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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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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