Primality proof for n = 13043:

Take b = 2.

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

6521 is prime.
b^((n-1)/6521)-1 mod n = 3, which is a unit, inverse 4348.

(6521) divides n-1.

(6521)^2 > n.

n is prime by Pocklington's theorem.