MathNet 2020 — Math Olympiad Vault — Part 05 Question 117

For an integer $n \gt 1$, let $gpf(n)$ denote the greatest prime factor of $n$. A *strange pair* is an unordered pair of distinct primes $p$ and $q$ such that $\{p, q\} = \{gpf(n), gpf(n + 1)\}$ for no integer $n \gt 1$. Prove that there exist infinitely many strange pairs.

Russia, Dmitry Krachun