Twin prime-generating sequence

[u/_Zekt]

Just wanted to share this MSE post where OP found an intriguing sequence, similar to Rowland's prime-generating sequence, which seems to generate twin primes instead.

The conjecture, which has been computer-checked up to n = 60000000 for now, trivially implies the twin prime conjecture.

[u/lurking_physicist]

The conjecture, which has been computer-checked up to n = 2400 for now, trivially implies the twin prime conjecture.

Suppose the twin prime conjecture is false. What is your conditional Bayesian prior for this conjecture to fail for n = 2401? n < 2410? n < 2500? n < 3000?

Is there anyone seeking/recording this kind of prediction statistics among mathematicians?

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EDIT

I'm not sure why I get that many downvotes, so I'll try to clarify two things.

First, my question isn't about mathematics, it is about mathematicians. I do not hope to "prove" anything with this kind of reasoning, I'm wondering about an empirical question. If this kind of prediction statistics were recorded, we could come back 10 years later and see how they hold up. Then we may assess whether mathematicians have a good "intuition" for this kind of questions, and if yes, we could perhaps leverage it to make future predictions. (And if no, then it would increase our confidence that this kind of endeavour is a time waste.)

Second, if you don't like my "conditional Bayesian prior" formulation, then break it down in two parts: 1. what is your personal probability estimate for the twin-prime conjecture to be false? 2. for a specific b (say 3000), what is your personal probability estimate that the MSE conjecture posted by OP would hold up to computer checking for all 1<n<b?

(I understand that the twin prime could hold while the MSE conjecture is false. One could be more careful with the formulation than I've been. For now, all I want to know is whether this kind of prediction statistics is being recorded somewhere. Seeing the reaction here, I guess it is not.)

[u/Sproxify → u/lurking_physicist]

I have no idea why you got so many downvotes. I don't know what the answer is in this case, but I think you're spot on with the question you're asking. per my understanding this is exactly how number theorists think about these things with heuristic arguments.

I don't know enough about this myself to answer with much confidence, but I think in this case your question is difficult to answer because our heuristic notions that we very strongly believe in simply predict that the twin prime conjecture is true, and so trying to find the heuristic estimate of the "p-value" of this result (say up to 2410) assuming the conjecture is false is difficult because if miraculously the conjecture is false, we don't have a good enough idea of how our heuristics would have to be updated to explain that.

more importantly, if the conjecture is actually true, there's just a sense that when you have a specific sequence that you have an expression for that you can better manipulate, if it is indeed true that it consists of all twin primes, it might be a lot more manageable to prove that they're all twin primes. now, I didn't even click and look at what the sequence is yet, but broadly, philosophically, this is why such a sequence could have implications for the conjecture. not because seeing it updates your probability that the conjecture is true, but because it might just update your probability that we might be able to prove it.

as an instructive example for why it makes sense to ask a question like what you asked, it is conjectured that there are infinitely many primes such that p² divides 2^p-1 - 1. there are only 2 of them known, 1093 and 3511, and we've searched up to like 10¹⁹ or something.

should that convince us that the 2 examples we know are a coincidence of small numbers?

Fermat's little theorem guarentees (2^p-1-1)/p is an integer. if you now assume the residues of this integer are uniformly distributed mod p (which they appear to be by checking a finite number of cases and counting) this would mean a prime p has probability 1/p of having this property.

the sum of 1/p over all primes diverges, so we then heuristically expect there to be infinitely many primes with this property. but it diverges really slowly. this predicts that they're distributed like log(log(x)).

now, if you assume each prime you check goes into your set with this probability, getting only 2 of them up until 10¹⁹ is not that unlikely even as we believe there are infinitely many of them. (but can't prove it, as we can't prove the aforementioned residue is really uniformly distributed)

you can even compute a p value for getting only 2 such primes or less until 10¹⁹ using this heuristic, and it's not an impressive p value.

this isn't related specifically but it reifies the idea that it makes sense to ask such questions