Euler Phi Function of Prime

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Theorem

Let $p$ be a prime number $p > 1$.


Then:

$\map \phi p = p - 1$

where $\phi: \Z_{>0} \to \Z_{>0}$ is the Euler $\phi$ function.


Proof

From the definition of a prime number, the only (strictly) positive integer less than or equal to a prime $p$ which is not prime to $p$ is $p$ itself.

Thus it follows directly that:

$\map \phi p = p - 1$

$\blacksquare$


Sources