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.References
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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
- 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
- © Copyright 2006 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 135 (2007), 1329-1333
- MSC (2000): Primary 11A07, 11A41
- DOI: https://doi.org/10.1090/S0002-9939-06-08777-6
- MathSciNet review: 2276641