Primality proof for n = 816170061203776523993317:

Take b = 2.

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

11716083324473119 is prime.
b^((n-1)/11716083324473119)-1 mod n = 758727006810694901801173, which is a unit, inverse 270721868540983870126083.

(11716083324473119) divides n-1.

(11716083324473119)^2 > n.

n is prime by Pocklington's theorem.