A note on the simultaneous Pell equations \(x^2-(a^2-1)y^2=1\) and \(y^2-bz^2=1\) (Q2221019)

Let \(a,b \in \mathbb{N}\) with \(a>1\) and the square-free part of \(b\) equal to \(2pq\), where \(p\) and \(q\) are distinct odd primes. The author explicitly solves the system of Pell equations \[x^2-(a^2-1)y^2=1,\quad y^2-bz^2=1.\] This completes a former result of \textit{M. Cipu} [Proc. Am. Math. Soc. 146, No. 3, 983--992 (2018; Zbl 1420.11060)], where some exceptional cases could not be managed. The proof is elementary and uses results on square solutions of Pell equations.