006

Received by editor(s) in revised form:
February 26, 2007

Published electronically:
February 12, 2008

Additional Notes:
This work was supported by Computing and Communications Center, Kyushu University

The second author was supported in part by JSPS Grant-in-Aid No. 15740025 and No. 18740020

Article copyright:
© Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.