Primality proof for n = 39286345187279:
Take b = 2.
b^(n-1) mod n = 1.
904340159 is prime.
b^((n-1)/904340159)-1 mod n = 7887706830485, which is a unit, inverse 21596437816407.
(904340159) divides n-1.
(904340159)^2 > n.
n is prime by Pocklington's theorem.