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.