Definition:Carmichael Number

Definition
An 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 $n$ which satisfies $a^n \equiv a \pmod n$ for all integers $a$ which are coprime to it.

Also defined as
Some sources insist that the definition of a Carmichael number presupposes that $n$ is odd, but this follows from Korselt's Theorem so it is not necessary to state this.

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.

Because this property holds for all $a$ coprime to $n$, it is also referred to as an absolute pseudoprime.

Also see

 * Korselt's Theorem


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