Handling a large bound for a problem on the generalized Pillai equation $\pm r a^x \pm s b^y = c$

TL;DR

This paper proves that within certain large bounds, no new solutions with four or more solutions exist for a generalized Pillai equation, extending previous results by allowing zero exponents and different sign combinations.

Contribution

The authors show that no additional solutions with at least four solutions exist within specified bounds, considering zero exponents and all sign combinations, which was not addressed before.

Findings

No further cases with N ≥ 4 within the bounds.

N = 3 solutions occur infinitely often, even after exclusions.

Extended the analysis to include zero exponents and all sign choices.

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 $s>0$. Previous work showed that there are nine essentially distinct $(a,b,c,r,s)$ for which $N \ge 4$, except possibly for cases in which the solutions have $r$, $a$, $x$, $s$, $b$, and $y$ each bounded by $8 \cdot 10^{14}$ or $2 \cdot 10^{15}$. In this paper we show that there are no further cases with $N \ge 4$ within these bounds. We note that $N = 3$ for an infinite number of $(a,b,c,r,s)$, even if we eliminate from consideration cases which are directly derived from other cases in one of several completely designated ways. Our work differs from previous work in that we allow $x$ and $y$ to be zero and also allow choices of $(u,v)$ other than $(0,1)$.