On Pillai’s problem with X-coordinates of Pell equations and powers of 2 II

In this paper, we show that if [Formula: see text] is the [Formula: see text]th solution of the Pell equation [Formula: see text] for some non-square [Formula: see text], then given any integer [Formula: see text], the equation [Formula: see text] has at most [Formula: see text] integer solutions [Formula: see text] with [Formula: see text] and [Formula: see text], except for the only pair [Formula: see text]. Moreover, we show that this bound is optimal. Additionally, we propose a conjecture about the number of solutions of Pillai’s problem in linear recurrent sequences.