Primality proof for n = 5991180651865208777371442029237375648904065304322611579:

Take b = 2.

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

298407043007230668716584089637 is prime.
b^((n-1)/298407043007230668716584089637)-1 mod n = 1914247336552663614035981549126475133752444325780677352, which is a unit, inverse 3402621935471121824054329726406801895518948557337526673.

(298407043007230668716584089637) divides n-1.

(298407043007230668716584089637)^2 > n.

n is prime by Pocklington's theorem.