Do perfect powers repel partition numbers?
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arXiv
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| Format: | Preprint |
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2025
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| author | Merca, Mircea Ono, Ken Tsai, Wei-Lun |
| author_facet | Merca, Mircea Ono, Ken Tsai, Wei-Lun |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_03754 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Do perfect powers repel partition numbers? Merca, Mircea Ono, Ken Tsai, Wei-Lun Combinatorics Number Theory 05A17. 05A20, 11P82, 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. |
| title | Do perfect powers repel partition numbers? |
| topic | Combinatorics Number Theory 05A17. 05A20, 11P82, |
| url | https://arxiv.org/abs/2501.03754 |