Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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$.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 11A25, 11Y70

Retrieve articles in all journals with MSC (2000): 11A25, 11Y70


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

DOI: http://dx.doi.org/10.1090/S0025-5718-07-01933-3
PII: S 0025-5718(07)01933-3
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.