![carmichaelLambda](https://upload.wikimedia.org/wikipedia/commons/3/3d/CarmichaelLambda.svg)
In
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mat ...
, a branch of
mathematics, the Carmichael function of a
positive integer is the smallest positive integer such that
:
holds for every integer
coprime
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
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
Robert Daniel Carmichael (March 1, 1879 – May 2, 1967) was an American mathematician.
Biography
Carmichael was born in Goodwater, Alabama. He attended Lineville College, briefly, and he earned his bachelor's degree in 1898, while he was s ...
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