Talk:Prime gap

Should the statements below the dashed line be placed in a separate section from "Upper bounds"? I ask because these statements don't imply anything with an upper bound, instead, they imply something with the lower bound count. If I am wrong, can some statement be placed in the article which makes this use clear with the upper bounds?

In 2005, Daniel Goldston, János Pintz and Cem Yıldırım proved that

${\displaystyle \liminf _{n\to \infty }{\frac {g_{n}}{\log p_{n}}}=0}$

and 2 years later improved this^[1] to

${\displaystyle \liminf _{n\to \infty }{\frac {g_{n}}{{\sqrt {\log p_{n}}}(\log \log p_{n})^{2}}}<\infty .}$

In 2013, Yitang Zhang proved that

${\displaystyle \liminf _{n\to \infty }g_{n}<7\cdot 10^{7},}$

meaning that there are infinitely many gaps that do not exceed 70 million.^[2] A Polymath Project collaborative effort to optimize Zhang’s bound managed to lower the bound to 4680 on July 20, 2013.^[3] In November 2013, James Maynard introduced a new refinement of the GPY sieve, allowing him to reduce the bound to 600 and show that for any m there exists a bounded interval containing m prime numbers.^[4] Using Maynard's ideas, the Polymath project improved the bound to 246;^[3][5] assuming the Elliott–Halberstam conjecture and its generalized form, N has been reduced to 12 and 6, respectively.^[3] — Preceding unsigned comment added by Reddwarf2956 (talk • contribs)

• I cannot understand: is there a conjecture (or, maybe, that is a trivial fact) that gaps could be enorm. large and, in fact, they could not have any certain bound? (in other words: some gap could be as large as you wish). --Tamtam90 (talk) 22:47, 6 January 2018 (UTC)

First occurrence prime gaps has two 20-digit first known occurrence prime gaps listed that might be large enough to be maximal. They are:

 gap = 1530   C?C   Bertil Nyman  2014   m = 34.52     20 decimal digits
 17678654157568190587  upper prime
 17678654157568189057  lower prime

 gap = 1550   C?C   Bertil Nyman  2014   m = 34.94     20 decimal digits
 18361375334787048247  upper prime
 18361375334787046697  lower prime

C = conventional gap; ? = first known occurrence; C = confirmed bounding primes

(Last updated 0800 GMT 03 January 2018.)

These gaps are near the lower edge of the expected range of possible maximal gaps. As they are quite a bit beyond the search limit, they could easily be pushed aside by others. They might be maximal with additional intermediate maximal gaps.

agb — Preceding unsigned comment added by 173.233.167.63 (talk) 19:45, 6 January 2018 (UTC) '

While they might be maximal gaps, they are not currently. At the current rate of progress, the exhaustive search by the PGS group will cover these by mid-2018, answering conclusively. Is there a pressing need to show the speculation? DAJ NT (talk) 09:45, 7 January 2018 (UTC)

Not "pressing", but might be helpful. That is why I put it only on the Talk page. I am not requesting that it be added to the main article, which I could do myself if I thought it was that important. Someone interested in maximal prime gaps might not have run across this information or its source. Note that as no one knows any formula for the locations of maximal prime gaps then studying the empirical patterns is one possible way to guess a formula. As the search is painfully slow and requires access to a modern very fast machine (which many people don't), such information just might help someone solve the problem. Also, the "Prime Gap Searches project at the Mersenne Forum" seems to be merely a somewhat haphazard collection of hundreds of blogs without any overall chart of progress (such as what ToS had for the Goldbach's Conjecture search). If they do have such a chart then they need to put it more prominently on their site. agb — Preceding unsigned comment added by 173.233.167.63 (talk) 18:49, 9 January 2018 (UTC)

I am thinking this is starting to get unmanagable. Which implies that a different table or tables should be made. I am thinking Smallest Ten Maximal Gaps and Largest Ten Known Maximal Gaps.

