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 :

**1.**Chun-Gang Ji,*A simple proof of a curious congruence by Zhao*, Proc. Amer. Math. Soc.**133**(2005), no. 12, 3469–3472 (electronic). MR**2163581**, 10.1090/S0002-9939-05-07939-6**2.**J.W.L. Glaisher,*On the residues of the sums of products of the first p-1 numbers, and their powers, to modulus or .**Quart. J. Pure Appl. Math.,***31**(1900): 321-353.**3.**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.**4.**Emma Lehmer,*On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson*, Ann. of Math. (2)**39**(1938), no. 2, 350–360. MR**1503412**, 10.2307/1968791**5.**Zhi-Hong Sun,*Congruences concerning Bernoulli numbers and Bernoulli polynomials*, Discrete Appl. Math.**105**(2000), no. 1-3, 193–223. MR**1780472**, 10.1016/S0166-218X(00)00184-0**6.**Lawrence C. Washington,*Introduction to cyclotomic fields*, 2nd ed., Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1997. MR**1421575****7.**Jiangqiang Zhao,*Bernoulli numbers, Wolstenholme's Theorem, and variations of Lucas' Theorem,*arxiv.org/abs/math.NT/0303332**8.**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:
http://dx.doi.org/10.1090/S0002-9939-06-08777-6

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