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

TL;DR

This paper investigates the solutions to a generalized Pillai equation involving sums and differences of exponential terms, establishing bounds on the number of solutions under various gcd and positivity conditions.

Contribution

It provides new bounds on the number of solutions to the generalized Pillai equation, including finiteness results and classifications based on gcd conditions.

Findings

When d(ra, sb)=1 and \u2205(x,y)>0, solutions are at most two, with finitely many exceptions.

For arbitrary gcd(ra, sb), solutions with (u,v)=(0,1) are at most three, with infinitely many such cases.

The paper characterizes the solution counts and finiteness properties of the generalized Pillai equation.

Abstract

In this paper 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 $s>0$. We show that $N \le 2$ when $\gcd(ra, sb) =1$ and $\min(x,y)>0$, except for a finite number of cases that can be found in a finite number of steps. For arbitrary $\gcd(ra, sb)$ and $\min(x,y) \ge 0$, we show that when $(u,v) = (0,1)$ we have $N \le 3$, with an infinite number of cases for which N=3.