Solution of Certain Pell Equations

Abstract: Let $a,b,c$ be any positive integers such that $c\mid ab$ and $d_i<sup>\pm$ is a square free positive integer of the form $$d_i^{\pm=a^{2k}} b^{2l}\pm i c<sup>m$$ where $k,l \geq m$ and $i=1,2.$ The main focus of this paper to find the fundamental solution of the equation $x^{2-d_i^\pm} y<sup>2=1$ with the help of the continued fraction of $\sqrt{d_i<sup>\pm}.$ 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 $$F_{d_i^\pm}(x, y) = x^{2-d_i^\pm} y²$$ 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].