# Carmichael function

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

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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

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