Primality proof for n = 12672138976649:

Take b = 2.

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

1584017372081 is prime.
b^((n-1)/1584017372081)-1 mod n = 255, which is a unit, inverse 9740153880091.

(1584017372081) divides n-1.

(1584017372081)^2 > n.

n is prime by Pocklington's theorem.