Asymptotic Fermat's Last Theorem for a family of equations of signature $(2, 2n, n)$
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912023618519040 |
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| author | García, Pedro-José Cazorla |
| author_facet | García, Pedro-José Cazorla |
| contents | In this paper, we study the integer solutions of a family of Fermat-type equations of signature $(2, 2n, n)$, $Cx^2 + q^ky^{2n} = z^n$. We provide an algorithmically testable set of conditions which, if satisfied, imply the existence of a constant $B_{C, q}$ such that if $n > B_{C,q}$, there are no solutions $(x, y, z, n)$ of the equation. Our methods use the modular method for Diophantine equations, along with level lowering and Galois theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_14098 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asymptotic Fermat's Last Theorem for a family of equations of signature $(2, 2n, n)$ García, Pedro-José Cazorla Number Theory 11D61 (Primary), 11D41, 11F80, 11F11 (Secondary) In this paper, we study the integer solutions of a family of Fermat-type equations of signature $(2, 2n, n)$, $Cx^2 + q^ky^{2n} = z^n$. We provide an algorithmically testable set of conditions which, if satisfied, imply the existence of a constant $B_{C, q}$ such that if $n > B_{C,q}$, there are no solutions $(x, y, z, n)$ of the equation. Our methods use the modular method for Diophantine equations, along with level lowering and Galois theory. |
| title | Asymptotic Fermat's Last Theorem for a family of equations of signature $(2, 2n, n)$ |
| topic | Number Theory 11D61 (Primary), 11D41, 11F80, 11F11 (Secondary) |
| url | https://arxiv.org/abs/2404.14098 |