Talk:Bernoulli number

A small revert and why (II)

Concerning the changes by 24.7.187.190 on 25 May 2009.

The name of the man who introduced the numbers discussed in this entry is 'Jakob Bernoulli'.
Many writings of Jakob have his name printed like this:
> Jacobi Bernoulli Prof. Basil. & design. Acad. Rect. Ad Fratrem suum Johannem Bernoulli Professorem Groninganum Epistola ....

'Jacobi' is just the latinization of the name of Jakob. Is there any need to use a latinization or an anglicized name build on this latinization today? I do not think so. I think it is best to use the full civilian name of a person in Wikipedia.
>> What do other modern sources in the world use?
Jakob on Wikipedia: da, de, eo, es, hu, gl, it, pl, pt, sv, tr.
>> What do other modern sources in the English speaking community use?

> The Encyclopædia Britannica, writes 'Jakob'.
http://www.britannica.com/EBchecked/topic/62599/Jakob-Bernoulli
> Eric W. Weisstein (Wolfram, scienceworld) writes 'Jakob'.
http://scienceworld.wolfram.com/biography/BernoulliJakob.html
> The most authoritive bibliography of Bernoulli Numbers writes 'Jakob'.
http://www.mscs.dal.ca/~dilcher/bernoulli.html

Sure, there is no universal agreement. But there are good reasons to support the natural modern convention and not an out-dated one. Wirkstoff (talk) 08:59, 25 May 2009 (UTC)

A small revert and why

Concerning the changes by 71.169.78.55 on 7 March 2009.

The formula reads after the changes:

Let n > 2, even.

${\frac {1}{n+1}}\sum _{k=1}^{n-1}{\binom {n}{k}}B_{k}B_{n-k}=-B_{n}\quad {\text{(L. Euler, with a different notaton)}}$

Previously the formula read:

Let n > 0.

${\frac {1}{n}}\sum _{k=1}^{n}B_{k}B_{n-k}+B_{n-1}=-B_{n}\quad {\text{(L. Euler)}}$

There are many formulas on this page were the notation is different from those given by the original author. The section on Bernoulli's text gives a good illustration on that. However it does not make sense to add such an comment to every formula which is written today in a different notation than, say, in 1713, 1819 or 1877.

The original formula is true for
n=1,2,3,4,5,.. : 1/2, -1/6, 0, 1/30, 0, -1/42, ....
and implies the formula given by 71.169.78.55. 71.169.78.55's formula is true only for
n=4,6,8,10,.. : 1/30, -1/42, 1/30, -5/66, ...

Mathematics looks for formulas which are true in general, not for special cases. Therefore I revert the changes of 71.169.78.55. If this person thinks that the special cases deserve special attention then please write them as an additional formula, indicate the relation to the general formula and say why they deserve special attention compared to the general formula.

Two general remarks: The Bernoulli numbers B_3, B_5,... etc. are not of minor value because they are 0. Look at the combinatorial interpretations to see that this '0' means something which should whenever possible be taken into account. So don't kick them out of the formulas.

Second, notation. Writing B_{2n} instead of B_{n} clutters in most cases the formulas up. The formulas are more readable in the B_{n} form. But more importantly, note what Graham, Knuth and Patashnik wrote in a similar context:

"Somehow it seems more efficient to add up [n] terms instead of [2n] terms. But such temptations should be resisted; efficiency of computation is not the same as efficiency of understanding!" "Zero-valued terms cause no harm, and they often save a lot of trouble." (Concrete Mathematics, 2.1) Wirkstoff (talk) 16:37, 9 March 2009 (UTC)

Inconsistency

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.

There is no inconsistency at all. The article says:

The Bernoulli numbers also appear in the Taylor series expansion of the tangent...

and that is exactly correct; they appear in the Taylor series expansion. The Taylor series for tan(x) is:

$\tan x=\sum _{n=1}^{\infty }{\frac {B_{2n}(-4)^{n}(1-4^{n})}{(2n)!}}x^{2n-1}$

in which $B_{2n}$ denotes the 2nth Bernoulli number; so the Bernoulli numbers do indeed appear in the Taylor series expansion. The multiplication by (-4)ⁿ ensures that all the coefficients are positive. I hope this clears things up for you. -- Dominus 12:40, 29 July 2005 (UTC)

Yes it does, thanks.

