Proofs of two conjectures on congruences of overcubic partition triples
Jiayu Chen, Jing Jin, Olivia X. M. Yao
TL;DR
The paper addresses congruence properties of the overcubic partition triple function $\overline{bt}(n)$ and confirms two infinite-family conjectures modulo $64$ and $128$ proposed by Saikia and Sarma. It develops a theta-function framework, deriving key congruence lemmas for generating functions $\sum \overline{bt}(4n)q^n$, $\sum \overline{bt}(8n)q^n$, $\sum \overline{bt}(16n)q^n$, and $\sum \overline{bt}(32n)q^n$ in terms of $f_k$, $\varphi$, and $\psi$, and uses modular-style identities to manipulate these series modulo powers of two. The authors then combine these preliminaries with a binomial-mod-$p$ principle to establish an induction that yields $\overline{bt}(2^{\alpha}(8n+7)) \equiv 0 \pmod{64}$ and a second family $\overline{bt}(2^{\alpha+4}n) \equiv \overline{bt}(16n) \pmod{128}$, together with corollaries such as $\overline{bt}(144n+42) \equiv 0 \pmod{128}$. These results extend the landscape of partition-congruence phenomena using theta-function identities and modular-form techniques, offering a systematic verification of the conjectured infinite families and their implications.
Abstract
Let $\overline{bt}(n)$ denote the number of overcubic partition triples of $n$. Nayaka, Dharmendra and Kumar proved some congruences modulo 8, 16 and 32 for $\overline{bt}(n)$. Recently, Saikia and Sarma established some congruences modulo 64 for $\overline{bt}(n)$ by using both elementary techniques and the theory of modular forms. In their paper, they also posed two conjectures on infinite families of congruences modulo 64 and 128 for $\overline{bt}(n)$. In this paper, we confirm the two conjectures.