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Mathematics of Computation

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Odd perfect numbers have a prime factor exceeding $ 10^8$

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: Jenkins in 2003 showed that every odd perfect number is divisible by a prime exceeding $ 10^7$. Using the properties of cyclotomic polynomials, we improve this result to show that every perfect number is divisible by a prime exceeding $ 10^8$.

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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

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

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.