Non-trivial Integer Solutions of $x^r+y^r=Dz^p$
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866917905232297984 |
|---|---|
| author | Kara, Yasemin Mocanu, Diana Özman, Ekin |
| author_facet | Kara, Yasemin Mocanu, Diana Özman, Ekin |
| contents | In this paper, we use the modular method over totally real fields together with some standard conjectures (the Weak Frey--Mazur Conjecture and the Eichler--Shimura Conjecture) to prove that infinitely many equations of the type $x^r+y^r=Dz^p$ do not have any non-trivial primitive integer solutions, where $r \geq 5$ is a fixed prime, whenever $p$ is large enough. For $r \equiv 3 \pmod 4$, we get the same result with only assuming the Weak Frey--Mazur Conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_07319 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-trivial Integer Solutions of $x^r+y^r=Dz^p$ Kara, Yasemin Mocanu, Diana Özman, Ekin Number Theory 11D41, 11G05 In this paper, we use the modular method over totally real fields together with some standard conjectures (the Weak Frey--Mazur Conjecture and the Eichler--Shimura Conjecture) to prove that infinitely many equations of the type $x^r+y^r=Dz^p$ do not have any non-trivial primitive integer solutions, where $r \geq 5$ is a fixed prime, whenever $p$ is large enough. For $r \equiv 3 \pmod 4$, we get the same result with only assuming the Weak Frey--Mazur Conjecture. |
| title | Non-trivial Integer Solutions of $x^r+y^r=Dz^p$ |
| topic | Number Theory 11D41, 11G05 |
| url | https://arxiv.org/abs/2404.07319 |