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A simple proof of a curious congruence by Zhao

Author(s): Chun-Gang Ji
Journal: Proc. Amer. Math. Soc. 133 (2005), 3469-3472.
MSC (2000): Primary 11A07, 11A41
Posted: June 8, 2005
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Abstract | References | Similar articles | Additional information

Abstract: The author gives a simple proof of the following curious congruence for odd prime which was established by Jianqiang Zhao:

$\begin{displaymath}\sum _{\substack{{i+j+k=p} {i, j, k>0}}}\frac{1}{ijk}\equiv -2B_{p-3}(\text{mod} p).\end{displaymath}$

References:

1.

D. F. Baily, Two $p^{3}$ variations of Lucas' theorem, Jour. Number Theory 35 (1990), 208-215. MR 1057323 (91f:11008)

2.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Clarendon Press, Oxford, 1979. MR 0568909 (81i:10002)

3.

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

4.

Zhihong Sun, Congruence concerning Bernoulli numbers and Bernoulli polynomials, Discrete Applied Mathematics 105 (2000),

193-223. MR 1780472 (2001m:11022)

5.

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

6.

Jianqiang Zhao, Partial sums of multiple zeta value series I: generalizations of Wolstenholme's theorem, xxx.lanl.gov/abs/math.NT/0301252, 19pages.

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Additional Information:

Chun-Gang Ji
Affiliation: Department of Mathematics, Nanjing Normal University, Nanjing 210097, People's Republic of China
Address at time of publication: Department of Mathematics, Nanjing University, Nanjing 210093, People's Republic of China
Email: jichungang@njnu.edu.cn

DOI: 10.1090/S0002-9939-05-07939-6
PII: S 0002-9939(05)07939-6
Keywords: Wolstenholme's theorem, Kummer congruence, prime number
Received by editor(s): December 1, 2003
Received by editor(s) in revised form: August 13, 2004
Posted: June 8, 2005
Additional Notes: This work was supported by the National Natural Science Foundation of China, Grant Nos. 10171046 and 10201013
Communicated by: Wen-Ching Winnie Li
Copyright of article: Copyright 2005, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.


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