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

Abstract

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. When (ra, sb) = 1, we show that N ≤ 3 except for a finite number of cases all of which satisfy max(a, b, r, s, x, y) < 2 • 10 15 for each solution; when (a, b) > 1, we show that N ≤ 3 except for three infinite families of exceptional cases. We find several different ways to generate an infinite number of infinite families of cases giving N = 3 solutions.