Primality proof for n = 343559:

Take b = 2.

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

9041 is prime.
b^((n-1)/9041)-1 mod n = 130192, which is a unit, inverse 91814.

(9041) divides n-1.

(9041)^2 > n.

n is prime by Pocklington's theorem.