Carmichael function

From AoPSWiki

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 $2^{\alpha-1}$ .