Proof of some conjectural congruences of da Silva and Sellers

TL;DR

This paper proves four Ramanujan-like congruences modulo 5 for the number of 3-regular partitions in three colours, confirming conjectures by da Silva and Sellers through modular form theory.

Contribution

It establishes the conjectured congruences of da Silva and Sellers using modular forms, advancing understanding of partition congruences.

Findings

Confirmed four conjectural Ramanujan-like congruences modulo 5

Demonstrated the application of modular forms to partition congruences

Extended the theory of arithmetic properties of colored partitions

Abstract

Let $p_{\{3, 3\}}(n)$ denote the number of $3$-regular partitions in three colours. In a very recent paper, da Silva and Sellers studied certain arithmetic properties of $p_{\{3, 3\}}(n)$. They further conjectured four Ramanujan-like congruences modulo $5$ satisfied by $p_{\{3, 3\}}(n)$. In this article, we confirm the conjectural congruences of da Silva and Sellers using the theory of modular forms.