Prove there do exist infinitely many positive integers $n$ such that if a prime $p$ divides $n(n+1)$ then $p^2$ also divides it (all primes dividing $n(n+1)$ bear exponent at least two). Exhibit (at least) two values, one even and one odd, for such numbers $n>8$. (Pál Erdös & Kurt Mahler)
Problem
Source: Stars of Mathematics 2011 - Seniors - Problem 2
Tags: number theory proposed, number theory