Primality proof for n = 288467:

Take b = 2.

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

6271 is prime.
b^((n-1)/6271)-1 mod n = 61076, which is a unit, inverse 100323.

(6271) divides n-1.

(6271)^2 > n.

n is prime by Pocklington's theorem.