Primality proof for n = 766223:

Take b = 2.

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

16657 is prime.
b^((n-1)/16657)-1 mod n = 9968, which is a unit, inverse 581816.

(16657) divides n-1.

(16657)^2 > n.

n is prime by Pocklington's theorem.