Primality proof for n = 1110318119:
Take b = 2.
b^(n-1) mod n = 1.
12527 is prime.
b^((n-1)/12527)-1 mod n = 1065496752, which is a unit, inverse 584671775.
487 is prime.
b^((n-1)/487)-1 mod n = 989411564, which is a unit, inverse 922745957.
(487 * 12527) divides n-1.
(487 * 12527)^2 > n.
n is prime by Pocklington's theorem.