TITLE: Where is the problem in this proposed elementary proof of Fermat's Last Theorem
QUESTION [5 upvotes]: I stumbled across a website by a chap called Tom Ballard in which he presents his proof of FLT based on elementary techniques: http://www.fermatproof.com
The style is rather 'non-standard', shall we say, and makes it difficult to assess. I have checked through it and have a couple of points to investigate further, but certainly the first part on pythagorean triples is interesting, and correct.
Has anybody else seen it and put in some effort to see if it is correct?

REPLY [4 votes]: The problem is that (4) is used but not proved. (1) through (3) are merely three different versions of a definition of $r$: (1) and (2) are rearrangements of each other, and (3) is obtained from (1) or (2) by adding $z-x$ or $z-y$ to (1) or (2), respectively. So the only "Pythagorean" content is in (4). While a lot of effort is expended on showing that (1) to (3) obtain in the cubic case (which is unsurprising since they merely express the definition of $r$), (4) is just pulled out of thin air and used to claim that $x,y,z$ have to form a Pythagorean triple, when in fact (4) can only be derived by assuming that they do.