1. Prove that no power of a prime is a perfect number.

proof. Let p be prime and k be an integer. Consider $\displaystyle p^k $

$\displaystyle O(p^k ) = \frac {p^{k+1} - 1 }{p-1} $, now, how do I get this to not equal to $\displaystyle 2 p^k $?

2. Prove that a perfect square is never a perfect number.

proof. Let $\displaystyle n = (p_{1}^{k_{1}}p_{2}^{k_{2}}...p_{r}^{k_{r}} )^2 $

Then $\displaystyle O (n) = O(p_{1}^{2k_{1}} )... O(p_{r}^{2k_{r}} ) $ I have some trouble trying to continue from here, I