Solutions of ternary quadratic Diophantine equations $x^2+y^2 \pm \dot{\lambda} y=z^2$

Abstract

The infinite integer solutions of the ternary quadratic Diophantine equations $x^2+y^2+\lambda y=z^2$ and $x^2+y^2-\lambda y=z^2$ are investigated in this study. It is shown that when $\lambda=2 \beta, \beta \in Z_{+}, x^2+y^2 \pm \lambda y=z^2$ has infinitely many pure integer solutions but the equations $x^2+y^2 \pm \lambda y=z^2$ has infinitely many mixed integer solutions when $\lambda=2 \beta+1, \beta \in Z_{+}$. A few interesting relations between solutions are also exhibited in this work.