On the generalized Fermat equation $x^{13} + y^{13} = z^n$
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866918160142172160 |
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| author | Best, Alex J. Dahmen, Sander R. Freitas, Nuno |
| author_facet | Best, Alex J. Dahmen, Sander R. Freitas, Nuno |
| contents | Let $n \in \mathbb{Z}_{\geq 2}$. We study the generalized Fermat equation \[x^{13}+y^{13}=z^n, \quad x,y,z \in \mathbb{Z}, \quad \gcd(x,y,z)=1.\] Using a combination of techniques, including the modular method, classical descent, unit sieves, and Chabauty and Mordell--Weil sieve methods over number fields, we show that for $n=5$ all its solutions $(a,b,c)$ are trivial, i.e. satisfy $abc=0$. Under the assumption of GRH, we also show that for $n=7$ there are only trivial solutions. Furthermore, we provide partial results towards solving the equation for general $n \in \mathbb{Z}_{\geq 2}$, in particular that any solution $(a,b,c)$ with $13\mid c$ is trivial. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_12092 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the generalized Fermat equation $x^{13} + y^{13} = z^n$ Best, Alex J. Dahmen, Sander R. Freitas, Nuno Number Theory Primary 11D41, Secondary 11F80, 11G05, 11G10, 11G30 Let $n \in \mathbb{Z}_{\geq 2}$. We study the generalized Fermat equation \[x^{13}+y^{13}=z^n, \quad x,y,z \in \mathbb{Z}, \quad \gcd(x,y,z)=1.\] Using a combination of techniques, including the modular method, classical descent, unit sieves, and Chabauty and Mordell--Weil sieve methods over number fields, we show that for $n=5$ all its solutions $(a,b,c)$ are trivial, i.e. satisfy $abc=0$. Under the assumption of GRH, we also show that for $n=7$ there are only trivial solutions. Furthermore, we provide partial results towards solving the equation for general $n \in \mathbb{Z}_{\geq 2}$, in particular that any solution $(a,b,c)$ with $13\mid c$ is trivial. |
| title | On the generalized Fermat equation $x^{13} + y^{13} = z^n$ |
| topic | Number Theory Primary 11D41, Secondary 11F80, 11G05, 11G10, 11G30 |
| url | https://arxiv.org/abs/2510.12092 |