Fundamental units for a family of totally real cubic orders and the diophantine equation u(u + a)(u + 2a) = v(v + 1)

Abstract

We determine a system of fundamental units of the totally real cubic orders [Formula: see text], where [Formula: see text] is a root of [Formula: see text], [Formula: see text], which arise in our study here of the equation [Formula: see text]. For potentially infinitely many values of [Formula: see text], we reduce the resolution of [Formula: see text] to the resolution of the Thue equations of the form [Formula: see text]. In particular, we solve [Formula: see text] for [Formula: see text] [Formula: see text] [Formula: see text] [Formula: see text] [Formula: see text] [Formula: see text] [Formula: see text] [Formula: see text] [Formula: see text] [Formula: see text] [Formula: see text] [Formula: see text] [Formula: see text] among [Formula: see text]. This extends and corrects the work of Godinho, Porto and Togbé for [Formula: see text], [Formula: see text], as well as extending earlier work of Mordell for [Formula: see text].