Solutions of some quadratic Diophantine equations
In this study, we investigate positive integer solutions of the Diophantine equations x2?kxy?y2?x=0 x 2 ? k x y ? y 2 ? x = 0 and x2?kxy?y2?y=0 x 2 ? k x y ? y 2 ? y = 0 . It is shown that when k>3 k > 3 , x2?kxy+y2+x=0 x 2 ? k x y + y 2 + x = 0 has no positive integer solutions but the equation x2?kxy+y2?x=0 x 2 ? k x y + y 2 ? x = 0 has positive integer solutions. Moreover, it is shown that the equations x2?kxy?y2?x=0 x 2 ? k x y ? y 2 ? x = 0 and x2?kxy?y2?y=0 x 2 ? k x y ? y 2 ? y = 0 have positive solutions when k?1 k ? 1 .
diophantine equations fibonacci and lucas numbers generalized fibonacci numbers