Partition functions that repel perfect-powers
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915573363900416 |
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| author | Ono, Ken |
| author_facet | Ono, Ken |
| contents | 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 $\geq 1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_19164 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Partition functions that repel perfect-powers Ono, Ken Number Theory Combinatorics 11P82, 05A17, 05A20 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 $\geq 1$. |
| title | Partition functions that repel perfect-powers |
| topic | Number Theory Combinatorics 11P82, 05A17, 05A20 |
| url | https://arxiv.org/abs/2510.19164 |