Definition:Carmichael Number

Definition
An odd integer $n > 0$ is a Carmichael number :
 * $(1): \quad n$ is composite
 * $(2): \quad \forall a \in \Z: a \perp n: a^n \equiv a \pmod n$, or, equivalently, that $a^{n - 1} \equiv 1 \pmod n$.

That is, a Carmichael number is a composite number which satisfies $a^n \equiv a \pmod n$ for all integers coprime to it.

Also known as
A Carmichael number is also referred to as a pseudoprime (or Fermat liar), as it exhibits the same properties as a prime when Fermat's Little Theorem is applied.

Properties
The characterization of Carmichael Numbers was given by in what is known as Korselt's Theorem, which states the following:

An odd integer $n>0$ is a Carmichael number both of the following conditions hold for each prime factor of $n$:
 * $(1): \quad p^2 \nmid n$
 * $(2): \quad \left({p - 1}\right) \mathrel \backslash \left({n - 1}\right)$

Also see

 * Definition:Riesel Number
 * Definition:Sierpiński Number of the Second Kind