Primality proof for n = 72371792138339:

Take b = 2.

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

27213313 is prime.
b^((n-1)/27213313)-1 mod n = 4466101998437, which is a unit, inverse 66153038314064.

(27213313) divides n-1.

(27213313)^2 > n.

n is prime by Pocklington's theorem.