We all know Wilson's theorem :
$n$ is prime if $\frac {\Gamma(n)+1}{n}$ is an integer .
We can call it a prime checking method .
There are other various trivial methods .
The main thing to notice about this method is that $\Gamma$ is analytic , but also inefficient ( as grows very rapidly )
Are there any similar method with analytic flavor to it which can be more efficient than this one ?