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.