Primality proof for n = 7057776167979264311:

Take b = 2.

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

2663886769 is prime.
b^((n-1)/2663886769)-1 mod n = 4160691322857289275, which is a unit, inverse 833551685794679818.

(2663886769) divides n-1.

(2663886769)^2 > n.

n is prime by Pocklington's theorem.