On simultaneous Pell equations x2 − (a2 − 1)y2 = 1 and y2 − pz2 = 1

Let p be a prime number. In this paper, for the simultaneous Pell equations x2−(a2−1)y2=1 and y2−pz2=1, we prove that there is no positive solution to the system for a=2,3 and the system only has a positive integer solution (x,y,z,p)=(31,8,3,7) for a=4.