Primality proof for n = 19813:

Take b = 2.

b^(n-1) mod n = 1.

127 is prime.
b^((n-1)/127)-1 mod n = 70, which is a unit, inverse 6510.

13 is prime.
b^((n-1)/13)-1 mod n = 9354, which is a unit, inverse 10292.

(13 * 127) divides n-1.

(13 * 127)^2 > n.

n is prime by Pocklington's theorem.