Primality proof for n = 24004372753289:

Take b = 2.

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

1418023 is prime.
b^((n-1)/1418023)-1 mod n = 1885917427470, which is a unit, inverse 13701827413197.

124471 is prime.
b^((n-1)/124471)-1 mod n = 16991633004856, which is a unit, inverse 19954874831253.

(124471 * 1418023) divides n-1.

(124471 * 1418023)^2 > n.

n is prime by Pocklington's theorem.