On the solutions of the generalized Fermat equation over totally real number fields
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910016924024832 |
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| author | Sahoo, Satyabrat |
| author_facet | Sahoo, Satyabrat |
| contents | Let $K$ be a totally real number field and $\mathcal{O}_K$ be the ring of integers of $K$. In this article, we study the asymptotic solutions of the generalized Fermat equation $Ax^p+By^p+Cz^p=0$ over $K$ with prime exponent $p$, where $A,B,C \in \mathcal{O}_K \setminus \{0\}$ with $ABC$ is even. For certain class of fields $K$, we prove that the equation $Ax^p+By^p+Cz^p=0$ has no asymptotic solution $(a,b,c) \in \mathcal{O}_K^3$ with $2|abc$. Then, under some assumptions on $A,B,C$, we also prove that $Ax^p+By^p+Cz^p=0$ has no asymptotic solution in $K^3$. Finally, we give several purely local criteria of $K$ such that $Ax^p+By^p+Cz^p=0$ has no asymptotic solutions in $K^3$, and calculate the density of such fields $K$ when $K$ is a real quadratic field. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_09171 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the solutions of the generalized Fermat equation over totally real number fields Sahoo, Satyabrat Number Theory Primary 11D41, 11R80, Secondary 11F80, 11G05, 11Y40 Let $K$ be a totally real number field and $\mathcal{O}_K$ be the ring of integers of $K$. In this article, we study the asymptotic solutions of the generalized Fermat equation $Ax^p+By^p+Cz^p=0$ over $K$ with prime exponent $p$, where $A,B,C \in \mathcal{O}_K \setminus \{0\}$ with $ABC$ is even. For certain class of fields $K$, we prove that the equation $Ax^p+By^p+Cz^p=0$ has no asymptotic solution $(a,b,c) \in \mathcal{O}_K^3$ with $2|abc$. Then, under some assumptions on $A,B,C$, we also prove that $Ax^p+By^p+Cz^p=0$ has no asymptotic solution in $K^3$. Finally, we give several purely local criteria of $K$ such that $Ax^p+By^p+Cz^p=0$ has no asymptotic solutions in $K^3$, and calculate the density of such fields $K$ when $K$ is a real quadratic field. |
| title | On the solutions of the generalized Fermat equation over totally real number fields |
| topic | Number Theory Primary 11D41, 11R80, Secondary 11F80, 11G05, 11Y40 |
| url | https://arxiv.org/abs/2404.09171 |