Primality proof for n = 719569513687:

Take b = 2.

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

3686357 is prime.
b^((n-1)/3686357)-1 mod n = 150719717655, which is a unit, inverse 45408564781.

(3686357) divides n-1.

(3686357)^2 > n.

n is prime by Pocklington's theorem.