On the determination of solutions of simultaneous Pell equations x2- (a2- 1) y2= y2- pz2= 1

Abstract

In this paper, we consider the simultaneous Pell equations x2-(a2-1)y2=1,y2-pz2=1,where p is prime and a> 1. Assuming the solutions of the Pell equation x2- (a2- 1) y2= 1 are x= xm and y= ym with m≥ 2 , we prove that the system (0.1) has solutions only when m= 2 or m= 3. In the case of m= 3 , we show that p= 2 and give the solutions of (0.1) in terms of Pell and Pell–Lucas sequences. When m= 2 and p≡3(mod4), we determine the values of a, x, y, and z. Lastly, we show that (0.1) has no solutions when p≡1(mod4). © 2017, Akadémiai Kiadó, Budapest, Hungary.