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?