Solutions of Some Diophantine Equations Using Generalized Fibonacci and Lucas Sequences

Abstract

In this study, we investigate Diophantine equations using the generalized Fibonacci and Lucas sequences. We obtain all integer solutions for several Diophantine equations such as \(x^2 -kxy- y^2 = \mp 1,\) \(x^2 -kxy+ y^2 = 1,\) \(x^2 – kxy-y^2 = \mp (k^2+4),\)
\(x^2 – (k^2 + 4)xy + (k^2+4)y^2 =\mp k^2,\) \(x^2 – kxy +y^2 = -(k^2-4)\). and \(x^2-(k^2-4)xy-(k^2-4)y^2=k^2\)
Some of these results are previously known, but we provide new and distinct proofs using generalized Fibonacci and Lucas sequences.