On quartic Diophantine equations with trivial solutions in the Gaussian integers

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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