Solutions of some quadratic Diophantine equations

In this study, we investigate positive integer solutions of the Diophantine equations x(2) - kxy(sic)y(2)(sic)x = 0 and x(2)-kxy-y(2)(sic)y = 0. It is shown that when k > 3,x2-kxy+y(2)+x = 0 has no positive integer solutions but the equation x(2) - kxy + y(2) - x = 0 has positive integer solutions. Moreover, it is shown that the equations x(2) - kxy - y(2) (sic) x = 0 and x(2) - kxy - y(2) (sic) y = 0 have positive solutions when k >= 1. (C) 2010 Elsevier Ltd. All rights reserved. https://doi.org/10.1016/j.camwa.2010.08.012