That argument has been in the literature for quite some time. It is a good heuristic probabilistic argument. If ever someone proves the Collatz Conjecture false, then that would be an answer to my question.

@Wojowu, I am now reading the paper by Ian Richards that I linked to above, and it appears that it addresses the issue that Greg Martin raised. Anyway, the paper is a good read.

@Wolfgang, I was not aware of that link. It seems that my question is only about probabilistic arguments, while that question includes all types of heuristics. I am not sure what to do. Should I delete this question?

Actually, negative numbers are not allowed to be prime in that particular conjecture. See the link.

@GregMartin, I don't see how the original probabilistic heuristic is flawed, even if in retrospect it is. After all, the density of primes decreases as the numbers get larger. Even the possibility that it is flawed is shocking.

Spencer, as for statistics being the answer to Hume's Problem of Induction, we cannot know if nature is uniform, but if it is, then using statistics is the best strategy to make predictions about nature.

see my question and also my answer: philosophy.stackexchange.com/questions/39545/…

I like to approach statistics as the answer to Hume's Problem of Induction.

Let us continue this discussion in chat.

@WillSawin 1. That is correct, but only if the two particles interact. 2. OK, I will assume that is what you mean when you say "computational approximation". You said above that you did not see how the interference pattern of the double slit experiment arises from a "computational approximation" to classical mechanics. But I'm claiming that it does, since it satisfies Schrodinger's equation and it is easy to compute. (If it were difficult to compute, the computer generating the universe would have to compute something different than Schrodinger's equation, perhaps just noise.)

@WillSawin, you said, "your claim requires in addition to that, that the interference pattern arise from a reasonable computational approximation to classical mechanics, which you have not demonstrated." My claim is that if one assumes that the universe is a digital computer that attempts to simulate the laws of Newtonian mechanics as well as it can, then the laws of physics will be an approximation of QM; I never say anything about a "computational approximation to classical mechanics".

@WillSawin, When you said, "individual particles don't follow the Schrodinger equation", I assume you mean that one can only see the probability distribution described by the Schrodinger equation when one observes an ensemble of particles.

I claimed above that "particle spin is a consequence of special relativity combined with quantum mechanics." But it is actually not so simple. See quantumchymist.blogspot.com/2014/04/…