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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 preciselythose 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 $\omega$ be the number of distinctprimes dividing $c$. Standard elementary approaches use an upper bound of $2^n$for the number of possible $D$, and an upper bound of $2^{\omega-1}$ for thenumber of ideal factorizations of $c$ in the field $\ratQ(\sqrt{-D})$ which cancorrespond (in a standard designated way) to a solution in which $(XY/D)^{1/2}\in \intZ$, and obtain $N \le 2^{n+\omega-1}$. Here we improve this by findingan 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 ofideal 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 severalcases with exactly $2^{n-1} + 1$ solutions. For higher values of $n$ the boundbecomes unrealistic, but is nevertheless an improvement on bounds obtained byboth elementary and non-elementary methods.

Two terms with known prime divisors adding to a power: REVISED with APPENDICES