The generalized Pillai equation $\pm r a^x \pm s b^y = c$

Abstract

In this paper we consider N , the number of solutions (x, y, u, v) to the equation (−1) u ra x +(−1) v sb y = c in nonnegative integers x, y and integers u, v ∈ {0, 1}, for given integers a > 1, b > 1, c > 0, r > 0 and s > 0. We show that N ≤ 2 when gcd(ra, sb) = 1 and min(x, y) > 0, except for a finite number of cases that can be found in a finite number of steps. For arbitrary gcd(ra, sb) and min(x, y) ≥ 0, we show that when (u, v) = (0, 1) we have N ≤ 3, with an infinite number of cases for which N = 3.