Solutions of Some Quadratic Diophantine Equations (European Journal of Pure and Applied Mathematics, 2023)
Let $P(t)_i^{±}=t^{2k} ± i t^m$ be a non square polynomial and $Q(t)_i^{±}=4k^2t^{4k-2}+i^2m^2t^{2m-2} ± 4imkt^{2k+m-2} -4t^{2k} \mp 4it^m -1$ be a polynomial, such that $k ≥ 2m$ and $i \in \left\lbrace 1,2 \right\rbrace $. In this paper, we consider the number of integer solutions of Diophantine equation $$E\ :\ x^2-P(t)_i^{±}y^2-2P'(t)_i^{±}x+4 P(t)_i^{±} y +Q(t)_i^{±}=0.$$ 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.