## On the largest prime divisor of an odd harmonic number

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- by Yusuke Chishiki, Takeshi Goto and Yasuo Ohno PDF
- Math. Comp.
**76**(2007), 1577-1587 Request permission

## 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
- 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
- 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.
- © Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp.
**76**(2007), 1577-1587 - MSC (2000): Primary 11A25, 11Y70
- DOI: https://doi.org/10.1090/S0025-5718-07-01933-3
- MathSciNet review: 2299789