On a variant of Pillai’s problem with binary recurrences and S-units

Abstract

Let [Formula: see text] be a fixed binary recurrence with real characteristic roots [Formula: see text] satisfying [Formula: see text] and let [Formula: see text] be fixed distinct prime numbers. In this paper, we show that there exist effectively computable, positive constants [Formula: see text] and [Formula: see text] such that the Diophantine equation [Formula: see text] has at most [Formula: see text] solutions [Formula: see text] if [Formula: see text] and at most [Formula: see text] solutions [Formula: see text] if [Formula: see text]. In order to demonstrate the strength of our method we show that for the binary recurrence sequence [Formula: see text], with [Formula: see text] for [Formula: see text], [Formula: see text] and [Formula: see text], the Diophantine equation [Formula: see text] has at most six solutions [Formula: see text] unless [Formula: see text] in which case it has exactly seven solutions.