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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A generalization of a curious congruence on harmonic sums


Authors: Xia Zhou and Tianxin Cai
Journal: Proc. Amer. Math. Soc. 135 (2007), 1329-1333
MSC (2000): Primary 11A07, 11A41
Published electronically: December 28, 2006
MathSciNet review: 2276641
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Abstract | References | Similar Articles | Additional Information

Abstract: Zhao established a curious harmonic congruence for prime $ p>3$:

$\displaystyle \sum_{\substack{i+j+k=p\\ i,j,k>0}}\frac{1}{ijk}\equiv -2B_{p-3}(\mathrm{mod} p).$

In this note the authors extend it to the following congruence for any prime $ p > 3$ and positive integer $ n \le p-2$:

\begin{displaymath}\sum_{\substack{l_{1}+l_{2}+\cdots+l_{n}=p\\ l_{1}, \cdots,l_... ..._{p-n-1} p ({\rm mod}\; p^2) &\text{if}\; 2\vert n. \end{cases}\end{displaymath}

Other improvements on congruences of harmonic sums are also obtained.


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Additional Information

Xia Zhou
Affiliation: Department of Mathematics, Zhejiang University, Hangzhou 310027, People’s Republic of China
Email: unitqq@zju.edu.cn

Tianxin Cai
Affiliation: Department of Mathematics, Zhejiang University, Hangzhou 310027, People’s Republic of China
Email: txcai@mail.hz.zj.cn

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08777-6
PII: S 0002-9939(06)08777-6
Keywords: Bernoulli numbers, congruences of harmonic sums, partitions
Received by editor(s): December 14, 2005
Received by editor(s) in revised form: February 6, 2006
Published electronically: December 28, 2006
Additional Notes: This work was supported by the National Natural Science Foundation of China, Project 10371107
Communicated by: Wen-Ching Winnie Li
Article copyright: © Copyright 2006 American Mathematical Society