91Ó°ÊÓ

A positive integer \(n\) is called square-full, or powerful, if \(p^{2} \mid n\) for every prime factor \(p\) of \(n\) (there are 992 square-full numbers less than 250,000 ). If \(n\) is square-full, show that it can be written in the form \(n=a^{2} b^{3}\), with \(a\) and \(b\) positive integers.

Determine all twin primes \(p\) and \(q=p+2\) for which \(p q-2\) is also prime.