Talk:Cramér's conjecture

A reference for the fact that it can be proved that the above conjecture holds true with probability one would be nice. Hottiger 19:43, 9 March 2006 (UTC)

I added a quick ref. It's a fairly basic fact, so it actually wasn't that easy to find a good mention. Feel free to replace this with a better reference if you find one. I found a better reference, the original I think. CRGreathouse (t | c) 12:11, 19 October 2006 (UTC)

Actually it goes back to the original Cramér paper. CRGreathouse (t | c) 17:09, 19 August 2009 (UTC)

Comment by Eagle1111

Eagle1111 (talk · contribs) left the following comment within the article, but since it's discussing the correctness of a previous point within the article it belongs better here:

The statement "For the largest known maximal gaps, R has remained near 1.13,” showing that, at least within the range of his calculation, the Granville refinement of Cramér's conjecture seems to be a good fit to the data." is in part incorrect. Since the criterion R is derived on the basis of the Cramer's model $R={\frac {\log p_{n}}{\sqrt {M}}}$ where M is the gap size, its limit in accordance with the Cramer's conjecture is equal to: $R={\frac {\log p_{n}}{\sqrt {(\log p_{n})^{2}}}}=1$. Therefore in order to support the Granville refinement of Cramer's model, the criterion R must be strictly less than 1. The case when R is strictly less than 1, implies that a counterexample to the Cramer's conjecture has been found, which accordingly becomes false.

Is this a valid concern? What if anything should we do to update the article in light of it? —David Eppstein (talk) 01:34, 11 March 2010 (UTC)

The comment in the article suggests not that Cramér's model is good, but that Granville's $2e^{-\gamma }$ is good. But just finding a value of R above 1 would not disprove Cramér's conjecture; that would require an infinite sequence of p with R bounded away from 1.

CRGreathouse (t | c) 03:05, 11 March 2010 (UTC)

My concern is that what the above mentioned line copied from the article says. Maybe I did not say it clearly enough, as long as R remains above or equal to 1 Cramer's model is fine, does not need any refinement to hold. To prove it false one need to find a maximal prime gap longer than $(\log p_{n})^{2}$, this is equivalent to demonstrating that the Shank's criterion (as D. Shanks came up with it) is strictly less than 1. The Granville' refinement is an attempt to salvage the Cramer's conjecture in the event that the maximal gaps would exceed $(\log p_{n})^{2}$. However, it will not support the conjecture indefinitely, it becomes false when R reaches R < 0.89053.

The line "showing that, at least within the range of his calculation, the Granville refinement of Cramér's conjecture seems to be a good fit to the data." Has not much to do with reality. Cramer' conjecture holds fine WITHIN THE RANGE OF HIS CALCULATIONS, and the Granville's refinement to the Cramer's conjecture does not come into question. Consequently, strictly speaking we can not talk about good data fit in the case when we do include the Granville's refinement.

If anything, the data calculated (here I mean the R values alone) may actually rather indicate, that in fact the Granville refinement may never be neccessary. But we now getting off the track into the topic of the convergence of the R sequence. There is no question that the sequence R converges, the real question is what is its limit? According to Cramer it is 1, according to Granville it is approx. 0.89054. Today its just sort of take your pick situation, I however, have a reason to believe that the true limit is approx 1.02. Time will tell, no need to stress over that. —Preceding unsigned comment added by Eagle1111 (talk • contribs) 14:02, 11 March 2010 (UTC)

It does seem evident that R converges -- but this certainly has not been proved. It's not even known that there exists a k with $g_{n}\ll \log ^{k}p_{n}$. Actually we're extremely far from a proof: AFAIK, we don't even know if there is a k with

$g_{n}\ll {\sqrt {p_{n}}}\log ^{k}p_{n}$

so we're not yet in the right order of magnitude...

CRGreathouse (t | c) 16:50, 11 March 2010 (UTC)

Heuristic justification and Heuristics?

