Primality proof for n = 109829:

Take b = 2.

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

27457 is prime.
b^((n-1)/27457)-1 mod n = 15, which is a unit, inverse 7322.

(27457) divides n-1.

(27457)^2 > n.

n is prime by Pocklington's theorem.