Binomial coefficients and the ring of $p$-adic integers

Abstract:

Let $k>1$ be an integer and let $p$ be a prime. We show that if $p^{a}\leqslant k<2p^{a}$ or $k=p^{a}q+1$ (with $q<p/2$) for some $a=1,2,3,\ldots$, then the set $\{\binom nk: n=0,1,2,\ldots \}$ is dense in the ring $\mathbb {Z}_{p}$ of $p$-adic integers; i.e., it contains a complete system of residues modulo any power of $p$.