Primality proof for n = 128262069029:

Take b = 2.

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

1687658803 is prime.
b^((n-1)/1687658803)-1 mod n = 2663518125, which is a unit, inverse 100001539811.

(1687658803) divides n-1.

(1687658803)^2 > n.

n is prime by Pocklington's theorem.