论文标题
二项式系数，团结根和质数的力量
Binomial coefficients, roots of unity and powers of prime numbers
论文作者
论文摘要
令$ t \ in \ mathbb {n} _+$。在本文中，我们有兴趣表征这些$ d \ in \ mathbb {n} _+$，以使一致性$$ \ frac {1} {t} {t} {t} \ sum_ {s = 0}^{t-1} {t-1} {n+dbive n+dbes_t^s \ for for for for for for d d d equiv d d d d d d d d d d d d d d d d d d d d d-1} $ equiv equiv equil $ n \ in \ mathbb {z} $。特别是，假设$ d $的素数大于$ t $，我们表明上述一致性为\ mathbb {z} $ in \ mathbb {z} $ in \＆d当$ d = p^r $时，其中$ p $是$ p $是$ t $和$ t $和$ r \ in \ in \ in \ in \ {1，\ ldots，t \ ldots，
Let $t\in\mathbb{N}_+$ be given. In this article we are interested in characterizing those $d\in\mathbb{N}_+$ such that the congruence $$\frac{1}{t}\sum_{s=0}^{t-1}{n+dζ_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 $r\in\{1,\ldots ,t\}$.
