Primality proof for n = 1609403:

Take b = 2.

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

593 is prime.
b^((n-1)/593)-1 mod n = 518584, which is a unit, inverse 1355953.

59 is prime.
b^((n-1)/59)-1 mod n = 860232, which is a unit, inverse 488642.

(59 * 593) divides n-1.

(59 * 593)^2 > n.

n is prime by Pocklington's theorem.