Solutions of Some Quadratic Diophantine Equations

Abstract

Let $P(t)_i^{\pm}=t^{2k} \pm i t^m$ be a non square polynomial and $Q(t)_i^{\pm}=4k^2t^{4k-2}+i^2m^2t^{2m-2} \pm 4imkt^{2k+m-2} -4t^{2k} \mp 4it^m -1$ be a polynomial, such that $k \geq 2m$ and $i \in \left\lbrace 1,2 \right\rbrace $. In this paper, we consider the number of integer solutions of Diophantine equation $$E\ :\ x^2-P(t)_i^{\pm}y^2-2P'(t)_i^{\pm}x+4 P(t)_i^{\pm} y +Q(t)_i^{\pm}=0.$$ We extend a previous results given by A. Tekcan and A. Chandoul et al. . We also derive some recurrence relations on the integer solutions of a Pell equation.