Some remarks on the diophantine equation $(x^2 - 1)(y^2 - 1) = (z^2 - 1)^2$

Abstract

In this paper, among other results, we prove that the title equation has finitely many solutions when $x - y = lz$ and $l$ is a fixed integer $\ne 2$. Moreover, all solutions $(x, y, z)$ satisfy $l < z < l^2 / 2$, $1 < y < l^2 / 2$ and $l^2+1 < x < (l^3 + l^2) / 2$. As a consequence, we extend a result of Cao.