Euler Phi Function of Prime Power/Corollary
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Corollary to Euler Phi Function of Prime Power
Let $\phi: \Z_{>0} \to \Z_{>0}$ be the Euler $\phi$ function.
Then:
- $\map \phi {2^k} = 2^{k - 1}$
Proof
We have that:
- $1 - \dfrac 1 2 = \dfrac {2 - 1} 2 = \dfrac 1 2$
It follows from Euler Phi Function of Prime Power:
- $\map \phi {2^k} = \paren {\dfrac 1 2} 2^k = 2^{k - 1}$
$\blacksquare$