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 $k,l\ge m$ 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].

16 pages