Primality proof for n = 13309:

Take b = 2.

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

1109 is prime.
b^((n-1)/1109)-1 mod n = 4095, which is a unit, inverse 13296.

(1109) divides n-1.

(1109)^2 > n.

n is prime by Pocklington's theorem.