and other similar equations. Firstly, we will find all positive integer solutions of the equation
x2 − 3xy + y2 ∓ x = 0.
Then we will find all positive integer solutions of the equations
x2 − xy − y2 ∓ 5x = 0 (1.4)
and
x2 − 3xy + y2 ∓ 5x = 0. (1.5)
Moreover, we will find all positive integer solutions of the equations
x2 − xy − y2 ∓ x = 0
and
x2 − xy − y2 ∓ y = 0.
2n−1, F2n−1F2n+1) where Fn is the nth Fibonacci number.
In this study, we consider the equations
x2 − kxy + y2 ∓ x = 0, (1.2