The generalized Pillai equation ±rax±sby=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 positive 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, except for a finite number of cases that can be found in a finite number of steps. For arbitrary gcd(ra, sb) with (u, v) = (0, 1), we show that N 3 with an infinite number of cases for which N = 3.