Proofs of power sum and binomial coefficient congruences via Pascal's identity
Number Theory
2011-03-23 v1 History and Overview
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.
Cite
@article{arxiv.1011.0076,
title = {Proofs of power sum and binomial coefficient congruences via Pascal's identity},
author = {Kieren MacMillan and Jonathan Sondow},
journal= {arXiv preprint arXiv:1011.0076},
year = {2011}
}
Comments
4 pages, to appear in Amer. Math. Monthly