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.