Primality proof for n = 16183:
Take b = 2.
b^(n-1) mod n = 1.
31 is prime.
b^((n-1)/31)-1 mod n = 8886, which is a unit, inverse 2872.
3 is prime.
b^((n-1)/3)-1 mod n = 10972, which is a unit, inverse 1736.
(3^2 * 31) divides n-1.
(3^2 * 31)^2 > n.
n is prime by Pocklington's theorem.