Primality proof for n = 8779:
Take b = 2.
b^(n-1) mod n = 1.
19 is prime.
b^((n-1)/19)-1 mod n = 8557, which is a unit, inverse 8344.
11 is prime.
b^((n-1)/11)-1 mod n = 8768, which is a unit, inverse 798.
(11 * 19) divides n-1.
(11 * 19)^2 > n.
n is prime by Pocklington's theorem.