Definition:Euler Phi Function

Definition

Let $n \in \Z_{>0}$, that is, a strictly positive integer.

The Euler $\phi$ (phi) function is the arithmetic function $\phi: \Z_{>0} \to \Z_{>0}$ defined as:

$\map \phi n =$ the number of strictly positive integers less than or equal to $n$ which are prime to $n$

That is:

$\map \phi n = \card {S_n}: S_n = \set {k: 1 \le k \le n, k \perp n}$

Examples

The values of the Euler $\phi$ function for the first few integers are as follows:

$n$ $\map \phi n$ $m$ not coprime: $1 \le m \le n$
$1$ $1$ $\varnothing$
$2$ $1$ $2$
$3$ $2$ $3$
$4$ $2$ $2, 4$
$5$ $4$ $5$
$6$ $2$ $2, 3, 4, 6$
$7$ $6$ $7$
$8$ $4$ $2, 4, 6, 8$
$9$ $6$ $3, 6, 9$
$10$ $4$ $2, 4, 5, 6, 8, 10$

Also known as

The Euler $\phi$ function is also known as the totient function or the indicator function

Some sources render it with a hyphen: Euler $\phi$-function.

Some sources merely refer to it as Euler's function.

Also see

• Results about the Euler $\phi$ function can be found here.

Source of Name

This entry was named for Leonhard Paul Euler.

Historical Note

The Euler $\phi$ function was invented by Leonhard Paul Euler in order to generalise Fermat's Little Theorem to non-prime indices.