John W. Nicholson (talk) 14:37, 26 February 2018 (UTC)

My opinion is this isn't yet too much, as these are primary data points and the rate they're added is fairly slow. For severe data hoarding, see Fermat pseudoprime, Euler–Jacobi pseudoprime, and some related pages -- I got tired of fighting to remove the cruft and just let all those random tables get in the way of the article. Culling big lists early isn't a bad thing if we can reference a source instead. Since the OEIS links give us all the data, I'm ok with trimming it if you think it's best. DAJ NT (talk) 20:40, 27 February 2018 (UTC)

There is a lot of interest in the growth and limits of maximal gaps, and a lot of computation was used to find this data. I think we should keep the full list of known maximal gaps. It fits on around a single normal screen and only three have been found since 2006. It may not become unmanagable for centuries if computers continue to improve at current rates. PrimeHunter (talk) 22:33, 27 February 2018 (UTC)

As an example of the interest, https://scholar.google.com/scholar?q=1693182318746371 currently shows 27 hits on a maximal gap found in 1999. PrimeHunter (talk) 22:48, 27 February 2018 (UTC)

Immediately after the maximal gap table in "Numerical results" is a paragraph starting: "In 1931, E. Westzynthius proved that maximal prime gaps grow more than logarithmically."...

This should be in the next section "Further results", but I am not sure just where it should go. Perhaps at the very start of "Lower bounds", but someone who is more sure should decide.

agb

I agree it should be moved and "Further results" seems to make the most sense, but am also not sure where. DAJ NT (talk) 00:22, 27 January 2018 (UTC)

In lower bounds, before Rankin's result, which supercedes this one. —David Eppstein (talk) 02:28, 27 January 2018 (UTC)

In row 30 of "The 77 known maximal prime gaps" these 4 numbers appear

30; 282; 436,273,009; 32,162,398;

The last (4th) number is incorrect. The correct number is 23,162,398. Note the transposition of digits.

It is also likely that the 4th number of row 65 is incorrect.

65; 1,184; 43,841,547,845,541,059; 1,175,662,926,421,598;

There appears to be an error in the 6th or 7th significant figure, although I can't compute the correct answer -- the numbers are just too big.

My method was to subtract the logarithmic integral up to the n'th prime (column 3) from n (column 4) and plot this difference (the ordinate) versus the row number in the table (column 1). The above two errors stand out dramatically as spikes in this plot.

108.20.242.117 (talk) 02:07, 3 February 2018 (UTC)

Well spotted. http://primes.utm.edu/nthprime/index.php#piofx says pi(436,273,009) = 23,163,298, so 32,162,398 has multiple errors. I haven't checked the other values but I guess OEIS:A005669 is correct. The linked https://oeis.org/A005669/b005669.txt says 65 1175661926421598. PrimeHunter (talk) 02:55, 3 February 2018 (UTC)

I have checked all values now and there were no other differences from OEIS:A005669. I have made the corrections. Thanks. PrimeHunter (talk) 14:26, 3 February 2018 (UTC)

First occurrence prime gaps points out that the search limit for maximal prime gaps has reached up to 2^64, and that this is the limit for programs using hardware register calculations. From now on, programs will be forced to use software calculations, which can be expected to slow the search "by one or two orders of magnitude".

Using the data in the time list, one might guess (if the current rate were to continue) that it could be 10 to 15 years before 10^20 is fully checked. The slowing would imply that it might be hundreds of years.

Since there average just over 4 maximal prime gaps per power of 10 number size, the new rate might be about 1 new maximal gap per 50 or more years.

Also, looking at other lists on the trnicely site, it is noticed that there are NO first occurrence prime gaps with either 21 or 22 digit primes. Maybe the hardware\software calculation problem affects the choice of where to look for large prime gaps.

Some of us wish there would be an improvement in hardware.

agb — Preceding unsigned comment added by 173.233.167.63 (talk) 22:43, 25 August 2018 (UTC)

Thomas Ray Nicely died September 11, 2019.

His internet site (trnicely.net) stayed up through November, but is defunct now. That site was the main source for news about prime gaps.

Is anyone planning to take over the dissemination of prime gap results?

In particular, the files allgaps.dat and merits.zip need to be kept updated, perhaps quarterly, at least annually, and made available at least on an ftp internet site.

