Proofs of power sum and binomial coefficient congruences via Pascal's identity

TL;DR

This paper provides an elementary proof of a classical prime modulus sum theorem using Pascal's identity, and applies it to binomial coefficient sum congruences, highlighting historical and mathematical insights.

Contribution

It introduces a new elementary proof of a well-known theorem using Pascal's identity and applies this to binomial coefficient sum congruences.

Findings

Elementary proof of power sum congruences using Pascal's identity

Simplified proof of Hermite and Bachmann's binomial sum congruence

Clarification of proof techniques for prime modulus sums

Abstract

A frequently cited theorem says that for n > 0 and prime p, the sum of the first p n-th powers is congruent to -1 modulo p if p-1 divides n, and to 0 otherwise. We survey the main ingredients in several known proofs. Then we give an elementary proof, using an identity for power sums proven by Pascal in 1654. An application is a simple proof of a congruence for certain sums of binomial coefficients, due to Hermite and Bachmann.