A simple proof of a curious congruence by Zhao
HTML articles powered by AMS MathViewer
- by Chun-Gang Ji PDF
- Proc. Amer. Math. Soc. 133 (2005), 3469-3472 Request permission
Abstract:
The author gives a simple proof of the following curious congruence for odd prime $p>3$ which was established by Jianqiang Zhao: \begin{equation*}\sum _{\substack {{i+j+k=p} {i,\ j,\ k>0}}}\frac {1}{ijk}\equiv -2B_{p-3}(\text {mod}\ p).\end{equation*}References
- D. F. Bailey, Two $p^3$ variations of Lucas’ theorem, J. Number Theory 35 (1990), no. 2, 208–215. MR 1057323, DOI 10.1016/0022-314X(90)90113-6
- G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 5th ed., The Clarendon Press, Oxford University Press, New York, 1979. MR 568909
- 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, DOI 10.2307/1968791
- Zhi-Hong Sun, Congruences concerning Bernoulli numbers and Bernoulli polynomials, Discrete Appl. Math. 105 (2000), no. 1-3, 193–223. MR 1780472, DOI 10.1016/S0166-218X(00)00184-0
- Lawrence C. Washington, Introduction to cyclotomic fields, 2nd ed., Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1997. MR 1421575, DOI 10.1007/978-1-4612-1934-7
- Jianqiang Zhao, Partial sums of multiple zeta value series I: generalizations of Wolstenholme’s theorem, xxx.lanl.gov/abs/math.NT/0301252, 19pages.
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
- Received by editor(s): December 1, 2003
- Received by editor(s) in revised form: August 13, 2004
- Published electronically: 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 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 133 (2005), 3469-3472
- MSC (2000): Primary 11A07, 11A41
- DOI: https://doi.org/10.1090/S0002-9939-05-07939-6
- MathSciNet review: 2163581