Primality proof for n = 21143:

Take b = 2.

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

31 is prime.
b^((n-1)/31)-1 mod n = 6040, which is a unit, inverse 17618.

(31^2) divides n-1.

(31^2)^2 > n.

n is prime by Pocklington's theorem.