Fermat's primality test

Fermat's primality test is an algorithm. It can test if a given number p is probably prime. There is a flaw however: There are numbers that pass the test, and that are not prime. These numbers are called Carmichael numbers.

Concept 
Fermat's little theorem states that if p is prime and , then 

.

If we want to test if n is prime, then we can pick random a'''s in the interval and see if the equation above holds.  If the equality does not hold for a value of a, then n is composite (not prime).  If the equality does hold for many values of a, then we can say that n is probably prime, or a pseudoprime.

It may be in our tests that we do not pick any value for a such that the equality fails.  Any a such that 

when n is composite is known as a Fermat liar.  If we do pick an a such that 

then a is known as a Fermat witness for the compositeness of n.'' is the modulo operation. Its result is what remains, if p is divided by n. As an example,

.

What this test is used for
The RSA algorithm for public-key encryption can be done in such a way that it uses this test. This is useful in cryptography.

Category:Computer science
Category:Prime numbers