Primality proof for n = 289450259140102639080071069:

Take b = 2.

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

479208894057626693 is prime.
b^((n-1)/479208894057626693)-1 mod n = 137428353945403922792031019, which is a unit, inverse 217927885242789779280994159.

(479208894057626693) divides n-1.

(479208894057626693)^2 > n.

n is prime by Pocklington's theorem.