Proof of some conjectural congruences of da Silva and Sellers

Abstract: Let $p_{{3, 3}}(n)$ denote the number of $3$-regular partitions in three colours. In a very 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.