Definition
Fermat's little theorem states that if p is prime and a is coprime to p, then ap−1 − 1 is divisible by p. If a composite integer x is coprime to an integer a > 1 and x divides ax−1 − 1, then x is called a Fermat pseudoprime to base a. In other words, a composite integer is a Fermat pseudoprime to base a if it successfully passes Fermat primality test for the base a.
The smallest base-2 Fermat pseudoprime is 341. It is not a prime, since it equals 11·31, but it satisfies Fermat's little theorem: 2340 ≡ 1 (mod 341) and thus passes Fermat primality test for the base 2.
Pseudoprimes to base 2 are sometimes called Poulet numbers, Sarrus numbers, or Fermatians (sequence A001567 in OEIS).
An integer x that is a Fermat pseudoprime for all values of a that are coprime to x is called a Carmichael number.
Read more about this topic: Fermat Pseudoprime
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