Primality proof for n = 12791:

Take b = 2.

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

1279 is prime.
b^((n-1)/1279)-1 mod n = 1023, which is a unit, inverse 3651.

(1279) divides n-1.

(1279)^2 > n.

n is prime by Pocklington's theorem.