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.