Do perfect powers repel partition numbers?

Abstract

In 2013 Zhi-Wei Sun conjectured that $p(n)$ is never a power of an integer when $n>1.$ We confirm this claim in many cases. We also observe that integral powers appear to repel the partition numbers. If $k>1$ and $Δ_k(n)$ is the distance between $p(n)$ and the nearest $k$th power, then for every $d\geq 0$ we conjecture that there are at most finitely many $n$ for which $Δ_k(n)\leq d.$ More precisely, for every $\varepsilon>0,$ we conjecture that $$M_k(d):=\max\{n \ : \ Δ_k(n)\leq d\}=o( d^{\varepsilon}).$$ In $k$-power aspect with $d$ fixed, we also conjecture that if $k$ is sufficiently large, then $$ M_k(d)=\max \left\{ n \ : \ p(n)-1\leq d\right\}. $$ In other words, $1$ generally appears to be the closest $k$th power among the partition numbers.

Figures (2)

• Figure 1: Growth of $M_k(d)$ with $d\leq 10^{70}$
• Figure 2: Growth of $M_k(d)$ with $k\in \{2, 3, 4, 5\}$

Theorems & Definitions (11)

• Conjecture : Sun
• Theorem 1
• proof : Proof of Theorem 1
• Example
• Remark
• Conjecture 1
• Conjecture 2
• Remark
• Conjecture 3
• Conjecture 4