Primality proof for n = 3415219:

Take b = 2.

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

13883 is prime.
b^((n-1)/13883)-1 mod n = 2028732, which is a unit, inverse 2683604.

(13883) divides n-1.

(13883)^2 > n.

n is prime by Pocklington's theorem.