Question related to pseudoprimes and Carmichael numbers We know by Fermat's little theorem that if $m$ is a prime then for any $n \in \mathbb{Z}$, then 
$$
n^m \equiv n (mod \ m).
$$
I was wondering if there was a composite $m$ that satisfies this condition?
(Maybe this condition is equivalent to being a Carmichael number, but I wasn't sure...)
Thank you!

Sure there is. It's a Carmichael number by definition. $561$ is the smallest Carmichael number.
