Primality proof for n = 811:
Take b = 3.
b^(n-1) mod n = 1.
5 is prime.
b^((n-1)/5)-1 mod n = 211, which is a unit, inverse 123.
3 is prime.
b^((n-1)/3)-1 mod n = 679, which is a unit, inverse 43.
(3^4 * 5) divides n-1.
(3^4 * 5)^2 > n.
n is prime by Pocklington's theorem.