Congruences of Power Sums

Abstract

The following congruence for power sums, $S_n(p)$, is well known and has many applications: $1^n+2^n+\cdots +p^n\equiv \{\begin{array}{llc}-1\text{ mod }p,& \text{ if }p-1|n;0\text{ mod }p,& \text{ if }p-1\nmid n,\end{array}$ where $n\in \mathbb{N}$ and $p$ is prime. We extend this congruence, in particular, to the case when $p$ is any power of a prime. We also show that the sequence $(S_n(m)\text{ mod }k{)}_{m\ge 1}$ is periodic and determine its period.

12 pages