Primality proof for n = 9063623:

Take b = 2.

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

4531811 is prime.
b^((n-1)/4531811)-1 mod n = 3, which is a unit, inverse 3021208.

(4531811) divides n-1.

(4531811)^2 > n.

n is prime by Pocklington's theorem.