On the solutions of $x^p+y^p=2^r z^p$, $x^p+y^p=z^2$ over totally real fields
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2022
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914767264808960 |
|---|---|
| author | Kumar, Narasimha Sahoo, Satyabrat |
| author_facet | Kumar, Narasimha Sahoo, Satyabrat |
| contents | In this article, we study the non-trivial primitive solutions of a certain type for the Diophantine equations $x^p+y^p=2^rz^p$ and $x^p+y^p=z^2$ of prime exponent $p$, $r \in \mathbb{N}$, over a totally real field $K$. Then for $r=2,3$, we study the non-trivial primitive solutions over $\mathcal{O}_K$ for the equation $x^p+y^p=2^rz^p$ of prime exponent $p$. Finally, we give several purely local criteria for $K$ such that the equation $x^p+y^p=2^rz^p$ has no non-trivial primitive solutions over $\mathcal{O}_K$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_10930 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the solutions of $x^p+y^p=2^r z^p$, $x^p+y^p=z^2$ over totally real fields Kumar, Narasimha Sahoo, Satyabrat Number Theory 11D41 (Primary), 11F80, 11R04, 11R80 (Secondary) In this article, we study the non-trivial primitive solutions of a certain type for the Diophantine equations $x^p+y^p=2^rz^p$ and $x^p+y^p=z^2$ of prime exponent $p$, $r \in \mathbb{N}$, over a totally real field $K$. Then for $r=2,3$, we study the non-trivial primitive solutions over $\mathcal{O}_K$ for the equation $x^p+y^p=2^rz^p$ of prime exponent $p$. Finally, we give several purely local criteria for $K$ such that the equation $x^p+y^p=2^rz^p$ has no non-trivial primitive solutions over $\mathcal{O}_K$. |
| title | On the solutions of $x^p+y^p=2^r z^p$, $x^p+y^p=z^2$ over totally real fields |
| topic | Number Theory 11D41 (Primary), 11F80, 11R04, 11R80 (Secondary) |
| url | https://arxiv.org/abs/2207.10930 |