Primality proof for n = 889917602181551:

Take b = 2.

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

278486521 is prime.
b^((n-1)/278486521)-1 mod n = 594923623531668, which is a unit, inverse 849903074800490.

(278486521) divides n-1.

(278486521)^2 > n.

n is prime by Pocklington's theorem.