Primality proof for n = 11711184643015782903697616449:
Take b = 2.
b^(n-1) mod n = 1.
352576608953991537322303 is prime.
b^((n-1)/352576608953991537322303)-1 mod n = 313246917351246334821809670, which is a unit, inverse 11302977211897168811983564210.
(352576608953991537322303) divides n-1.
(352576608953991537322303)^2 > n.
n is prime by Pocklington's theorem.