On the generalized Pillai equation ±ax±by=c

Abstract

We show that the equation ±a x ± b y = c (where the ± signs are independent) has at most two solutions (x, y) for given integers a and b both greater than one and c greater than zero, except for listed specific cases. For any prime a > 5 and b = 2, we show that there are at most two values of c allowing more than one solution to this equation, not counting trivial rearrangements; further restricting a to be a non-Wieferich prime, we improve this result: we show that there are no values of c allowing more than one solution, apart from designated exceptional cases. Finally, we give all solutions to the equation |a x 1 − b y 1 | = |a x 2 − b y 2 | for b = 2 or 3 and prime a not a base-b Wieferich prime.