Binomial Coefficients, Roots of Unity and Powers of Prime Numbers

Abstract

Let \(t\in {\mathbb {N}}_+\) be given. In this article, we are interested in characterizing those \(d\in {\mathbb {N}}_+\) such that the congruence $$\begin{aligned}\frac{1}{t}\sum _{s=0}^{t-1}{n+d\zeta _t^s\atopwithdelims ()d-1}\equiv {n\atopwithdelims ()d-1}\pmod {d}\end{aligned}$$is true for each \(n\in {\mathbb {Z}}\). In particular, assuming that d has a prime divisor greater than t, we show that the above congruence holds for each \(n\in {\mathbb {Z}}\) if and only if \(d=p^r\), where p is a prime number greater than t and \(r\in \{1,\ldots ,t\}\).