Talk:Firoozbakht's conjecture

Relation with Cramér–Granville heuristic

I have reverted comments on the relationship of Firoozbakht's conjecture with the Cramér–Granville heuristic as they appear to be original research. Any comments about the degree of belief in the research community in the various conjectures needs to be support by explicit citations to independent reliable sources. Expressions of personal belief in one or the other by individual editors are not acceptable. Deltahedron (talk) 16:17, 4 May 2014 (UTC)

I noticed that since my last edit here these references have been removed:

• 1)

• 2) Daniel Shanks conjectured asymptotic equality of record gaps

I know the second one is not WP:OR and the first is there for just for citing the conjecture. So, why were they removed and should they place back? I am sure better wording can be done, but that is not a reason to remove the cites.

John W. Nicholson (talk) 18:20, 4 May 2014 (UTC)

Added to pick up the references

I reverted what appears to be original research which had deleted several previous citations. Of the two references you cite, one is an Arxiv paper (which is not necessarily a reliable source) and the other is puzzling as it does not appear in the text I reverted. Deltahedron (talk) 18:28, 4 May 2014 (UTC)

An IP user in the 37.254/255 range seems to be engaged in an edit war over this material. I am asking for the page to be restricted to confirmed users only so that we can get a discussion going on this page and consensus on the content. It's not possible to do it with a series of edit summaries. Deltahedron (talk) 21:09, 4 May 2014 (UTC)

Good. But, I think my issue is separate, but relate in the opposite direction -- taking cited material away. I might agree with the reliable source issue with the 1) Arxiv paper if you can show how it is in error with the part talking about the conjecture. The 2) paper does not have this reliable sources issue, but it has been removed by user CRGreathouse. I will agree that it needs some better wording as to what the Shanks conjecture is, but something needs to be said about its relationship to this conjecture. Also, should there be a full article on the Shanks conjecture? John W. Nicholson (talk) 21:48, 4 May 2014 (UTC)

I removed the Arxiv reference as it was being used for original research by synthesis by IP 37.254/255. The preprint does indeed comment on the relationship between Granville's heuristic and Firoozbakht's conjecture, and a reliable source for that would be welcome. However, Arxiv preprints are not reliable sources as such: this has been thrashed out a number of times, see for example, the archives of Wikipedia:Reliable sources/Noticeboard. The Shanks paper from 1964 does not mention the Firoozbakht conjecture. Deltahedron (talk) 07:01, 5 May 2014 (UTC)

Of course not, Shanks paper is from 1964, so it came out before the conjecture here was even founded, but both conjectures "contradicts the Cramér–Granville heuristic" which means that it supports Shanks statement. Also, because there is no representation at Cramér–Granville heuristic, there is no real statement about the Shank conjecture. So, should there be a page? John W. Nicholson (talk) 11:48, 5 May 2014 (UTC)

We need an independent reliable source that compares the various conjectures. Do you know of one? Deltahedron (talk) 11:59, 5 May 2014 (UTC)

www.emis.ams.org/journals/JIS/VOL6/Nicely/nicely2.pdf

I am sure we can find something else which might be more fitting for a Shanks conjecture page, but for this page it looks like both the Cramér–Granville heuristic support and the Shanks heuristic support are use with this page but both are lacking support other than what someone or another will call unreliable sources. Because one of the first published books with the Firoozbakht's conjecture is Paulo Ribenboim 2004 book, I doubt there will be much on it to find connecting the these two more directly than what has already been stated as unreliable sources, so I do not know what is the best thing to do. I do not like this all or nothing bit that the sources are forcing. — Preceding unsigned comment added by Reddwarf2956 (talk • contribs) 16:28, 5 May 2014‎

The Nyman and Nicely paper does indeed Cramer, Shanks and Granville, but not Firoozbakht, which is what we need here. The point about insisting on independent reliable sources is to verify the material added to the encyclopaedia. Whereof one cannot speak, thereof one must be silent. Deltahedron (talk) 16:41, 5 May 2014 (UTC)

