# Difference between revisions of "Carmichael function"

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== First Definition == | == First Definition == | ||

− | + | The Carmichael function <math>\lambda</math> is defined at <math>n</math> to be the smallest [[positive integer]] <math>\lambda(n)</math> such that <math>a^{\lambda(n)} \equiv 1\pmod {n}</math> for all positive [[integer]]s <math>a</math> [[relatively prime]] to <math>n</math>. The [[order]] of <math>a\pmod {n}</math> always divides <math>\lambda(n)</math>. | |

This function is also known as the ''reduced totient function'' or the ''least universal exponent'' function. | This function is also known as the ''reduced totient function'' or the ''least universal exponent'' function. | ||

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− | Suppose < | + | Suppose <math>n=p_1^{\alpha_1}\cdot p_2^{\alpha_2}\cdots p_k^{\alpha_k}</math>. We have |

− | <center><p>< | + | <center><p><math>\lambda(n) = \begin{cases} |

\phi(n) & | \phi(n) & | ||

\mathrm {for}\ n=p^{\alpha},\ \mathrm {with}\ p=2\ \mathrm {and}\ \alpha\le 2,\ \mathrm {or}\ p\ge 3\\ | \mathrm {for}\ n=p^{\alpha},\ \mathrm {with}\ p=2\ \mathrm {and}\ \alpha\le 2,\ \mathrm {or}\ p\ge 3\\ | ||

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\mathrm{lcm} (\lambda(p_1^{\alpha_1}), \lambda(p_2^{\alpha_2}), \ldots, \lambda(p_k^{\alpha_k})) & | \mathrm{lcm} (\lambda(p_1^{\alpha_1}), \lambda(p_2^{\alpha_2}), \ldots, \lambda(p_k^{\alpha_k})) & | ||

\mathrm{for}\ \mathrm{all}\ n. | \mathrm{for}\ \mathrm{all}\ n. | ||

− | \end{cases}<math></p></center | + | \end{cases}</math></p></center> |

=== Examples === | === Examples === |

## Revision as of 00:45, 10 August 2019

There are two different functions called the **Carmichael function**. Both are similar to Euler's totient function .

## First Definition

The Carmichael function is defined at to be the smallest positive integer such that for all positive integers relatively prime to . The order of always divides .

This function is also known as the *reduced totient function* or the *least universal exponent* function.

Suppose . We have

### Examples

*This article is a stub. Help us out by expanding it.*

Evaluate . [1]

## Second Definition

The second definition of the Carmichael function is the least common multiples of all the factors of . It is written as . However, in the case , we take as a factor instead of .

### Examples

*This article is a stub. Help us out by expanding it.*