Primality proof for n = 39504432196131312905021:

Take b = 2.

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

18141104598658771 is prime.
b^((n-1)/18141104598658771)-1 mod n = 33455913209282560998740, which is a unit, inverse 38874397261369135798588.

(18141104598658771) divides n-1.

(18141104598658771)^2 > n.

n is prime by Pocklington's theorem.