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| NOTES ON GLAISHER'S CONGRUENCES |
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Citation： |
HONG Shaofang.NOTES ON GLAISHER'S CONGRUENCES[J].Chinese Annals of Mathematics B,2000,21(1):33~38 |
| Page view： 1346
Net amount： 765 |
Authors： |
HONG Shaofang; |
Foundation： |
Project supported by the Postdoctoral Foundation of China. |
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| Abstract： |
Let $p$ be an odd prime and let $n\ge 1, k\ge 0$ and $r$ be integers. Denote by $B_k$ the $k$-th Bernoulli number. It is proved that (i) If $r\ge 1$ is odd and suppose $p\ge r+4,$ then $\sum^{p-1}_{j=1}\limits {1\over (np+j)^r}\equiv -{{(2n+1)r(r+1)}\over {2(r+2)}} B_{p-r-2}p^2 \ (\hbox {mod}\, p^3).$ (ii) If $r\ge 2$ is even and suppose $p\ge r+3,$ then $\sum_{j=1}^{p-1}\limits {1\over (np+j)^r}\equiv {r\over {r+1}}B_{p-r-1}p\ (\hbox {mod}\, p^2).$ (iii) $\sum^{p-1}_{j=1}\limits {1\over (np+j)^{p-2}}\equiv -(2n+1)p\ (\hbox {mod}\, p^2).$ This result generalizes the Glaisher's
congruence. As a corollary, a generalization of the Wolstenholme's theorem is obtained. |
Keywords： |
Glaisher’s congruence, kth Bernoulli number, Teichmuller character, p-adic L function |
Classification： |
11A41, 11S80 |
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