Carmichaelgetal

A Carmichael number is a composite number n which satisfies the modular arithmetic congruence relation b^(n-1) = 1 (mod n) for all integers b which are relatively prime to n. In other words, a composite number n is a Carmichael number if it is a Fermat pseudoprime to every base a with gcd(a, n) = 1.

The smallest Carmichael number is 561. The next ones are 1105, 1729, 2465, 2821, 6601, and 8911. Carmichael numbers are very rare. There are infinitely many Carmichael numbers, but their density is very low.

Carmichael numbers are named after the American mathematician Robert Carmichael, who described them in 1910. However, they were known before Carmichael. In 1899, the English mathematician J. J. Sylvester mentioned them and gave an example, 1729.

A composite integer n is a Carmichael number if and only if it is square-free, and for all prime divisors p of n, it is true that p-1 divides n-1. This property is known as Korselt's criterion, named after the Danish mathematician Axel Thue Korselt, who stated it in 1899.

Carmichael numbers are important in number theory and cryptography because they are composite numbers that behave like prime numbers in the context of Fermat's Little Theorem. This means that they can fool primality tests that rely on this theorem, such as the Fermat primality test.