Primality proof for n = 34381:

Take b = 2.

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

191 is prime.
b^((n-1)/191)-1 mod n = 2370, which is a unit, inverse 18264.

(191) divides n-1.

(191)^2 > n.

n is prime by Pocklington's theorem.