Product representations of perfect powers
Fuente:
arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866912817320296448 |
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| author | Pach, Péter Pál Sándor, Csaba |
| author_facet | Pach, Péter Pál Sándor, Csaba |
| contents | Let $ρ_k(N)$ denote the maximum size of a set $A\subseteq \{1,2,\dots,N\}$ such that no product of $k$ distinct elements of $A$ is a perfect $d$-th power. In this short note, we prove that $ρ_d(N)=\sum\limits_{k=1}^{d-1}π\left( \frac{N}{k} \right) +O_d(π(N^{1/2}))$, furthermore, for prime power $d$ and sufficiently large $N$ we have $ρ_d(N)=\sum\limits_{k=1}^{d-1}π\left( \frac{N}{k} \right)$. This answers a question of Verstraëte. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_07000 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Product representations of perfect powers Pach, Péter Pál Sándor, Csaba Combinatorics Number Theory Let $ρ_k(N)$ denote the maximum size of a set $A\subseteq \{1,2,\dots,N\}$ such that no product of $k$ distinct elements of $A$ is a perfect $d$-th power. In this short note, we prove that $ρ_d(N)=\sum\limits_{k=1}^{d-1}π\left( \frac{N}{k} \right) +O_d(π(N^{1/2}))$, furthermore, for prime power $d$ and sufficiently large $N$ we have $ρ_d(N)=\sum\limits_{k=1}^{d-1}π\left( \frac{N}{k} \right)$. This answers a question of Verstraëte. |
| title | Product representations of perfect powers |
| topic | Combinatorics Number Theory |
| url | https://arxiv.org/abs/2601.07000 |