Primality proof for n = 8423:

Take b = 2.

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

4211 is prime.
b^((n-1)/4211)-1 mod n = 3, which is a unit, inverse 2808.

(4211) divides n-1.

(4211)^2 > n.

n is prime by Pocklington's theorem.