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Archive 1 | Archive 2 |
Inconsistency here: Bernoulli numbers as defined on Bernouilli number page are alternately negative and positive.
But Taylor series for Tan(x) and Cot(x) use Benouilli numbers that are all positive. Formula for Taylor series should use absolute value |Bn| and not Bn.
Or Bernouilli numbers should be defined as all positive.
Yes it does, thanks.
Another question:
for various fixed values of n. The closed forms are always polynomials in m of degree n + 1
But term of highest degree appears to be which has a degree of 'n'.
Thanks for the timely revert, Dmharvey. I mistakenly thought you were using the omega notation in its historical sense, equivalent to big O. Elroch 22:21, 10 February 2006 (UTC)
I could be mistaken, but I don't see the recursive definition as being recursive. Maybe the 0 on the right hand side should be Bm+1? Psellus 23:19, 7 July 2006
I was afraid it would turn out to be something like this. OK, I will look at it harder. Thanks very much. Psellus 23:54, 7 July 2006 (UTC)
Hi -
I've got another set of identities based on an recursive definition, which I found recently, and which is not much known.
However, scanning some internet resources, I found the basic idea also mentioned in Zhi Wei Sun's article about "courious results concerning Bernoulli and Euler-polynomials", where he cites this relation according to von Ettinghaus in the early 19'th century.
This recursive definition relates some basic number-theoretic sequences in a very simple scheme. Would it be appropriate to link to this article of mine?
GeneralizedBernoulliRecursion.pdf
--Gotti 21:08, 18 February 2007 (UTC)
I made a check about that formula, it is ok but not especially good. The ordinary formula which is B(n) = 2*n!/Pi^n/2^n (shown just above) is much better : for n=1000 the approximation is good for the first 300 digits(!) and that one is good to something like 20 digits. I don't see the point of showing that formula. Maybe is it new but by far less efficient than the usual one. In my opinion that formula should be removed and we could maybe put a reference to that guy that found it. The litterature on bernoulli numbers is quite large and I am sure many authors looked at these pages.
Plouffe 08:29, 18 February 2007 (UTC)
A recent edit removed the assertion that the Bernoulli numbers were first studied by Bernoulli, and instead attributed them to the great Japanese mathematician Seki "in 1683", and asserting that Bernoulli did not study them until "the 18th Century". The implication here is that Bernoulli was greatly anticipated by Seki. But Bernoulli (1654-1705) and Seki (1642-1708) were contemporaries, and without two dates, I am reluctant to believe any claims of priority.
If anyone has any real information, I would be glad to hear it. Meantime, I am going to change the article again to note Seki's discovery. -- Dominus 01:15, 28 March 2006 (UTC)
the great JAPANESE mathematician did it FIRST! KOWASHITA! Washowasho! 128.226.162.163 ( talk)japniggerfromhell —Preceding comment was added at 17:39, 27 November 2007 (UTC)
After the words
There follows a sequence of expressions in which the pattern is clear EXCEPT for the pattern in the plus and minus signs. I left a message on the talk page of the user who wrote that passage inquiring about it, but he's been away from us for six days now, and left no email address. Can anyone explain that pattern? Michael Hardy ( talk) 02:26, 8 July 2008 (UTC)
No, that is inconsistent with what appears on the page. The cases of B1 and B3 should have a minus sign in the term with the empty set. Either you're wrong or what user:Wirkstoff wrote there is wrong. Michael Hardy ( talk) 04:01, 8 July 2008 (UTC)
I remove this:
I hold this to be (unfounded) opinion. If all the v. Staudt sums were 1, it would be differenct, but they are, including all the odd values, infinity, (0 or 1), 1, .5, 1, .5, 1, .5, 1, .5, 1, ,5, 1, .5, 2, .5, -6... Who shall say that 0 or 1 is more natural for the second member of this series? Septentrionalis PMAnderson 00:50, 9 July 2008 (UTC)
Look at this formula:
Take n=1.
Hence, B_1 is -1/2.
However, in the change you reverted, you now have B_1 = 1/2.
I.e., (1 Choose 0) is |{\emptyset}|=1, not |\emptyset|=0.
Loisel ( talk) 15:50, 9 July 2008 (UTC)
All right. I stand corrected. Loisel ( talk) 16:47, 9 July 2008 (UTC)
I tried to check the table below with the asymptotic formula now given on the content page. Some digits are true, some not. I think the table has to be put under quarantine here until someone has checked it. Hopefully there is a better asymptotic expansion so that we can decide this question. Don't forget to tell us how you did! And please give a checkable reference.
n | N | D | Bn = N / D |
102 | -9.4598037819... × 1082 | 33330 | -2.8382249570... × 1078 |
103 | -1.8243104738... × 101778 | 342999030 | -5.3187044694... × 101769 |
104 | -2.1159583804... × 1027690 | 2338224387510 | -9.0494239636... × 1027677 |
105 | -5.4468936061... × 10376771 | 9355235774427510 | -5.8222943146... × 10376755 |
106 | -2.0950366959... x 104767553 | 936123257411127577818510 | -2.2379923576... × 104767529 |
107 | -4.7845869850... × 1057675291 | 9601480183016524970884020224910 | -4.9831764414... × 1057675260 |
108 | -1.8637053533... × 10676752608 | 394815332706046542049668428841497001870 | -4.7204482677... × 10676752569 |
The displayed values for n = 107 and n = 108 were computed in less than one second with the von Staudt-Clausen formula and the asymptotic formula given below.
