  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.