Another question:

$\sum _{k=0}^{m-1}k^{n}=0^{n}+1^{n}+2^{n}+\cdots +{(m-1)}^{n}$

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 $(m-1)^{n}$ which has a degree of 'n'.

there is a sum of m terms involved here. If you take in the sum of powers only those summands say with k   >   m/2− 1 then the cut-off sum will be already bigger than

$(m/2)(m/2)^{n}={1 \over {2^{n+1}}}m^{n+1}\ .$

Therefore the order of magnitude (for fixed n) is asymptotically $O(m^{n+1})$ and not $O(m^{n})$ (the exact order is ${1 \over {n+1}}\,m^{n+1}+O(m^{n})).$.

Big O notation

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)

Relationship of Bernouilli numbers to Riemann zeta function

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)

Kowa Seki

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)

Recursive Definition

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 B_m+1? Psellus 23:19, 7 July 2006

It's recursive but not phrased in an explicitly recursive manner. For example, try substituting m = 3 and then solve for B₃. If you like you can rearrange the equation to show the recursion more explicitly, but it's quite elegant the way it's written currently. Dmharvey 23:48, 7 July 2006 (UTC)

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)

Asymptotic expansion

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)

It is apparently the ordinary asymptotic formula with a Stirling-like approximation for the factorial. Perhaps it could be useful for computing Bernoulli numbers with double (16-digit) precision? Nevertheless, it seems redundant to mention in this article since anyone who'd be interested in an n-digit approximation for large Bernoulli numbers could simply take the original asymptotic formula and substitute an n-digit factorial approximation of his own choice. Maybe the section could be reworded to say something to this effect? By the way, the article should also mention how to compute Bernoulli numbers exactly with the zeta function and the von Staudt-Clausen theorem. Fredrik Johansson 19:50, 18 February 2007 (UTC)

An asymptotic expansion using the factorial n! is in many cases not so useful as an expansion without n!. This is perceived for example on Mathworld [Bernoulli Number, formula (35)] which displays the 'standard' asymptotic form. However, the formula on Mathworld is by far not as efficient as the formula given here. Moreover, not every user of the Bernoulli numbers is ready to substitute first an approximation of the factorial in some other approximation of the Bernoulli numbers. Overall the formula looks like a good balance between simplicity and efficiency in a great many practical cases. —The preceding unsigned comment was added by 84.136.133.103 (talk) 23:41, 3 April 2007 (UTC).

Another set of identities

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)

No neutral point of view - removed paragraphs

There is no place for such remarks:

So Pavlyk could have saved his employer Wolfram a lot of resources had he used free and open source software. Pavlyk wrote: "The numerator begins with 571642756... and ends with ...597039303." This result is false as a quick check with the asymptotic formula in the next section reveals. Interestingly, none of the math-professors in the newsgroup seem to have noticed.

The following removed paragraphs are personal opinions of Peter Luschny, see an essay of Luschny The Bernoulli Confusion. This guy (is he an expert on number theory or a graduate mathematician at all? I don't think so!) rigidly claims, against the consensus of experts, to change the definition of Bernoulli numbers so that B(1) = 1/2. For both cases, B(1) = 1/2 or B(1) = -1/2, there are advantages and disadvantages. There is no favored viewpoint! One can start with the summation of powers up to n-1 (simple) or the Riemann zeta function on the negative x-axis (advanced).

This convention is looked at as an unhappy choice by many mathematicians as it is not in accordance with the viewpoint suggested by the Riemann zeta function. Therefore more and more modern writers depart from it. John H. Conway for example uses the convention B_n = B_n(1) in his books. He remarked that this convention also makes "certain formulas more aesthetically pleasing to our eyes".

and is often preferred by number theorists

In conclusion: The Bernoulli numbers B_n admit a variety of different representations. Since not all of these agree with one another conventions are introduced. Which among them should be adopted may depend not only on mathematical insight. But mathematical insight, simplicity and aesthetics suggest that the convention supporting the concordance is the best one in almost all of the cases.

Indeed for many mathematicians there is no valid alternative to the above concordance as they agree with G. H. Hardy: "The mathematician's patterns, like the painter's or the poet's must be beautiful; the ideas like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics."

