Proving $11! + 1$ is prime

Question

Prove that:

$$11! + 1$$ is a prime number. Without computing the number (or factorial).

Obviously, from Wilson's theorem, a number $n$ is prime if,

$$(n-1)! + 1 \equiv 0 \pmod{n}$$

Since $n = 11! + 1 \in \mathbb{N}$, it is prime iff

$$(11!)! + 1 \equiv 0 \pmod{11! + 1}$$

I have a problem here, how do I use Wilson's theorem with factorials?

For a beginning,

Multiples of 11:

$$11, 22$$

$11! = 11*10*9...2*1 = 22*10!$

Next,

$$(11!)! = (22*10!)! $$

I need help at this point..

Answer 1

It can be shown to be prime using the Pratt certificate $g = 26$.

Specifically $p = 11!+1$ is prime if (A) $g^{11!} = 1 \pmod p$ and (B) $g^{11!/q} \not= 1 \pmod p$ for each $q$ in $2,3,5,7,11$.

We would expect both these facts to hold for any prime p by Fermats little theorem and existence of primitive roots. Condition (A) is easily verified.

With some computation it is found that $13$ and $26$ are the first two numbers $g$ that satisfy (B) with $q=2$, but $13$ is quickly ruled out as $13^{11!/3} = 1 \pmod p$. Condition (B) can be verified quickly with $g=26$ for all $q$ proving that $p$ is prime.