Primality proof for n = 5:

Take b = 2.

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

2 is prime.
b^((n-1)/2)-1 mod n = 3, which is a unit, inverse 2.

(2^2) divides n-1.

(2^2)^2 > n.

n is prime by Pocklington's theorem.