On p-adic approximation of sums of binomial coefficients

TL;DR

This paper introduces advanced p-adic approximation techniques for binomial coefficients, providing explicit formulas for linear combinations divisible by high powers of a prime, extending Jacobsthal's work.

Contribution

It presents higher-order generalizations of Jacobsthal's p-adic approximation, offering new explicit formulas for binomial coefficient sums with divisibility properties.

Findings

Derived explicit formulas for binomial sums divisible by large powers of p

Extended Jacobsthal's p-adic approximation to higher orders

Established divisibility results for linear combinations of binomial coefficients

Abstract

We propose higher-order generalizations of Jacobsthal's $p$-adic approximation for binomial coefficients. Our results imply explicit formulae for linear combinations of binomial coefficients $\binom{ip}{p}$ ($i=1,2,\dots$) that are divisible by arbitrarily large powers of prime $p$.