Nontrivial solutions to the Fermat equation $x^3+y^3=kz^3$ over quadratic number fields
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866909616236920832 |
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| author | Argaez-Garcia, Alejandro Diaz-Vargas, Javier Pech-Moreno, Luis Eli |
| author_facet | Argaez-Garcia, Alejandro Diaz-Vargas, Javier Pech-Moreno, Luis Eli |
| contents | We give sufficient conditions to determine the existence of nontrivial solutions to the Fermat equation $x^3+y^3=kz^3$ over $\mathbb{Q}(\sqrt{d})$ by constructing a relationship with the points on the elliptic curve $y^2=x^3-432d^3k^2$ over $\mathbb{Q}$ for certain $k\in\mathbb{N}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_07339 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Nontrivial solutions to the Fermat equation $x^3+y^3=kz^3$ over quadratic number fields Argaez-Garcia, Alejandro Diaz-Vargas, Javier Pech-Moreno, Luis Eli Number Theory We give sufficient conditions to determine the existence of nontrivial solutions to the Fermat equation $x^3+y^3=kz^3$ over $\mathbb{Q}(\sqrt{d})$ by constructing a relationship with the points on the elliptic curve $y^2=x^3-432d^3k^2$ over $\mathbb{Q}$ for certain $k\in\mathbb{N}$. |
| title | Nontrivial solutions to the Fermat equation $x^3+y^3=kz^3$ over quadratic number fields |
| topic | Number Theory |
| url | https://arxiv.org/abs/2304.07339 |