Primality proof for n = 1869236796843064056413:

Take b = 2.

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

5195037399650551 is prime.
b^((n-1)/5195037399650551)-1 mod n = 815890344222225492706, which is a unit, inverse 267328296273347899544.

(5195037399650551) divides n-1.

(5195037399650551)^2 > n.

n is prime by Pocklington's theorem.