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.