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arXiv · 1801.01542

Congruences of Power Sums

Nicholas J. Newsome·Maria S. Nogin·Adnan H. Sabuwala
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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.

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Nicholas J. Newsome, Maria S. Nogin, Adnan H. Sabuwala. 2017-12-20. Congruences of Power Sums. https://arxiv.org/abs/1801.01542

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