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 $$\frac{1}{t}\sum _{s=0}^{t-1}\left(\genfrac{}{}{0ex}{}{n+d{\zeta }_t^s}{d-1}\right)\equiv \left(\genfrac{}{}{0ex}{}{n}{d-1}\right)(\mathrm{mod}d)$$ 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 $r\in \{1,\dots ,t\}$.

Accepted to Bulletin of the Malaysian Mathematical Sciences Society