Is min exponents of three positive integers $n$, $n+1$ and $n+2$ $=1$ true or false?

Question

Given a positive integer $P>1$, let its prime factorization be written $$P=p_1^{a_1}p_2^{a_2}p_3^{a_3}\cdots p_k^{a_k}$$

Define the functions $h(P)$ by $h(1)=1$ and $h(P)=\min(a_1, a_2,\ldots,a_k)$

Is the follows property true or false?

The property: Let $n$ is a positive integer then $\min(h(n), h(n+1), h(n+2)) = 1$

PS: The property was checked up to $n=5.10^7$

Answer 1

It is actually an old conjecture of Erdős, Mollin, and Walsh that the pattern you have noticed does indeed go on forever, i.e., there are no three consecutive powerful numbers.