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

Author(s): Takeshi Goto; Yasuo Ohno.
Journal: Math. Comp. 77 (2008), 1859-1868.
MSC (2000): Primary 11A25, 11Y70
Posted: February 12, 2008
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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
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: 10.1090/S0025-5718-08-02050-9
PII: S 0025-5718(08)02050-9
Keywords: Odd perfect numbers, cyclotomic numbers
Received by editor(s): December 13, 2006
Received by editor(s) in revised form: February 26, 2007
Posted: 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
Copyright of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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