Primality proof for n = 115263316429:

Take b = 2.

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

43462789 is prime.
b^((n-1)/43462789)-1 mod n = 94235078530, which is a unit, inverse 39838661826.

(43462789) divides n-1.

(43462789)^2 > n.

n is prime by Pocklington's theorem.