Show that a number of the form (p^m)(q^n), where p and q are Distinct Odd Primes,can never be a perfect number. Also show that a number of the form (3^m)(5^n)(11^k) can never be a perfect number.
We want
$\displaystyle \sigma(p^nq^m) = 2p^nq^m$
but
$\displaystyle \sigma(p^nq^m) = \sum_{i=0}^n \sum_{j=0}^m p^iq^j = \sum_{i=0}^n p^i \sum_{j=0}^m q^j = \frac{p^{n+1}+1}{p-1}\frac{q^{m+1}-1}{q-1}$
Clear denominators, see what has to divide what, and figure out the possible values of m and n.
The proof is similar for the second case.