Peter Luschny —Preceding unsigned comment added by 85.179.164.224 ( talk) 18:32, 22 December 2008 (UTC)
I affirm that I published the approximation formula and the inclusion formulas cited in the article. To the best of my knowledge they were new when published in January 2007. I think they should be published by Wikipedia if due attribution to the author is made. I think they should not be published without attribution.
Silly Rabbit removed the attribution to the author of the formulas but not the formulas nor did he gave any reason why he did so.
Why? Notability? Worthy of notice? I think they are worthy of notice.
Verifiability? It might very well be the case that this requirement is not met. Not in the sense of correctness, even Simon Plouffe says on this page that the formula is correct (in fact every freshman in mathematics should be able to verify them) but in the sense of 'source for quotations'.
But in this case the formulas are to be deleted, not the attribution only! The editorial practice which Silly Rabbit shows is more than questionable.
I respect the requirement of verifiability put forward by the rules of Wikipedia. But I also think that their is room for common sense. Is their a quote for each mathematical formula displayed on Wikipedia? I doubt that. I think it would be absurd to require this in such a narrow sense.
However, I can not agree with the arbitrariness shown in the editing practice of Silly Rabbit. Publish a result and give due credit or don't publish the result for whatever reason. Therefore I revert the paragraphs containing the formulas to something which is similar to what it was before they were included.
(*) I replace in the section 'Asymptotic approximation'
Substituting an asymptotic approximation for the factorial function in this formula gives an asymptotic approximation for the Bernoulli numbers. For example
This formula (Peter Luschny, 2007) is based on an approximation of the factorial function given by Gergo Nemes in 2007. For example this approximation gives
which is off only by three units in the least significant digit displayed.
(*) by
(**) And I replace in the section 'Inequalities'
The following two inequalities (Peter Luschny, 2007) hold for n > 8 and the arithmetic mean of the two bounds is an approximation of order n−3 to the Bernoulli numbers B2n.
Deleting the squared brackets on both sides and replacing on the right hand side the factor 4 by 5 gives simple inequalities valid for n > 1. These inequalities can be compared to related inequalities for the Euler numbers.
For example the low bound for 2n = 1000 is 5.31870445... × 101769, the high bound is 5.31870448... × 101769 and the mean is 5.31870446942... × 101769.
(**) by
Peter Luschny
P.S.
http://www.luschny.de/math/primes/bernincl.html
http://www.luschny.de/math/factorial/approx/SimpleCases.html —Preceding
unsigned comment added by
85.179.164.224 (
talk)
16:30, 22 December 2008 (UTC)
The reason I removed the pseudo-referenced material was that there was no reference in the References section, so that the needs of WP:V were clearly not met. As a reader, I was unable to verify the material. If you are able to provide a proper reference to the material, including a properly peer-reviewed journal, then it can stay in the article. Otherwise, Wikipedia is not the place to publish original research, even if it appears elsewhere in a self-published resource. Is this published in a peer-reviewed source or isn't it? The links you have provided, while interesting and probably suitable for an external links section of the article, do not seem to me suitable to base sections of the article on, per Wikipedia policy. At any rate, if you give a Harvard reference like (Luschny, 2007) then you really ought also to say where the material was published, just like if you were to try to write an article in a scientific journal. siℓℓy rabbit ( talk) 22:32, 22 December 2008 (UTC)
(unindent) The material at stake is a trivial application of the Stirling formula that anyone can derive on the back of an enveloppe. That doesn't warrant either publication in a peer-reviewed journal (try it and you'll be politely directed to a community college mathematical gazette at best) or formal citation. Bikasuishin ( talk) 13:23, 31 December 2008 (UTC)
Either there's a mistake in the png, or there's a mistake in the Ars Conjectandi itself. The last term of the sum(n9) should be -3/20 instead of -1/12. An easy way to verify this is to put in 1 for n. All the coefficients in that row, when added, should equal 1. Epte ( talk) 06:18, 27 May 2009 (UTC)
(I read in J.H. Conway's "The Book of Numbers", page 108:) Those constants in Faulhaber's formula are known as Bernoulli numbers because they are discussed at length in Ars Conjectandi (1713), Jacob Bernoulli's posthumous masterpiece, in which the latter points out that they were originally discovered by Johann Faulhaber.
