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arXiv · 1011.0076

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

Kieren MacMillan·Jonathan Sondow
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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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Kieren MacMillan, Jonathan Sondow. 2010-10-30. Proofs of power sum and binomial coefficient congruences via Pascal's identity. https://arxiv.org/abs/1011.0076

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