广义Pillai方程$\pm r a^x \pm s b^y = c$
The generalized Pillai equation $\pm r a^x \pm s b^y = c$
摘要：
本文研究了方程$ (-1)^u r a^x + (-1)^v s b^y = c$在非负整数$x, y$和整数$u, v \in \{0,1\}$中的解的个数$N$，对于给定的整数$a>1$，$b>1$，$c>0$，$r>0$和$s>0$ .我们表明，$N \le 2$时，$\gcd(ra, sb) =1$和$\min(x,y)>0$，除了有限数量的情况下，可以发现在有限数量的步骤。对于任意的$\gcd(ra, sb)$和$\min(x,y) \ge 0$，我们表明，当$(u,v) = (0,1)$，我们有$N \le 3$，与无限数量的情况下，N=3。
英文摘要：
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.
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