A Carmichael number is named after Robert Carmichael [here]. It is a composite number \(n\) (a number that is not a prime number) that is defined as:
\(a^{n-1} \equiv 1 \pmod {n}\)
for all integers \(a\) which are relatively prime to \(n\). The smallest Carmichael number is 561, and which is made up of \(3 \times 11 \times 17\). In fact, every Carmichael number is made up of at least three factors. If we try 561, we cannot use a value which a factor of 3, 11 or 17. So if we try \(a=4\), we get \(4^{560} \pmod {561}\) and which equals 1. A value of \(a=8\) will also work as its factors are 2. But \(a=6\) will not work as it shares a factor of 3 with 561. Overall, Carmicheal numbers are numbers which will pass the Fermat test for primality: \(a^{n-1} \equiv 1 \pmod {n}\)