Binomial coefficients and the ring of p-adic integers

TL;DR

This paper investigates the density of binomial coefficients in the ring of p-adic integers, establishing conditions under which these coefficients form a complete residue system modulo any power of p.

Contribution

It provides new criteria for when binomial coefficients are dense in Z_p, expanding understanding of their distribution in p-adic number theory.

Findings

Binomial coefficients are dense in Z_p under specified conditions.

The set of binomial coefficients contains a complete residue system modulo any power of p.

Conditions involve inequalities relating k, p, and powers of p.

Abstract

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