(This is not quite in line with what the Wikipedia section says.) —Preceding unsigned comment added by Puddington ( talk • contribs) 21:46, 19 April 2009 (UTC)
There have been a few news reports of a 16 y.o Iraqi boy in Sweden with a "formula to explain the calculation of Bernoulli numbers." References [1] or Google translation of swedish news site. The second reference includes the formula
—Preceding unsigned comment added by Zeimusu ( talk • contribs) 09:21, 28 May 2009
This page is currently in the "Integer sequence" category. Since Bernoulli numbers are not integers, maybe it should be moved to the parent category, "Sequences and series". (I didn't see a "rational number sequences" category.) -- Spiffy sperry ( talk) 19:27, 31 May 2009 (UTC)
I just read about these numbers in the news (mentioned above) and came here hoping to learn why they were important, but other than a mention of number theory my question remains unanswered. To all the smart people who write the math wikis, could you please remember us simple folk who don't already know about these subjects and what to learn the basics of "what" and "why?." Thank you. —Preceding unsigned comment added by Skintigh ( talk • contribs) 14:44, 1 June 2009 (UTC)
I am the strongest of all advocates of informality, and also a strong advocate of formality. They both need to be there. Michael Hardy ( talk) 17:59, 2 June 2009 (UTC)
Agreed, that topology article is basically useless for 99% of people reading it. I already know a bit about topology but that first paragraph could have been describing anything. In fact, I first learned about topology in 5th grade, and the Boston Museum of Science has an exhibit on it. I wonder how many excited young minds come home from there, look up the subject on Wikipedia, and instantly lose interest and go back to the Wii. (Not that I'm saying we should write at a 5th grade level...) As for formality, I'm all for it. But "formal" != "only elites and PhDs are allowed to understand what we are talking about and our article should be useless to all others." It reminds me of some undergrad texts I've read that can't be understood until you have a PhD. Not helpful. All I ask is just a brief paragraph on "what" and "why" before launching into the gobbledygook.-- Skintigh ( talk) 18:29, 3 June 2009 (UTC)
I have redirected Mohamed Altoumaimi to this page, due to extensive media coverage (primarily in Swedish language media), eg:
-- Mais oui! ( talk) 20:18, 28 May 2009 (UTC)
I know, I can't believe it either. Teens nowadays are really starting to upstage us adults.
(here is the full article)
STOCKHOLM (AFP) – A 16-year-old Iraqi immigrant living in Sweden has cracked a maths puzzle that has stumped experts for more than 300 years, Swedish media reported on Thursday.
In just four months, Mohamed Altoumaimi has found a formula to explain and simplify the so-called Bernoulli numbers, a sequence of calculations named after the 17th century Swiss mathematician Jacob Bernoulli, the Dagens Nyheter daily said.
Altoumaimi, who came to Sweden six years ago, said teachers at his high school in Falun, central Sweden were not convinced about his work at first.
"When I first showed it to my teachers, none of them thought the formula I had written down really worked," Altoumaimi told the Falu Kuriren newspaper.
He then got in touch with professors at Uppsala University, one of Sweden's top institutions, to ask them to check his work.
After going through his notebooks, the professors found his work was indeed correct and offered him a place in Uppsala.
But for now, Altoumaimi is focusing on his school studies and plans to take summer classes in advanced mathematics and physics this year.
"I wanted to be a researcher in physics or mathematics; I really like those subjects. But I have to improve in English and social sciences," he told the Falu Kuriren. —Preceding unsigned comment added by Philemmons ( talk • contribs) 06:29, 30 May 2009 (UTC)
What was his contribution? The yahoo blurb doesn't actually include or explain the formula. Also, I second the view that a simple redirect is insufficient; I actually think it's worse than nothing. Without reference to the kids name anywhere in the article, it seems more mocking than helpful. -(anonymous)
Reference 3 links to an article on Naharnet. The link is temporary and no longer goes to the correct story. This link to the same story may be permanent: http://www.naharnet.com/domino/tn/NewsDesk.nsf/AwayPolitics/5DAFD8132D1DFCC5C22575C4002BAD8F?OpenDocument Xot ( talk) 20:02, 30 May 2009 (UTC)
I'm just here to say that Jmk and Dcmq are right. Melchoir ( talk) 22:43, 30 May 2009 (UTC)
One of the references for this is incorrect. Currently citation #3, to [2] should go to [3]. Atonix ( talk) 13:12, 1 June 2009 (UTC)
Well, the current (frozen) article text has things backwards: it states the "correctly simplifying the computation" as a fact, while demoting the "discovered before and well known" to something that mathematicians "claim". According to the sources, it should be other way round.
Based on these sources, a more faithful representation of the events is: A high-school student re-discovered a well-known formula. There was a short media hype based on the misunderstanding that a 300-year-old problem has now been solved. But such a short-lived media hype is hardly notable enough for Wikipedia (Wikinews might be a better place), so I would say it does not belong to this article. If someone disagrees, please show a respectable source for the novelty, otherwise it is just gossip. -- Jmk ( talk) 08:03, 2 June 2009 (UTC)
The story has now been removed from the article as non-notable. However, eager contributors keep [6] [7] bringing it back. Presumably these contributors are in good faith, and sincerely believe that a 300-year-old problem was now solved for the first time. After all, the original (false) story got much better media coverage than its meager correction. Should we give in and include a sentence or two about the media circus, just to make things clear? -- Jmk ( talk) 07:52, 4 June 2009 (UTC)
AMorozov edited the lead. It says now: "They [the Bernoulli numbers] were first studied by the Swiss mathematician Jakob Bernoulli and the Japanese mathematician Seki Kōwa at around the same time." This needs some citation.
