Primality proof for n = 743104567:
Take b = 2.
b^(n-1) mod n = 1.
11351 is prime.
b^((n-1)/11351)-1 mod n = 586497401, which is a unit, inverse 39408624.
3637 is prime.
b^((n-1)/3637)-1 mod n = 200893089, which is a unit, inverse 444174805.
(3637 * 11351) divides n-1.
(3637 * 11351)^2 > n.
n is prime by Pocklington's theorem.