Primality proof for n = 4244885885521:

Take b = 2.

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

34871 is prime.
b^((n-1)/34871)-1 mod n = 1757770627752, which is a unit, inverse 1463581284075.

97 is prime.
b^((n-1)/97)-1 mod n = 3297520073297, which is a unit, inverse 3374400983609.

(97 * 34871) divides n-1.

(97 * 34871)^2 > n.

n is prime by Pocklington's theorem.