Talk:Fibonacci number/Archive 3

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Citation to Binet's vs. Abraham de Moivre's formula

In paragraph Fibonacci_number#Closed-form_expression citation is needed for disambiguation that closed-form formula was introduces by Abraham de Moivre and not Jacques Philippe Marie Binet. It can be found in the book The_Art_of_Computer_Programming and I think this book should be cited. — Preceding unsigned comment added by Milikicn (talkcontribs) 18:30, 19 August 2011 (UTC)

Simple is best

To initially demonstrate the relationship between the Fibonacci sequence and the Golden Ratio, the Kepler solution is clearly the best. It is the simplest, clearest and most obvious therefore the most elegant solution. The other solutions are definitely worthy of mention but they are needlessly complex answers where a direct answer to a very simple question is already available. The Kepler solution should be the first listed followed by the Binet. Wading through the Binet solution only to find the obvious and to the point Kepler solution leads the reader to conclude that he has stumbled upon an Asperger's self stroking fest rather than an encyclopedia. (talk) 11:00, 4 September 2011 (UTC)

Editors here are unlikely to take your suggestions seriously if you cannot express them without throwing in gratuitous playground insults. Gandalf61 (talk) 12:47, 4 September 2011 (UTC)

Identities and combinatorial interpretations

There are two problems with the beginning of the "Identities" section. (1) The first sentence of this section asserts that "Most identities involving Fibonacci numbers draw from combinatorial arguments." This statement sounds subjective; unless reinforced by strong evidence I would remove it. In any case it's irrelevant to the statement of identities. (2) The first identity cannot be proved, as it is the definition. The proper way to handle it is to prove the "interpretation" given (without proof) in the previous section. That should be in a separate section on "Combinatorial interpretations of the Fibonacci numbers". Zaslav (talk) 01:29, 24 October 2011 (UTC)


The linked page misleadingly suggests that a certain Cody Birsner discovered the relationship between the series and the fraction, whereas it had been known for a considerable time before. Perhaps it would be better to link to another page, e.g. or or Dadge (talk) 20:50, 31 December 2011 (UTC)

I agree. Thanks for pointing this out. I changed it to:
Köhler, Günter (1985). "Generating functions of Fibonacci-like sequences and decimal expansions of some fractions" (PDF). The Fibonacci Quarterly 23 (1): 29–35. Retrieved December 31, 2011.  Unknown parameter |month= ignored (|date= suggested) (help)
which in turn cites some earlier papers from FQ. —Mark Dominus (talk) 22:04, 31 December 2011 (UTC)

add some formulas and proofs

put the even,odd,odd,even pattern on the article.and proofs.and before you do this:is there a pattern like this?yes or no and why?John kaiser (talk) 06:08, 30 January 2012 (UTC)

This pattern is mentioned in the section headed "Divisibility properties" and is described more generally in our article on Pisano periods. Gandalf61 (talk) 08:53, 30 January 2012 (UTC)

Recognizing Fibonacci Number

According to the Fibonacci number article "a positive integer  z is a Fibonacci number if and only if one of  5z^2 + 4 or  5z^2 - 4 is a perfect square."

However, the statement  5z^2+4=25^2 or  5z^2-4=25^2 does not imply that  z is a Fibonacci number. — Preceding unsigned comment added by Yonizilpa (talkcontribs) 19:10, 30 January 2012 (UTC)

You are starting at the wrong end. If z is an integer such that 5z^2+4 or 5z^2-4 is a perfect square then z is a Fibonacci number. So 5x1^2+4 =9, 5x2^2-4=16 and 5x3^2+4=49. It doesn't say there is a Fibonacci number corresponding to every square. Gandalf61 (talk) 19:31, 30 January 2012 (UTC)
In fact, the only squares that give Fibonacci numbers in the reverse direction are the squares of Lucas numbers. 1,2,3,4 and 7 are Lucas numbers, but 5 and 6 are not, so we have
which are all squares of Fibonacci numbers, but \frac{5^2 \pm 4}{5}\, and \frac{6^2 \pm 4}{5}\, are not squares of Fibonacci numbers. Gandalf61 (talk) 09:04, 31 January 2012 (UTC)

Thanks for the explanation I now realize my mistake. — Preceding unsigned comment added by Yonizilpa (talkcontribs) 13:10, 11 February 2012 (UTC)

