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.