Timeline for Consequences of Legendre's conjecture
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 2, 2013 at 20:23 | comment | added | Andrew Granville | Thanks for getting those changes made! I am not keen on making conjectures if we have little or no evidence for any particular answer. What is evident from calculations is that $\max_{n\leq N} (p_{n+1}−p_n)/(\log p_n)^2$ grows slowly to its limit. It is possible that the limit is $2e^{-\gamma}$, but perhaps it is $\infty$? Who knows? There is no convincing heuristic, and it is evident that this is a delicate question. | |
| Feb 26, 2013 at 16:42 | comment | added | Emil Jeřábek | I see, thanks for looking into this. I have added something on the terminological ambiguity in the lead section. In any case, please don’t hesitate to edit it yourself if you have an idea how to improve it. | |
| Feb 26, 2013 at 14:17 | comment | added | user9072 | Yet, now having also looked at Cramér's paper I can see why the person editing this made this edit with that justification. In eq (4) of Cramérs paper, where he is explicit, it is written only p_(n+1)-p_n = O((log p_n)^2). | |
| Feb 26, 2013 at 14:10 | comment | added | user9072 | @Emil Jeřábek: In Andrew Granville's paper before eq (14) a relevant part of Cramér's paper is quoted. The issue is somehow, I think, that Cramér says "something similar should hold" (where the thing referred to is the statement that the limsup is 1 for 'probabilistic primes', roughly speaking). In this paper (Granville's) also both statements with constant 1 and just finite, are called Cramér's conjecture; the latter mentioned in a way suggesting it is less common (see again around eq 14). | |
| Feb 26, 2013 at 13:57 | comment | added | Emil Jeřábek | I gave the WP page another go. One remaining issue is that while the world seems to think that Cramér’s conjecture states limsup = 1 (or $\le1$), on WP it is formulated only as $< \infty$. This was introduced in this edit en.wikipedia.org/w/… “based on re-read of Cramér's paper”. Since I have not read Cramér’s paper, I have no idea whether this is accurate. | |
| Feb 26, 2013 at 13:34 | comment | added | user9072 | An occassion where this came up is mathoverflow.net/questions/90327/… There the situation is a bit different; whether you are happy with the phrasing in that question, I asked already indirectly. In my answer there I think I was more careful what I actually attributed to you; in any case I you could have a look to check if that is alright, I'd be greatful. | |
| Feb 26, 2013 at 12:27 | comment | added | user9072 | Somebody already changed the Wikipedia page, but I am not completely convinced this change fully addresses the problem. | |
| Feb 26, 2013 at 11:39 | comment | added | user9072 | In this answer I now simply removed the mention of this more precise form, as it is not crucial; I believe however to be aware of some other places on MO where this comes up, and as far as I oversee it will try to correct these over a shorter period of time. (I am however not active on Wikipedia.) | |
| Feb 26, 2013 at 11:27 | comment | added | user9072 | Thank you for bringing this to my (or our) attention! As you mention this (mis)attribution is wider-spread (and does not originate with me). On "Cramér-Granville conjecture": it is also mentioned in a different form on MathWorld mathworld.wolfram.com/Cramer-GranvilleConjecture.html (there is is just the O((log p_n)^2) conjecture with some constant strictly greater than 1. It is now not clear to me if you are comfortable this weaker conjecture being attributed to you, so I mention it. Also, if I may ask: do you just not conjecture the limsup is this or do you not believe it? | |
| Feb 26, 2013 at 2:54 | comment | added | Gerhard Paseman | Welcome to MathOverflow, Prof. Granville At the moment (assuming you are indeed Andrew Granville), the best we can do is make references to this post; we can't assume any influence over Wikipedia. However, it is likely the community can help make the changes you request. If you also register at tea.mathoverflow.net, you can make a post asking for assistance in making such corrections. In any case, I am confident the MathOverflow community can help. Gerhard "Ask Me About System Design" Paseman, 2013.02.25 | |
| Feb 26, 2013 at 1:58 | history | answered | Andrew Granville | CC BY-SA 3.0 |