Primality proof for n = 97:

Take b = 5.

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

3 is prime.
b^((n-1)/3)-1 mod n = 34, which is a unit, inverse 20.

2 is prime.
b^((n-1)/2)-1 mod n = 95, which is a unit, inverse 48.

(2^5 * 3) divides n-1.

(2^5 * 3)^2 > n.

n is prime by Pocklington's theorem.