On a system of equations with primes

TL;DR

This paper investigates the existence of primes dividing certain products minus signs, proving conditions under which a set of primes must contain all primes, with implications for prime divisibility patterns.

Contribution

The paper provides a positive answer to a specific prime divisibility question for prime power variables under certain restrictions, and applies this to characterize sets of primes.

Findings

Existence of primes dividing products minus signs under certain conditions

Sets of primes containing all prime divisors of specific forms must include all primes

Results apply to prime power variables and particular subset restrictions

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.