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

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

2010-10-30

arxiv.org/abs/1011.0076v1

Amer. Math. Monthly 118 (2011) 549-551

Mathematics

Number Theory

4 pages, to appear in Amer. Math. Monthly

Scientific paper

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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