Complete solutions of the simultaneous Pell equations x2−24y2=1 and y2−pz2=1

We prove that the simultaneous Pell equations{x2−24y2=1y2−pz2=1, where p is a prime, have positive integer solutions only in the cases of p=11 and p=2. Furthermore, the only solutions are (x,y,z,p)=(49,10,3,11) and (x,y,z,p)=(485,99,70,2).