Primality proof for n = 8097347:

Take b = 2.

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

4048673 is prime.
b^((n-1)/4048673)-1 mod n = 3, which is a unit, inverse 2699116.

(4048673) divides n-1.

(4048673)^2 > n.

n is prime by Pocklington's theorem.