A pyramid relating the golden ratio and the tetrahedron

If you take a regular tetrahedron and truncate(cut) it so that you keep the three original 60degree angles at one vertex but change the three lengths from that vertex to any three successive terms of the Fibonacci series then the base of the new pyramid will be the two internal diagonals of a pentagon and the corresponding side.The face with the corresponding side has the other sides Fib(n)and Fib(n+1),the angle opposite Fib(n) is 37.76124degrees and opposite Fib(n+1) is 82.23876degrees. One face of the pyramid with one of the internal diagonals as a side has other sides Fib(n+1) and Fib(n+2) and is similar to the previously mentioned face. The third face having the other internal diagonal as a side has other sides Fib(n)-angle opposite being 22.23876degrees- and Fib(n+2)-angle opposite being 97.76124degrees. Sabastianblak (talk) 23:20, 12 February 2012 (UTC)Bradley J. Grantham February 12,2012

The fibonacci numbers used for multiplication

The Fibonacci numbers are meso-American multiplication method. It may use any type of numbers I think.... — Preceding unsigned comment added by (talk) 01:26, 29 March 2012 (UTC)

Nobel Prize ?

Excuse me but, i win a nobel prize if i say that the fibonacci phenomenon is due to rotation of earth "plus" growth factor ???

for example... for the spiral of sunflowers, and sea animal shells, just draw a straight line...very slowly...maybe following the sun light... and the rotation of earth make (plus growth factor) it become a spiral... does anybody before me understand it ???

thank you.

M.G. — Preceding unsigned comment added by (talk) 15:39, 11 June 2012 (UTC)

See golden spiral. Gandalf61 (talk) 15:47, 11 June 2012 (UTC)

ty i mention to relate it to the rotation of earth? (or taking the sun as a polar star)? — Preceding unsigned comment added by (talk) 16:09, 11 June 2012 (UTC)

Well, you have a hypothesis, which is that phyllotaxis is caused by the movement of the sun across the sky. But that is just the first part of the scientific method. Next you need to make a prediction - for example, if the sun did not move across the sky, then plants would not show phyllotaxis. Then you need to devise and carry out an experiment that tests your prediction - for example, you could grow plants under artificial sun lamps so that the light under which they grow comes from a constant direction. If you did all this, and got a positive result from the experiment, then it would be interesting, but I really doubt it would qualify for a Nobel Prize. Gandalf61 (talk) 08:56, 12 June 2012 (UTC)

aaaa!! ty Gandalf61!! :) mmm i already have a certain idea..:) but i will make some experiment anyway. ty so much for your interest on my topic anyway! thank you. and...if i suppose is due to the earth magnetic field..? kinda like make grow a broccoflower inside a solenoid? i will try.

-mmm..nice reading comes out that golden ratio have something to do with pentagon, and pentagon with dodecahedron, a platonic solid.(and the that-time-believed shape of universe). Bye. — Preceding unsigned comment added by (talk) 18:34, 12 June 2012 (UTC)

first list of fibonacci numbers

in the image shouldn't the 7 be an 8? — Preceding unsigned comment added by JaysnArr (talkcontribs) 18:55, 5 October 2012 (UTC)

It says 8 for me but I see 7 in The article was recently changed to a wrong sequence with 7. It was fixed 10 minutes later.[1] PrimeHunter (talk) 19:02, 5 October 2012 (UTC)

Initial Fibonacci numbers

The original Fibonacci numbers began 1, 1, as everyone knows. In modern times mathematicians have often found it useful to start with 0, 1, 1, but often they have not found that useful and still begin with 1, 1. The article ought not to be prescribing usage to mathematicians, and I have revised the introduction accordingly. Some time ago, a person objected to any use of the beginning 1, 1, so I provide a reference for the 1, 1 sequence:

Matthias Beck and Ross Geoghegan, The Art of Proof: Basic Training for Deeper Mathematics. New York: Springer, 2010.

Zaslav (talk) 06:46, 9 October 2012 (UTC)

Addendum (not merely beating a dead horse, I'm sorry to say): In applications of Fibonacci numbers in science, the initial numbers seem to be 1, 1, not 0, 1, 1 (0 being inappropriate for them). That is certainly true in biology. Users of Fibonacci numbers in science (or anywhere else), not only professional mathematicians, ought to be taken account of in deciding what "the" initial numbers are. Zaslav (talk) 06:16, 23 October 2012 (UTC)
Starting with 0 makes both the generating function and the closed-form expression in terms of the golden ratio nicer.
It's not really an issue, though, since the mathematics and computer science references I see tend to refer to F_0=0, F_1=1, F_2=1, F_3=2, etc. That makes terms 1,2 of the sequence 1,1 for the life sciences, and "term 0" is there for those who want to start with 0. Husoski (talk) 18:45, 25 January 2013 (UTC)


From the page:

Variations of two earlier meters [is the variation]... For example, for [a meter of length] four, variations of meters of two [and] three being mixed, five happens. [works out examples 8, 13, 21]... In this way, the process should be followed in all mAtrA-vr.ttas (prosodic combinations).

