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.