Definition:Euler Phi Function

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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:

$\phi \left({n}\right) = $ the number of strictly positive integers less than or equal to $n$ which are prime to $n$


That is:

$\phi \left({n}\right) = \left|{S_n}\right|: S_n = \left\{{k: 1 \le k \le n, k \perp n}\right\}$


Examples

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

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

This sequence is A000010 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


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.


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.


Sources