The main problem would be in the checking of new gaps. A modern fast multicore machine would probably be needed.

If someone does this then please add a date code to the filename, perhaps merit201.zip and algap201.dat for the first ones in the year 2020. That way someone would not accidently overwrite older data.

If there is already another internet site with this information, then what is the URL?

agb — Preceding unsigned comment added by 143.43.144.168 (talk) 19:32, 10 December 2019 (UTC)

The last version of trnicely.net is mirrored at https://faculty.lynchburg.edu/~nicely/. https://gjhiggins.github.io/ is apparently planning to continue the record keeping. PrimeHunter (talk) 01:02, 11 December 2019 (UTC)

I conjectured that the sum of all partial sums (Pn+Pn+2-2Pn)/2.=0 as n increases. I tested this sum for primes up to 9,973 and found the sum to be zero. As a example, P7=17, P8=19, P9=23, the partial sum is +1. Since I cannot find this in this talk I am curious to know if this has been observed before. — Preceding unsigned comment added by 70.19.60.34 (talk) 02:19, 9 May 2020 (UTC) Stanley Kravitz, Ph.D. — Preceding unsigned comment added by 70.19.60.34 (talk) 02:23, 9 May 2020 (UTC)

This talk page is for discussing improvements to the article. General questions about mathematics can be asked at Wikipedia:Reference desk/Mathematics. Your formula is unclear to me (it reduces to the constant 1 if taken literally) so I don't know what you are asking. PrimeHunter (talk) 14:40, 9 May 2020 (UTC)

Oops: I understand your interpretation and thanks for pointing out an error in the formula. Pn is the nth prime, Pn+1 is the next prime, Pn+2 is the next prime. The corrected formula is (Pn+(Pn+2)-2(Pn+1))/2 where Pn is for example 17, +(Pn+2) is 23, and -2(Pn+1) i.e -2x19=-38. The total is 2 this divided by 2 is the partial sum 1. If all partial sums are added the total appears to be zero. — Preceding unsigned comment added by 70.19.60.34 (talk)

Every prime from P3 is added once when it's Pn+2, subtracted twice when it's Pn+1, and added once when it's Pn. This cancels out. If n goes from 1 to m then the only primes which don't cancel out are P1, P2, Pm+1, Pm+2. The total sum becomes (P1-P2-(Pm+1)+(Pm+2))/2, where P1-P2 = 2-3 = -1. Prime gaps after 3 are even so the sum is never an integer. PrimeHunter (talk) 17:26, 9 May 2020 (UTC)

I was fascinated by the graph plotted in the section "Conjectures about gaps between primes". May I know how the graph was plotted? Thank you😊 ISHANBULLS (talk) 13:17, 9 May 2020 (UTC)

@ISHANBULLS: commons:File:Wikipedia primegaps.png says "Excel diagram of relation between primegaps and primes", but that description is from the original 2011 version. The current 2013 version has the upload comment: "New version created with data from English-language Wikipedia. Own work. Created with OpenOffice.org Calc. Theoretical curves (parabolae) show maximal gaps according to conjectures due to Cramér (g = (log p)^2), to Cramér-Granville (g = 2exp(-γ)(lo...". It was made by User:Jeppesn who can be contacted at User talk:Jeppesn, or may reply here. He only has two edits since 2018. PrimeHunter (talk) 14:37, 9 May 2020 (UTC)

"For any integer n, the factorial n! is the product of all positive integers up to and including n. Then in the sequence n! + 2 , n! + 3 , … , n! + n the first term is divisible by 2, the second term is divisible by 3, and so on."

