Two perfect number proofs

**tttcomrader** · March 18th 2008, 06:26 AM

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

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

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

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

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

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

**CaptainBlack** · March 18th 2008, 07:02 AM

Originally Posted by tttcomrader

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

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

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

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

and:

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

hence:

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

RonL

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