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

TL;DR

The paper investigates how close partition-counting functions can come to being perfect powers by introducing and studying $\Delta_{f,k}(n)$ and $M_{f,k}(d)$ for a broad class of $f(n)$, notably $p(n)$. It develops a probabilistic heuristic model around the main growth term $A(n)$ to predict polylogarithmic growth in $d$ for $M_{f,k}(d)$ (in particular $M_k(d) \asymp (\log d)^2$ for $p(n)$). Rigorous bounds link $\Delta_{f,k}(n)$ to $M_{f,k}(d)$, establish stabilization indices $N_{f,d}$, and provide equidistribution evidence for fractional parts of $p(n)^{1/k}$, all complemented by substantial computations. The authors also extend the framework to other partition functions (overpartitions, plane partitions, etc.) and propose conjectures mirroring Sun's in these contexts, supported by numerical data. Overall, the work offers a coherent probabilistic-analytic approach that suggests polylogarithmic growth as a universal feature under a random-model perspective and lays groundwork for broader Sun-type conjectures.

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 $M_k(d) = o(d^ε)$ for every $ε> 0$. 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.

Figures (5)

• Figure 1: Histograms showing distribution of fractional parts of $p(n)^{1/k}$ for $k=2,3,4$ and increasing values of $n$.
• Figure 2: Empirical distribution functions of fractional parts of $p(n)^{1/k}$ for $k=2,3,4$ and increasing values of $n$, compared with the uniform distribution on $[0,1)$.
• Figure 3: Kolmogorov–Smirnov statistics for fractional parts of $p(n)^{1/k}$ for $k=2,3,4$, and $N=100\,000$.
• Figure 4: Computed values of $M_k(d)$ for the partition function $p(n)$, based on checking all $n \le 10^7$.
• Figure 5: Conjectured values of $M_{f,k}(d)$ for some partition functions $f$, based on checking all $n \leq 10^5$.

Theorems & Definitions (54)

• Conjecture A: Sun
• Conjecture B
• Conjecture C
• Theorem 1.1
• Conjecture D
• Conjecture E
• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Remark 2.1