Talk:Fibonacci's identity

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There is an article called "Brahmagupta's identity" about the same equality. Shouldn't this one be just a REDIRECT to the original one? Alex 04:28, 26 April 2006 (UTC)

The MathWorld article called "Brahmagupta's identity" makes it appear that that is a different identity. Could we settle that question before merging or redirecting? Michael Hardy 21:03, 26 April 2006 (UTC)

0. Oh, let's not move or merge anything before reaching a consensus.
1. I checked MathWorld article on Brahmagupta Identity. They are talking about exactly this one, (just presented in more generilized form). To see it for yourself, follow the link to "Brahmagupta matrix" and take a look at line 2 with t=-1.
2. I also checked MathWorld for "Fibonacci Identity". I found 2 things: first, line (3) gives you Brahmagupta identity (of course, you substitute a, b, c, d for x1, y1, x2, y2. Secondly, and more importantly, the reference (http://www.cis.upenn.edu/~wilf/AeqB.html) there points to a book "A=B", where on page 12 in chapter 1.16 the authors use term "Fibonacci identity" for the absolutely different identity regarding Fibonacci numbers, while identity we are discussing is not called that, but rather is mentioned unnamed on page 9 (chapter 1.3 "Polynomial identities") and in more generalized form on page 13 (chapter 1.17 "Symmetric function identities").
   Conclusion.
MathWorld confirms that identity in this article can be called Brahmagupta's and also MathWorld article on Fibonacci identity has a wrong reference. Add to this that Brahmagupta not only lived before Fibonacci, but that namely Fibonacii was translating his work and you may agree that this article should be merged into Brahmagupta's.--Alex 05:23, 27 April 2006 (UTC)

It seems to me, by clicking on the links, that this identity is called "Fibonacci's Identity" by the community (or at least by MathWorld) and "Brahmagupta's identity" seems to be something entirely different. Assuming that MathWorld is correct, it seems that the community calls this identity Fibonacci's identity, and therefore Wikipedia should respect that (otherwise we're doing WP:OR). I'd be very much in support of keeping the title named after Fibonacci, but then putting a note in the article that it is due to Brahmagupta, and then changing the text at Brahmagupta's identity to be in line with MathWorld. (Of course, if MathWorld is wrong and the community has yet different usage, this this changes everything.)--Deville (Talk) 22:02, 28 April 2006 (UTC)

Thank you for your comment, but let me repeat more clearly what I said above.

• 1. Community calls Fibonacci Identity an identity about Fibonacci numbers, namely
     F(n-1) * F(n+1) = (-1)^n + (Fn)^2.
     I can present zillion of for good references,

• • but one of them is right there on MathWorld in the "Fibonacci identity" article (http://www.cis.upenn.edu/~wilf/AeqB.html page 12 chapter 1.16).
  • Second online reference is here (http://mathforum.org/library/drmath/view/52718.html).
  • Third one is here (M. Werman and D. Zeilberger, A bijective proof of Cassini's Fibonacci identity, Discrete Math. 58 (1986), 109.)
  • One more, quite authoritive and very clear is here on page 6 (http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Simons/simons53.ps)
  • see also Halton, J. H. "On a Generalised Fibonacci Identity", The Fibonacci Quarterly (Feb 1965) 3, pp31-43.
  • or Rose, N.J. “A note on a Fibonacci Identity", The Fibonacci Quarterly 23, No.2 (1985)pp151-152.

In fact, no one exept for erroneous (check for yourself) refence in MathWorld (which they, hopefully will correct soon) calls sum of two squares identity by the name of Fibonacci.

2. On the other hand, sum of two squares identity (a^2 + b^2)(c^2 + d^2) = (ac-bd)^2 + (ad+bc)^2 (and its generalizations) are called Brahmagupta's

• • by the very same MathWorld in their article "Brahmagupta's identity"
  • by Weil in (Weil (1983) Number Theory: an approach through the history from Hammurapi to Legendre).

Interestingly, Weil has the following passage (pp.10-11): identity records of are not found in Europe "..until we find one, with an elaborate proof for it, in Fibonacci's Liber Quadratorum of 1225 (Leon.II.257-260 = LVE.prop.IV). Fibonacci claims no credit for it, and seems rather to treat it as a result, well-known to specialists but deserving wider diffusion."
Later Weil mentions Brahmagupta's name in relevance to "Brahmagupta's identity for the composition of the slightly more general forms (x^2 +- N y^2)(z^2 +- N t^2)". Of, course when N=-1 we get our identity in question.

I suggest, we a)notify MathWorld about their mistake, b)Change the Fibonacci identity article to talk about F(n-1) * F(n+1) = (-1)^n + (Fn)^2 and c)mention more generalized form(s) of Brahmagupta's identity in Brahmagupta's identity article as well as add Weil's note about Fibonacci proof.

--Alex 05:03, 29 April 2006 (UTC)

In fact, no one exept for erroneous (check for yourself) refence in MathWorld (which they, hopefully will correct soon) calls sum of two squares identity by the name of Fibonacci.

That is incorrect. Norman Wildberger calls it that in his book Divine Proportions: Rational Trigonometry and Universal Geometry. Michael Hardy 19:32, 29 April 2006 (UTC)

1. General note. You are (formally speaking) correct. My statement about "no one" is, formally speaking, incorrect. I didn't check all the references in mathematical literature. I did check, however, quite a few serious, reputable, authoritive and reliable sources (I mentioned a few above). No one in this sources, including the only article referred by MathWorld calls "sum of two squares identity" Fibonacci's. It does not mean that someone somewhere wouldn't attribute this identity to Fibonacci (read below).

2. Your particular source. Norman Wildberger's book. I didn't read it, and there is a reason for that: this book was published in 2005 [1]. Hm-m-m. The article in MathWorld [2] is copyrighted (and, I presume, published) by 1999. As we established, it has a single reference, which it uses erroneously . Given that Google search for "Fibonacci Identity" spits out that MathWorld's article first (and Wikipedia's second), I would be suspicious about any fresh source created in post Google IPO world. Wouldn't you? To illustrate my point take a look at (http://www.math.rutgers.edu/~erowland/pythagoreantriples.html), find "Fibonacci Identity" and see where it refers to.

3. I think it is time to correct both Wikipedia and MathWorld articles. And your reference just enforces the urgency.

--Alex 05:55, 30 April 2006 (UTC)

Cassini's identity

1. Let me adjust my previous comment about identity F(n-1) * F(n+1) = (-1)^n + (Fn)^2. Only some people call it Fibonacci's (because of Fibonacci's numbers), majority calls it Cassini's. Thus, until I (or anyone) find a definitely clarifying source I have to admit that there exist justification for calling "sum of two squares identity" Fibonacci's.

2. On merging of two articles (Fibonacci and Brahmagupta). I think we have 3 valid choices:

• either we can leave both articles as they are
• or we can expand Brahmagupta article to include generalized form of his identity and add a reference to Brahmagupta article from Fibonacci's
• or (better yet) we can make both articles to be just redirects to combo article "Brahamagupta and Fibonacci identities" in the same fashion as "Cassini and Catalan identities".

What do you think?

--Alex^talk 14:13, 2 May 2006 (UTC)