Primality proof for n = 108598647457:

Take b = 2.

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

5113 is prime.
b^((n-1)/5113)-1 mod n = 15367512069, which is a unit, inverse 12768722298.

61 is prime.
b^((n-1)/61)-1 mod n = 67140502316, which is a unit, inverse 52606230643.

31 is prime.
b^((n-1)/31)-1 mod n = 7011395121, which is a unit, inverse 12728058339.

(31 * 61 * 5113) divides n-1.

(31 * 61 * 5113)^2 > n.

n is prime by Pocklington's theorem.