Primality proof for n = 14221:
Take b = 2.
b^(n-1) mod n = 1.
79 is prime.
b^((n-1)/79)-1 mod n = 4927, which is a unit, inverse 8558.
5 is prime.
b^((n-1)/5)-1 mod n = 1229, which is a unit, inverse 7151.
(5 * 79) divides n-1.
(5 * 79)^2 > n.
n is prime by Pocklington's theorem.