Take b = 2.

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

110244005101 is prime. b^((n-1)/110244005101)-1 mod n = 173878787309271, which is a unit, inverse 100905488817268.

(110244005101) divides n-1.

(110244005101)^2 > n.

n is prime by Pocklington's theorem.