On the system of Pell equations $$x^2-(a^2b^2 {\pm } a)y^2=1$$ and $$y^2-pz^2=4b^2$$

Abstract

In this paper, we consider the title simultaneous Pell equations, where a ! 2; b ! 1 are positive integers and p is an odd prime. Particularly, we give all the solutions when 2 a 1000 and 1 b 1000.