Primality proof for n = 10748006189:

Take b = 2.

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

121271 is prime.
b^((n-1)/121271)-1 mod n = 2683161203, which is a unit, inverse 3580907642.

(121271) divides n-1.

(121271)^2 > n.

n is prime by Pocklington's theorem.