Carmichael Function

Inputs

Integer

n

A positive integer, arbitrarily large.

Factorization difficulty depends on the size of the factors, not just the size of the input. The calculator may return a partial factorization for difficult inputs. Computation runs locally in your browser.

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Result

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Prime factorization

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How it works

$\lambda(n)$ is the smallest positive integer $m$ such that $a^{m}\equiv 1 \pmod n$ for every $a$ coprime to $n$ . It is the exponent of the group $(\mathbb{Z}/n\mathbb{Z})^{\times}$ , and it always divides $\varphi(n)$ — the two are not the same function in general.

For odd prime powers $\lambda(p^{a})=\varphi(p^{a})$ ; for powers of 2,

\lambda(2^{a})=\begin{cases}1, & a=1\\ 2, & a=2\\ 2^{a-2}, & a\ge 3.\end{cases}

For $n=p_1^{a_1}\cdots p_r^{a_r}$ , the prime-power values are combined with the least common multiple:

\lambda(n)=\operatorname{lcm}\left(\lambda(p_1^{a_1}),\dots,\lambda(p_r^{a_r})\right).

$\lambda(1)=1$ .

Inputs

Result

Prime factorization

λ(n) — Carmichael Function

How it works

$\lambda(n)$ — Carmichael Function