Nonexistence of Constant Congruences for the Cumulative Partition Function

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 $a\ge 1$, $b\ge 0$, and $m\ge 2$, 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 ${\mathrm{S}\mathrm{L}}_2(\mathbb{Z})$ with Voloch’s theorem on the transcendence of elliptic modular functions in positive characteristic.