Primality proof for n = 181913:

Take b = 2.

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

22739 is prime.
b^((n-1)/22739)-1 mod n = 255, which is a unit, inverse 9274.

(22739) divides n-1.

(22739)^2 > n.

n is prime by Pocklington's theorem.