Primality proof for n = 1176529195798022443468570606581645973:

Take b = 2.

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

3625906222327272823 is prime.
b^((n-1)/3625906222327272823)-1 mod n = 479344850446506592468932476099613208, which is a unit, inverse 180917623039453839019773823233037933.

(3625906222327272823) divides n-1.

(3625906222327272823)^2 > n.

n is prime by Pocklington's theorem.