Euler Phi Function of Prime

Theorem

Let $p$ be a prime number.

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

• 1964: Walter Ledermann: Introduction to the Theory of Finite Groups (5th ed.) ... (previous) ... (next): Chapter $\text {I}$: The Group Concept: $\S 6$: Examples of Finite Groups: $\text{(iii)}$: $(1.27)$
• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.4$: Integer Functions and Elementary Number Theory: Exercise $27$