Carmichael function

Carmichael $λ$ function: $λ (n)$ for $1 \leq n \leq 1000$ (compared to Euler $φ$ function)

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

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

holds for every integer $a$ coprime to $n$ . In algebraic terms, $λ (n)$ is the exponent of the multiplicative group of integers modulo $n$ . As this is a finite abelian group, there must exist an element whose order equals the exponent, $λ (n)$ . Such an element is called a primitive $λ$ -root modulo $n$ .

The Carmichael function is named after the American mathematician Robert Carmichael who defined it in 1910.^[1] It is also known as Carmichael's λ function, the reduced totient function, and the least universal exponent function.

The order of the multiplicative group of integers modulo $n$ is $φ (n)$ , where $φ$ is Euler's totient function. Since the order of an element of a finite group divides the order of the group, $λ (n)$ divides $φ (n)$ . The following table compares the first 36 values of $λ (n)$ (sequence A002322 in the OEIS) and $φ (n)$ (in bold if they are different; the $n$ s such that they are different are listed in OEIS: A033949).

$n$	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36
$λ (n)$	1	1	2	2	4	2	6	2	6	4	10	2	12	6	4	4	16	6	18	4	6	10	22	2	20	12	18	6	28	4	30	8	10	16	12	6
$φ (n)$	1	1	2	2	4	2	6	4	6	4	10	4	12	6	8	8	16	6	18	8	12	10	22	8	20	12	18	12	28	8	30	16	20	16	24	12

^ Carmichael, Robert Daniel (1910). "Note on a new number theory function". Bulletin of the American Mathematical Society. 16 (5): 232–238. doi:10.1090/S0002-9904-1910-01892-9.

[1] Carmichael, Robert Daniel (1910). "Note on a new number theory function". Bulletin of the American Mathematical Society. 16 (5): 232–238. doi:10.1090/S0002-9904-1910-01892-9.

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