Verification

The text says Using a table of maximal gaps, the inequality can be verified for all primes below 4×10^18. The reference is to a web page Gaps between consecutive primes which does not mention the conjecture and while it might provide the data from which the conjecture might be verified up to this limit (this is not quite clear) there is no assertion that anyone has actually done so. The best I can find is a private communication of Marek Wolf that he has done so up to 3.495×10^16 quoted in Sun, Zhi-Wei (2013). "Conjectures involving arithmetical sequences". In Kanemitsu, Shigeru; Li, Hongze; Liu, Jianya (eds.). Number theory. Arithmetic in Shangri-La. Proceedings of the 6th China-Japan seminar, Shanghai, China, August 15–17, 2011. Series on Number Theory and Its Applications. Vol. 8. Hackensack, NJ: World Scientific. pp. 244–258. ISBN 978-981-4452-44-1. Zbl 1263.11001. Deltahedron (talk) 19:01, 5 May 2014 (UTC)

Before the CRGreathouse edits, that section read "Currently, the largest stated verification was done by using "maximal gaps between consecutive primes less than 4.444 × 10¹² by Firoozbakht. Using a table of maximal gaps and the above gap inequality, the confirmation value can be extended to all primes below 4×10¹⁸." (with the same link to the web page)

This allowed the reader to see how the confirmation was done, and I personally have confirmed it to the 4×10¹⁸ value using the table (worse case was the 64th prime of the table), it is easy. I think the paper you state is an update to the "less than 4.444 × 10¹²" statement or keep the original as the "first" and add this new one as the more one current. What do you feel needs to be done? John W. Nicholson (talk) 21:58, 5 May 2014 (UTC)

Could you please explain/prove why is it enough to verify only the maximal gaps? — Preceding unsigned comment added by 46.121.232.249 (talk) 16:25, 2 September 2014 (UTC)

I suspect, although I don't know, that verification using the maximal gaps relies on showing that FC in a range of primes p_a to p_b would follow from showing that that all primes gaps are less than some bound Z, which depends only on the values of a,b,p_a and p_b; and then checking that the largest gap in that range is already known to be less than Z. Deltahedron (talk) 16:45, 2 September 2014 (UTC)

Material being removed

I'm not sure why the reddwarf SPA is removing a section from this article. It's properly sourced, has been in the article for a long time, and represents the consensus of the mathematical community. - CRGreathouse (t | c) 12:45, 16 May 2014 (UTC)

You might like to refer to the discussion above, where the issue was, whether the material in question, namely this is relevant, correct and supported the references cited. As I said above, "Any comments about the degree of belief in the research community in the various conjectures needs to be support[ed] by explicit citations to independent reliable sources. Expressions of personal belief in one or the other by individual editors are not acceptable." Let's unpick this material

• The conjecture is believed to be false, as it contradicts the Cramér–Granville heuristic

• Granville, A. (1995), "Harald Cramér and the distribution of prime numbers" (PDF), Scandinavian Actuarial Journal, 1: 12–28
• Granville, Andrew. "Consequences of Legendre's conjecture".
• Granville, Andrew (1995), "Unexpected irregularities in the distribution of prime numbers" (PDF), Proceedings of the International Congress of Mathematicians, 1: pp. 388–399 {{citation}}: |pages= has extra text (help)

None of these references mentions Firoozbakht's conjecture and therefore none of them can be used to support either the proposition that it is believed to be false, or that it contradicts the Cramér–Granville heuristic.

• Pintz refers to this as MCM, the modified Cramér model

• Pintz, János (2007), "Cramér vs. Cramér: On Cramér's probabilistic model for primes", Funct. Approx. Comment. Math., 37 (2): pp. 232–471 {{citation}}: |pages= has extra text (help)

Again, does not mention Firoozbakht's conjecture and hence again cannot be used to support the text.

It might be possible to make an argument that Firoozbakht's conjecture implies Cramér's conjecture and that the latter conjecture with a constant of 1 is inconsistent with a refined form of the conjecture with a constant of 1.1229. This would be merely original research but might be acceptable as obvious to the mathematically sophisticated readership likely for this topic. Even if it were possible, it would still be irrelevant (and so unaccaptable) unless it can be established that Firoozbakht's conjecture implies Carmer with an explicit constant.

