Primality proof for n = 114430077720284863765762092766225391679591020381:

Take b = 2.

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

190455129210693390224369129 is prime.
b^((n-1)/190455129210693390224369129)-1 mod n = 107972223486331334895858625313247482326361346609, which is a unit, inverse 99463431654920979124871871238534949953466966039.

(190455129210693390224369129) divides n-1.

(190455129210693390224369129)^2 > n.

n is prime by Pocklington's theorem.