Carmichael number

In number theory, a Carmichael number is a composite number ${\displaystyle n}$, which in modular arithmetic satisfies the congruence relation:

${\displaystyle b^{n}\equiv b{\pmod {n}}}$

for all integers ${\displaystyle b}$. The relation may also be expressed in the form:

${\displaystyle b^{n-1}\equiv 1{\pmod {n}}}$.

for all integers ${\displaystyle b}$ which are relatively prime to ${\displaystyle n}$. Carmichael numbers are named after American mathematician Robert Carmichael, the term having been introduced by Nicolaas Beeger in 1950 (Øystein Ore had referred to them in 1948 as numbers with the "Fermat property", or "F numbers" for short). They are infinite in number.

They constitute the comparatively rare instances where the strict converse of Fermat's Little Theorem does not hold. This fact precludes the use of that theorem as an absolute test of primality.

The Carmichael numbers form the subset K₁ of the Knödel numbers.