Congruences of Power Sums

TL;DR

This paper extends known congruences of power sums to prime powers and analyzes the periodicity of the sequence of power sums modulo k, providing new insights into their algebraic properties.

Contribution

It generalizes the classical congruence for power sums to prime powers and characterizes the periodicity of the sequence of power sums modulo k.

Findings

Extended congruence to prime power moduli.

Proved periodicity of power sum sequences modulo k.

Determined the explicit period of these sequences.

Abstract

The following congruence for power sums, $S_n(p)$, is well known and has many applications: $1^n+2^n +\dots +p^n \equiv\begin{cases} -1 \text{ mod } p, & \text{ if } \ p-1 \ | \ n; 0 \text{ mod } p, & \text{ if } \ p-1 \ \not| \ n, \end{cases}$ 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 $(S_n(m) \text{ mod } k )_{m \geq 1}$ is periodic and determine its period.