Primality proof for n = 479216833:

Take b = 2.

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

5791 is prime.
b^((n-1)/5791)-1 mod n = 130839220, which is a unit, inverse 355834915.

431 is prime.
b^((n-1)/431)-1 mod n = 478388572, which is a unit, inverse 321309679.

(431 * 5791) divides n-1.

(431 * 5791)^2 > n.

n is prime by Pocklington's theorem.