Continued Fractions Expansion of $\sqrt{D}$ and Pell Equation $x^{2}-Dy^{2}=1$

Ahmet Tekcan

Let $D\neq 1$ be a positive non-square integer. In the first section, we give some preliminaries from Pell equations and simple continued fraction expansion. In the second section, we give a formula for the continued fraction expansion of $\sqrt{D}$ for some specific values of $D$ and then we consider the integer solutions of Pell equations $x^{2}-Dy^{2} = 1$ for these values of $D$ including recurrence relations on the integer solutions of it.