# Carmichael function

## Function in mathematical number theory / From Wikipedia, the free encyclopedia

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In number theory, a branch of mathematics, the **Carmichael function** *λ*(*n*) of a positive integer n is the smallest member of the set of positive integers m having the property 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 following table compares the first 36 values of *λ*(*n*) (sequence A002322 in the OEIS) with Euler's totient function φ (in **bold** if they are different; the ns such that they are different are listed in OEIS: A033949).

**More information**n, λ(n) ...

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 |