Why is 2 not a Carmichael Number?

Question

I know I'm missing something super basic here but:

From the definition in my textbook, a Carmichael is any number n such that $(\forall a)\; \gcd(a,n) = 1$:

$a^n \equiv a \pmod n$ or
$a^{n-1} \equiv 1 \pmod n$

So by this definition, how come $2$ is not a Carmichael number? $(\forall a) \; \gcd(a,2) = 1$ (i.e. $a$ is odd),

$a \equiv 1 \pmod 2$ or
$a^2 \equiv a \pmod 2 \equiv 1 \pmod 2$

But obviously $2$ isn't a Carmichael number, so I'm not sure where I went wrong.

Answer 1

By definition, Carmichael numbers are composite. Check your book. The point is that they are composite integers that behave arithmetically like primes ("pseudoprimes"). Thus they prove to be stumbling blocks for certain naive (probabilistic) primality tests, e.g. they are universal liars for the Fermat primality test.