The number of nonzero binomial coefficients modulo p^alpha

TL;DR

This paper generalizes Fine's 1947 result on counting nonzero binomial coefficients modulo a prime p to prime powers p^alpha, using Kummer's theorem and expressing the count via integer partitions and base-p representations.

Contribution

It introduces a new formula for the number of nonzero binomial coefficients modulo p^alpha, extending previous results to prime power moduli with explicit base-p representation dependence.

Findings

Derived a sum-over-partitions formula for nonzero binomial coefficients modulo p^alpha

Revealed explicit dependence on base-p digit patterns in the count

Extended classical results from prime moduli to prime power moduli

Abstract

In 1947 Fine obtained an expression for the number of binomial coefficients on row n of Pascal's triangle that are nonzero modulo p. In this paper we use Kummer's theorem to generalize Fine's theorem to prime powers, expressing the number of nonzero binomial coefficients modulo p^alpha as a sum over certain integer partitions. For fixed alpha, this expression can be rewritten to show explicit dependence on the number of occurrences of each subword in the base-p representation of n.