Euler Phi Function of Prime Power/Corollary

Corollary to Euler Phi Function of Prime Power
Let $\phi: \Z_{>0} \to \Z_{>0}$ be the Euler $\phi$ function.

Then:
 * $\phi \left({2^k}\right) = 2^{k-1}$

Proof
We have that:
 * $\displaystyle 1 - \frac 1 2 = \frac {2 - 1} 2 = \frac 1 2$

It follows from Euler Phi Function of Prime Power:
 * $\phi \left({2^k}\right) = \left({\dfrac 1 2}\right) 2^k = 2^{k-1}$