Partition functions that repel perfect-powers

TL;DR

The paper proves analogs of Sun and Merca–Ono–Tsai's conjectures for the truncated partition functions $p_B(n)$ by establishing finiteness results for exact and near-matching $k$th powers when $B\ge4$, $k\ge3$, and $k\nmid (B-1)$. The approach exploits a quasipolynomial description of $p_B(n)$ on residue classes modulo $L=\mathrm{lcm}(1,\dots,B)$, writing $p_B(Ln+r)=Q_r(n)$ with deg$(Q_r)=B-1$ and positive leading coefficient, and reduces the finiteness questions to Diophantine problems on curves of genus at least one. A key ingredient is a dichotomy between generic and power-type $Q_r$, including a finite set of exceptional shifts $T_r(d)$ and the use of Siegel’s theorem, Thue–Mahler theory, and Pell-type analyses for controlling integral points. The results reinforce Sun’s conjecture in the limit $B\to\infty$ (since $p_B(n)\to p(n)$) and clarify the distinct behavior of small $B$ (e.g., $p_2(n)$ and $p_3(n)$).

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 $\geq 1$.

Theorems & Definitions (12)

• Conjecture : Sun, Merca et al.
• Remark
• Theorem 1
• Example
• Example
• Lemma 2: QP structure, degree, and spacing
• proof
• Lemma 3
• proof
• Corollary 4