# On a system of equations with primes

•  Università Bocconi via Sarfatti 25 20100 Milan, Italy
•  Texas A&M University at Qatar PO Box 23874 Education City DOHA, 5825 QATAR
• Volume: 26, Issue: 2, page 399-413
• ISSN: 1246-7405

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## Abstract

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Given an integer $n\ge 3$, let ${u}_{1},...,{u}_{n}$ be pairwise coprime integers $\ge 2$, $𝒟$ a family of nonempty proper subsets of $\left\{1,...,n\right\}$ with “enough” elements, and $\epsilon$ a function $𝒟\to \left\{±1\right\}$. Does there exist at least one prime $q$ such that $q$ divides ${\prod }_{i\in I}{u}_{i}-\epsilon \left(I\right)$ for some $I\in 𝒟$, 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 $𝒟$ are subjected to certain restrictions.We use the result to prove that, if ${\epsilon }_{0}\in \left\{±1\right\}$ 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.

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