# Difference between revisions of "Carmichael function"

(→Examples) |
(→First Definition) |
||

Line 2: | Line 2: | ||

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

− | The Carmichael function <math>\lambda< | + | <math>\boxed{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. | ||

Line 8: | Line 8: | ||

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

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

Line 17: | Line 17: | ||

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

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

## Revision as of 00:44, 10 August 2019

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

## First Definition

$\boxed{The Carmichael function$ (Error compiling LaTeX. ! File ended while scanning use of \boxed.)\lambdan\lambda(n)a^{\lambda(n)} \equiv 1\pmod {n}ana\pmod {n}\lambda(n)$.

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

Suppose$ (Error compiling LaTeX. ! Missing $ inserted.)n=p_1^{\alpha_1}\cdot p_2^{\alpha_2}\cdots p_k^{\alpha_k}$. We have

<center><p>$ (Error compiling LaTeX. ! Missing $ inserted.)\lambda(n) = \begin{cases}

\phi(n) & \mathrm {for}\ n=p^{\alpha},\ \mathrm {with}\ p=2\ \mathrm {and}\ \alpha\le 2,\ \mathrm {or}\ p\ge 3\\ \frac{1}{2}\phi(n) & \mathrm {for}\ n=2^{\alpha}\ \mathrm {and}\ \alpha\ge 3\\ \mathrm{lcm} (\lambda(p_1^{\alpha_1}), \lambda(p_2^{\alpha_2}), \ldots, \lambda(p_k^{\alpha_k})) & \mathrm{for}\ \mathrm{all}\ n.

\end{cases}$</p></center>}$ (Error compiling LaTeX. ! Extra }, or forgotten $.)

### 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.*