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On the largest prime divisor of an odd harmonic number

Authors: Yusuke Chishiki, Takeshi Goto and Yasuo Ohno
Journal: Math. Comp. 76 (2007), 1577-1587
MSC (2000): Primary 11A25, 11Y70
Published electronically: January 30, 2007
MathSciNet review: 2299789
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Abstract: A positive integer is called a (Ore’s) harmonic number if its positive divisors have integral harmonic mean. Ore conjectured that every harmonic number greater than $1$ is even. If Ore’s conjecture is true, there exist no odd perfect numbers. In this paper, we prove that every odd harmonic number greater than $1$ must be divisible by a prime greater than $10^5$.

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Additional Information

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

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: Harmonic numbers, perfect numbers, cyclotomic numbers
Received by editor(s): September 29, 2005
Received by editor(s) in revised form: February 15, 2006
Published electronically: January 30, 2007
Additional Notes: The third author was supported in part by JSPS Grant-in-Aid No. 15740025.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.