Primality proof for n = 108140989558681:

Take b = 3.

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

23609 is prime.
b^((n-1)/23609)-1 mod n = 27458299228746, which is a unit, inverse 10679667100567.

1433 is prime.
b^((n-1)/1433)-1 mod n = 13658565625703, which is a unit, inverse 104633227636071.

(1433 * 23609) divides n-1.

(1433 * 23609)^2 > n.

n is prime by Pocklington's theorem.