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NONTRIVIAL SOLUTIONS TO THE FERMAT EQUATION $x^3+y^3=k z^3$ OVER QUADRATIC NUMBER FIELDS
Keywords:
Fermat equations, elliptic curvesDOI:
https://doi.org/10.17654/0972555525018Abstract
We give sufficient conditions to determine the existence of nontrivial solutions to the Fermat equation $x^3+y^3=k z^3$ over $\mathbb{Q}(\sqrt{d})$ by constructing a relationship with the points on the elliptic curve $y^2=x^3-432 d^3 k^2$ over $\mathbb{Q}$ for certain $k \in \mathbb{N}$.
Received: January 14, 2025
Accepted: April 2, 2025
References
M. Jones and J. Rouse, Solutions of the cubic Fermat equation in quadratic fields, Int. J. Number Theory 9 (2013), 1579-1591.
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.
H. Cohen, Number Theory Volume II: Analytic and Modern Tools, Springer- Verlag, New York, 2007.
D. Husemöler, Elliptic Curves, 2nd ed., Springer-Verlag, New York, 2004, p. 35.
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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