Carmichael number

In number theory, a Carmichael number is a composite number n {displaystyle n} which satisfies the modular arithmetic congruence relation: In number theory, a Carmichael number is a composite number n {displaystyle n} which satisfies the modular arithmetic congruence relation: for all integers b {displaystyle b} which are relatively prime to n {displaystyle n} .They are named for Robert Carmichael.The Carmichael numbers are the subset K1 of the Knödel numbers. Equivalently, a Carmichael number is a composite number n {displaystyle n} for which for all integers b {displaystyle b} . Fermat's little theorem states that if p is a prime number, then for any integer b, the number bp − b is an integer multiple of p. Carmichael numbers are composite numbers which have this property. Carmichael numbers are also called Fermat pseudoprimes or absolute Fermat pseudoprimes. A Carmichael number will pass a Fermat primality test to every base b relatively prime to the number, even though it is not actually prime.This makes tests based on Fermat's Little Theorem less effective than strong probable prime tests such as the Baillie–PSW primality test and the Miller–Rabin primality test. However, no Carmichael number is either an Euler–Jacobi pseudoprime or a strong pseudoprime to every base relatively prime to itso, in theory, either an Euler or a strong probable prime test could prove that a Carmichael number is, in fact, composite. Arnaultgives a 397-digit Carmichael number N {displaystyle N} that is a strong pseudoprime to all prime bases less than 307: