Primality proof for n = 8081:

Take b = 2.

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

101 is prime.
b^((n-1)/101)-1 mod n = 6309, which is a unit, inverse 7137.

(101) divides n-1.

(101)^2 > n.

n is prime by Pocklington's theorem.