A Note on Some Diophantine Equations

Abstract

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 x²+y²-3xy+x=0, x²+y²-3xy-x=0, and x²+y²-7xy-x=0. Moreover, it is shown that the equation x²+y²-7xy+x=0 has no positive integer solutions.