Handling a large bound for a problem on the generalized Pillai equation \(\pm r a^{x} \pm sb^{y}=c\) (Q2926275)

Consider the equation \((-1)^ura^x+(-1)^vsb^y=c\) in nonnegative integers \(x,y\) and \(u,v\in\{0,1\}\), where \(a,b,c,r,s\) are given positive integers with \(a>1\), \(b>1\). Write \(N\) for the number of solutions in \((x,y,u,v)\).NEWLINENEWLINEIn a previous paper the authors proved that there are essentially nine distinct tuples \((a,b,c,r,s)\) allowing \(N\geq 4\) solutions, except possibly the cases where \(a,b,r,s,x,y\) are all smaller than \(2\cdot 10^{15}\). In the present paper the authors prove that no new solutions arise for these small values of the parameters. The present paper is more general than the previous work in that now \(x\) and \(y\) can also be zero, and \((u,v)\) may differ from \((0,1)\).