The introduction says: "At approximately the same time in Japan an equivalent method for calculating sums of powers was discovered by Seki Kōwa. However, Seki did not present his method as a formula based on a sequence of constants." Note the difference: 'method for calculating sums of powers' versus 'a single sequence of constants B0, B1, B2, … which provide a uniform formula for all sums of powers'. This article is about these constants, not about 'methods for calculating sums of powers'.
AMorozov, please give a reliable reference which affirms that the constants in Bernoulli's formula were known to Seki Kōwa. This is absolutely necessary for such a claim. Wirkstoff ( talk) —Preceding undated comment added 18:59, 16 August 2009 (UTC).
{{
citation}}
: Missing or empty |title=
(
help)CS1 maint: unflagged free DOI (
link) —
Dominus (
talk)
17:28, 18 August 2009 (UTC)The article began with an extremely long historical discussion, including the section "Reconstruction of 'Summae Potestatum'", before getting into any of the mathematical issues. I moved the history down a long way. The first section in the article now discusses what I think is the most salient fact about the Bernoulli numbers, which is that they appear in the formulas for the sums of the first nth powers. This is the context in which they were first discovered by both Bernoulli and Seki, and the context in which they are most likely to be of interest to the general reader. — Dominus ( talk) 14:32, 31 August 2009 (UTC)
I undid the following contribution by Yahord:
Rodrigues-like formula. This formula was invented by V.M. Kalinin.
This is why: The identity says:
where Tk is (up to sign) the (2k+1)th coefficient of the exponential expansion of tan, (the tangent (or "Zag") numbers). This is explained in detail in the article (section "An algorithmic view: the Seidel triangle"). Obviously the tangent numbers can be computed in various ways; if this is of interest this can be stated in an article on the tangent numbers. Wirkstoff ( talk) 23:11, 31 August 2009 (UTC)
I was dismayed to see that the actual values of the Bernoulli numbers are not mentioned anywhere in the article before section 17. Is there some reason why there is no tabulation of the numbers earlier, perhaps in the lede section, or in a sidebar adjacent to the lede section? This seems very strange to me. — Dominus ( talk) 14:11, 31 August 2009 (UTC)
I do not understand your last changes. For example it now reads: "Bernoulli's formula was generalized by V. Guo and J. Zeng to a q-analog (Guo & Zeng 2005)." Before it read: "Faulhaber's formula was generalized by V. Guo and J. Zeng to a q-analog (Guo & Zeng 2005)." The paper which is referenced (did you look at it?) is: "Guo, Victor J. W.; Zeng, Jiang (2005), "A q-Analogue of Faulhaber's Formula for Sums of Powers". For me your changes make little sense. Please, can you comment on it? Wirkstoff ( talk) 21:26, 3 September 2009 (UTC)
Faulhaber's formula
Bernoulli's formula is sometimes called Faulhaber's formula. There is no evidence which justifies this nomenclature. Johann Faulhaber found remarkable ways to calculate sum of powers but he never stated Bernoulli's formula.
Faulhaber realized that for odd m, Sm(n) is not just a polynomial in n but a polynomial in the triangular number N = n(n + 1)/2. For example Faulhaber's formulas read as follows:
To call Bernoulli's formula Faulhaber's formula does injustice to Bernoulli and simultaneously hides the genius of Faulhaber as Faulhaber's formula is in fact more efficient than Bernoulli's formula. According to Knuth ( Knuth 1993) a rigorous proof of Faulhaber’s formula was first published by Carl Jacobi in 1834 ( Jacobi 1834) . Donald E. Knuth's in-depth study of Faulhaber's formula concludes:
“Faulhaber never discovered the Bernoulli numbers; i.e., he never realized that a single sequence of constants B0, B1, B2, … would provide a uniform
for all sums of powers. He never mentioned, for example, the fact that almost half of the coefficients turned out to be zero after he had converted his formulas for from polynomials in N to polynomials in n.” ( Knuth 1993, p. 14)
Faulhaber's formula was generalized by V. Guo and J. Zeng to a q-analog ( Guo & Zeng 2005) . Wirkstoff ( talk) 10:55, 4 September 2009 (UTC)
I decided to change the language used to describe the relationship of the Bernouilli numbers to the Riemann zeta function, which grated with me as it stood. As I understand it, two sequences are the same "up to a factor" if one is a constant multiple of the other, and describing one sequence as "essentially" another sequence was wooly language at best. Elroch 22:29, 10 February 2006 (UTC)
I would say that the Bernoulli numbers are related to the Riemann zeta function for positive integers, and not for negative integers, as stated here. The usual convention of the zeta function is to write fractions. ( Gio74 ( talk) 12:21, 12 September 2009 (UTC))
The formula for sum-of-powers can be generalized to a real power:
is the bernoulli numbers with .
