Pell Equations

This thesis consists of fundamentally four chapters and these chapters consist of subchapters in itself. In the first chapter, first of all fundamental definitions and theorems concerning number theory are given. In the second part of this chapter, the information about continued fractions is given. Here it is shown that how to get the continued fraction expansion of ?d.In the second chapter, after the Pell equations are described briefly, all positive integer solutions to the equations x^2-dy^2=±1 and x^2-dy^2=±N are given. Defininition of the solution to the Pell equations is given and the fundamental solutions to some Pell equations are calculated by means of the convergent of continued fraction. All positive integer solutions to the Pell equations are given by means of fundamental solutions. In order to calculate the solutions of Pell equation x^2-dy^2=1 an alternative method called Bhaskara?s method is used.In the third chapter, it is given that how the fundamental solution to the equations x^2-dy^2=±4 can be obtained. Moreover, the formulas are given to obtain all the positive integer solution to the equations x^2-dy^2=±4 with the help of fundamental solution.Finally, in the fourth chapter, if the equations x²-dy² =±4 and x²-dy²=±1 have a solution, then all positive integer solutions of them are given in terms of the generalized Fibonacci and Lucas numbers when d?{k²±4,k²±1} and k is a positive integer. Especially, all positive integer solutions of the equations x²-5y² =±4 and x²-5y²=±1 are determined in terms of the Fibonacci and Lucas numbers.