Primality proof for n = 479208894057626693:

Take b = 2.

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

119802223514406673 is prime.
b^((n-1)/119802223514406673)-1 mod n = 15, which is a unit, inverse 415314374849943134.

(119802223514406673) divides n-1.

(119802223514406673)^2 > n.

n is prime by Pocklington's theorem.