论文标题
两个术语具有已知的主要除数，从而增加了力量：用附录修订
Two terms with known prime divisors adding to a power: REVISED with APPENDICES
论文作者
论文摘要
让$ c $是一个正奇数整数，$ r $ a $ n $ primes coprime带有$ c $。我们考虑方程$ x + y = c^z $在三个整数未知$ x $，$ y $，$ z $中，其中$ z> 0 $，$ y> x> 0 $，而划分$ xy $的Primes恰好是$ r $中的$。我们考虑$ n $，即这种方程的解决方案数量。给定解决方案$（x，y，z）$，让$ d $是最小的正整数，因此$（xy/d）^{1/2} $是整数。此外，令$ω$为划分$ c $的不同素数的数量。标准基本方法对于可能的$ d $数量使用$ 2^n $的上限，以及$ 2^{ω-1} $的$ 2^{ω-1} $用于现场$ \ ratq（\ sqrt {-d}）的理想因数数量，该$ \ ratq（\ sqrt {-d} $以标准指定的方式）与该解决方案相对应，以获取$（$/d）的$（xy/d）^1/d^2）^{1/2） $ n \ le 2^{n+ω-1} $。在这里，我们通过在解决方案中可能发生的$ d $数量与$ d $（独立于$ d $）的$ d $的数量之间找到反比的关系来改善这一点，这在$ c $的理想因素数量数量上可以与给定$ d $的解决方案相对应。我们获得$ n \ le 2^{n-1}+1 $。对于$ n <4 $：有几种情况，有几种限制的限制是$ 2^{n-1} + 1 $解决方案。对于$ n $的较高值，界限变得不现实，但仍然是基础和非元素方法获得的界限的改进。
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.
