Primality proof for n = 2221:
Take b = 2.
b^(n-1) mod n = 1.
37 is prime.
b^((n-1)/37)-1 mod n = 979, which is a unit, inverse 152.
5 is prime.
b^((n-1)/5)-1 mod n = 329, which is a unit, inverse 2194.
(5 * 37) divides n-1.
(5 * 37)^2 > n.
n is prime by Pocklington's theorem.