Solution of Certain Pell Equations

Abstract

Let $a,b,c $ be any positive integers such that $c\mid ab$ and $d_i^\pm$ is a square free positive integer of the form $d_i^\pm=a^{2k} b^{2l}\pm i c^m$ where $k,l \geq m$ and $i=1,2.$ The main focus of this paper to find the fundamental solution of the equation $ x^2-d_i^\pm y^2=1$ with the help of the continued fraction of $\sqrt{d_i^\pm}.$ We also obtain all the positive solutions of the equations $ x^2-d_i^\pm y^2=\pm 1$ and $ x^2-d_i^\pm y^2=\pm 4$ by means of the Fibonacci and Lucas sequences. Furthermore, in this work, we derive some algebraic relations on the Pell form $ F_{d_i^\pm}(x, y) = x^2-d_i^\pm y^2 $ including cycle, proper cycle, reduction and proper automorphism of it. We also determine the integer solutions of the Pell equation $ F_{\Delta_{d_i^\pm}} (x, y) = 1 $ in terms of $d_i^\pm. We generalized all the results of the papers [2], [9], [26], and [37].