I'm sorry that I didn't remember it off the top of my head.

Factoring is not a huge burden compared to the earlier steps, and also not necessary a lot of the time, especially if your goal is to keep feeding the number back into the system to create numbers made out of new primes.

I agree with this. My point is just that it's not required for the proof, and usually isn't included -- so the proof is not constructive. You can extend it to be constructive in a very straightforward manner, but you don't have to and you usually don't.

The reason I'm calling this a nonconstructive proof is by analogy to the following proof, often given as an example of a nonconstructive proof:

Theorem: it is possible for an irrational number raised to an irrational power to result in a rational number.

0. For typographic convenience, let r = sqrt(2). r is known to be irrational.

1. r^r might or might not be rational -- who knows?

2a. If r^r is rational, the theorem is proved.

2b. If r^r is not rational, then (r^r)^r is an irrational number raised to an irrational power, but (r^r)^r = r^(r⋅r) = r^2 = 2, which is rational.

3. Therefore, there exists some pair a,b for which the following is true: (I) a and b are both irrational; (II) a^b is rational.

I don't see any way to consider this proof nonconstructive while considering the proof of the inifinitude of the primes that I've described as constructive. Do you?