On 3 and 9-regular cubic partitions

TL;DR

This paper establishes infinite families of congruences modulo powers of 3 for 3 and 9-regular cubic partition functions, expanding understanding of their divisibility properties.

Contribution

It derives new infinite congruences for 3 and 9-regular cubic partitions modulo powers of 3, which were previously unknown.

Findings

Established congruences for a_3(n) modulo 3^α.

Established congruences for a_9(n) modulo 3^{α+1}.

Extended the theory of partition congruences for cubic partitions.

Abstract

Let $a_3(n)$ and $a_9(n)$ are 3 and 9-regular cubic partitions of $n$. In this paper, we find the infinite family of congruences modulo powers of 3 for $a_3(n)$ and $a_9(n)$ such as \[a_3\left (3^{2\alpha}n+\frac{3^{2\alpha}-1}{4}\right )\equiv 0 \pmod{3^{\alpha}}\] and \[a_9\left (3^{\alpha+1}n+3^{\alpha+1}-1\right )\equiv 0 \pmod{3^{\alpha+1}}.\]