Solutions of some quadratic Diophantine equations

Abstract

In this study, we investigate positive integer solutions of the Diophantine equations x 2 − k x y ∓ y 2 ∓ x = 0 and x 2 − k x y − y 2 ∓ y = 0 . It is shown that when k > 3 , x 2 − k x y + y 2 + x = 0 has no positive integer solutions but the equation x 2 − k x y + y 2 − x = 0 has positive integer solutions. Moreover, it is shown that the equations x 2 − k x y − y 2 ∓ x = 0 and x 2 − k x y − y 2 ∓ y = 0 have positive solutions when k ≥ 1 .