Primality proof for n = 213441916511:

Take b = 2.

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

292386187 is prime.
b^((n-1)/292386187)-1 mod n = 47282696164, which is a unit, inverse 209712463866.

(292386187) divides n-1.

(292386187)^2 > n.

n is prime by Pocklington's theorem.