- International Electronic Journal of Algebra
- Vol: 31 Issue: 31
- On quartic Diophantine equations with trivial solutions in the Gaussian integers
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 : 13 | Download : 9
Publication Date : 2022-01-17
Article Type : Research
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