A Carmichael number is a special type of composite number that satisfies certain mathematical properties, making it appear prime under specific tests. These numbers are significant because they can fool some primality tests, such as the Fermat primality test, which can incorrectly identify them as prime. Carmichael numbers are defined by the condition that for every prime divisor p of the number n , it holds that n \equiv 1 \mod p . The smallest Carmichael number is 561, and they are important in number theory and cryptography for understanding the limitations of certain algorithms.