# Carmichael number

In number theory a Carmichael number is a composite positive integer ${\displaystyle n}$, which satisfies the congruence ${\displaystyle b^{n-1}\equiv 1{\pmod {n}}}$ for all integers ${\displaystyle b}$ which are relatively prime to ${\displaystyle n}$. Being relatively prime means that they do not have common divisors, other than 1. Such numbers are named after Robert Carmichael.
All prime numbers ${\displaystyle p}$ satisfy ${\displaystyle b^{p-1}\equiv 1{\pmod {p}}}$ for all integers ${\displaystyle b}$ which are relatively prime to ${\displaystyle p}$. This has been proven by the famous mathematician Pierre de Fermat. In most cases if a number ${\displaystyle n}$ is composite, it does not satisfy this congruence equation. So, there exist not so many Carmichael numbers. We can say that Carmichael numbers are composite numbers that behave a little bit like they would be a prime number.