Primality proof for n = 10111:

Take b = 2.

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

337 is prime.
b^((n-1)/337)-1 mod n = 4178, which is a unit, inverse 5888.

(337) divides n-1.

(337)^2 > n.

n is prime by Pocklington's theorem.