A Kernel-Certified Verification of the Erdős-Mollin-Walsh Conjecture below $10^{14}$

Erdős problem 364 asks whether three consecutive powerful numbers exist, where $n$ is powerful if $p \mid n$ implies $p^2 \mid n$. Erdős (1976) and, independently, Mollin and Walsh (1986) conjectured that none do; the $abc$ conjecture implies at most finitely many. The conjecture remains open. We pr