On quartic Diophantine equations with trivial solutions in the Gaussian integers

Authors : Mahnaz AHMADI, Ali S JANFADA

Pages : 134-142

Doi:10.24330/ieja.964819

View : 103 | Download : 12

Publication Date : 2022-01-17

Article Type : Research Paper

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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