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Volume 64 (2025)

	Volume 64, Issue 5 (In Progress) Pg 451 - 545 (September 2025)
	Volume 64, Issue 4 Pg 329 - 450 (July 2025)
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JP Journal of Algebra, Number Theory and Applications

JP Journal of Algebra, Number Theory and Applications
Volume 64, Issue 4, Pages 329 - 351 (July 2025)
http://dx.doi.org/10.17654/0972555525018

NONTRIVIAL SOLUTIONS TO THE FERMAT EQUATION x³ + y³ = k^z OVER QUADRATIC NUMBER FIELDS

Alejandro Argáez-García, Javier Diaz-Vargas and Elí Pech-Moreno

Abstract:

We give sufficient conditions to determine the existence of nontrivial solutions to the Fermat equation x³ + y³ = k^z over by constructing a relationship with the points on the elliptic curve over for certain

Keywords and phrases:

Fermat equations, elliptic curves

Received: January 14, 2025; Accepted: April 2, 2025; Published: May 13, 2025

How to cite this article: Alejandro Argáez-García, Javier Diaz-Vargas and Elí Pech-Moreno, Nontrivial solutions to the Fermat equation over quadratic number fields, JP Journal of Algebra, Number Theory and Applications 64(4) (2025), 329-351. https://doi.org/10.17654/0972555525018

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

References:

[1]M. Jones and J. Rouse, Solutions of the cubic Fermat equation in quadratic fields, Int. J. Number Theory 9 (2013), 1579-1591.

[2]P. Morton, Solutions of the cubic Fermat equation in ring class fields of imaginary quadratic fields (as periodic points of a 3-adic algebraic function), Int. J. Number Theory 12(4) (2016), 853-902.

[3]H. Cohen, Number Theory Volume II: Analytic and Modern Tools, Springer- Verlag, New York, 2007.

[4]D. Husemöler, Elliptic Curves, 2nd ed., Springer-Verlag, New York, 2004, p. 35.

[5]E. Liverance, A formula for the root number of a family of elliptic curves, J. Number Theory 51 (1995), 288-305.

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