The Power of Powerful Numbers

In this note we discuss recent progress concerning powerful numbers, raise new questions and show that solutions to existing open questions concerning powerful numbers would yield advancement of solutions to deep, long-standing problems such as Fermat's Last Theorem. This is primarily a survey article containing no new, unpublished results. A powerful number is a positive integer n which is divisible by p2 whenever n is divisible by the prime p; i.e., in the canonical prime factorization of n, no prime appears with ex2ponent one.