It follows from the Euler–Maclaurin formula with and
-- 77.127.51.123 ( talk) 08:29, 7 October 2009 (UTC)
![]() | This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
Archive 1 | Archive 2 |
Inconsistency here: Bernoulli numbers as defined on Bernouilli number page are alternately negative and positive.
But Taylor series for Tan(x) and Cot(x) use Benouilli numbers that are all positive. Formula for Taylor series should use absolute value |Bn| and not Bn.
Or Bernouilli numbers should be defined as all positive.
Yes it does, thanks.
Another question:
for various fixed values of n. The closed forms are always polynomials in m of degree n + 1
But term of highest degree appears to be which has a degree of 'n'.
Thanks for the timely revert, Dmharvey. I mistakenly thought you were using the omega notation in its historical sense, equivalent to big O. Elroch 22:21, 10 February 2006 (UTC)
I could be mistaken, but I don't see the recursive definition as being recursive. Maybe the 0 on the right hand side should be Bm+1? Psellus 23:19, 7 July 2006
I was afraid it would turn out to be something like this. OK, I will look at it harder. Thanks very much. Psellus 23:54, 7 July 2006 (UTC)
Hi -
I've got another set of identities based on an recursive definition, which I found recently, and which is not much known.
However, scanning some internet resources, I found the basic idea also mentioned in Zhi Wei Sun's article about "courious results concerning Bernoulli and Euler-polynomials", where he cites this relation according to von Ettinghaus in the early 19'th century.
This recursive definition relates some basic number-theoretic sequences in a very simple scheme. Would it be appropriate to link to this article of mine?
GeneralizedBernoulliRecursion.pdf
--Gotti 21:08, 18 February 2007 (UTC)
I made a check about that formula, it is ok but not especially good. The ordinary formula which is B(n) = 2*n!/Pi^n/2^n (shown just above) is much better : for n=1000 the approximation is good for the first 300 digits(!) and that one is good to something like 20 digits. I don't see the point of showing that formula. Maybe is it new but by far less efficient than the usual one. In my opinion that formula should be removed and we could maybe put a reference to that guy that found it. The litterature on bernoulli numbers is quite large and I am sure many authors looked at these pages.
Plouffe 08:29, 18 February 2007 (UTC)
A recent edit removed the assertion that the Bernoulli numbers were first studied by Bernoulli, and instead attributed them to the great Japanese mathematician Seki "in 1683", and asserting that Bernoulli did not study them until "the 18th Century". The implication here is that Bernoulli was greatly anticipated by Seki. But Bernoulli (1654-1705) and Seki (1642-1708) were contemporaries, and without two dates, I am reluctant to believe any claims of priority.
If anyone has any real information, I would be glad to hear it. Meantime, I am going to change the article again to note Seki's discovery. -- Dominus 01:15, 28 March 2006 (UTC)
the great JAPANESE mathematician did it FIRST! KOWASHITA! Washowasho! 128.226.162.163 ( talk)japniggerfromhell —Preceding comment was added at 17:39, 27 November 2007 (UTC)
After the words
There follows a sequence of expressions in which the pattern is clear EXCEPT for the pattern in the plus and minus signs. I left a message on the talk page of the user who wrote that passage inquiring about it, but he's been away from us for six days now, and left no email address. Can anyone explain that pattern? Michael Hardy ( talk) 02:26, 8 July 2008 (UTC)
No, that is inconsistent with what appears on the page. The cases of B1 and B3 should have a minus sign in the term with the empty set. Either you're wrong or what user:Wirkstoff wrote there is wrong. Michael Hardy ( talk) 04:01, 8 July 2008 (UTC)
I remove this:
I hold this to be (unfounded) opinion. If all the v. Staudt sums were 1, it would be differenct, but they are, including all the odd values, infinity, (0 or 1), 1, .5, 1, .5, 1, .5, 1, .5, 1, ,5, 1, .5, 2, .5, -6... Who shall say that 0 or 1 is more natural for the second member of this series? Septentrionalis PMAnderson 00:50, 9 July 2008 (UTC)
Look at this formula:
Take n=1.
Hence, B_1 is -1/2.
However, in the change you reverted, you now have B_1 = 1/2.
I.e., (1 Choose 0) is |{\emptyset}|=1, not |\emptyset|=0.
Loisel ( talk) 15:50, 9 July 2008 (UTC)
All right. I stand corrected. Loisel ( talk) 16:47, 9 July 2008 (UTC)
I tried to check the table below with the asymptotic formula now given on the content page. Some digits are true, some not. I think the table has to be put under quarantine here until someone has checked it. Hopefully there is a better asymptotic expansion so that we can decide this question. Don't forget to tell us how you did! And please give a checkable reference.
n | N | D | Bn = N / D |
102 | -9.4598037819... × 1082 | 33330 | -2.8382249570... × 1078 |
103 | -1.8243104738... × 101778 | 342999030 | -5.3187044694... × 101769 |
104 | -2.1159583804... × 1027690 | 2338224387510 | -9.0494239636... × 1027677 |
105 | -5.4468936061... × 10376771 | 9355235774427510 | -5.8222943146... × 10376755 |
106 | -2.0950366959... x 104767553 | 936123257411127577818510 | -2.2379923576... × 104767529 |
107 | -4.7845869850... × 1057675291 | 9601480183016524970884020224910 | -4.9831764414... × 1057675260 |
108 | -1.8637053533... × 10676752608 | 394815332706046542049668428841497001870 | -4.7204482677... × 10676752569 |
The displayed values for n = 107 and n = 108 were computed in less than one second with the von Staudt-Clausen formula and the asymptotic formula given below.
