A positive integer $n$ is called *powerful* (OEIS: A001694) if $p^2$ divides $n$ whenever $p$ is a prime that divides $n$. Equivalently, $n$ is powerful if $n = a^2b^3$, where $a$ and $b$ are positive integers.

**Conjecture**: If $n$ is a powerful integer greater than one, then there is a prime $p$ and a powerful number $m$ such that $n=m+p$.

This conjecture holds for every powerful number $\leq 1000$. Given its similarity to the Goldbach conjecture, my suspicion is that this might be quite difficult to prove. Any references/ideas/partial results on this would be greatly appreciated.

9more comments