RMM 2020 Problem 6

For each integer $n \geq 2$ , let $F(n)$ denote the greatest prime factor of $n$ . A strange pair is a pair of distinct primes $p$ and $q$ such that there is no integer $n \geq 2$ for which $F(n)F(n+1)=pq$ .

Prove that there exist infinitely many strange pairs.