Primality proof for n = 34367:

Take b = 2.

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

17183 is prime.
b^((n-1)/17183)-1 mod n = 3, which is a unit, inverse 11456.

(17183) divides n-1.

(17183)^2 > n.

n is prime by Pocklington's theorem.