On a conjecture about the 3-adic valuation of a binomial sum

Abstract

Given any [Formula: see text] define the binomial sum [Formula: see text] A fundamental question regarding the binomial sum is to understand its [Formula: see text]-adic valuation. While the case [Formula: see text] has been completely resolved in recent work, it was conjectured for [Formula: see text] that the valuation depends on the sum of digits in the base-[Formula: see text] expansion of [Formula: see text]. In this paper, we confirm this conjecture and also make a general remark applicable to an infinite family of [Formula: see text]