Carmichael function

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

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

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

In other words, in more algebraic terms, it defines the exponent of the multiplicative group of residues modulo n.

1 Carmichael's theorem
2 Hierarchy of results
3 Properties of the Carmichael function
4 See also
5 References

[edit] Carmichael's theorem

This function can also be defined recursively, as follows.

For prime $p$ and positive integer $k$ such that $p \ge 3$ or $k \le 2$ :

$\lambda(p^k) = p^{k-1}(p-1).\,$ (This is equal to $\phi(p^k)\,$ )

For integer $k \ge 3$ ,

$\lambda(2^k) = 2^{k-2}\,$ .

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 $lcm$ denotes the least common multiple.

Carmichael's theorem states that if a is coprime to n, then

$a^{\lambda(n)} \equiv 1 \pmod{n}$ ,

where $λ$ is the Carmichael function defined recursively. In other words, it asserts the correctness of the recursion.

[edit] Hierarchy of results

The classical Euler's theorem implies that λ(n) divides φ(n), the Euler's totient function. In fact Carmichael's theorem is related to Euler's theorem, because the exponent of finite abelian group must divide the order of the group, by elementary group theory. The two functions differ already in small cases: λ(15) = 4 while φ(15) = 8.

Fermat's little theorem is the special case of Euler's theorem in which n is a prime number p. Carmichael's theorem for a prime p adds nothing to Fermat's theorem, because the group in question is a cyclic group for which the order and exponent are both p − 1.

[edit] Properties of the Carmichael function

[edit] Average and typical value

Theorem 3 in [1]: For any $x>16\,$ , and a constant $B \approx 0.34537$ :

$\frac{1}{x} \sum_{n \leq x} \lambda (n) = \frac{x}{\ln x} e^{\frac{B\ln\ln x}{\ln\ln\ln x} (1+o(1))}$ .

Theorem 2 in [1]: For all numbers $N$ and all except $o (N)$ positive integers $n \leq N$ :

$\lambda(n) = n / (\ln n)^{\ln\ln\ln n + A + o(1)}\,$

where $A$ is a constant, $A \approx 0.226969$ .

[edit] Lower bounds

Theorem 5 in [2]: For any sufficiently large number $N$ and for any $\Delta \geq (\ln\ln N)^3$ , there are at most

$Ne^{-0.69(\Delta\ln\Delta)^{1/3}}$

positive integers $n \leq N$ such that $\lambda(n) \leq ne^{-\Delta}$ .

Theorem 1 in [1]: For any sequence $n_1<n_2<n_3<\cdots$ of positive integers, any constant $0 < c < 1 / ln2$ , and any sufficiently large $i$ :

$\lambda(n_i) > (\ln n_i)^{c\ln\ln\ln n_i}$ .

[edit] Small values

Theorem 1 in [1]: For a constant $c$ and any sufficiently large positive $A$ , there exists an integer $n > A$ such that $λ(n) < (ln A) c lnlnln A$ . Moreover, $n$ is of the form

n =	∏	q
	(q − 1) \| m and q is prime

for some square-free integer $m < (ln A) c lnlnln A$ .

[edit] See also

Carmichael number

[edit] References

[1] Paul Erdős, Carl Pomerance, Eric Schmutz, Carmichael's lambda function, Acta Arithmetica, vol. 58, 363--385, 1991

[2] John Friedlander, Carl Pomerance, Igor E. Shparlinski, Period of the power generator and small values of the Carmichael function, Mathematics of Computation, vol. 70 no. 236, pp. 1591-1605, 2001

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Category: Modular arithmetic