A pseudoprime is a probable prime (an integer that shares a property common to all prime numbers) which is not actually prime. Pseudoprimes can be classified according to which property they satisfy.
Pseudoprimes are of primary importance in public-key cryptography, which makes use of the difficulty of factoring large numbers into their prime factors. Carl Pomerance estimated in 1988 that it would cost $10 million to factor a number with 144 digits, and $100 billion to factor a 200-digit number.[1] However, finding and factoring the proper prime numbers for this use is correspondingly expensive, so various shortcuts are used to find primes amongst large numbers, some of which incorrectly identify composite numbers as prime in rare cases. Some primality tests, such as the AKS primality test work well on all numbers, so there are no pseudoprimes with respect to them.
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. Some sources use variations of this definition, for example to only allow odd numbers to be pseudoprimes.[2]
An integer x that is a Fermat pseudoprime to all values of a that are coprime to x is called a Carmichael number.