Partition functions that repel perfect-powers

TL;DR

This paper proves that for fixed bounds on partition parts, the number of partitions close to perfect powers is finite, supporting conjectures that the partition function rarely equals perfect powers, especially as the bound grows.

Contribution

It generalizes previous conjectures by proving finiteness results for partition functions with bounded largest parts, linking the problem to Siegel's Theorem and extending the understanding of perfect-power avoidance.

Findings

Finiteness of pairs where $p_B(n)$ is close to perfect powers for fixed $B$, $k$, and $d$.

Supports conjecture that $p(n)$ rarely equals perfect powers as $B o

Reduces the problem to Siegel's Theorem on integral points on curves.

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 $\geq 1$.