Primality proof for n = 11716083324473119:

Take b = 2.

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

22842497 is prime.
b^((n-1)/22842497)-1 mod n = 7325182935956143, which is a unit, inverse 6570772928389180.

2084989 is prime.
b^((n-1)/2084989)-1 mod n = 2360467097553465, which is a unit, inverse 4799401598133071.

(2084989 * 22842497) divides n-1.

(2084989 * 22842497)^2 > n.

n is prime by Pocklington's theorem.