Primality proof for n = 19603:
Take b = 2.
b^(n-1) mod n = 1.
11 is prime.
b^((n-1)/11)-1 mod n = 3727, which is a unit, inverse 5002.
3 is prime.
b^((n-1)/3)-1 mod n = 9870, which is a unit, inverse 16312.
(3^4 * 11^2) divides n-1.
(3^4 * 11^2)^2 > n.
n is prime by Pocklington's theorem.