Primality proof for n = 8039:

Take b = 2.

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

4019 is prime.
b^((n-1)/4019)-1 mod n = 3, which is a unit, inverse 2680.

(4019) divides n-1.

(4019)^2 > n.

n is prime by Pocklington's theorem.