Asymptotic solutions of the generalized Fermat-type equation of signature $(p,p,3)$ 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_ | 1866915180651216896 |
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| author | Sahoo, Satyabrat Kumar, Narasimha |
| author_facet | Sahoo, Satyabrat Kumar, Narasimha |
| contents | In this article, we study the asymptotic solutions of the generalized Fermat-type equation of signature $(p,p,3)$ over totally real number fields $K$, i.e., $Ax^p+By^p=Cz^3$ with prime exponent $p$ and $A,B,C \in \mathcal{O}_K \setminus \{0\}$. For certain class of fields $K$, we prove that $Ax^p+By^p=Cz^3$ has no asymptotic solutions over $K$ (resp., solutions of certain type over $K$) with restrictions on $A,B,C$ (resp., for all $A,B,C \in \mathcal{O}_K \setminus \{0\}$). Finally, we present several local criteria over $K$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_14640 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asymptotic solutions of the generalized Fermat-type equation of signature $(p,p,3)$ over totally real number fields Sahoo, Satyabrat Kumar, Narasimha Number Theory Primary 11D41, 11R80, Secondary 11F80, 11G05, 11R04 In this article, we study the asymptotic solutions of the generalized Fermat-type equation of signature $(p,p,3)$ over totally real number fields $K$, i.e., $Ax^p+By^p=Cz^3$ with prime exponent $p$ and $A,B,C \in \mathcal{O}_K \setminus \{0\}$. For certain class of fields $K$, we prove that $Ax^p+By^p=Cz^3$ has no asymptotic solutions over $K$ (resp., solutions of certain type over $K$) with restrictions on $A,B,C$ (resp., for all $A,B,C \in \mathcal{O}_K \setminus \{0\}$). Finally, we present several local criteria over $K$. |
| title | Asymptotic solutions of the generalized Fermat-type equation of signature $(p,p,3)$ over totally real number fields |
| topic | Number Theory Primary 11D41, 11R80, Secondary 11F80, 11G05, 11R04 |
| url | https://arxiv.org/abs/2403.14640 |