Primality proof for n = 251:
Take b = 3.
b^(n-1) mod n = 1.
5 is prime. b^((n-1)/5)-1 mod n = 218, which is a unit, inverse 38.
(5^3) divides n-1.
(5^3)^2 > n.
n is prime by Pocklington's theorem.