states that if p is prime and a is not divisible by p, then If one wants to test whether p is prime, then we can pick random integers a not divisible by p and see whether the equality holds. If the equality does not hold for a value of a, then p is composite. This congruence is unlikely to hold for a random a if p is composite. Therefore, if the equality does hold for one or more values of a, then we say that p is probably prime. However, note that for, the above congruence holds trivially. It also holds trivially if p is odd and. For this reason, one usually chooses a number a in the interval. Any a such that when n is composite a is known as a Fermat liar. In this case n is called Fermat pseudoprime to base a. If we do pick an a such that then a is known as a Fermat witness for the compositeness of n.
Example
Suppose we wish to determine whether n = 221 is prime. Randomly pick 1 < a < 221, say a = 38. We check the above equality and find that it holds: Either 221 is prime, or 38 is a Fermat liar, so we take another a, say 24: So 221 is composite and 38 was indeed a Fermat liar. Furthermore, 24 is a Fermat witness for the compositeness of 221.
Algorithm
The algorithm can be written as follows: The a values 1 and n-1 are not used as the equality holds for all n and all odd n respectively, hence testing them adds no value.
First, there are infinitely many Fermat pseudoprimes. A more serious flaw is that there are infinitely many Carmichael numbers. These are numbers for which all values of with are Fermat liars. For these numbers, repeatedapplication of the Fermat primality test performs the same as a simple random search for factors. While Carmichael numbers are substantially rarer than prime numbers there are enough of them that Fermat's primality test is not often used in the above form. Instead, other more powerful extensions of the Fermat test, such as Baillie–PSW, Miller–Rabin, and Solovay–Strassen are more commonly used. In general, if is a composite number that is not a Carmichael number, then at least half of all are Fermat witnesses. For proof of this, let be a Fermat witness and,,..., be Fermat liars. Then and so all for are Fermat witnesses.
Applications
As mentioned above, most applications use a Miller–Rabin or Baillie–PSW test for primality. Sometimes a Fermat test is performed first to improve performance. GMP since version 3.0 uses a base-210 Fermat test after trial division and before running Miller–Rabin tests. Libgcrypt uses a similar process with base 2 for the Fermat test, but OpenSSL does not. In practice with most big number libraries such as GMP, the Fermat test is not noticeably faster than a Miller–Rabin test, and can be slower for many inputs. As an exception, OpenPFGW uses only the Fermat test for probable prime testing. The program is typically used with multi-thousand digit inputs with a goal of maximum speed with very large inputs. Another well known program that relies only on the Fermat test is PGP where it is only used for testing of self-generated large random values.
