Proofs of Power Sum and Binomial Coefficient Congruences Via Pascal's Identity

2011/06/01 by Kieren MacMillan, Jonathan Sondow · 4 citations
Mathematics · #Analytic Number Theory Research #Advanced Mathematical Identities #History and Theory of Mathematics

paper · doi:10.4169/amer.math.monthly.118.06.549

openalex publication_date 2011/06/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29

Abstract

A well-known and frequently cited congruence for power sums is where n ≥ 1 and p is prime. 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.

Cited by

• On difference-differential Lax pairs and integrals of Painlevé equations in finite characteristic
• On integrals of non-autonomous dynamical systems in finite characteristic
• Blaise Pascal [wikipedia]
• Faulhaber's formula [wikipedia]

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