Primality proof for n = 183429412511632473752903671:

Take b = 2.

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

6114313750387749125096789 is prime.
b^((n-1)/6114313750387749125096789)-1 mod n = 1073741823, which is a unit, inverse 73639481190466771590872284.

(6114313750387749125096789) divides n-1.

(6114313750387749125096789)^2 > n.

n is prime by Pocklington's theorem.