Explicit formula for the solution of simultaneous Pell equations 𝑥²-(𝑎²-1)𝑦²=1, 𝑦²-𝑏𝑧²=1

Abstract

For b b an odd integer whose square-free part has at most two prime divisors, it is shown that the equations in the title have a common solution in positive integers precisely when b b divides 4 a 2 − 1 4a^2-1 and the quotient is a perfect square. The proof provides an explicit formula for the common solution, known to be unique. Similar results are obtained assuming the square-free part of b b is even or has three prime divisors.