However, the assertion "believed to be false" requires an independent reliable source which explicitly addresses that and demonstrates in a scholarly way that the academic consensus is against belief in the conjecture. Where is this claimed consensus published? So far, the references completely fail to address this latter point and so this part of the text completely fails verification. Unverified material which has been repeatedly challenged, as this has, simply cannot be used, and repeatedly inserting it against consensus is Wikipedia:Edit warring. Deltahedron (talk) 16:58, 16 May 2014 (UTC)

Firoozbakht's conjecture isn't particularly well-known, and it's rare to see a reference to it in peer-reviewed journal articles. But the link to prime gaps is both obvious and in the article already. Perhaps an overly legalistic reading of Wikipedia's policies would suggest a phrasing like "1. F -> gap < log^2 x; 2. references a, b, and c suggest gap > 1.12 log^2 x" and leaving the application of modus ponens to the reader. But this would be a disservice: it obscures the clarity of the article.

Or perhaps you are claiming that conclusion (1) itself is unacceptable. But then why have you not removed that claim from the article? I don't think you should -- it falls under what WikiProject Mathematics has long considered WP:OR#Routine calculations, and its removal would similarly disimprove the article.

CRGreathouse (t | c) 07:20, 18 May 2014 (UTC)

It seems to be accepted that the sources cited do not directly support the text inserted, and that any support comes from some degree of original research or synthesis. This is a prima facie case against its inclusion. There might be a discussion about the acceptable element of routine calculation.

But. It is very far from a "routine calculation" to deduce that The conjecture is believed to be false. This is an assertion about the state of the world, not of mathematics. Is this asserted of Firoozbakht's conjecture in any source whatsoever? None has been produced. It we accept for the sake of discussion the synthesis that Firoozbakht's conjecture implies gap < log^2 and so contradicts the 1.12 heuristic, is there an reliable source that states that gap < log^2 is believed to be false by any significant number of mathematicians, or that the falsehood of gap < log^2 is "the consensus of the mathematical community"? Again, none has been produced. The statement The conjecture is believed to be false is simply not supported by the sources cited, or any other sources that have yet been identified, and is not even synthesis of other sourced statement. Hence we cannot include it. To insist on verification by independent reliable sources is not "overly legalistic", it is the core policy of Wikipedia, which is supposed to be an encyclopaedia, not a compendium of personal opinions.

There might be an acceptable consensus for saying that Firoozbakht's conjecture implies gap < log^2 as a routine calculation, but even so right now I want to see a citation.

It is clear that at present there is no consensus to include any of this material. Deltahedron (talk) 09:14, 18 May 2014 (UTC)

Under the circumstances, it may be better to be explicit about what the sources actually assert about the prime gap, and leave the implication to the reader. I would prefer this to the outright removal of the content. Sławomir Biały (talk) 12:10, 18 May 2014 (UTC)

I think there are several assertions here which need to be unpacked.

• A. FC implies Cramer in the form O(log^2), which is stated in the article in its present form
• B1. FC implies Cramer in the strong form of a constant multiplier of 1+epsilon
• B2. FC implies Cramer in the strong form of a constant multiplier of 1 for n>>1
• C. Cramer in the strong form is inconsistent with Granville's modified heuristic constant of 1.12
• D. Cramer in the string form is not believed, or at least the academic consensus is against it
• E. FC is not believed, or at least the academic consensus is against it

There might be a case for including (A) or even (B1) in the text without sources, as a routine calculation, though right now I'm not convined. I think that B2 is much harder to prove, if true, and would definitely need to see a citation. If (B1) or (B2) is included, it might well be proportionate to mention (C) and let readers draw their own conclusions. So far I have seen no reason to believe that (D) or (E) is true, and would definitely insist on verification by independent reliable sources. Deltahedron (talk) 14:57, 18 May 2014 (UTC)

A, B1, B2, C, D, and E are all correct. B2 is not needed for C, B1 suffices. (And B12 is already in the article, not just A.) I'll edit in Sławomir Biały's compromise of (in your terms) B1 + C. CRGreathouse (t | c) 18:03, 18 May 2014 (UTC)

