Primality proof for n = 11369:

Take b = 2.

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

29 is prime.
b^((n-1)/29)-1 mod n = 1237, which is a unit, inverse 1397.

7 is prime.
b^((n-1)/7)-1 mod n = 5419, which is a unit, inverse 4068.

(7^2 * 29) divides n-1.

(7^2 * 29)^2 > n.

n is prime by Pocklington's theorem.