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Math Help - Two perfect number proofs

  1. #1
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    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  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
    Last edited by CaptainBlack; March 18th 2008 at 07:55 AM. Reason: correcting LaTeX errors
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  2. #2
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    Quote Originally Posted by tttcomrader View Post
    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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