Take b = 2.

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

27301809421 is prime. b^((n-1)/27301809421)-1 mod n = 63, which is a unit, inverse 59803963494.

(27301809421) divides n-1.

(27301809421)^2 > n.

n is prime by Pocklington's theorem.