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Asymptotic Fermat's Last Theorem for a family of equations of signature $(2, 2n, n)$

Authors :
García, Pedro-José Cazorla
Source :
Mathematika, Volume 70, Issue 4, October 2024
Publication Year :
2024

Abstract

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.<br />Comment: 20 pages

Details

Database :
arXiv
Journal :
Mathematika, Volume 70, Issue 4, October 2024
Publication Type :
Report
Accession number :
edsarx.2404.14098
Document Type :
Working Paper
Full Text :
https://doi.org/10.1112/mtk.12279