In number theory, a branch of mathematics, the Carmichael function of a positive integer is the smallest positive integer such that holds for every integer coprime to . In algebraic terms, is the exponent of the multiplicative group of integers modulo . The Carmichael function is named after the American mathematician Robert Carmichael who defined it in 1910. 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 with Euler's totient function (in bold if they are different; the s such that they are different are listed in ). Numerical examples Carmichael's function at 5 is 4, , because for any number coprime to 5, i.e. there is with namely, , , and . And this is the smallest exponent with this property, because (and as well.)Moreover, Euler's totient function at 5 is 4, , because there are exactly 4 numbers less than and coprime to 5 (1, 2, 3, and 4). Euler's theorem assures that for all coprime to 5, and 4 is the smallest such exponent. Carmichael's function at 8 is 2, , because for any number coprime to 8, i.e. it holds that . Namely, , , and .Euler's totient function at 8 is 4, , because there are exactly 4 numbers less than and coprime to 8 (1, 3, 5, and 7). Moreover, Euler's theorem assures that for all coprime to 8, but 4 is not the smallest such exponent. Computing with Carmichael's theorem By the unique factorization theorem, any can be written in a unique way as where are primes and are positive integers. Then is the least common multiple of the of each of its prime power factors: This can be proved using the Chinese remainder theorem. Carmichael's theorem explains how to compute of a prime power : for a power of an odd prime and for 2 and 4, is equal to the Euler totient ; for powers of 2 greater than 4 it is equal to half of the Euler totient: Euler's function for prime powers is given by Properties of the Carmichael function In this section, an integer is divisible by a nonzero integer if there exists an integer such that . This is written as Order of elements modulo Let and be coprime and let be the smallest exponent with , then it