Problem

Source: ELMO 2019 Problem 5, 2019 ELMO Shortlist N3

Tags: number theory



Let $S$ be a nonempty set of positive integers such that, for any (not necessarily distinct) integers $a$ and $b$ in $S$, the number $ab+1$ is also in $S$. Show that the set of primes that do not divide any element of $S$ is finite. Proposed by Carl Schildkraut