On a system of equations with primes

Abstract

Given an integer $n \ge 3$, let $u_1, \ldots, u_n$ be pairwise coprime integers $\ge 2$, $\mathcal D$ a family of nonempty proper subsets of $\{1, \ldots, n\}$ with "enough" elements, and $\varepsilon$ a function $ \mathcal D \to \{\pm 1\}$. Does there exist at least one prime $q$ such that $q$ divides $\prod_{i \in I} u_i - \varepsilon(I)$ for some $I \in \mathcal D$, but it does not divide $u_1 \cdots u_n$? We answer this question in the positive when the $u_i$ are prime powers and $\varepsilon$ and $\mathcal D$ are subjected to certain restrictions. We use the result to prove that, if $\varepsilon_0 \in \{\pm 1\}$ and $A$ is a set of three or more primes that contains all prime divisors of any number of the form $\prod_{p \in B} p - \varepsilon_0$ for which $B$ is a finite nonempty proper subset of $A$, then $A$ contains all the primes.