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A frequently cited theorem says that for n > 0 and prime p, the sum of thefirst p n-th powers is congruent to -1 modulo p if p-1 divides n, and to 0otherwise. We survey the main ingredients in several known proofs. Then we givean 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 binomialcoefficients, due to Hermite and Bachmann.

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