Problem

Source: ELMO Shortlist 2024/N2

Tags: Elmo, number theory



Call a positive integer emphatic if it can be written in the form $a^2+b!$, where $a$ and $b$ are positive integers. Prove that there are infinitely many positive integers $n$ such that $n$, $n+1$, and $n+2$ are all emphatic. Allen Wang