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A generalization of a curious congruence on harmonic sums

Author(s): Xia Zhou; Tianxin Cai
Journal: Proc. Amer. Math. Soc. 135 (2007), 1329-1333.
MSC (2000): Primary 11A07, 11A41
Posted: December 28, 2006
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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.

References:

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Chun-Gang Ji, A Simple Proof of A Curious Congruence By Zhao. Proceedings of The American Mathematical Society, 133 (2005):3469-3472. MR 2163581 (2006d:11005)

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J.W.L. Glaisher, On the residues of the sums of products of the first p-1 numbers, and their powers, to modulus $ p^2$ or $ p^3$. Quart. J. Pure Appl. Math., 31 (1900): 321-353.

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J.W.L. Glaisher, On the residues of the inverse powers of numbers in arithmetic progression. Quart. J. Pure Appl. Math., 32 (1901):271-305.

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E. Lehmer, On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson. Ann. Math., 39 (1938):350-360. MR 1503412

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Zhihong Sun, Congruence concerning Bernoulli numbers and Bernoulli polynomials, Discrete Applied Mathematics, 105 (2000):193-223. MR 1780472 (2001m:11022)

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L.C. Washington, Introduction to Cyclotomic Fields, 2nd ed., Springer-Verlag, New York, 1997. MR 1421575 (97h:11130)

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Jiangqiang Zhao, Bernoulli numbers, Wolstenholme's Theorem, and $ p^5$ variations of Lucas' Theorem, arxiv.org/abs/math.NT/0303332

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Jiangqiang Zhao, Multiple Harmonic Sums I: Generalizations of Wolstenholme's Theorem, xxx.lanl.gcv/abs/math.NT/0301252

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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: 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
Posted: 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 of article: Copyright 2006, American Mathematical Society

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Jianqiang Zhao, Bernoulli numbers, Wolstenholme's theorem, and p^5 variations of Lucas' theorem, Journal of Number Theory 123 (2007), 18-26. (English)

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