Polynomial Parents of Some Binomial Congruences and Fermat Quotients

Abstract

The main purpose of this article is to observe and use some rather simple polynomial identities to deduce some congruences involving binomial coefficients and Fermat quotients. Thus, the consequential identities or congruences may consider the polynomial identity as a parent. We point out that our purpose here is modest, in observing how some simple polynomial identities can lead to congruences for Fermat quotients, etc. Indeed, some more general congruences are known in literature which connect Fermat quotients with Bernoulli numbers and harmonic sums, etc. In what follows, we do not spell out which of our identities are known in more general forms by other methods. This article is partly expository.