Repellent properties of perfect powers on partition functions: a heuristic approach

TL;DR

This paper explores the rarity of perfect powers in partition functions, providing heuristic and theoretical evidence that such occurrences are extremely sparse, and establishes bounds on related growth functions.

Contribution

It introduces new bounds and heuristics for the growth of the largest integers where partition functions are near perfect powers, extending previous conjectures.

Findings

M_k(d) likely grows polylogarithmically in d

Set of integers with partition functions as perfect powers is finite with probability 1

Provides sharp lower bounds and heuristic arguments for growth rates

Abstract

In 2013, Sun conjectured that the partition function $p(n)$ is never a perfect power for $n \geq 2$. Building on this, Merca, Ono, and Tsai recently observed that for any fixed integers $d \geq 0$ and $k \geq 2$, there appear to be only finitely many integers $n$ such that $p(n)$ differs from a perfect $k$th power by at most $d$. Denoting by $M_k(d)$ the largest such $n$, they conjectured that $M_k(d) = o(d^\epsilon)$ for every $\epsilon > 0$. In this paper, we investigate the asymptotic growth of analogs of $M_k(d)$ for a wide class of partition functions. We establish sharp lower bounds and provide heuristics which suggest that $M_k(d)$ in fact grows polylogarithmically in $d$, i.e. of order $\log^2(d)$. More generally, we prove that if $f(n)$ is a suitably random chosen function with asymptotic growth rate similar to that of $p(n)$, then the set of integers $n$ for which $f(n)$ is a perfect power is finite with probability 1.