On the Diophantine equation x(2) - kxy plus y(2)-2(n)=0

Abstract

In this study, we determine when the Diophantine equation x (2)-kxy+y (2)-2 (n) = 0 has an infinite number of positive integer solutions x and y for 0 a (c) 1/2 n a (c) 1/2 10. Moreover, we give all positive integer solutions of the same equation for 0 a (c) 1/2 n a (c) 1/2 10 in terms of generalized Fibonacci sequence. Lastly, we formulate a conjecture related to the Diophantine equation x (2) - kxy + y (2) - 2 (n) = 0.