On the generalized Pillai equation $\pm a^x\pm b^y=c$

Abstract

We show that the equation $\pm a^x\pm b^y=c$ (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.