New Congruences for 9-Regular Cubic Partition Pairs

Let \(b_9(n)\) $b_9(n)$ denote the number of 9-regular cubic partition pairs of n. Naika and Nayaka established some congruences modulo 27 and 81 for \(b_9(n)\) $b_9(n)$ . In this paper, we generalize their results and derive some congruences modulo powers of 2 and 3 for \(b_9(n)\) $b_9(n)$ . For example, for any integer \(n\ge 0\) $n\ge 0$ and \(\alpha \ge 2\) $\alpha \ge 2$ , \(\begin{aligned} b_9(3^\alpha n+3^\alpha -2)\equiv 0\pmod {3^{2\alpha -2}}. \end{aligned}\) $$\begin{array}{c}\hfill b_9(3^{\alpha }n+3^{\alpha }-2)\equiv 0(mod{3}^{2\alpha -2}).\end{array}$$