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.