Primality proof for n = 411679:

Take b = 2.

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

22871 is prime.
b^((n-1)/22871)-1 mod n = 262143, which is a unit, inverse 346308.

(22871) divides n-1.

(22871)^2 > n.

n is prime by Pocklington's theorem.