Primality proof for n = 816763:

Take b = 2.

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

691 is prime.
b^((n-1)/691)-1 mod n = 84713, which is a unit, inverse 110598.

197 is prime.
b^((n-1)/197)-1 mod n = 89027, which is a unit, inverse 288551.

(197 * 691) divides n-1.

(197 * 691)^2 > n.

n is prime by Pocklington's theorem.