Erdős–Mollin–Walsh conjecture on three consecutive powerful numbers

A positive integer $n$ is powerful if $p^2\mid n$ for every prime $p$ dividing $n$. Three positive integers are consecutive powerful numbers if they have the form $n,n+1,n+2$ and each is powerful. Erdős–Mollin–Walsh conjecture. No three consecutive powerful numbers exist. The paper studies this longstanding non-existence problem using Pell equations, elliptic curves, and second-order recurrences; the conjecture remains unresolved in the supplied source.