Primality proof for n = 8336320627:

Take b = 2.

b^(n-1) mod n = 1.

1389386771 is prime.
b^((n-1)/1389386771)-1 mod n = 63, which is a unit, inverse 1455548046.

(1389386771) divides n-1.

(1389386771)^2 > n.

n is prime by Pocklington's theorem.