Primality proof for n = 11243:
Take b = 2.
b^(n-1) mod n = 1.
73 is prime.
b^((n-1)/73)-1 mod n = 9601, which is a unit, inverse 1075.
11 is prime.
b^((n-1)/11)-1 mod n = 7610, which is a unit, inverse 9903.
(11 * 73) divides n-1.
(11 * 73)^2 > n.
n is prime by Pocklington's theorem.