# Thread: Two perfect number proofs

1. ## Two perfect number proofs

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

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 $\displaystyle 2 p^k$?
$\displaystyle \frac {p^{k+1} - 1 }{p-1}=p^k+p^{k-1}+ ... + 1$

and:

$\displaystyle p^{k-1}+ ... + 1=\frac{p^k-1}{p-1} < p^k$

hence:

$\displaystyle \frac {p^{k+1} - 1 }{p-1}<2p^k$.

RonL