On the largest prime divisor of an odd harmonic number

Chishiki, Yusuke; Goto, Takeshi; Ohno, Yasuo

doi:10.1090/S0025-5718-07-01933-3

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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