Primality proof for n = 10882031:

Take b = 2.

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

12227 is prime.
b^((n-1)/12227)-1 mod n = 1661812, which is a unit, inverse 5160873.

(12227) divides n-1.

(12227)^2 > n.

n is prime by Pocklington's theorem.