Primality proof for n = 715138273065985889:

Take b = 2.

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

24004372753289 is prime.
b^((n-1)/24004372753289)-1 mod n = 618721898860463124, which is a unit, inverse 55506206237878790.

(24004372753289) divides n-1.

(24004372753289)^2 > n.

n is prime by Pocklington's theorem.