On the unique solvability of the simultaneous Pell equations x2 − a y2 = 1 and z2 − b x2 = 1

We consider the simultaneous Pell equations x 2 − a y 2 = 1 , z 2 − b x 2 = 1 , where a ≥ b ≥ 2 are positive integers. We describe a procedure which, for any fixed b , either confirms that the simultaneous Pell equations have at most one solution in positive integers, or finds all exceptions for which we have two solutions.