Cubic partitions modulo powers of 5

Abstract

Let $p^{*}(n)$ be the number of partitions of n in which even parts come in two colours (so-called “cubic partitions”). It is known that if $\alpha \ge 2$ and $\delta _\alpha$ is the reciprocal of 8 modulo $5^\alpha$ then for $n\ge 0$,

$$\begin{aligned} {p^*}({5^\alpha }n + {\delta _\alpha }) \equiv 0\;(\mathrm{{mod }}\;{\mathrm{{5}}^{ \lfloor \alpha \mathrm{{/2}} \rfloor }}). \end{aligned}$$

We give a completely elementary proof of this fact.