Is this correct? E.g., for n = 2, the second term (2! + 3 = 5) isn't divisible by 3. --Ex-prodigy (talk) 03:47, 7 July 2020 (UTC)

@Ex-prodigy: The last term in the mentioned sequence is n! + n. For n = 2 this is 2! + 2, so there is no second term 2! + 3. It's standard math language to speak of the second term or more in a sequence which may be shorter in some cases. There is no need to change the formulation, and doing so would be unnecessarily clumsy. The sequence can also be empty for n = 1. PrimeHunter (talk) 09:27, 7 July 2020 (UTC)

these both appear in the article.--142.163.195.46 (talk) 20:59, 27 May 2021 (UTC)

It seems that the figure peaks at multiples of 3 and not 6, and yet my modifications is reverted! Please have a look.

https://en.wikipedia.org/w/index.php?title=Prime_gap&diff=1168585045&oldid=1168575199 Mehdiabbasi (talk) 18:51, 3 August 2023 (UTC)

Note that the bars are at 2, 4, 6, ..., not 1, 2, 3, .... - CRGreathouse (t | c) 14:17, 4 August 2023 (UTC)

Do we really need to include a column for the prime index of each of the maximal gaps? With Kim Walisch's primecount, one may compute (for example) the number of primes under the largest known maximal gap in just a few minutes on a personal computer. The numbers don't seem relevant to much else in the page.

Also, the table is currently too wide to fit on the screen, on both mobile and web, and removing the column should fix that issue. 2001:56A:7060:8200:15F:4A11:CE0E:F8A5 (talk) 22:40, 22 November 2024 (UTC)

The trouble with Kim Walisch's primecount program is that is for the very small fraction of people who use C or C++. It also sounds as if it needs ultafast recent multicore computers. Finally (perhaps Firstly) it is on github with the usual multiple files and little if any instructions on what the heck to do. Since some people need the pi(x) function, my suggestion is to add a footnote saying "To seen the pi(p_n ) values, go to the history page 16:55, 20 October 2024 ." Hitting F11 gets the entire table except for the main title line. (It is hard to believe that someone with a machine that can do primecount doesn't have a modern widescreen monitor that does this.) Then PrintScreen and paste into Paint and save as .png for posting on your bulletin board. Or highlight and copy and paste into a word processor if you need to reenter the numerical values into other programs. agb 173.233.167.51 (talk) 20:07, 13 January 2025 (UTC)

The article mentions that the values of n (i.e. corresponding to the column that was removed) can be found in OEIS A005669, so it is not necessary to use primesieve at all. And it is true that the values of n do not really have any relevance to the article. Overall, on balance, I think the reasons for removing the column are much more convincing than the reasons for keeping it. 2001:56A:7060:8200:F54C:4CA3:54E3:7844 (talk) 20:35, 25 February 2025 (UTC)

The reason I'm looking at the article is that I was hoping to understand what maximum value of the prime gap would suffice to prove the Riemann Hypothesis. However I am not a mathematician and I think I have been confused by fuzzy writing in Things to Make and Do in the Fourth Dimension by Matt Parker. (In the form of a joke: "Is an amateur psychologist less or more dangerous than a sloppy mathematician?") It seems that if the maximum value of the gap can be bounded, then the Riemann Hypothesis would be proven (or easily disproven). On pages 151-2 he actually appears to be saying that the maximum gap between primes was proven (in 2013) and soon reduced to 5,414. Clearly I'm not understanding something here, and I was hoping this article would help me understand what it is. Shanen (talk) 20:23, 17 May 2025 (UTC)

Not sure if replying to myself is the best way to help other people understand the confusion, but... After more rereading I think this is what is going on: What Parker was talking about was actually the smallest known prime gap for which there are an infinite number of prime numbers that are separated by that gap, not the maximum gap between primes. (Some confusion because of the way it was separated from the twin primes question.) I was also partly misled by the algorithm that is frequently given for finding a prime with an arbitrarily large gap in front of it, and I'm still confused on that one because it appears that such an algorithm would be a royal road to the generation of arbitrarily long prime numbers... Shanen (talk) 20:31, 17 May 2025 (UTC)

You can generate arbitrarily large primes. There's always a prime between x and 2x (for x > 1, say), so just test each one until you find a prime.

Yes, the gap (now at 246) is not an upper bound on gaps between primes, but rather an upper bound on the smallest gap that occurs infinitely often. We don't know if there are infinitely many primes followed by a gap of 246, but there must be some g in {2, 4, 6, ..., 244, 246} such that there are infinitely many gaps of length g.

CRGreathouse (t | c) 19:02, 20 May 2025 (UTC)