Primality proof for n = 15467:
Take b = 2.
b^(n-1) mod n = 1.
37 is prime.
b^((n-1)/37)-1 mod n = 13418, which is a unit, inverse 11270.
11 is prime.
b^((n-1)/11)-1 mod n = 12092, which is a unit, inverse 3680.
(11 * 37) divides n-1.
(11 * 37)^2 > n.
n is prime by Pocklington's theorem.