Primality proof for n = 82163:

Take b = 2.

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

41081 is prime.
b^((n-1)/41081)-1 mod n = 3, which is a unit, inverse 27388.

(41081) divides n-1.

(41081)^2 > n.

n is prime by Pocklington's theorem.