Carmichael number

A Carmichael number is a composite number, who is named after the mathematician Robert Daniel Carmichael. A Carmichael number $\scriptstyle c\$ satisfies for every integer $\scriptstyle a\$ , that $\scriptstyle a^{c}-a\$ is divisible by $\scriptstyle c\$ . A Carmichael number c satisfies also the conrgruence $\scriptstyle a^{c-1}\equiv 1{\pmod {c}}$ , if $\scriptstyle \operatorname {gcd} (a,c)=1$ . In 1994 Pomerance, Alford and Granville proved that there exist infinitely many Carmichael numbers.

Properties of a Carmichael number

Every Carmichael number is squarefree and has at least three different prime factors
For every Carmichael number c is true, that $c-1$ is divisible by $p_{n}-1$ for every of its prime factors $p_{n}$ .
Every Carmichael number is an Euler pseudoprime.
Every absolute Euler pseudoprime is a Carmichael number.

Chernicks Carmichael numbers

J. Chernick found in 1939 a way to construct Carmichael numbers^[1]. If, for a natural number n, the three numbers $\scriptstyle 6n+1\$ , $\scriptstyle 12n+1\$ and $\scriptstyle 18n+1\$ are prime numbers, the product $\scriptstyle (6n+1)\cdot (12n+1)\cdot (18n+1)$ is a Carmichael number. Equivalent to this is that if $\scriptstyle m\$ , $\scriptstyle 2m-1\$ and $\scriptstyle 3m-2$ are prime numbers, then the product $\scriptstyle m\cdot (2m-1)\cdot (3m-2)$ is a Carmichael number.