I don't see that Sławomir Biały supports the insertion of B1, at least not without sources (explicit about what the sources actually assert) and I currently don't either. Explain why B1 is a routine calculation (I can do it, but can the likely reader?) or provide sources: we don't have any consensus yet. As far as D and E are concerned, you persist in those assertions but have failed to provide any support from independent reliable sources in spite of several requests to do so. I am going to assume that they are nothing more than your own personal views. Deltahedron (talk) 18:11, 18 May 2014 (UTC)

I have written a version which seems to give due weight to the contrary heuristics on the assumption that we agree (which we have not done so far) that A, B1 or B2 can be asserted without citation. In particular I have removed the word "wrong" which was tendentious and unsupported. Deltahedron (talk) 18:25, 18 May 2014 (UTC)

I noticed two things with the changes.

1. The lack of the contrary supporting reference, namely the Shanks, Daniel reference above with the "Relation with Cramér–Granville heuristic" section. Maybe add this after the Granville and Pintz sentence with the Shanks link?:

On the contrary, conjecture would be consistent with the heuristics of Shanks conjecture by [Daniel Shanks].

2. The 2. Rivera, Carlos reference in the article does have a "proof" of the <\math>g_n < (\log p_n)^2 - \log p_n<\math> part where the [citation needed] is. It was done by Luis Rodríguez. I would put it there, and I know there is a way as to not have the same references twice, but I don't know how to make it happen. If someone would help me I would do it. One question with this. Should the easy computable n > 4, be included? John W. Nicholson (talk) 22:26, 18 May 2014 (UTC)

I have added the Rivera citation but note that it is a personal web site and may not be a reliable source. I have also added a comment on Shanks's heuristic for consideration. Deltahedron (talk) 06:42, 19 May 2014 (UTC)

This

In the last sentence, it is not clear what "this" is. — Preceding unsigned comment added by 158.58.172.33 (talk) 13:46, 16 May 2014 (UTC)

A clear mistake in article Firoozbakht's conjecture

There is a absolute mistake in article Firoozbakht's conjecture, the current 5th line should be changed to: "Equivalently: $p_{n+1}<p_{n}^{(n+1)/n},$ for all n ≥ 1, see..."; Hope someone help to correct it soon. JohnAu2000 (talk) 10:51, 6 June 2014 (UTC)

$p_{n+1}<p_{n}^{n+1/n},$ would make the statement incorrect.

$1/n*(n+1)={\frac {n+1}{n}}=1+{\frac {1}{n}}$ which is what the article shows. John W. Nicholson (talk) 11:51, 6 June 2014 (UTC)

No, the statement (in the article) looks incorrect, I meant it should be changed to (better suggestions- I corrected the above comment): $p_{n+1}<p_{n}^{(n+1)/n},$ or $p_{n+1}<p_{n}^{1+{\frac {1}{n}}}$ JohnAu2000 (talk) 12:47, 6 June 2014 (UTC)

Correct statement?

I am looking at the phrase "which suggest that $g_{n}>{\frac {2-\varepsilon }{e^{\gamma }}}(\log p_{n})^{2}$ infinitely often for any $\varepsilon >0$." and doing the following.

$g_{n}{\frac {e^{\gamma }}{(\log p_{n})^{2}}}>2-\varepsilon$. 

Everything on the LHS is positive and everything on the RHS is negative for $\varepsilon >2$. So, what is the point of the statement?

Also, $e^{\gamma }$ is described where in this inequality?

--John W. Nicholson (talk) 01:10, 11 February 2015 (UTC)

One can imagine such 'simple' questions should not take more than a month to answer, specially for the persistent editor of this page, CGH. A bit harder question is shortly on the way regarding the basis of this statement, and disregarding Sinha's important heuristic conclusion on the two cnjectures: FC vs CGM. — Preceding unsigned comment added by 37.254.101.10 (talk) 08:11, 7 April 2015 (UTC)

When $\varepsilon$ is large, the inequality holds trivially. The interesting case is for small $\varepsilon$, as then $(2-\varepsilon )/e^{\gamma }>1$.—Emil J. 17:32, 4 May 2015 (UTC)

