On a system of equations with primes
Résumé
Given an integer $n \ge 3$, let $u_1, \ldots, u_n$ be pairwise coprime integers for which $2 \le u_1 < \cdots < u_n$, and let $\mathcal D$ be a family of nonempty proper subsets of $\{1, \ldots, n\}$ with ''enough'' elements and $\varepsilon$ a map $ \mathcal D \to \{\pm 1\}$. Does there exist at least one $q \in \mathbb P$ such that $q$ divides $\prod_{i \in I} u_i - \varepsilon(I)$ for some $I \in \mathcal D$ and $q \nmid u_1 \cdots u_n$? We answer this question in the positive in the case where the integers $u_i$ are prime powers and some restrictions hold on $\varepsilon$ and $\mathcal D$. We use the result to prove that, if $\varepsilon_0 \in \{\pm 1\}$ and $A$ is a set of three or more primes with the property that $A$ contains all prime divisors of any product 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.
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