Primality proof for n = 166823:

Take b = 2.

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

349 is prime.
b^((n-1)/349)-1 mod n = 122817, which is a unit, inverse 148964.

239 is prime.
b^((n-1)/239)-1 mod n = 85940, which is a unit, inverse 120804.

(239 * 349) divides n-1.

(239 * 349)^2 > n.

n is prime by Pocklington's theorem.