Continued fractions and applications to Pell equations

Our study consists of 4 main sections. The first section is the introduction. In this section, the history of continued fractions and the Pell equation is discussed. In the second section, under the title of Continued Fractions, the approximations of finite and infinite continued fractions, some theorems related to these approximations, the periodic expansion of √d, periodic continued fractions, and completely periodic continued fractions are discussed. Also, the theorems related to the approximations of infinite continued fractions that assist in solving Pell equations are given. In the third section, using the approximations of infinite continued fractions, it is discussed how to find the fundamental solution and all integer solutions of the equation x^2-dy^2=±1, where d is a non-square positive integer. The fourth section discusses the results and recommendations.