Primality proof for n = 1109324011:

Take b = 2.

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

10303 is prime.
b^((n-1)/10303)-1 mod n = 390703367, which is a unit, inverse 324371717.

97 is prime.
b^((n-1)/97)-1 mod n = 696175262, which is a unit, inverse 491306362.

(97 * 10303) divides n-1.

(97 * 10303)^2 > n.

n is prime by Pocklington's theorem.