OPEN
Are there any triples of consecutive positive integers all of which are powerful (i.e. if $p\mid n$ then $p^2\mid n$)?
Erdős originally asked Mahler whether there are infinitely many pairs of consecutive powerful numbers, but Mahler immediately observed that the answer is yes from the infinitely many solutions to the Pell equation $x^2=8y^2+1$.
Erdős [Er76d] believed the answer to this question is no, and in fact if $n_k$ is the $k$th powerful number then
\[n_{k+2}-n_k > n_k^c\]
for some constant $c>0$.
It is trivial that there are no quadruples of consecutive powerful numbers since one must be $2\pmod{4}$.
By OEIS A060355 there are no such $n$ for $n<10^{22}$.
See also [137], [365], and [938].