I assume "mAtrA-vr.ttas" is some sort of transliteration, but I've never seen it before (and originally interpreted it as vandalism). I think we should mention what transliteration was used to make this a little less confusing. Cheers, — sligocki (talk) 15:44, 21 October 2012 (UTC)

Looks like a seven-bit way of writing mātrā-vṛttas, though I don't know if that's a word. —Tamfang (talk) 17:04, 21 October 2012 (UTC)
Looking at [2] these (mātrā-vṛttas) seem to be Metre used in Sanskrit poetry. There seem to be a few character conversion errors in the references both numbers 11 and 12 look dodgy. --Salix (talk): 18:09, 21 October 2012 (UTC)


The article Gopala (mathematician) redirects here (presumably because there is little to no (easily locatable) information regarding him, other than that he was involved in the development of the "Fibonacci" numbers. However, wouldn't it be intuitively better (controversial as it may be) to permit Gopala his own stub (given that it would likely be a complete article in terms of available information)? Additionally, the sources mentioned in the article, [4] and [5], may contain more information on him - perhaps someone with access to these could confirm this, and if so whether there is sufficient information available to justify a separate article for Gopala. - R160K (talk) 19:45, 3 November 2012 (UTC)


The photo in the icon (with the page from Liber Abaci) is broken. However, when is clicked, the original photo is right. Can someone restore the photo in the icon? (I don't know how to do this). Thanks. — Preceding unsigned comment added by (talk) 03:39, 24 November 2012 (UTC)

GA Review

This review is transcluded from Talk:Fibonacci number/GA1. The edit link for this section can be used to add comments to the review.

Reviewer: TonyTheTiger (talk · contribs) 14:26, 4 April 2013 (UTC)

I'll take a look at this one.--TonyTheTiger (T/C/BIO/WP:CHICAGO/WP:FOUR) 14:26, 4 April 2013 (UTC)

The external link tool to the right reveals one dead link that needs to be addressed.--TonyTheTiger (T/C/BIO/WP:CHICAGO/WP:FOUR) 23:18, 8 April 2013 (UTC)
The disambiguation link tool to the right reveals 4 issues to be addressed.--TonyTheTiger (T/C/BIO/WP:CHICAGO/WP:FOUR) 23:18, 8 April 2013 (UTC)
  • "saying that the cases for m beats (Fm+1) is obtained by adding a [S] to Fm cases and [L] to the Fm−1 cases" Makes little sense to me.--TonyTheTiger (T/C/BIO/WP:CHICAGO/WP:FOUR) 05:03, 12 April 2013 (UTC)
  • Revise each line of the first, second, third and fourth month to say how many rabbits and thus how many pairs. It gets confusing and I think some of your explaintions make it more so.--TonyTheTiger (T/C/BIO/WP:CHICAGO/WP:FOUR) 05:03, 12 April 2013 (UTC)
List of Fibonacci numbers
  • Whereas there seems to be a useful relationship with the bunnies justifying considering Fibonacci numbers, you present no reason for the relevance of the negafibonaccis.--TonyTheTiger (T/C/BIO/WP:CHICAGO/WP:FOUR) 05:09, 12 April 2013 (UTC)
Occurrences in mathematics
  • Most of this section is unsourced, leaving the reader to question the veracity of the claim and to wonder if these were selected from a much larger list of examples.--TonyTheTiger (T/C/BIO/WP:CHICAGO/WP:FOUR) 20:22, 14 April 2013 (UTC)
  • I am going to FAIL this article. There are too many subsections that are entirely unsourced. I am so far removed from my Masters in Stats that I keep looking for references that are not there. Any reader who wants to read this article will be at a loss to WP:V many important elements of the article.--TonyTheTiger (T/C/BIO/WP:CHICAGO/WP:FOUR) 20:31, 14 April 2013 (UTC)
  • This is so filled with jargon, I can barely read it. (talk) 03:17, 12 June 2014 (UTC)

