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Math Help - Multiplicative Question

  1. #1
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    Multiplicative Question

    Assume  \sigma is multiplicative

    a) Let p be a prime and a \in Z^+. Prove that for an odd prime p, \sigma(p^a) is odd if, and only f, a is even. Also prove that for p = 2, \sigma(2^a) is odd for all a  \in Z^+.

    b) Let n  \in Z^+. Prove that \sigma(n) is odd if, and only if, either n is a perfect square or twice a perfect square (i.e. n \in {(m^2 \mid m \in Z^+)} \cup {(2m^2 \mid m \in Z})
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  2. #2
    MHF Contributor Also sprach Zarathustra's Avatar
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    \sigma(5^2)=6\cdot6=36 (even)
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  3. #3
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    Quote Originally Posted by Also sprach Zarathustra View Post
    \sigma(5^2)=6\cdot6=36 (even)
    Wait, I have to prove that if p is an odd prime, then \sigma(p^a) is odd when a is even. But \sigma(5^2) is even which does not follow....
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