Arithmetic properties of bipartitions with 3-cores

Abstract

Recently, Lin discovered several nice congruences modulo 4, 5, 7 and 8 for $A_3(n)$, where $A_3(n)$ is the number of bipartitions with 3-cores of $n$. For example, Lin proved that for all $\alpha \ge 0$ and $n\ge 0$, $A_3(16^{\alpha +1}n+\frac{2^{4\alpha +3}-2}{3})\equiv 0 \ (\mathrm{mod }\ 5)$. Yao also established several infinite families of congruences modulo 3 and 9 for $A_9(n)$. In this paper, several infinite families of congruences modulo 4, 8 and $\frac{4^k-1}{3}\ (k\ge 2)$ for $A_3(n)$ are established. We generalize some results due to Lin and Yao. For example, we prove that for $n\ge 0, \ \alpha \ge 0$ and $k\ge 2$, $A_3\left( 4^{k(\alpha +1)} n+\frac{2^{2k(\alpha +1)-1}-2}{3}\right) \equiv 0 \ (\mathrm{mod\ } \frac{4^{k}-1}{3})$.