To J W Nicholson, please add the rest of the articles minus what was a false statement. There was a problem with the connection. Sorry. — Preceding unsigned comment added by 37.255.146.122 (talk) 13:49, 4 May 2015 (UTC) When epsilon is greater than 2 the inequality fails as it is noted by J W Nicholson above. Is there not an editor to compare the basis of of Pintz heuristics versus that of Sinha's via the above statemet? It seems like I am writing this for 3rd graders! — Preceding unsigned comment added by 37.254.26.152 (talk) 18:23, 4 May 2015 (UTC)

You are misreading Nicholson’s post and failing to work out elementary algebra at the same time. If $\epsilon >2$, the inequality holds. Every positive number is greater than every negative number.—Emil J. 14:16, 5 May 2015 (UTC)

Let's get to the gist of the question of the User 37.254.26.152 regarding the statement in question: Sinha's heuristics shows Cramer-Granville's conjecture is false while Firoozbakht's conjecture is true. On the other hand, Pintz heuristics, on the basis of Cramer-Granville's conjecture being true, which is false by the previous statement, shows the statement in question is inconsistent with Firoozbakht' conjecture. Therfore, one can conclude heuristically that the statement in question is true without the huristics of Sinha's. But we do have the Sinha's heuristics. So the statement in question is false. Under these conditions, perhaps, the best thing to do is to delete the statement in question and open a new page on the comparison of all prime gaps conjectures.There are enough articles and books already in circulation as up now.

Attention all editors: Materials and references are being removed by CRG that he himself has had also contributions to its formation!! Removal of the "List of prime numbers conjectures" of OEIS from this page is one of his recent blunders not to mentioning here on this page and other forums about Firoozbakht's conjecture that the "mathematical community thinks that the conjecture is believed to be false" or "oh well it is not a well-known conjecture". These comments and these reasoning it's not even worth a penny! He also has rudly calls the originator of the above mentioned sight on OEIS a "high school kid!" to add to the discredibility of the OEIS sight mentioned above. This comment is absolutely a pitty, shameful, and insult to the intelligence of all high schoolers who are users or editors. Please put back the "List of Prime numbers conjectures" from the OEIS back to where it it was on this page along with its reference, as you did other materials that he had removed from this page previously with bogous reasoning. I truely believe he must be on some sort of evaluation or probation for his eddditing on this page alone. — Preceding unsigned comment added by 5.219.238.182 (talk) 22:15, 29 January 2016 (UTC)

Verifying up to 84th maximal prime gap

Visser's paper, which was published when only 80 maximal prime gaps were known, clearly explains the procedure for testing a new prime gap. He's simplified the tests to g_n < f(n), where f(n) is a monotone increasing function of the prime number n, and g_n is the gap after the nth prime. It suffices to do these tests only when a new maximial prime gap is found.

This simplified tests are slightly stricter than the actual conjectures, but subsumes them; if the test passes, the conjectures are confirmed.

The functions are:

$f_{1}(n)=(\ln(n\ln n)-1)\ln(n\ln n),\quad {\text{ and}}$

$f_{2}(n)=(\ln(n\ln n)-1)(\ln(n\ln n)+\ln \ln n-\ln \ln(n\ln(n\ln n)),$

Where g_n < f₁(n) verifies the Firoozbakht and Nicholson conjectures, while g_n < f₂(n) verifies the Farhadian conjecture.

Now that three more maximal gaps are known, we can extend the test to them:

Gap #	gn	n	f1(n)	f2(n)
81	1552	426,181,820,436,140,029	1917.95	1914.08
82	1572	428,472,240,920,394,477	1918.43	1914.56
83	1676	477,141,032,543,986,017	1928.10	1924.23

The conjectures are confirmed for the additional maximal prime gaps, and any future gaps will certainly have f₂(n) > f₁(n) > 1924, so they are confirmed up to the first prime gap of size > 1924, at least. Which may be the 84th or some future maximal prime gap.

The 83rd and currently largest known maximal prime gap begins at 0x11FBD109969027C2F = 1.124 × 2⁶⁴. This is close enough to 2⁶⁴ that updating the page is not pressing. 97.102.205.224 (talk) 04:37, 7 November 2024 (UTC)