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:

$\displaystyle 1 - \frac 1 2 = \frac {2 - 1} 2 = \frac 1 2$


It follows from Euler Phi Function of Prime Power:

$\map \phi {2^k} = \paren {\dfrac 1 2} 2^k = 2^{k - 1}$

$\blacksquare$