Primality proof for n = 9535423:
Take b = 2.
b^(n-1) mod n = 1.
2843 is prime.
b^((n-1)/2843)-1 mod n = 2032800, which is a unit, inverse 1672325.
43 is prime.
b^((n-1)/43)-1 mod n = 1227293, which is a unit, inverse 3088149.
(43 * 2843) divides n-1.
(43 * 2843)^2 > n.
n is prime by Pocklington's theorem.