A positive integer is called cool if it is divisible by the square of each of its prime divisors. Prove that $n$ and $n+1$ are simultaneously cool for infinitely many $n$.
Source: Stars of Mathematics 2024 P2 (junior level)
Tags: number theory
A positive integer is called cool if it is divisible by the square of each of its prime divisors. Prove that $n$ and $n+1$ are simultaneously cool for infinitely many $n$.