Primality proof for n = 4348819:

Take b = 2.

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

241601 is prime.
b^((n-1)/241601)-1 mod n = 262143, which is a unit, inverse 517476.

(241601) divides n-1.

(241601)^2 > n.

n is prime by Pocklington's theorem.