Peter Luschny —Preceding unsigned comment added by 85.179.164.224 ( talk) 18:32, 22 December 2008 (UTC)
I affirm that I published the approximation formula and the inclusion formulas cited in the article. To the best of my knowledge they were new when published in January 2007. I think they should be published by Wikipedia if due attribution to the author is made. I think they should not be published without attribution.
Silly Rabbit removed the attribution to the author of the formulas but not the formulas nor did he gave any reason why he did so.
Why? Notability? Worthy of notice? I think they are worthy of notice.
Verifiability? It might very well be the case that this requirement is not met. Not in the sense of correctness, even Simon Plouffe says on this page that the formula is correct (in fact every freshman in mathematics should be able to verify them) but in the sense of 'source for quotations'.
But in this case the formulas are to be deleted, not the attribution only! The editorial practice which Silly Rabbit shows is more than questionable.
I respect the requirement of verifiability put forward by the rules of Wikipedia. But I also think that their is room for common sense. Is their a quote for each mathematical formula displayed on Wikipedia? I doubt that. I think it would be absurd to require this in such a narrow sense.
However, I can not agree with the arbitrariness shown in the editing practice of Silly Rabbit. Publish a result and give due credit or don't publish the result for whatever reason. Therefore I revert the paragraphs containing the formulas to something which is similar to what it was before they were included.
(*) I replace in the section 'Asymptotic approximation'
Substituting an asymptotic approximation for the factorial function in this formula gives an asymptotic approximation for the Bernoulli numbers. For example
This formula (Peter Luschny, 2007) is based on an approximation of the factorial function given by Gergo Nemes in 2007. For example this approximation gives
which is off only by three units in the least significant digit displayed.
(*) by
(**) And I replace in the section 'Inequalities'
The following two inequalities (Peter Luschny, 2007) hold for n > 8 and the arithmetic mean of the two bounds is an approximation of order n−3 to the Bernoulli numbers B2n.
Deleting the squared brackets on both sides and replacing on the right hand side the factor 4 by 5 gives simple inequalities valid for n > 1. These inequalities can be compared to related inequalities for the Euler numbers.
For example the low bound for 2n = 1000 is 5.31870445... × 101769, the high bound is 5.31870448... × 101769 and the mean is 5.31870446942... × 101769.
(**) by
Peter Luschny
P.S.
http://www.luschny.de/math/primes/bernincl.html
http://www.luschny.de/math/factorial/approx/SimpleCases.html —Preceding
unsigned comment added by
85.179.164.224 (
talk)
16:30, 22 December 2008 (UTC)
The reason I removed the pseudo-referenced material was that there was no reference in the References section, so that the needs of WP:V were clearly not met. As a reader, I was unable to verify the material. If you are able to provide a proper reference to the material, including a properly peer-reviewed journal, then it can stay in the article. Otherwise, Wikipedia is not the place to publish original research, even if it appears elsewhere in a self-published resource. Is this published in a peer-reviewed source or isn't it? The links you have provided, while interesting and probably suitable for an external links section of the article, do not seem to me suitable to base sections of the article on, per Wikipedia policy. At any rate, if you give a Harvard reference like (Luschny, 2007) then you really ought also to say where the material was published, just like if you were to try to write an article in a scientific journal. siℓℓy rabbit ( talk) 22:32, 22 December 2008 (UTC)
(unindent) The material at stake is a trivial application of the Stirling formula that anyone can derive on the back of an enveloppe. That doesn't warrant either publication in a peer-reviewed journal (try it and you'll be politely directed to a community college mathematical gazette at best) or formal citation. Bikasuishin ( talk) 13:23, 31 December 2008 (UTC)
Either there's a mistake in the png, or there's a mistake in the Ars Conjectandi itself. The last term of the sum(n9) should be -3/20 instead of -1/12. An easy way to verify this is to put in 1 for n. All the coefficients in that row, when added, should equal 1. Epte ( talk) 06:18, 27 May 2009 (UTC)
(I read in J.H. Conway's "The Book of Numbers", page 108:) Those constants in Faulhaber's formula are known as Bernoulli numbers because they are discussed at length in Ars Conjectandi (1713), Jacob Bernoulli's posthumous masterpiece, in which the latter points out that they were originally discovered by Johann Faulhaber.