Cryptic passage

After the words

The combinatorics of this representation can be seen from:

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)

My understanding is that the sign is (−1)^n+k multiplying the term containing the number of words of length n on k distinct symbols, as in the summation formula which precedes the passage in question. I don't know if there is an easy way to make this more visually apparent in the formula. siℓℓy rabbit (talk) 03:00, 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)

...OK, maybe he's just neglecting the sign in the case where the term is zero. Not the clearest way of writing. Michael Hardy (talk) 04:16, 8 July 2008 (UTC)

Sign

I remove this:

The von Staudt-Clausen representation of the Bernoulli numbers gives in a natural way B₁ = 1/2.This confirms the observation, that the deeper arithmetical properties of the Bernoulli numbers do not support the convention B₁ = −1/2. (See also the section on conventions above.)

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)

Silly Rabbit, I think the change you reverted was correct

Look at this formula:

$B_{n}=\sum _{k=0}^{n}(-1)^{k+n}{\frac {k!}{k+1}}\left\{{\begin{matrix}n\\k\end{matrix}}\right\}\ .$

Take n=1.

For k=0, the summand is (-1)^1 0!/1 (1 Choose 0), which is -1.

For k=1, the summand is (-1)^2 1!/2 (1 Choose 1), which is +1/2.

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)

Except that $\left\{{\begin{matrix}n\\k\end{matrix}}\right\}$ is not a binomial coefficient, but a Stirling number of the second kind. It measures the number of distinct words of length n that can be formed out of a set of k symbols. Thus in particular

$\left\{{\begin{matrix}1\\0\end{matrix}}\right\}=0$

since exactly zero words of length one can be formed out of a set of zero symbols. siℓℓy rabbit (talk) 16:38, 9 July 2008 (UTC)

All right. I stand corrected. Loisel (talk) 16:47, 9 July 2008 (UTC)

Verifiability and fair attribution.

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

$|B_{2n}|\sim 4{\sqrt {\pi n}}\left({\frac {n}{\pi e}}\cdot {\frac {480n^{2}+9}{480n^{2}-1}}\right)^{2n}.$

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

$|B(1000)|\approx 5.318704469415522033\ldots \times 10^{1769}\,$

which is off only by three units in the least significant digit displayed.

(*) by

$|B_{2n}|\sim 4{\sqrt {\pi n}}\left({\frac {n}{\pi e}}\right)^{2n}.$

(**) 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 B_2n.

$4{\sqrt {\pi n}}\left({\frac {n}{\pi e}}\right)^{2n}\left[1+{\frac {1}{24n}}\right]<|B_{2n}|<4{\sqrt {\pi n}}\left({\frac {n}{\pi e}}\right)^{2n}\left[1+{\frac {1}{24n}}\left(1+{\frac {1}{24n}}\right)\right]$

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

$4{\sqrt {\pi n}}\left({\frac {n}{\pi e}}\right)^{2n}<|B_{2n}|<5{\sqrt {\pi n}}\left({\frac {n}{\pi e}}\right)^{2n}$

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)

-

>> 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?

What are telling me? I wrote: "It might very well be the case that this

requirement is not met." And I wrote "I respect the requirement of verifiability

put forward by the rules of Wikipedia." So the type of your answer is absolutely

misplaced. You do not need to tell me what I can read and obviously did read on

the respective pages of Wikipedia policy. You are ignoring what I wrote but give

the impression that there is a need to tell me about. This is quite impertinent.

And at the same time you do not answer my main question: Why did you leave the

formulas on the page and deleted the attribution only? This is what I really want

to know. This is why I object your editorial practice. This does not meet the needs

of Wikipedia in any way. Your answer is still missing.

>> 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.

How can you say that? I did not give "a Harvard reference like (Luschny, 2007)"

nor any other kind of reference. There was a discussion in the newsgroup

de.sci.mathematik where I spoke about this formula.

http://groups.google.com/group/de.sci.mathematik/browse_frm/thread/4a980ba9ca71f0df/

I looked up the editing history. From what I see I conclude that I published the

formula on my website and that someone put this formula to Wikipedia.

I also often wrote about integer sequences in newsgroups and there are quite

a few of these sequences to be found on OEIS published by people who read these

threads and care more to publish these sequences then I do.

I reiterate the question:

As a reader, are you able to verify the material which is attributed to Ramanujan?

