On quartic Diophantine equations with trivial solutions in the Gaussian integers

Year 2022, Volume: 31 Issue: 31, 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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