(This is not quite in line with what the Wikipedia section says.) —Preceding unsigned comment added by Puddington ( talk • contribs) 21:46, 19 April 2009 (UTC)
There have been a few news reports of a 16 y.o Iraqi boy in Sweden with a "formula to explain the calculation of Bernoulli numbers." References [1] or Google translation of swedish news site. The second reference includes the formula
—Preceding unsigned comment added by Zeimusu ( talk • contribs) 09:21, 28 May 2009
This page is currently in the "Integer sequence" category. Since Bernoulli numbers are not integers, maybe it should be moved to the parent category, "Sequences and series". (I didn't see a "rational number sequences" category.) -- Spiffy sperry ( talk) 19:27, 31 May 2009 (UTC)
I just read about these numbers in the news (mentioned above) and came here hoping to learn why they were important, but other than a mention of number theory my question remains unanswered. To all the smart people who write the math wikis, could you please remember us simple folk who don't already know about these subjects and what to learn the basics of "what" and "why?." Thank you. —Preceding unsigned comment added by Skintigh ( talk • contribs) 14:44, 1 June 2009 (UTC)
I am the strongest of all advocates of informality, and also a strong advocate of formality. They both need to be there. Michael Hardy ( talk) 17:59, 2 June 2009 (UTC)
Agreed, that topology article is basically useless for 99% of people reading it. I already know a bit about topology but that first paragraph could have been describing anything. In fact, I first learned about topology in 5th grade, and the Boston Museum of Science has an exhibit on it. I wonder how many excited young minds come home from there, look up the subject on Wikipedia, and instantly lose interest and go back to the Wii. (Not that I'm saying we should write at a 5th grade level...) As for formality, I'm all for it. But "formal" != "only elites and PhDs are allowed to understand what we are talking about and our article should be useless to all others." It reminds me of some undergrad texts I've read that can't be understood until you have a PhD. Not helpful. All I ask is just a brief paragraph on "what" and "why" before launching into the gobbledygook.-- Skintigh ( talk) 18:29, 3 June 2009 (UTC)
I have redirected Mohamed Altoumaimi to this page, due to extensive media coverage (primarily in Swedish language media), eg:
-- Mais oui! ( talk) 20:18, 28 May 2009 (UTC)
I know, I can't believe it either. Teens nowadays are really starting to upstage us adults.
(here is the full article)
STOCKHOLM (AFP) – A 16-year-old Iraqi immigrant living in Sweden has cracked a maths puzzle that has stumped experts for more than 300 years, Swedish media reported on Thursday.
In just four months, Mohamed Altoumaimi has found a formula to explain and simplify the so-called Bernoulli numbers, a sequence of calculations named after the 17th century Swiss mathematician Jacob Bernoulli, the Dagens Nyheter daily said.
Altoumaimi, who came to Sweden six years ago, said teachers at his high school in Falun, central Sweden were not convinced about his work at first.
"When I first showed it to my teachers, none of them thought the formula I had written down really worked," Altoumaimi told the Falu Kuriren newspaper.
He then got in touch with professors at Uppsala University, one of Sweden's top institutions, to ask them to check his work.
After going through his notebooks, the professors found his work was indeed correct and offered him a place in Uppsala.
But for now, Altoumaimi is focusing on his school studies and plans to take summer classes in advanced mathematics and physics this year.
"I wanted to be a researcher in physics or mathematics; I really like those subjects. But I have to improve in English and social sciences," he told the Falu Kuriren. —Preceding unsigned comment added by Philemmons ( talk • contribs) 06:29, 30 May 2009 (UTC)
What was his contribution? The yahoo blurb doesn't actually include or explain the formula. Also, I second the view that a simple redirect is insufficient; I actually think it's worse than nothing. Without reference to the kids name anywhere in the article, it seems more mocking than helpful. -(anonymous)
Reference 3 links to an article on Naharnet. The link is temporary and no longer goes to the correct story. This link to the same story may be permanent: http://www.naharnet.com/domino/tn/NewsDesk.nsf/AwayPolitics/5DAFD8132D1DFCC5C22575C4002BAD8F?OpenDocument Xot ( talk) 20:02, 30 May 2009 (UTC)
I'm just here to say that Jmk and Dcmq are right. Melchoir ( talk) 22:43, 30 May 2009 (UTC)
One of the references for this is incorrect. Currently citation #3, to [2] should go to [3]. Atonix ( talk) 13:12, 1 June 2009 (UTC)
Well, the current (frozen) article text has things backwards: it states the "correctly simplifying the computation" as a fact, while demoting the "discovered before and well known" to something that mathematicians "claim". According to the sources, it should be other way round.
Based on these sources, a more faithful representation of the events is: A high-school student re-discovered a well-known formula. There was a short media hype based on the misunderstanding that a 300-year-old problem has now been solved. But such a short-lived media hype is hardly notable enough for Wikipedia (Wikinews might be a better place), so I would say it does not belong to this article. If someone disagrees, please show a respectable source for the novelty, otherwise it is just gossip. -- Jmk ( talk) 08:03, 2 June 2009 (UTC)
The story has now been removed from the article as non-notable. However, eager contributors keep [6] [7] bringing it back. Presumably these contributors are in good faith, and sincerely believe that a 300-year-old problem was now solved for the first time. After all, the original (false) story got much better media coverage than its meager correction. Should we give in and include a sentence or two about the media circus, just to make things clear? -- Jmk ( talk) 07:52, 4 June 2009 (UTC)
AMorozov edited the lead. It says now: "They [the Bernoulli numbers] were first studied by the Swiss mathematician Jakob Bernoulli and the Japanese mathematician Seki Kōwa at around the same time." This needs some citation.
