Notes on Glaisher’s Congruences

Abstract

Let p be an odd prime and let n ≥ 1, k ≥ 0 and r be integers. Denote by B _k the k-th Bernoulli number. It is proved that (i) If r ≥ 1 is odd and suppose p ≥ r + 4, then ${άthop♪m⌜mits_{j=1}^{p-1}} {1⩈er{(np+j)^r}}≡uiv-{(2n+1)r(r+1)⩈er{2(r+2)}}B_{p-r-2}p^2 ({⤪ mod} p^3)$. (ii) If r ≥ 2 is even and suppose p ≥ r + 3, then ${άthop♪m⌜mits_{j=1}^{p-1}} {1⩈er{(np+j)^r}}≡uiv-{r⩈er{r+1}}B_{p-r-1}p ({⤪ mod} p^2)$. (iii) ${άthop♪m⌜mits_{j=1}^{p-1}} {1⩈er{(np+j)}^{p-2}}≡uiv-(2n+1)p ({⤪ mod} p^2)$. This result generalizes the Glaisher’s congruence. As a corollary, a generalization of the Wolsten-holme’s theorem is obtained.

Keywords

Glaisher’s congruence / kth Bernoulli number / Teichmuller character / p-adic L function / 11A41 / 11S80 / O156.1

References

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