Primality proof for n = 443:
Take b = 2.
b^(n-1) mod n = 1.
17 is prime.
b^((n-1)/17)-1 mod n = 122, which is a unit, inverse 69.
13 is prime.
b^((n-1)/13)-1 mod n = 34, which is a unit, inverse 430.
(13 * 17) divides n-1.
(13 * 17)^2 > n.
n is prime by Pocklington's theorem.