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.