Primality proof for n = 111032963:

Take b = 2.

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

55516481 is prime.
b^((n-1)/55516481)-1 mod n = 3, which is a unit, inverse 37010988.

(55516481) divides n-1.

(55516481)^2 > n.

n is prime by Pocklington's theorem.