Primality proof for n = 8877889:

Take b = 2.

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

15413 is prime.
b^((n-1)/15413)-1 mod n = 7451399, which is a unit, inverse 3668240.

(15413) divides n-1.

(15413)^2 > n.

n is prime by Pocklington's theorem.