# Carmichael function

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

## First Definition

$\boxed{The Carmichael function$ (Error compiling LaTeX. ! File ended while scanning use of \boxed.)\lambda$is defined at$n$to be the smallest [[positive integer]]$\lambda(n)$such that$a^{\lambda(n)} \equiv 1\pmod {n}$for all positive [[integer]]s$a$[[relatively prime]] to$n$. The [[order]] of$a\pmod {n}$always divides$\lambda(n)$. This function is also known as the ''reduced totient function'' or the ''least universal exponent'' function. Suppose$ (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

Evaluate $2009^{2009}\pmod{1000}$. [1]

## 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 $2^{\alpha-1}$.