Primality proof for n = 17923:
Take b = 2.
b^(n-1) mod n = 1.
103 is prime.
b^((n-1)/103)-1 mod n = 8786, which is a unit, inverse 1634.
29 is prime.
b^((n-1)/29)-1 mod n = 5957, which is a unit, inverse 3102.
(29 * 103) divides n-1.
(29 * 103)^2 > n.
n is prime by Pocklington's theorem.