On members of Lucas sequences which are products of Catalan numbers

Abstract

We show that if [Formula: see text] is a Lucas sequence, then the largest [Formula: see text] suc that [Formula: see text] with [Formula: see text], where [Formula: see text] is the [Formula: see text]th Catalan number satisfies [Formula: see text]. In case the roots of the Lucas sequence are real, we have [Formula: see text]. As a consequence, we show that if [Formula: see text] is the sequence of the [Formula: see text] coordinates of a Pell equation [Formula: see text] with a nonsquare integer [Formula: see text], then [Formula: see text] implies [Formula: see text].