Computing solutions to the congruence ${1^n + 2^n + \dotsb + n^n\equiv p \pmod{n}}$

TL;DR

This paper characterizes solutions to a specific modular sum congruence involving prime p, providing an algorithm to efficiently compute solutions up to high bounds, surpassing naive methods.

Contribution

It offers a complete characterization of solutions for the sum congruence with prime p and introduces an efficient algorithm for their computation.

Findings

Characterization of solutions for the sum congruence with prime p.

Development of an algorithm to compute solutions efficiently.

Ability to find solutions up to higher bounds than naive search.

Abstract

It is well-known that the congruence $\sum_{i=1}^{ n} i^{ n} \equiv 1 \pmod{n}$ has exactly five solutions: $\{1,2,6,42,1806\}$. In this work, we characterize the solutions to the congruence $1^n + 2^n + \dotsb + n^n\equiv p \pmod{n}$ for every prime $p$. This characterization leads to an algorithm for computing all such solutions, when there is a finite number of them. More generally, our algorithm enables computing all the solutions below a much higher bound as compared to what can be achieved by a naive exhaustive search.