Carmichael function

In number theory, the Charmichael function of a positive integer $n$ , denoted $\lambda (n)$ , is defined as the smallest integer $m$ such that

a^{m}\equiv 1{\pmod {n}}

for every integer $a$ that is coprime to $n$ .

This function can also be defined recursively, as follows.

For prime $p$ and positive integer $k$ such that $p\geq 3$ or $k\leq 2$ :

\lambda (p^{k})=p^{k-1}(p-1)

For integer $k\geq 3$ ,

\lambda (2^{k})=2^{k-2}

For distinct primes $p_{1},p_{2},\ldots ,p_{t}$ and positive integers $k_{1},k_{2},\ldots ,k_{t}$ :

\lambda (p_{1}^{k_{1}}p_{2}^{k_{2}}\cdots p_{t}^{k_{t}})=\mathrm {lcm} (\lambda (p_{1}^{k_{1}})\lambda (p_{2}^{k_{2}})\cdots \lambda (p_{t}^{k_{t}}))

where $\mathrm {lcm}$ denotes the least common denominator.

See Also