Totient of a prime

Statement

Let $p$ be a prime number. Then $\varphi(p) = p - 1$, where $\varphi$ is Euler's totient function.

Remarks

The proof: the only divisors of $p$ are $1$ and $p$. So each of $1, \dots, p - 1$ is coprime to $p$, and $p$ is not. The converse holds: if $n > 1$ and $\varphi(n) = n - 1$, then $n$ is prime. The result is the case $k = 1$ of the totient of a prime power.

Explanations

L1 human

For a prime $p$, each number from $1$ to $p - 1$ has no common factor with $p$. For example, $\varphi(7) = 6$, because $1, 2, 3, 4, 5, 6$ are all coprime to $7$.