Positive Integer Solutions of the Pell Equation $x^{2}-dy^{2}=N,$ $% d\in \left\{k^{2}\pm 4,\text{}k^{2}\pm 1\right\} $ and $N\in \left\{\pm 1,\pm 4\right\}

Abstract

Let $\ k$ be a natural number and $d=k^{2}\pm 4$ or $k^{2}\pm 1$. In this paper, by using continued fraction expansion of $\sqrt{d},$ we find fundamental solution of the equations $x^{2}-dy^{2}=\pm 1$ and we get all positive integer solutions of the equations $x^{2}-dy^{2}=\pm 1$ in terms of generalized Fibonacci and Lucas sequences. Moreover, we find all positive integer solutions of the equations $x^{2}-dy^{2}=\pm 4$ in terms of generalized Fibonacci and Lucas sequences. Although some of the results are well known, we think our method is elementary and different from the others.