Primality proof for n = 7240555601:

Take b = 2.

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

15173 is prime.
b^((n-1)/15173)-1 mod n = 4761324069, which is a unit, inverse 3376787645.

1193 is prime.
b^((n-1)/1193)-1 mod n = 5989585659, which is a unit, inverse 4568713278.

(1193 * 15173) divides n-1.

(1193 * 15173)^2 > n.

n is prime by Pocklington's theorem.