Primality proof for n = 34871:

Take b = 2.

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

317 is prime.
b^((n-1)/317)-1 mod n = 16150, which is a unit, inverse 10837.

(317) divides n-1.

(317)^2 > n.

n is prime by Pocklington's theorem.