The introduction says: "At approximately the same time in Japan an equivalent method for calculating sums of powers was discovered by Seki Kōwa. However, Seki did not present his method as a formula based on a sequence of constants." Note the difference: 'method for calculating sums of powers' versus 'a single sequence of constants B0, B1, B2, … which provide a uniform formula for all sums of powers'. This article is about these constants, not about 'methods for calculating sums of powers'.
AMorozov, please give a reliable reference which affirms that the constants in Bernoulli's formula were known to Seki Kōwa. This is absolutely necessary for such a claim. Wirkstoff ( talk) —Preceding undated comment added 18:59, 16 August 2009 (UTC).
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Dominus (
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17:28, 18 August 2009 (UTC)The article began with an extremely long historical discussion, including the section "Reconstruction of 'Summae Potestatum'", before getting into any of the mathematical issues. I moved the history down a long way. The first section in the article now discusses what I think is the most salient fact about the Bernoulli numbers, which is that they appear in the formulas for the sums of the first nth powers. This is the context in which they were first discovered by both Bernoulli and Seki, and the context in which they are most likely to be of interest to the general reader. — Dominus ( talk) 14:32, 31 August 2009 (UTC)
I undid the following contribution by Yahord:
Rodrigues-like formula. This formula was invented by V.M. Kalinin.
This is why: The identity says:
where Tk is (up to sign) the (2k+1)th coefficient of the exponential expansion of tan, (the tangent (or "Zag") numbers). This is explained in detail in the article (section "An algorithmic view: the Seidel triangle"). Obviously the tangent numbers can be computed in various ways; if this is of interest this can be stated in an article on the tangent numbers. Wirkstoff ( talk) 23:11, 31 August 2009 (UTC)
I was dismayed to see that the actual values of the Bernoulli numbers are not mentioned anywhere in the article before section 17. Is there some reason why there is no tabulation of the numbers earlier, perhaps in the lede section, or in a sidebar adjacent to the lede section? This seems very strange to me. — Dominus ( talk) 14:11, 31 August 2009 (UTC)
I do not understand your last changes. For example it now reads: "Bernoulli's formula was generalized by V. Guo and J. Zeng to a q-analog (Guo & Zeng 2005)." Before it read: "Faulhaber's formula was generalized by V. Guo and J. Zeng to a q-analog (Guo & Zeng 2005)." The paper which is referenced (did you look at it?) is: "Guo, Victor J. W.; Zeng, Jiang (2005), "A q-Analogue of Faulhaber's Formula for Sums of Powers". For me your changes make little sense. Please, can you comment on it? Wirkstoff ( talk) 21:26, 3 September 2009 (UTC)
Faulhaber's formula
Bernoulli's formula is sometimes called Faulhaber's formula. There is no evidence which justifies this nomenclature. Johann Faulhaber found remarkable ways to calculate sum of powers but he never stated Bernoulli's formula.
Faulhaber realized that for odd m, Sm(n) is not just a polynomial in n but a polynomial in the triangular number N = n(n + 1)/2. For example Faulhaber's formulas read as follows:
To call Bernoulli's formula Faulhaber's formula does injustice to Bernoulli and simultaneously hides the genius of Faulhaber as Faulhaber's formula is in fact more efficient than Bernoulli's formula. According to Knuth ( Knuth 1993) a rigorous proof of Faulhaber’s formula was first published by Carl Jacobi in 1834 ( Jacobi 1834) . Donald E. Knuth's in-depth study of Faulhaber's formula concludes:
“Faulhaber never discovered the Bernoulli numbers; i.e., he never realized that a single sequence of constants B0, B1, B2, … would provide a uniform
for all sums of powers. He never mentioned, for example, the fact that almost half of the coefficients turned out to be zero after he had converted his formulas for from polynomials in N to polynomials in n.” ( Knuth 1993, p. 14)
Faulhaber's formula was generalized by V. Guo and J. Zeng to a q-analog ( Guo & Zeng 2005) . Wirkstoff ( talk) 10:55, 4 September 2009 (UTC)
I decided to change the language used to describe the relationship of the Bernouilli numbers to the Riemann zeta function, which grated with me as it stood. As I understand it, two sequences are the same "up to a factor" if one is a constant multiple of the other, and describing one sequence as "essentially" another sequence was wooly language at best. Elroch 22:29, 10 February 2006 (UTC)
I would say that the Bernoulli numbers are related to the Riemann zeta function for positive integers, and not for negative integers, as stated here. The usual convention of the zeta function is to write fractions. ( Gio74 ( talk) 12:21, 12 September 2009 (UTC))
The formula for sum-of-powers can be generalized to a real power:
is the bernoulli numbers with .
It follows from the Euler–Maclaurin formula with and
-- 77.127.51.123 ( talk) 08:29, 7 October 2009 (UTC)