Primality proof for n = 211191412011185201:

Take b = 2.

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

347582969077 is prime.
b^((n-1)/347582969077)-1 mod n = 185527529140536649, which is a unit, inverse 131012102865490614.

(347582969077) divides n-1.

(347582969077)^2 > n.

n is prime by Pocklington's theorem.