Timeline for What is the minimal range $[f(n),g(n)]$ that contains a prime number for every integer $n>0$?
Current License: CC BY-SA 3.0
20 events
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| Mar 3, 2014 at 9:16 | comment | added | GH from MO | @Jan-Christoph: I agree, but let us also emphasize that it has been very difficult to find larger than average gaps between primes (en.wikipedia.org/wiki/Prime_gap#Lower_bounds). | |
| Mar 3, 2014 at 7:46 | comment | added | Jan-Christoph Schlage-Puchta | Probably the probabilistic model does not give the correct answer when dealing with primes in short intervals. Maier's matrix method shows that on a scale up to a power of $\log x$ the distribution of primes is much more irregular than what probability would predict. | |
| Mar 1, 2014 at 22:56 | vote | accept | barak manos | ||
| Feb 27, 2014 at 14:54 | comment | added | GH from MO | @barak: Legendre's conjecture implies a slightly weaker variant of $L(1/2)$: there is a prime in $[n-5\sqrt{n},n]$. On the other hand, $L(1/2)$ implies Legendre's conjecture. It is subjective what is interesting and for what reasons. quid tried to say, and I agree with him, that the specific intervals $[n^2,(n+1)^2]$ are not too interesting in this problem. We don't really distinguish between the following intervals: $[n-\sqrt{n},n]$, $[n-100\sqrt{n},n]$, $[n-\sqrt{n}\log^{29} n,n]$, $[n-n^{1/2+o(1)},n]$. They are much the same, and they are on the edge what the Riemann Hypothesis can handle. | |
| Feb 27, 2014 at 12:13 | comment | added | barak manos | OK, @GH from MO,quid: So the 'prime between $n^2$ and $(n+1)^2$' conjecture implies $L(1/2)$ which implies $L(0.525)$, is that correct? Now, $L(0.525)$ has been proven, but $L(1/2)$ hasn't been proven, is that correct? If yes, doesn't that mean that the unproven 'prime between $n^2$ and $(n+1)^2$' conjecture is still interesting not only for historical reasons? | |
| Feb 27, 2014 at 10:43 | comment | added | GH from MO | @barak: Let me clarify and supplement quid's remarks. For $c>0$ let $L(c)$ denote the statement that "if $n$ is sufficiently large then there is always a prime between $n-n^c$ and $n$". Note that decreasing $c$ makes $L(c)$ stronger, and $L(1/2)$ is essentially the same as Legendre's conjecture. Currently $L(0.525)$ is known, the Riemann Hypothesis implies $L(c)$ for any $c>1/2$, while probably $L(c)$ is true for any $c>0$. | |
| Feb 27, 2014 at 10:33 | comment | added | user9072 | But then also to improve it to 90 would be relevant and so on, so that 100 does not really have any special role (intrinsic to the mathematics at hand), except for historical reasons. This is somehow the situation here, one knows something for .525, for .5 and something extra one would get the conjecture, but actually it should also be true for .4, for .0001 for everything positive. Sure, proving .5 would be relevant but it does not seem to have any very special role. I hope this clears up what I meant. If you want to continue perhaps comment on my answer so that GH does not get notified. | |
| Feb 27, 2014 at 10:28 | comment | added | user9072 | @barakmanos also I always said it was not know, please, see that I said if one can improve on BHP then it would follow. If somebody conjectures something is at most size 100, and then somebody else conjectures it is actually size at most 15 and now most everybody agrees this 15 should be alright then the original conjecture of at most 100 is not really relevant anymore, even if 105 is the best anybody can prove at the moment. At least this is what I think and meant. Of course if somebody could improve the result from 105 to 100 this would be relevant. [...] | |
| Feb 27, 2014 at 10:16 | comment | added | barak manos |
@quid, sorry but I'm not sure I understand. First you say that there is a better approximation than the range $n^2$ and $(n+1)^2$, so it is interesting only due to historical reasons. But then you say that the result of Baker-Harman-Pintz is actually weaker (well, GH from MO says that, but you appear to agree with it). Can you please clarify?
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| Feb 27, 2014 at 10:09 | comment | added | user9072 | @barakmanos yes of course it is tighter than what is known (otherwise I would not have answered that it is true that this is a conjecture as you said). What I mean is that the conjecture now seems a bit arbitrary as a lot stronger things should be true and these days one would rather not phrase a conjecture like this to begin with. Or put differently the conjecture got superseeded by other stronger/better conjectures, so it is not that relevant anymore. | |
| Feb 27, 2014 at 9:47 | comment | added | barak manos | @quid, can you please have a look at the comment above? | |
| Feb 27, 2014 at 9:46 | comment | added | barak manos | Quid, can you please have a look at the comment above, made by @GH from MO? It suggests that the 'prime between $n^2$ and $(n+1)^2$' conjecture actually puts a tighter bound than the current bound inferred by the Baker-Harman-Pintz conjecture. So doesn't this make the former conjecture still "relevant"? | |
| Feb 27, 2014 at 9:39 | comment | added | user9072 | @barakmanos well in the end I came around to answer it rather then close it but point taken. Yes, Granville showing up to clarify some points regarding conjectures was interesting. I think it was really helpful this information came up here. | |
| Feb 27, 2014 at 9:35 | comment | added | GH from MO | @barak: Having a prime between $n^2$ and $(n+1)^2$ is much the same as having a prime in $[n-n^{1/2},n]$. The result of Baker-Harman-Pintz is somewhat weaker, because the exponent there is 0.525 instead of 1/2. The (unproven) Riemann Hypothesis almost implies the exponent 1/2, but as things stand today, we need an extra factor of $\log n$ there. More optimistic conjectures reduce the gap dramatically, cf. my response. | |
| Feb 27, 2014 at 9:06 | comment | added | barak manos | @quid: Thanks, interesting post (including the notion about that guy who claims he did not make a specific conjecture, and it's also wrong on Wikipedia)... BTW, I would not vote to close it (as you did)... | |
| Feb 27, 2014 at 8:55 | comment | added | user9072 | @barakmanos yes in my opinion this conjecture is almost purely relevant for historical reasons (Legendre was a pioneer in that field an the conjecture dates from around 1800 where these questions were much less understood), see my comments on mathoverflow.net/questions/114399/… | |
| Feb 27, 2014 at 8:52 | comment | added | barak manos | So my question to both you and @quid: your answers make the conjecture of a prime number between $n^2$ and $(n+1)^2$ sound a little redundant (i.e., not so important to prove), as there is a "tighter bound" conjecture. Am I correct? | |
| Feb 27, 2014 at 8:50 | comment | added | GH from MO | @quid: Thanks for the feedback and your response as well! | |
| Feb 27, 2014 at 8:47 | comment | added | user9072 | Strangely I only got notified of your answer when it was already almost 15 mins old; so mine was almost done and I thus posted it and leave it, also as the symmetric difference is nonempty. And, thanks for bringing that preprint to my attention. | |
| Feb 27, 2014 at 8:25 | history | answered | GH from MO | CC BY-SA 3.0 |