Primality proof for n = 257:

Take b = 3.

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

2 is prime.
b^((n-1)/2)-1 mod n = 255, which is a unit, inverse 128.

(2^8) divides n-1.

(2^8)^2 > n.

n is prime by Pocklington's theorem.