PlanetMath: Euler phi function

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Euler phi function

(Definition)

For any positive integer , $\varphi(n)$ is the number of positive integers less than or equal to which are coprime to . The function $\varphi$ is known as the Euler $\varphi$ function. This function may also be denoted by $\phi$ .

Among the most useful properties of $\varphi$ are the facts that it is multiplicative (meaning if $\gcd(a,b)=1$ , then $\varphi(ab)=\varphi(a)\varphi(b)$ ) and that $\varphi(p^k)=p^{k-1}(p-1)$ for any prime and any positive integer . These two facts combined give a numeric computation of $\varphi$ for all positive integers:

$\displaystyle \varphi(n)$

$\displaystyle =n \prod_{p\vert n} \left( 1-\frac{1}{p} \right).$

(1)

For example,

$\displaystyle \varphi (2000)$	$\displaystyle = 2000 \prod_{p\vert 2000} \left(1-\frac{1}{p} \right)$
	$\displaystyle = 2000 \left( 1 - \frac{1}{2} \right) \left( 1 - \frac{1}{5} \right)$
	$\displaystyle = 2000 \left( \frac{1}{2} \right) \left( \frac{4}{5} \right)$
	$\displaystyle = \frac{8000}{10}$
	$\displaystyle = 800.$

From equation (1), it is clear that $\varphi(n)\le n$ for any positive integer with equality holding exactly when . This is because

$\displaystyle \prod_{p\vert n} \left( 1-\frac{1}{p} \right) \le 1,$

with equality holding exactly when

Another important fact about the Euler $\varphi$ function is that

$\displaystyle \sum_{d\vert n} \varphi(d)=n,$

where the sum extends over all positive divisors of

. Also, by definition, $\varphi(n)$ is the number of units in the ring $\mathbb{Z}/n\mathbb{Z}$ of integers modulo

The sequence $\{\varphi(n)\}$ appears in the OEIS as sequence A000010.

"Euler phi function" is owned by Wkbj79. [ full author list (2) | owner history (1) ]

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See Also: proof that Euler $\varphi$ function is multiplicative, values of $n$

for which $\varphi(n)=\tau(n)$ , prime residue class

Other names:

Euler totient function, Euler $\varphi$ function, Euler $\phi$ function

Keywords:

number theory

Attachments:: derivation of Euler phi-function (Derivation) by jwaixs

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Cross-references: OEIS, sequence, ring, units, divisors, sum, equality, clear, equation, prime, multiplicative, function, coprime, number, integer, positive
There are 39 references to this entry.

This is version 14 of Euler phi function, born on 2001-10-15, modified 2008-05-28.
Object id is 196, canonical name is EulerPhifunction.
Accessed 34432 times total.

Classification:

AMS MSC:	11-00 (Number theory :: General reference works )
	11A25 (Number theory :: Elementary number theory :: Arithmetic functions; related numbers; inversion formulas)

Pending Errata and Addenda

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