We would falsely conclude that 85 is prime. We don't want to just force the verifier to factor the number, so a better way to avoid this issue is to give primality certificates for each of the prime factors of "n" − 1 as well, which are just smaller instances of the original problem. We continue recursively in this manner until we reach a number known to be prime, such as 2. We end up with a tree of prime numbers, each associated with a witness "a". For example, here is a complete Pratt certificate for the number 229: