The generalized Pillai equation $\pm r a^x \pm s b^y = c$, II

TL;DR

This paper investigates the number of solutions to a generalized Pillai equation involving powers and signs, establishing bounds and characterizing exceptional cases with potential for infinite solutions.

Contribution

It provides bounds on the number of solutions and classifies exceptional cases, including infinite families, for the generalized Pillai equation with various parameter conditions.

Findings

Maximum of 3 solutions in most cases

Finite exceptions with solutions bounded by 2×10^15

Identification of three infinite families with solutions

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.