Two terms with known prime divisors adding to a power

Abstract

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 Q( √ −D) which can correspond (in a standard designated way) to a solution in which (XY /D) 1/2 ∈ Z, and obtain N ≤ 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 ≤ 2 n−1 + 1. The bound is precise for n < 4: there are several cases with exactly 2 n−1 + 1 solutions.where c is a fixed positive odd integer, gcd(XY, c) = 1, and the set of primes in the factorization of XY is prechosen. Previous work on this problem includes both strictly elementary treatments and deeper, non-elementary approaches.The most common type of non-elementary approach uses results on S-unit equations to obtain a bound which is exponential in s, where s is the number of primes dividing XY c. A familiar general result of Evertse [7] shows that there are at most exp(4s + 6) solutions to the equation x + y = z in coprime positive integers (x, y, z) each composed of primes from a given set of s primes. It follows Mathematics Subject Classification: 11D61.