Congruences on the Number of Restricted $m$-ary Partitions

TL;DR

This paper extends known prime-based congruences for restricted m-ary partitions to all positive integers m, confirming a conjecture and broadening the understanding of partition congruences.

Contribution

It generalizes existing prime-specific congruences to all positive integers m, resolving a conjecture in the theory of restricted m-ary partitions.

Findings

Congruences hold for all positive integers m, not just primes.

Confirmed the conjecture of Andrews et al. for arbitrary m.

Extended the applicability of previous prime-based results.

Abstract

Andrews, Brietzke, R\o dseth and Sellers proved an infinite family of congruences on the number of the restricted $m$-ary partitions when $m$ is a prime. In this note, we show that these congruences hold for arbitrary positive integer $m$ and thus confirm the conjecture of Andrews, et al.