Take b = 2.

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

84178317913 is prime.
b^((n-1)/84178317913)-1 mod n = 10767698974475, which is a unit, inverse 33565861275020.

(84178317913) divides n-1.

(84178317913)^2 > n.

n is prime by Pocklington's theorem.