Carmichael function
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
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).
Use in cryptography[edit]
The Carmichael function is important in cryptography due to its use in the RSA encryption algorithm.