Primality proof for n = 333814693:

Take b = 2.

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

6247 is prime.
b^((n-1)/6247)-1 mod n = 84522612, which is a unit, inverse 260870676.

73 is prime.
b^((n-1)/73)-1 mod n = 125608130, which is a unit, inverse 187263298.

(73 * 6247) divides n-1.

(73 * 6247)^2 > n.

n is prime by Pocklington's theorem.