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.