Primality proof for n = 11290956913871:

Take b = 2.

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

66417393611 is prime.
b^((n-1)/66417393611)-1 mod n = 11005834445931, which is a unit, inverse 4247528171144.

(66417393611) divides n-1.

(66417393611)^2 > n.

n is prime by Pocklington's theorem.