Primality proof for n = 2773320623:
Take b = 2.
b^(n-1) mod n = 1.
569003 is prime. b^((n-1)/569003)-1 mod n = 2039232612, which is a unit, inverse 2440407146.
(569003) divides n-1.
(569003)^2 > n.
n is prime by Pocklington's theorem.