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

Pedro-José Cazorla García

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

Pedro-José Cazorla García

Department of Mathematics

Research output: Preprint/Working paper › Preprint

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Abstract

In this paper, we study the integer solutions of a family of Fermat-type equations of signature (2, 2n, n), Cx² + q^ky²ⁿ = zⁿ. 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.

Original language	English
Publication status	Published - 22 Apr 2024

Keywords

Exponential Diophantine equation
Fermat equations
Galois representations
Frey-Hellegouarch curve
asymptotic Fermat's Last Theorem
modularity
level lowering

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2404.14098v1

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

abstract = " In this paper, we study the integer solutions of a family of Fermat-type equations of signature (2, 2n, n), Cx2 + qky2n = zn. We provide an algorithmically testable set of conditions which, if satisfied, imply the existence of a constant BC, q such that if n > BC, 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. ",

keywords = "Exponential Diophantine equation, Fermat equations, Galois representations, Frey-Hellegouarch curve, asymptotic Fermat's Last Theorem, modularity, level lowering",

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N2 - In this paper, we study the integer solutions of a family of Fermat-type equations of signature (2, 2n, n), Cx2 + qky2n = zn. We provide an algorithmically testable set of conditions which, if satisfied, imply the existence of a constant BC, q such that if n > BC, 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.

AB - In this paper, we study the integer solutions of a family of Fermat-type equations of signature (2, 2n, n), Cx2 + qky2n = zn. We provide an algorithmically testable set of conditions which, if satisfied, imply the existence of a constant BC, q such that if n > BC, 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.

KW - Exponential Diophantine equation

KW - Fermat equations

KW - Galois representations

KW - Frey-Hellegouarch curve

KW - asymptotic Fermat's Last Theorem

KW - modularity

KW - level lowering

M3 - Preprint

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

ER -

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

Abstract

Keywords

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