On the solutions of $x^2= By^p+Cz^p$ and $2x^2= By^p+Cz^p$ over totally real fields
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866917837987119104 |
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| author | Kumar, Narasimha Sahoo, Satyabrat |
| author_facet | Kumar, Narasimha Sahoo, Satyabrat |
| contents | In this article, we study the solutions of certain type over $K$ of the Diophantine equation $x^2= By^p+Cz^p$ with prime exponent $p$, where $B$ is an odd integer and $C$ is either an odd integer or $C=2^r$ for $r \in \mathbb{N}$. Further, we study the non-trivial primitive solutions of the Diophantine equation $x^2= By^p+2^rz^p$ ($r\in {1,2,4,5}$) (resp., $2x^2= By^p+2^rz^p$ with $r \in \mathbb{N}$) with prime exponent $p$, over $K$. We also present several purely local criteria of $K$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_09263 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the solutions of $x^2= By^p+Cz^p$ and $2x^2= By^p+Cz^p$ over totally real fields Kumar, Narasimha Sahoo, Satyabrat Number Theory Primary 11D41, 11R80, Secondary 11F80, 11G05, 11R04 In this article, we study the solutions of certain type over $K$ of the Diophantine equation $x^2= By^p+Cz^p$ with prime exponent $p$, where $B$ is an odd integer and $C$ is either an odd integer or $C=2^r$ for $r \in \mathbb{N}$. Further, we study the non-trivial primitive solutions of the Diophantine equation $x^2= By^p+2^rz^p$ ($r\in {1,2,4,5}$) (resp., $2x^2= By^p+2^rz^p$ with $r \in \mathbb{N}$) with prime exponent $p$, over $K$. We also present several purely local criteria of $K$. |
| title | On the solutions of $x^2= By^p+Cz^p$ and $2x^2= By^p+Cz^p$ over totally real fields |
| topic | Number Theory Primary 11D41, 11R80, Secondary 11F80, 11G05, 11R04 |
| url | https://arxiv.org/abs/2301.09263 |