Partition functions that repel perfect-powers

Abstract

A conjecture by Sun states that the partition function $p(n)$, for $n>1$, is never a perfect power. Recent work by Merca et al. proposes generalizations of perfect-power repulsion for $p(n)$. In this note, we prove these generalizations for the functions $p_B(n)$, which count the number of partitions of $n$ with the largest part $\leq B$. If $B\geq 4$ and $k\geq 3$, with $k\nmid (B-1)$, then we prove that there are only finitely many pairs $(n,m)$ for which $$\lvert p_B(n)-m^k\rvert\le d.$$ These results support Sun and Merca et al.'s conjectures, as $p_B(n) \rightarrow p(n)$ when $B \rightarrow +\infty.$ To prove this, we reduce the problem to Siegel's Theorem, which guarantees the finiteness of integral points on curves with genus $\geq 1$.