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Difference between revisions of "Carmichael function"

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There are two different [[function]]s that are both called '''Carmichael function'''.  Both are similar to [[Euler's totient function]] <math>\phi</math>.
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There are two different [[function]]s called the '''Carmichael function'''.  Both are similar to [[Euler's totient function]] <math>\phi</math>.
  
 
== First Definition ==
 
== First Definition ==

Revision as of 22:18, 16 March 2008

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

First Definition

The Carmichael function $\lambda$ is defined at $n$ to be the smallest positive integer $\lambda(n)$ such that $a^{\lambda(n)} \equiv 1\pmod {n}$ for all positive integers $a$ relatively prime to $n$. The order of $a\pmod {n}$ always divides $\lambda(n)$.

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


Suppose $n=p_1^{\alpha_1}\cdot p_2^{\alpha_2}\cdots p_k^{\alpha_k}$. We have

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

Examples

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Second Definition

The second definition of the Carmichael function is the least common multiples of all the factors of $\phi(n)$. It is written as $\lambda'(n)$. However, in the case $8|n$, we take $2^{\alpha-2}$ as a factor instead of $2^{\alpha-1}$.

Examples

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See also

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