A Note On Some Diophantine Equations

In this study, we solve the equations (x+y+1)^2 = 5xy, (x+y−1)^2 = 5xy, and (x−y+1)^2 = 5xy. We find all positive integer solutions of these equations in terms of Fibonacci and Lucas sequences. By using the solutions of these equations we give all positive integer solutions of the equations x^2 +y^2 −3xy+x = 0, x^2+y^2 −3xy−x = 0, and x^2+y^2 −7xy−x = 0. Moreover, it is shown that the equation x^2+y^2 −7xy+x = 0 has no positive integer solutions.