两个已知素除子因子的项加幂：REVISED与APPENDICES
Two terms with known prime divisors adding to a power: REVISED with APPENDICES
摘要：
设$c$是正奇整数，$R$是与$c$互质的$n$素数的集合.我们考虑方程$X + Y = c^z$在三个整数未知数$X$，$Y$，$z$，其中$z > 0$，$Y > X > 0$，和素数划分$XY$正是那些在$R$。我们考虑$N$，这样的方程的解决方案的数量。给定一个解$(X, Y, z)$，令$D$为最小正整数，使得$(XY/D)^{1/2}$为整数。此外，令$ω$为除$c$的不同素数的数目。标准初等方法使用$2^n$的上界作为可能的$D$的数目，使用$2^{ω-1}$的上界作为域$\ratQ(\sqrt{-D})$中$c$的理想分解的数目，域$\ratQ(\sqrt{-D})$可以对应于（以标准指定方式）其中有$(XY/D)^{1/2} \in \intZ$的解，并且获得$N \le 2^{n+ω-1}$。在这里，我们改善这一点，找到一个反比例关系之间的约束的数目$D$可以发生在解决方案和约束（独立的$D$）的数目理想分解的$c$可以对应于一个给定的$D$的解决方案。我们得到了$N \le 2^{n-1}+1$。对于$n<4$，该界是精确的：有几种情况具有精确的$2^{n-1} + 1$解。对于更高的$n$值的界限变得不切实际，但仍然是一个改进的边界获得的初等和非初等方法。
英文摘要：
Let $c$ be a positive odd integer and $R$ a set of $n$ primes coprime with $c$. We consider equations $X + Y = c^z$ in three integer unknowns $X$, $Y$, $z$, where $z > 0$, $Y > X > 0$, and the primes dividing $XY$ are precisely those in $R$. We consider $N$, the number of solutions of such an equation. Given a solution $(X, Y, z)$, let $D$ be the least positive integer such that $(XY/D)^{1/2}$ is an integer. Further, let $ω$ be the number of distinct primes dividing $c$. Standard elementary approaches use an upper bound of $2^n$ for the number of possible $D$, and an upper bound of $2^{ω-1}$ for the number of ideal factorizations of $c$ in the field $\ratQ(\sqrt{-D})$ which can correspond (in a standard designated way) to a solution in which $(XY/D)^{1/2} \in \intZ$, and obtain $N \le 2^{n+ω-1}$. Here we improve this by finding an inverse proportionality relationship between a bound on the number of $D$ which can occur in solutions and a bound (independent of $D$) on the number of ideal factorizations of $c$ which can correspond to solutions for a given $D$. We obtain $N \le 2^{n-1}+1$. The bound is precise for $n<4$: there are several cases with exactly $2^{n-1} + 1$ solutions. For higher values of $n$ the bound becomes unrealistic, but is nevertheless an improvement on bounds obtained by both elementary and non-elementary methods.
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