Article name

It seems to me the current name "Fibonacci number" is not the best option. Wouldn't "Fibonacci numbers" (note the s), "Fibonacci series" or specially "Fibonacci sequence" be a better alternative? --Götz (talk) 04:43, 2 July 2013 (UTC)

What's wrong with the current title? "Not the best option" is too vague to be helpful. I suspect that "number" better fits WP:COMMONNAME than "series" or "sequence" (which you also haven't provided any specific reason for preferring). "Numbers" does not match WP:SINGULAR. —David Eppstein (talk) 04:49, 2 July 2013 (UTC)
I was not aware of WP:SINGULAR, although I find it odd that when describing a series/sequence, the page title is in singular, and the rest of the article is in plural. But there is also WP:PLURAL, under the Exceptions section, "Similarly, one is much more likely to mention the Bernoulli numbers than a particular Bernoulli number."
Then, following WP:Search engine test and considering WP:SINGULAR, it seems that "sequence" fits better with WP:COMMONNAME, as it can be seen in Google Ngram and Google Trends. But, what should we do regarding WP:SINGULAR and WP:PLURAL? --Götz (talk) 17:42, 2 July 2013 (UTC)
I agree that Fibonacci sequence is a better name. Google Trends (Thanks Götz!) supports the idea that "fibonacci sequence" is most common. And when recently doing research on the Fibonacci sequence (how I got here), I always searched for the more familiar "fibonacci sequence." Are there any good reasons or precedents for leaving the title as Fibonacci number? Tedsanders (talk) 03:07, 21 August 2013 (UTC)
Most Wikipedia articles about an individual sequence of integers are named "X number" rather than "X sequence". An article named "X sequence" is usually about a family of integer sequences with similar characteristics (examples are Hofstadter sequence, fractal sequence, complete sequence). But there are exceptions to this "rule" - for example, Padovan sequence, Golomb sequence and Euclid–Mullin sequence are all articles about individual sequences. Renaming this article to Fibonacci sequence (which is currenctly a redirect) has been occassionally suggested in the past - see this talk page archive - but never actioned AFAIK. Gandalf61 (talk) 09:54, 21 August 2013 (UTC)

Divisibility properties Error

It look like there is an Error Displayed under the Section "Divisibility properties" which is under "Primes and divisibility"

Check it out here: Fibonacci_number#Divisibility_properties

I don't know to code "math" so if anyone how knows whats going on in that section please fix it immediately!

If there is no Error displayed it might have been fixed or removed.

Thanks in advance,

GideonWanna talk? 04:51, 12 October 2013 (UTC)

Can you state exactly what is the error? The only one that I have found is the "it follows" of the second paragraph: the result does not follows immediately from the previous one, but is an easy consequence of the basic recurrence relation. I have corrected this. On the other hand, if the error lies in the math display, it is possible that it is not an error in the article, but in the data transmission; it seems that Internet did not work correctly yesterday. D.Lazard (talk) 07:53, 12 October 2013 (UTC)
hey D.Lazard I', sorry I think its the internet problem (My Fault!) Really sorry, I should have reloaded the webpage!! Sorry, Thanks though... GideonWanna talk? 10:17, 12 October 2013 (UTC)

"Vandalization" of MacTutor history of Mathematics reference

User:Wcherowi removed my so-called "tongue in cheek" reference to a publication from the MacTutor history of Mathematics archive, claiming that it is not published. This is absurd. The url was given to the publication itself. MacTutor publishes refereed articles ONLINE! MacTutor is a reputable archive of mathematical history. I want to put this material back. It is instructive.TonyMath (talk) 01:40, 21 March 2014 (UTC)

