On a system of equations with primes

Given an integer n ≥ 3, let u1, …, un be pairwise coprime integers ≥ 2, D a family of nonempty proper subsets of {1, …, n} with “enough” elements, and ε a function D → {±1}. Does there exist at least one prime q such that q divides ∏i∈I ui− ε(I) for some I ∈ D, but it does not divide u1 · · · un? We answer this question in the positive when the ui are prime powers and " and D are subjected to certain restrictions. We use the result to prove that, if ε0 ∈ {±1} and A is a set of three or more primes that contains all prime divisors of any number of the form ∏p∈B p−ε0 for which B is a finite nonempty proper subset of A, then A contains all the primes.