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Math Help - Euler's totient function and its range (at odd n)

  1. #1
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    Euler's totient function and its range (at odd n)

    True or false (and of course why):
    Euler's totient function implies all its range for odd variables.

    I proved that for all n>1 odd numbers:
    φ(n)=φ(2n).
    But it's not enough. Because I should prove that for all even n I can find an odd k for which:
    φ(n)=φ(2k). But I can't see it.

    Please help me!
    Any help would be appreciated.
    Thanks!
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  2. #2
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    Quote Originally Posted by doug View Post
    True or false (and of course why):
    Euler's totient function implies all its range for odd variables.

    I proved that for all n>1 odd numbers:
    φ(n)=φ(2n).
    But it's not enough. Because I should prove that for all even n I can find an odd k for which:
    φ(n)=φ(2k). But I can't see it.

    Please help me!
    Any help would be appreciated.
    Thanks!

    What do you mean by "implies all its range for odd variables"? That the Euler function gets all its range when the variable runs over odd integers?

    If so then it can't be: \phi(2^n)=2^{n-1}, and there can't be an odd number whose Euler function value is this since
    \phi(n)=n\cdot\prod\limits_{p\mid n,\,p\,\,a\,\,prime}\left(1-\frac{1}{p}\right)

    Tonio
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