Problem

Source: 2019 Switzerland MO

Tags: number theory



Let $\mathbb{P}$ be the set of all primes and let $M$ be a subset of $\mathbb{P}$ with at least three elements. Suppose that for all $k \geq 1$ and for all subsets $A=\{p_1,p_2,\dots ,p_k \}$ of $M$ ,$A\neq M$ , all prime factors of $p_1p_2\dots p_k-1$ are in $M$ . Prove that $M=\mathbb{P}$.