Primality proof for n = 81371:
Take b = 2.
b^(n-1) mod n = 1.
103 is prime.
b^((n-1)/103)-1 mod n = 27717, which is a unit, inverse 320.
79 is prime.
b^((n-1)/79)-1 mod n = 76290, which is a unit, inverse 4340.
(79 * 103) divides n-1.
(79 * 103)^2 > n.
n is prime by Pocklington's theorem.