Nonexistence of Constant Congruences for the Cumulative Partition Function

Abstract

Let $p(n)$ be the ordinary partition function and let $c(n)=\sum_{k=0}^{n-1}p(k)$, with $c(0)=0$. We prove that, for every $a\geq1$, $b\geq0$, and $m\geq2$, the sequence $c(an+b)$ is not eventually constant modulo $m$, thereby answering a question posed by Chen. The proof combines the algebraicity over $\mathbb C(j)$ of modular functions for finite-index subgroups of $\mathrm{SL}_2(\mathbb Z)$ with Voloch's theorem on the transcendence of elliptic modular functions in positive characteristic.