Thread: Euler's totient function and its range (at odd n)

1. 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.

Any help would be appreciated.
Thanks!

2. Originally Posted by doug
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.

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
$\phi(n)=n\cdot\prod\limits_{p\mid n,\,p\,\,a\,\,prime}\left(1-\frac{1}{p}\right)$