Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2412.01068 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910932169392128 |
|---|---|
| author | Argáez-García, Alejandro Pech-Moreno, Luis Elí |
| author_facet | Argáez-García, Alejandro Pech-Moreno, Luis Elí |
| contents | We provide the necessary conditions for the existence of solutions $(x,y,z)$ to $Ax^p+By^p+Cz^p=0$ over any quadratic number field $K$ with $A,B,C$ pth powerfree integer numbers. We determine when $x$, $y$ and $z$ are rational numbers for pairwise coprime integers $A$, $B$ and $C$. Moreover, we prove that $x$, $y$ and $z$ are in $K\setminus\mathbb{Q}$ when $BC=\pm 1$ and $A\neq \pm 2$. Finally, we prove that no solutions $(x,y,z)$ to $Ax^p+By^p+Cz^p=0$ exist in $K\setminus\mathbb{Q}$ when $BC\neq \pm 1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_01068 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the solutions to $Ax^p+By^p+Cz^p=0$ over quadratic fields Argáez-García, Alejandro Pech-Moreno, Luis Elí Number Theory We provide the necessary conditions for the existence of solutions $(x,y,z)$ to $Ax^p+By^p+Cz^p=0$ over any quadratic number field $K$ with $A,B,C$ pth powerfree integer numbers. We determine when $x$, $y$ and $z$ are rational numbers for pairwise coprime integers $A$, $B$ and $C$. Moreover, we prove that $x$, $y$ and $z$ are in $K\setminus\mathbb{Q}$ when $BC=\pm 1$ and $A\neq \pm 2$. Finally, we prove that no solutions $(x,y,z)$ to $Ax^p+By^p+Cz^p=0$ exist in $K\setminus\mathbb{Q}$ when $BC\neq \pm 1$. |
| title | On the solutions to $Ax^p+By^p+Cz^p=0$ over quadratic fields |
| topic | Number Theory |
| url | https://arxiv.org/abs/2412.01068 |