Congruences for cubic partition pairs modulo powers of 3
In the paper under review, the authors establish congruences for \(b(n)\) modulo arbitrary powers of three, where the partition function \(b(n)\) enumerates the number of cubic partition pairs of \(n\) and has generating function \[ \sum_{n=0}^{\infty}b(n)q^n=\frac{1}{(q;q)_{\infty}^2(q^2;q^2)_{\infty}^2}. \] For instance, they show that for integers \(j\geq 2\) and \(n\geq 0\), \[ b\left(3^{2j}n+\frac{3^{2j+1}+1}{4}\right)\equiv 0\,\pmod {3^{2j+1}}. \] The authors also prove some conjectures for \(b(n)\) modulo \(81\) and \(243\), proposed by the first author in [J Number Theory 171, 31--42 (2017; Zbl 1357.11105)].
scientific article; zbMATH DE number 6908982
scientific article