Carmichael function

The Carmichael function in the field of Mathematics is a number-theoretic function that every natural number n to the smallest determined such that:

Is true for any A, which is relatively prime to n. In group-theoretic way of speaking is the group exponent of the residue class group.

The Carmichael function goes back to the mathematician Robert Daniel Carmichael. One meaning plays the function in Fermat primes and pseudo- primes.

Calculation

The Carmichael function can be calculated according to the following scheme:

Here are the different pairwise prime numbers and all positive numbers.

The following formula will produce the same result:

Be the prime factorization of ( with, if even):

• If
• If

It denotes the Euler φ - function. For powers of odd primes

Example

The Carmichael function and the Carmichael number

Since the Carmichael function to every natural number is the smallest determined, such that for each, which is to prime, and at any Carmichael number is divisible by, the following:

Also

For each there is a Carmichael number, such that for each. Take.

The Carmichael function and Euler's φ - function

The Carmichael function and Euler's φ - function are identical for the numbers one, two, four, for any odd prime power and for all doubles of odd prime powers. Just when, also exist primitive roots modulo. In general, both functions are different; however, is always a divisor of.

• Euler's φ - function:

• Carmichael function:

• Number theoretic function