# Difference between revisions of "Euler's totient function"

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=== Formulas === | === Formulas === | ||

− | The formal definition is <math> | + | The formal definition is <math>\phi(n):=\# \left\{ a \in \mathbb{Z}: 1 \leq a \leq n , \gcd(a,n)=1 \right\} </math>. |

Given the [[prime factorization]] of <math>{n} = {p}_1^{e_1}{p}_2^{e_2} \cdots {p}_n^{e_n}</math>, then another formula for <math>\phi(n)</math> is <math> \phi(n) = n\left(1-\frac{1}{p_1}\right)\left(1-\frac{1}{p_2}\right) \cdots \left(1-\frac{1}{p_n}\right) </math>. | Given the [[prime factorization]] of <math>{n} = {p}_1^{e_1}{p}_2^{e_2} \cdots {p}_n^{e_n}</math>, then another formula for <math>\phi(n)</math> is <math> \phi(n) = n\left(1-\frac{1}{p_1}\right)\left(1-\frac{1}{p_2}\right) \cdots \left(1-\frac{1}{p_n}\right) </math>. |

## Revision as of 04:09, 20 June 2006

**Euler's totient function**, , is defined as the number of positive integers less than or equal to a given positive integer that are relatively prime to that integer.

### Formulas

The formal definition is .

Given the prime factorization of , then another formula for is .

### Identities

For prime p, , because all numbers less than are relatively prime to it.

For relatively prime , .

In fact, we also have for any that .

For any , we have where the sum is taken over all divisors d of .