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 r a^x + (-1)^v s b^y = c $ in nonnegative integers $x, y$ and integers $u, v \in \{0,1\}$, for given integers $a>1$, $b>1$, $c>0$, $r>0$ and $s>0$. When $(ra,sb)=1$, we show that $N \le 3$ except for a finite number of cases all of which satisfy $\max(a,b,r,s, x,y) < 2 \cdot 10^{15}$ for each solution; when $(a,b)>1$, we show that $N \le 3$ 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.