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

2010/10/30 by Kieren MacMillan, Jonathan Sondow · 1 citation
Mathematics · #math.NT #math.HO #msc:11A07 #msc:11B65

paper · pdf

published as Amer. Math. Monthly 118 (2011) 549-551 · 4 pages, to appear in Amer. Math. Monthly

arxiv created 2010/10/30 · arxiv updated 2011/03/23

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.

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