Modulo $p^2$ congruences involving harmonic numbers
Volume 121 / 2018
Annales Polonici Mathematici 121 (2018), 263-278
MSC: Primary 11B75; Secondary 05A19.
DOI: 10.4064/ap180401-12-9
Published online: 9 November 2018
Abstract
The harmonic numbers $H_k=\sum _{j=1}^k1/j$ $(k=0,1,2,\ldots )$ play important roles in mathematics. Let $p \gt 3$ be a prime. We calculate $\sum _{k=1}^{p-1}k^mH_k^n\ ({\rm mod} p^2) $ for $m=1,\ldots ,p-2$ and $n=1,2,3$.