Why is there two section Heuristic justification and Heuristics? Should they be merged? John W. Nicholson (talk) 02:55, 16 November 2013 (UTC)

Disputed content: Wolf's conjecture

The following paragraph Wolf's conjecture has been the subject of dispute:

In the paper Marek Wolf has proposed the formula for the maximal gaps $G(x)$ expressed directly by the counting function of prime numbers $\pi (x)$:

$G(x)\sim {\frac {\pi (x)}{x}}(2\ln(\pi (x))-\ln(x)+c_{0}),$

where $c_{0}=\ln(C_{2})=0.2778769...$, here $C_{2}=1.3203236...$ is the twin primes constant. Putting Gauss's approximation $\pi (x)\sim x/\ln(x)$ gives

$G(x)\sim \ln(x)(\ln(x)-2\ln(\ln(x)))$

and for large $x$ it goes into the Cramer's conjecture $G(x)\sim \ln ^{2}(x)$. As it is seen on Fig. Prime gap function no one of conjectures of Cramer, Granville and Firoozbakht crosses the actual plot of maximal gaps while the Wolf's formula shows over 20 intersection with currently available actual data up to $1.43\times 10^{18}$.

Wolf, Marek (2014), "Nearest-neighbor-spacing distribution of prime numbers and quantum chaos", Phys. Rev. E, 89: 022922

The content might well be broadly acceptable, although consensus needs to be established, but the wording certainly needs improvement first, edit warring over it is not acceptable, and it seems likely that at least one of the contributors has a connexion with research. Deltahedron (talk) 22:31, 1 March 2014 (UTC)

How reliable should we consider to be a brand-new number theory article (too soon to have accumulated citations or other evidence of significance) published in a (good) physics journal? I am also not impressed by the spectacle of Wolf naming a conjecture after himself and inserting it into Wikipedia himself. —David Eppstein (talk) 23:22, 1 March 2014 (UTC)

The link to the referenced article appears to be broken. Also, is it a link to a the Physics Review or something that happens to have the sequence of letters P-h-y-s-R-e-v in the url? YohanN7 (talk) 23:53, 1 March 2014 (UTC)

The correct ref is [1]. It's Phys. Rev. E. --Mark viking (talk) 23:58, 1 March 2014 (UTC)

It's the real Phys. Rev. E. The correct link is doi:10.1103/PhysRevE.89.022922. The broken link has a spurious period at its end. There's also a preprint at arxiv:1212.3841. —David Eppstein (talk) 00:05, 2 March 2014 (UTC)

A t m, the article is a bit inconsistent since the Wolf picture is still there.

Also, the article seems legitimate to me, but I still doubt that (published) heuristic arguments and computer simulations (see preprint) are notable enough for the original author to insert his material into this article.

There is questionable material in Cousin prime that has been there for quite some time (and three recent edits by an IP reverted by me just now). YohanN7 (talk) 02:01, 2 March 2014 (UTC)

• I would propose the following text

Wolf [ref] has a proposed a formula for the maximal gaps

$G(x)\sim {\frac {\pi (x)}{x}}(2\ln(\pi (x))-\ln(x)+c_{0}),$

where $c_{0}=\ln(C_{2})=0.2778769...$ is the logarithm of the twin primes constant. This would imply the Conjecture.

Normally I would have inserted it boldly but thought it better at present to bring it here for discussion first. Deltahedron (talk) 17:53, 2 March 2014 (UTC)

Correction: $c_{0}=\ln(C_{2})=0.2778769...$ is NOT the logarithm of the twin primes constant, ln(A005597). It is the logarithm of twice the twin primes constant $c_{0}=\ln(2*C_{2})=0.2778769...$, ln(A114907).

I would suggest that both the current plot and the new one be presented as to show how these conjectures fit together. John W. Nicholson (talk) 02:32, 3 March 2014 (UTC)