Binomial coefficients, roots of unity and powers of prime numbers

TL;DR

This paper characterizes the integers d for which a specific binomial coefficient congruence involving roots of unity holds universally, showing it is true precisely when d is a prime power with certain conditions.

Contribution

It provides a complete characterization of d satisfying the congruence, linking roots of unity, binomial coefficients, and prime power structures.

Findings

The congruence holds for all n if and only if d=p^r with p > t and r ≤ t.

If d has a prime divisor greater than t, then d must be a prime power p^r with p > t.

The result connects combinatorial identities with number-theoretic properties of prime powers.

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 $r\in\{1,\ldots ,t\}$.