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Reputable on-line sources agree that Leonard 'Fibonacci' proved the nonexistence of positive-integer solutions to $c^4 - b^4 = a^2$ . Yet my change to Wikipedia to reflect this was reverted. I hope to get a consensus on what seems like an interesting historical question.

(Reputable sources include 1. http://www-history.mcs.st-andrews.ac.uk/Biographies/Fibonacci.html with "Fibonacci also proves many interesting number theory results such as: ... $x^4 - y^4$ cannot be a square."; and 2. http://books.google.com/books?id=dTVnPUl8OQ4C&pg=PA94 ... which I can't quote because "I've reached my viewing limit." ... J.D.A.)

I've prepared a webpage summarizing the question: http://fabpedigree.com/james/fibflt4.htm

Briefly, the issue seems to be about Leonardo's statement:

When $x > y$ ... then $x (x-y) \neq y (x+y)$ and from this it may be shown that no square number can be a congruum. For [then] ... the four factors $x$, $y$, $(x+y)$, $(x-y)$ must severally be squares which is impossible.

If he'd just added "by infinite descent" here, his "proof" would be valid, right?

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    Looking at your webpage, I note the following sentence: "Since it is minimal, x and y have no common factors and (possibly excepting 2), neither do x+y and x-y. For this reason, any squared odd factor of t occurs in only one of the terms x, y, x+y, x-y, so they are each themselves squares." However, x+y and x-y could each be divisible by an odd power of 2 (e.g., x+y is divisible by 2, and x-y is divisible by 8), and thus not be squares.2011-10-12
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    The first part of the highlighted "quote" follows from [the irrationality of square-root of two](http://en.wikipedia.org/wiki/Square_root_of_2), a secret "discovery" of the Pythagorean school: $x(x-y) = y(x+y)$ implies $2x^2 = (x+y)^2$. Although a full proof appears in Euclid's Elements, Book X, Prop. 117, historians believe it to be an interpolation. However it would seem to be the sort of "infinite descent" proof by contradiction that supports the possibility of a reasoned argument by Fibonacci.2011-10-12
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    Craig -- you're right; I did present that badly. I think one way around this is to note that for the squares (s-t, s, s+t) to be in lowest terms, one of (x, y) must be even, the other odd, so in fact neither x+y nor x-y is divisible by 2. (From the brief excerpt of Liber Quadratorum I have, I don't see how Leonardo argues, but this is not the part of Leonardo's "proof" that Anderson-Wilson objected to.)2011-10-12
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    You might find of interest my [speculative reconstruction](http://math.stackexchange.com/questions/27309/help-to-understand-a-proof-by-descent/27317#27317) of Fibonacci's Lost Theorem = FLT_4.2011-10-12
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    Thanks, Bill. It is interesting. I intend to mention Fibonacci's "proof" of FLT4 on Wikipedia, but am not sure how best to phrase the mention ....2011-10-19
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    Direct quotes and translations thereof are generally most likely to be accepted.2011-11-28
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    Might be good to verify this story, too: "In 1742, almost a century after Fermat's death, the Swiss mathematician Leonhard Euler asked his friend Clerot to search Fermat's house"2012-05-04

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