A generalization of a curious congruence on harmonic sums

Zhou, Xia; Cai, Tianxin

doi:10.1090/S0002-9939-06-08777-6

A generalization of a curious congruence on harmonic sums
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by Xia Zhou and Tianxin Cai PDF

Proc. Amer. Math. Soc. 135 (2007), 1329-1333 Request permission

Abstract:

Zhao established a curious harmonic congruence for prime $p>3$: \[ \sum _{\substack {i+j+k=pi,j,k>0}} \frac {1}{ijk} \equiv -2B_{p-3}(\operatorname {mod} p). \] In this note the authors extend it to the following congruence for any prime $p > 3$ and positive integer $n \le p-2$: \[ \sum _{\substack {l_{1}+l_{2}+\cdots +l_{n}=pl_{1}, \cdots ,l_{n}>0}} \frac {1}{l_{1}l_{2}\cdots l_{n}}\equiv \begin {cases} -(n-1)!\ B_{p-n} (\textrm {mod}\; p) & \text {if $2\nmid n$},\\ -\frac {n}{2(n+1)}\ n!\ B_{p-n-1}\ p\ (\operatorname {mod} p^2) &\text {if $2|n$}. \end {cases} \] Other improvements on congruences of harmonic sums are also obtained.

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