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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2501.17619 |
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| _version_ | 1866917905701011456 |
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| author | Koymans, Peter Paterson, Ross Santens, Tim Shute, Alec |
| author_facet | Koymans, Peter Paterson, Ross Santens, Tim Shute, Alec |
| contents | For every $n \geq 2$ we determine the asymptotic formula for the number of integer triples $(a,b,c)$ of bounded absolute value such that the generalised Fermat equation given by $ax^n+by^n+cz^n=0$ is everywhere locally soluble. We compute the leading constant, answering a question of Loughran--Rome--Sofos, and determine that the conjectures of Loughran--Smeets and Loughran--Rome--Sofos hold for such equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_17619 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Local solubility of generalised Fermat equations Koymans, Peter Paterson, Ross Santens, Tim Shute, Alec Number Theory For every $n \geq 2$ we determine the asymptotic formula for the number of integer triples $(a,b,c)$ of bounded absolute value such that the generalised Fermat equation given by $ax^n+by^n+cz^n=0$ is everywhere locally soluble. We compute the leading constant, answering a question of Loughran--Rome--Sofos, and determine that the conjectures of Loughran--Smeets and Loughran--Rome--Sofos hold for such equations. |
| title | Local solubility of generalised Fermat equations |
| topic | Number Theory |
| url | https://arxiv.org/abs/2501.17619 |