Primality proof for n = 17183:
Take b = 2.
b^(n-1) mod n = 1.
71 is prime.
b^((n-1)/71)-1 mod n = 2723, which is a unit, inverse 8683.
11 is prime.
b^((n-1)/11)-1 mod n = 9543, which is a unit, inverse 16569.
(11^2 * 71) divides n-1.
(11^2 * 71)^2 > n.
n is prime by Pocklington's theorem.