In 2013, Sun conjectured that the partition function p(n) is never a perfect power for <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$n \ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. Building on this, Merca, Ono, and Tsai recently observed that for any fixed integers <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$d \ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$k \ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, there appear to be only finitely many integers n such that p(n) differs from a perfect kth power by at most d. Denoting by <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$M_k(d)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>M</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> the largest such n, they conjectured that <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$M_k(d) = o(d^\epsilon )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>M</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>o</mi> <mrow> <mo stretchy="false">(</mo> <msup> <mi>d</mi> <mi>ϵ</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for every <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\epsilon > 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϵ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we investigate the asymptotic growth of analogs of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$M_k(d)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>M</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for a wide class of partition functions. We establish sharp lower bounds and provide heuristics which suggest that <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$M_k(d)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>M</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> in fact grows polylogarithmically in d, i.e., of order <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\log ^2(d)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mo>log</mo> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. 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.

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

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发表日期：

2026年4月29日

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Summer Haag,
Praneel Samanta,
Swati,
Holly Swisher,
Stephanie Treneer,
Robin Visser

摘要

In 2013, Sun conjectured that the partition function p(n) is never a perfect power for $n \ge 2$ $n \geq 2$ . Building on this, Merca, Ono, and Tsai recently observed that for any fixed integers $d \ge 0$ $d \geq 0$ and $k \ge 2$ $k \geq 2$ , there appear to be only finitely many integers n such that p(n) differs from a perfect kth power by at most d. Denoting by $M_k(d)$ $M_{k} (d)$ the largest such n, they conjectured that $M_k(d) = o(d^\epsilon )$ $M_{k} (d) = o (d^{ϵ})$ for every $\epsilon > 0$ $ϵ > 0$ . In this paper, we investigate the asymptotic growth of analogs of $M_k(d)$ $M_{k} (d)$ for a wide class of partition functions. We establish sharp lower bounds and provide heuristics which suggest that $M_k(d)$ $M_{k} (d)$ in fact grows polylogarithmically in d, i.e., of order $\log ^2(d)$ ${log}^{2} (d)$ . 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.