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 ⁠${\displaystyle b}$⁠.^[1] The relation may also be expressed^[2] in the form:

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

for all integers ${\displaystyle b}$ that are relatively prime to ⁠${\displaystyle n}$⁠. They are infinite in number.^[3]

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.^[4]

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

The Carmichael numbers were named after the American mathematician Robert Carmichael by Nicolaas Beeger, in 1950. Øystein Ore had referred to them in 1948 as numbers with the "Fermat property", or "F numbers" for short.^[5]