On quartic Diophantine equations with trivial solutions in the Gaussian integers

Year 2022, , 134 - 142, 17.01.2022

Mahnaz Ahmadı Ali S. Janfada

https://doi.org/10.24330/ieja.964819

Cited By: 1

Abstract

We show that the quartic Diophantine equations $ax^4+by^4=cz^2$ has only trivial solution in the Gaussian integers
for some particular choices of $a,b$ and $c$. Our strategy is by elliptic curves method. In fact, we exhibit
two null-rank corresponding families of elliptic curves over Gaussian field. We also determine the torsion groups of both families.

Keywords

Diophantine equation, elliptic curves, quartic equation, number of solutions of Diophantine equations

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