Primality proof for n = 409:

Take b = 2.

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

17 is prime.
b^((n-1)/17)-1 mod n = 35, which is a unit, inverse 187.

3 is prime.
b^((n-1)/3)-1 mod n = 354, which is a unit, inverse 290.

(3 * 17) divides n-1.

(3 * 17)^2 > n.

n is prime by Pocklington's theorem.