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Does anyone know an example of a prime-related situation which is only seen finitely many times before (provably) stopping? I realize that is incredibly open-ended, so I'll try to provide a little more guidance. I'm looking for properties that have plenty of examples but eventually stop completely at some large $n$, as opposed to counterexamples which you don't see at all until they start at some large $n$.

I know some polynomials only admit a handful of primes, but that's well understood and not what I'm after. What I'm really interested in is examples of properties that at first glance seem likely to occur infinitely many times, but where we know that's not the case. Fermat primes would be a good one, if we actually knew that there were only finitely many. Since Sierpinski/Riesel numbers and primeless recurrence sequences can be readily explained by divisibility, ignore 'em.

I increasingly get the sense that unless there are very definite reasons why a certain prime-related phenomenon can't happen, it's very likely that it eventually will, almost like anything that wouldn't ultimately make the whole system inconsistent is guaranteed to happen eventually.

However, I've yet to track down even one single prime behavior of the general type I'm trying to describe, something which happens a whole bunch for a while, but ultimately, only finitely many times. If there are examples of things that happen, say, a million times and then stop, that would be very helpful to know about.

Trevor
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  • A somewhat related question, that you may find interesting, is Conjectures that have been disproved with extremely large counterexamples?. Note it also has a few answers which I believe are pertinent to your question as they are "prime-related" (e.g., this one). Also, the "Linked" section on the right of this other question has several other related questions you may find interesting, & possibly pertinent to your question. – John Omielan Dec 04 '19 at 08:35
  • Thanks, that's the right ballpark, but I was on top of that one, I posted on that thread yesterday. :) The problem is most of those are cases where the first counterexamples to something appear late, and what I'm really after here are cases where examples stop appearing late. – Trevor Dec 04 '19 at 09:05
  • Would the following qualify ? We fix an integer $\ b>1\ $ and ask which numbers of the form $\ a!+b\ $ are prime. This can produce large primes, but we know that $\ a\ge b\ $ implies that $\ a!+b\ $ must be composite. – Peter Dec 04 '19 at 12:31
  • Good thought, but still not quite what I'm looking for. Any kind of example like that that can be easily constructed and scaled to an arbitrary bound doesn't tell me much about the primes; I'm hoping for an example where because of the changing density etc. of the primes themselves, something that worked before doesn't work for large enough $n$ anymore, ever. Like if your example no longer yielded primes ever for all $b>10^{50}$, that would be perfect. – Trevor Dec 04 '19 at 12:37
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    The question seems rather vague. What will be considered as not "easily constructed and scaled"? What exactly are you expecting it to tell you about the primes? (I do think this question is interesting, but perhaps it can use more specification.) – YiFan Tey Dec 04 '19 at 12:42
  • @YiFan I know it is, I've been struggling to define it more clearly. My basic objection is that there seems to be NO case whatsoever of a genuine, interesting observation about prime distribution which only has a finite lifetime. The prototypical ideal example would be if twin primes turned out to be finite at some enormous bound. That's clearly absurd, but I'll settle for anything sort of interesting that comes up repeatedly for a while, yet is provably finite and ends, and not because of a bound imposed arbitrarily in the description. – Trevor Dec 04 '19 at 12:54
  • One other would-be-good example is that "prime race" phenomenon, where one modulus is in the lead and it doesn't actually change until a relatively large value. However, it's known that that alternates back and forth infinitely many times, so it's not a real example either. At this point, I'm looking for any indication of finiteness in any behavior of the primes, aside from the obvious regular patterns and periods you see when you only watch a finite subset of them. – Trevor Dec 04 '19 at 12:57
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    The problem is that sequences you are looking for tend to increase slow, since they should produce many primes. But then, the expression tends to produce infinite many primes. So, I think there is not much hope to find such a lately vanishing pattern. – Peter Dec 04 '19 at 14:36

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