Czechoslovak Mathematical Journal, vol.63, no.3, pp.783-797, 2013 (SCI-Expanded)
In this study, we determine when the Diophantine equation x 2-kxy+y 2-2n = 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 - 2n = 0. © 2013 Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic.