Primality proof for n = 412771:

Take b = 2.

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

13759 is prime.
b^((n-1)/13759)-1 mod n = 124452, which is a unit, inverse 112453.

(13759) divides n-1.

(13759)^2 > n.

n is prime by Pocklington's theorem.