Primality proof for n = 347582969077:

Take b = 2.

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

10273 is prime.
b^((n-1)/10273)-1 mod n = 152760838303, which is a unit, inverse 301429130273.

6827 is prime.
b^((n-1)/6827)-1 mod n = 324937597743, which is a unit, inverse 108274241734.

(6827 * 10273) divides n-1.

(6827 * 10273)^2 > n.

n is prime by Pocklington's theorem.