Primality proof for n = 8199883:

Take b = 2.

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

211 is prime.
b^((n-1)/211)-1 mod n = 6678338, which is a unit, inverse 5217736.

127 is prime.
b^((n-1)/127)-1 mod n = 5058040, which is a unit, inverse 1319579.

(127 * 211) divides n-1.

(127 * 211)^2 > n.

n is prime by Pocklington's theorem.