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

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.