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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2208.05145 |
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| _version_ | 1866914998714892288 |
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| author | Santicola, Katerina |
| author_facet | Santicola, Katerina |
| contents | Let $\mathscr{P}_\mathbb{Q}=\{ α^n \; : \; α\in \mathbb{Q}, \; n \ge 2\}$ be the set of rational perfect powers, and let $S \subseteq \mathscr{P}_\mathbb{Q}$ be a finite subset. We prove the existence of a polynomial $f_S \in \mathbb{Z}[X]$ such that $f(\mathbb{Q}) \cap \mathscr{P}_\mathbb{Q}=S$. This generalizes a recent theorem of Gajović who recently proved a similar theorem for finite subsets of integer perfect powers. Our approach makes use of the resolution of the generalized Fermat equation of signature $(2,4,n)$ due to Ellenberg and others, as well as the finiteness of perfect powers in non-degenerate binary recurrence sequences, proved by Pethő and by Shorey and Stewart. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_05145 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Reverse Engineered Diophantine Equations over $\mathbb{Q}$ Santicola, Katerina Number Theory Let $\mathscr{P}_\mathbb{Q}=\{ α^n \; : \; α\in \mathbb{Q}, \; n \ge 2\}$ be the set of rational perfect powers, and let $S \subseteq \mathscr{P}_\mathbb{Q}$ be a finite subset. We prove the existence of a polynomial $f_S \in \mathbb{Z}[X]$ such that $f(\mathbb{Q}) \cap \mathscr{P}_\mathbb{Q}=S$. This generalizes a recent theorem of Gajović who recently proved a similar theorem for finite subsets of integer perfect powers. Our approach makes use of the resolution of the generalized Fermat equation of signature $(2,4,n)$ due to Ellenberg and others, as well as the finiteness of perfect powers in non-degenerate binary recurrence sequences, proved by Pethő and by Shorey and Stewart. |
| title | Reverse Engineered Diophantine Equations over $\mathbb{Q}$ |
| topic | Number Theory |
| url | https://arxiv.org/abs/2208.05145 |