On the determination of solutions of simultaneous Pell equations x 2 - ( a 2 - 1 ) y 2 = y 2 - p z 2 = 1

In this paper, we consider the simultaneous Pell equations x 2 - ( a 2 - 1 ) y 2 = 1 , y 2 - p z 2 = 1 , where p is prime and a > 1 . Assuming the solutions of the Pell equation x 2 - ( a 2 - 1 ) y 2 = 1 are x = x m and y = y m 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 ( mod 4 ) , we determine the values of a, x, y, and z. Lastly, we show that (0.1) has no solutions when p ≡ 1 ( mod 4 ) .