On a system of equations with primes

Abstract

Given an integer $n\ge 3$, let $u_1,\dots ,u_n$ be pairwise coprime integers $\ge 2$, $\mathcal{D}$ a family of nonempty proper subsets of $\{1,\dots ,n\}$ with "enough" elements, and $\epsilon$ a function $\mathcal{D}\to \{\pm 1\}$. Does there exist at least one prime $q$ such that $q$ divides ${\prod }_{i\in I}{u}_i-\epsilon (I)$ 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 $\epsilon$ and $\mathcal{D}$ are subjected to certain restrictions. We use the result to prove that, if ${\epsilon }_0\in \{\pm 1\}$ and $A$ is a set of three or more primes that contains all prime divisors of any number of the form ${\prod }_{p\in B}p-{\epsilon }_0$ for which $B$ is a finite nonempty proper subset of $A$, then $A$ contains all the primes.

13 pp, to appear in Journal de Théorie des Nombres de Bordeaux. Fixed a number of typos (particularly, in the proof of Theorem 1.3). Abridged the part on the lifting-the-exponent lemma. Slightly simplified the formulation of Question 1. Added a new question (viz., Question 4 in this version). "Generalized" Theorem 1.3 (a hypothesis in the old statement was "evidently" useless)