On the Diophantine equation x2 − kxy + y2 − 2n = 0

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 ⩽ n ⩽ 10. Moreover, we give all positive integer solutions of the same equation for 0 ⩽ n ⩽ 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.