Primality proof for n = 421:

Take b = 2.

b^(n-1) mod n = 1.

7 is prime.
b^((n-1)/7)-1 mod n = 369, which is a unit, inverse 170.

5 is prime.
b^((n-1)/5)-1 mod n = 278, which is a unit, inverse 368.

(5 * 7) divides n-1.

(5 * 7)^2 > n.

n is prime by Pocklington's theorem.