FYI, I would also like to point out that this published refereed paper is linked to the very biography of Fibonacci himself at the MacTutor historical archive in St-Andrews University. It's at the top of the list of "Additional material". I Also note that Ron Knott's material on the Fibonacci numbers is also online and cited in this very article. So what is this egregious comment "tongue-in-cheek" all about?
This preprint was submitted to MacTutor in March 2014. I hardly think it has been refereed in this short period. While I generally respect the MacTutor material, it can be of uneven quality. At best this could be considered an opinion piece (advancing a hypothesis) and would not be published in a legitimate scholastic journal. Its appearance on-line does not make it a reliable source. Even if I am wrong about it being a hoax, it is clearly a case of WP:TOOSOON. Bill Cherowitzo (talk) 04:51, 21 March 2014 (UTC)
I happen to know that the article had been seen long before the March date and was refereed by the editors themselves. Edmund F. Robertson had seen this article as far back as September 2013 i.e. several months before in fact. What is this stuff about MacTutor not being a legitimate scholastic journal? The editors of MacTutor material would not have linked the biography of Fibonacci to this paper had they not considered it acceptable.TonyMath (talk) 08:54, 21 March 2014 (UTC)
BTW, Reference to MacTutor appears in the Wikipedia site for Al-Karaji namely Note no. 3 and References_and_external_links. MacTutor is also cited in the general references of the Wikipedia site on the great Mathematicians Muḥammad ibn Mūsā al-Khwārizmī and Abū Kāmil Shujāʿ ibn Aslam. MacTutor is intensely involved in studying Mathematics of the middle-ages. So explain something to me: if MacTutor is an acceptable reference in these sites, then why not here?TonyMath (talk) 09:02, 21 March 2014 (UTC)
I checked the Wikipedia site MacTutor History of Mathematics archive and if you use the tools on the left and find what cites to that archive: you find a HUGE number of Wikipedia articles that cite MacTutor! It is so big, they have to be categorized in alphabetical order. I am all the more amazed at the claim that MacTutor is not a sufficiently credible citation for Wikipedia herein.TonyMath (talk) 10:03, 21 March 2014 (UTC)
You have to be careful with MacTutor. Their biographies are reliable but some of their other stuff is just student essays that should not be considered reliable. In particular, I do not believe that the source in question, [3], should be considered reliable. It is not one of the biographies, is labeled as a preprint, does not seem to have been reliably published, and contains what looks to me like decidedly fringey speculation. —David Eppstein (talk) 12:46, 22 March 2014 (UTC)
What I have to be careful about is this kind of bias but you and your colleagues will have your way, rest assured. Yes, the article was submitted as a preprint (of course, for the review process) but MacTutor would not have put it online and linked it to their very biography of Fibonacci even it had not been accepted for publication. I cannot buy these claims about MacTutor or some of MacTutor not being "reliable" subject to your evaluation. MacTutor is mentioned in the very Wikipedia site for Pythagoras. FYI, Do you know how many people out there insist that Wikipedia itself is not authoritative? But we can all get through this with discernment and objectivity. I can understand that something of an obscure speculative history from a recent publication might be premature for this site which focusses on the Mathematics of the Fibonacci sequence. I get that and I will no longer insist on mentioning this recent MacTutor content on this particular Wikipedia site but you and your colleagues can do yourself a favor by avoiding this ad-hominem bashing of the reliability of MacTutor's content (against Wikipedia's guidelines I might add). It is an affront to its editors and its authors especially since Wikipedia does cite MacTutor in so many cases. Moreover, I don't really believe that the true objection is really the Publisher of the material but rather its content and its implications. Some of the editors simply don't appreciate the content and its controversial implications e.g. how much Fibonacci virtually plagiarized Muslim scholarship. It would have been more honest to simply admit that. At any rate, I will no longer pursue the matter.TonyMath (talk) 03:36, 23 March 2014 (UTC)

Why are the sequence examples images?

The two examples at the top of the page are images, rather than simply text. Why is this preferable? Images with text are generally considered bad from a usability standpoint. Unlike Unicode characters with distant code points, these reside in ascii (except for the trailing ellipsis) so there's no compatibility issue. — Preceding unsigned comment added by Preinheimer (talkcontribs) 15:45, 8 September 2014 (UTC)

You mean the lists of numbers at the top of the article? It's because of Wikimedia's continued poor handling of mathematics markup. They are coded using the <math> tag in the source code of the article, and by default Wikimedia turns that into an image. If you turn on mathjax rendering in your preferences it will be formatted better. As for why they're coded that way: to make them look consistent with the other mathematical formulas in the article. —David Eppstein (talk) 16:15, 8 September 2014 (UTC)

Starting points

Can there be some explanation about the starting points 1,1 versus 0,1? Right now I am confused which one to pick, and why the "0,1" is called modern starting point. (talk) 11:48, 29 October 2014 (UTC)

They are both common and both have F1=1, F2=1, F3=2, F4=3, .... The only difference is whether you also have a term F0=0 before F1. That term is practical in many situations such as some formulas which would get more cumbersome without having F0. In some applications it is logical to include F0=0, while it can easily be omitted in others. Fibonacci himself started at 1 but that was in 1202. Modern mathematical texts usually include F0=0 which has no downside for mathematicians, but less formal presentations for lay people often omit it. PrimeHunter (talk) 12:34, 29 October 2014 (UTC)