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 $$\frac{1}{t}\sum_{s=0}^{t-1}{n+d\zeta_t^s\choose d-1}\equiv {n\choose d-1}\pmod{d}$$ 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 $r\in\{1,\ldots ,t\}$.