Primality proof for n = 309473862136357:

Take b = 2.

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

25789488511363 is prime.
b^((n-1)/25789488511363)-1 mod n = 4095, which is a unit, inverse 255212022572522.

(25789488511363) divides n-1.

(25789488511363)^2 > n.

n is prime by Pocklington's theorem.