Primality proof for n = 9347238719917243:

Take b = 2.

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

4244885885521 is prime.
b^((n-1)/4244885885521)-1 mod n = 221639729211885, which is a unit, inverse 7096897053525091.

(4244885885521) divides n-1.

(4244885885521)^2 > n.

n is prime by Pocklington's theorem.