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. Unknown error_msg)\lambdan
\lambda(n)
a^{\lambda(n)} \equiv 1\pmod {n}
a
n
a\pmod {n}
\lambda(n)$.
This function is also known as the ''reduced totient function'' or the ''least universal exponent'' function.
Suppose$ (Error compiling LaTeX. Unknown error_msg)n=p_1^{\alpha_1}\cdot p_2^{\alpha_2}\cdots p_k^{\alpha_k}$. We have
<center><p>$ (Error compiling LaTeX. Unknown error_msg)\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. Unknown error_msg)
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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