Why did you not delete the attribution to Ramanujan and left only his formulas?

"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."

Ok. Where are references to Ramanujan's work? Why did you not remove it?

As a reader, are you able to verify the material which is "pseudo-referenced"

to von Ettingshausen?

Why did you not delete the attribution to von Ettingshausen and left only his formulas?

"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."

Ok. Where are references to von Ettingshausen's work? Why did you not remove it?

And so on. It is not difficult to extend this list. As far as I can judge your

editing does not work on the standards of Wikipedia but on your own ones, and

these are quite arbitrary.

Peter Luschny

I deleted the attribution and replaced it with a {{citation needed}} tag. The material was not properly cited, and Google scholar did not turn up any hits for Luschny 2007. I'm sorry that you were unable to publish your original research, but Wikipedia is not the place to do it. I also acknowledge that there are other substantial problems with the referencing of the article: this was just the first one that I noticed, and so attempted to fix. I am now in the process of adding {{citation}} templates, so that other bogus references can be worked out appropriately. siℓℓy rabbit (talk) 01:23, 23 December 2008 (UTC)

I think there are two issues.

1. Mr. Luschny wants fair attribution. This is easy; just replace "(Peter Luschny, 2007)" with "due to Peter Luschny".
2. Wikipedia needs verifiable sources. This is easy and already done; just replace the broken citation with {{citation needed}}. If a citation is not provided eventually, then the fact will be removed, but there is no need to do this immediately if the fact itself is not disputed (it may be published later with attribution to Luschny by some other mathematician who read the estimate on the usenet newsgroup).

If there are no objections, I suggest adding back the sharper asymptotics and inequalities by Peter Luschny with the corresponding attribution and citation needed cleanup tags. Finding specific publications of the estimates is actually fairly hard, and often a decade goes by after the community knows of the sharper estimate before someone actually records it in a reliable source (for instance, how many subgroups does the symmetric group on n points have? I think there is 7 year gap between announcement of the sharp result in the community and the first reliable source stating that it is true, only the nearly sharp estimate is proven that I have seen). JackSchmidt (talk) 01:46, 23 December 2008 (UTC)

This seems to be a reasonable compromise. At any rate, this at least provides the desired attribution without masquerading as a citation, as the "(Luschny, 2007)" format certainly suggested. siℓℓy rabbit (talk) 02:05, 23 December 2008 (UTC)

Fair enough, but I am still quite skeptical. The previous version of this article said it relied on work of Nemes. Tracking down this reference I found it was published in a place called Stan's Library which boasts on its front page "Get published for free with the unique warranty that your work will receive immediate attention!" (the web sites emphasis not mine). This seems pretty far from from peer reviewed, and I am skeptical if any reasonable source will show up. If they don't appear in some reasonable time frame we should remove the material. If it turns out there is a 7 year gap, we should wait 7 years then include it. How many times was it announced that the Poincare conjecture had been proven? Thenub314 (talk) 14:40, 23 December 2008 (UTC)

I think the material should be removed. I tend towards including things which others would delete but I do think that the wiki notability guideline should be adhered to, especially in the case of an author with their own original research. Dmcq (talk) 21:26, 23 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)

True, then again I don't think WP need mention things if nobody else does. Does the approximation bounds section need to be in at all? - people will just wonder if they are missing something and wonder why it is so wide with that multiply by 5 in there. Dmcq (talk) 14:41, 31 December 2008 (UTC)

Actually I already cut the "inequality" section and some of the blatter in the "asymptotic" section. It might be OK to keep the asymptotic itself, though; some mathematical software packages like PARI can compute approximations of the Bernoulli numbers for large indices, so it's possible that they make use of a formula such as that one. One would need to check the relevant papers. Bikasuishin (talk) 15:09, 31 December 2008 (UTC)

Are these values correct?

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)

Who found the Bernoulli numbers first

(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)

This page says "Faulhaber did not discover the Bernoulli numbers but Jacob Bernoulli refers to Faulhaber in Ars Conjectandi published in Basel in 1713, eight years after Jacob Bernoulli died, where the Bernoulli numbers (so named by De Moivre) appear." McKay (talk) 04:04, 20 April 2009 (UTC)

Mistake in Ars Conjectandi

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)