Primality proof for n = 32309:

Take b = 2.

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

197 is prime.
b^((n-1)/197)-1 mod n = 15086, which is a unit, inverse 13229.

(197) divides n-1.

(197)^2 > n.

n is prime by Pocklington's theorem.