Abstract
In this paper, we show that if (X-n,Y-n) is the nth solution of the Pell equation X-2 - dY(2) = +/- 1 for some non-square d, then given any integer c, the equation c = X-n - 2(m) has at most 2 integer solutions (n,m) with n >= 0 and m >= 0, except for the only pair (c,d) = (-1, 2). 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.