Arithmetic properties of cubic and overcubic partition pairs

TL;DR

This paper investigates arithmetic properties of cubic and overcubic partition pairs, proving specific divisibility congruences and demonstrating that these partition counts are divisible by high powers of 2 for most integers, using modular forms.

Contribution

It proves a conjecture on divisibility of cubic partition pairs and establishes new congruences for overcubic partition pairs using modular form techniques.

Findings

Proved that b(49n+37) is divisible by 49.

Established two congruences modulo 256 for overcubic partition pairs.

Showed that overcubic partition counts are divisible by 2^k for almost all n.

Abstract

Let $b(n)$ denote the number of cubic partition pairs of $n$. We give affirmative answer to a conjecture of Lin, namely, we prove that $$b(49n+37)\equiv 0 \pmod{49}.$$ We also prove two congruences modulo $256$ satisfied by $\overline{b}(n)$, the number of overcubic partition pairs of $n$. Let $\overline{a}(n)$ denote the number of overcubic partition of $n$. For a fixed positive integer $k$, we further show that $\overline{b}(n)$ and $\overline{a}(n)$ are divisible by $2^k$ for almost all $n$. We use arithmetic properties of modular forms to prove our results.