Primality proof for n = 3167286967475404111215579779960013343298694729431415913444417037:

Take b = 2.

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

9533836766459662372116970697 is prime.
b^((n-1)/9533836766459662372116970697)-1 mod n = 154318801853022251295015190968958956314004248316389551849264740, which is a unit, inverse 1996320642578129881446866087629616338888490509704543666681665061.

22153296627978863 is prime.
b^((n-1)/22153296627978863)-1 mod n = 741588322980934153901532593185948478864044125556208390474674811, which is a unit, inverse 2993177587335983237114720435137133936166162015115981714418626676.

(22153296627978863 * 9533836766459662372116970697) divides n-1.

(22153296627978863 * 9533836766459662372116970697)^2 > n.

n is prime by Pocklington's theorem.