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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?

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    What exactly are you asking for?2012-02-08
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    I have not really read any details, but the fact that he mentions that his approach to pythagorean triplets is new, which has been confirmed by many "math people", and the fact that he at some point goes into detail of what a "reduction ad absurdum" proof involves, makes me very skeptical.2012-02-08
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    I do get, on average, about two purpoted elementary proofs of FLT in the mail every year. All of them start with considerations about Pythagorean triplets (none of them seems to be aware of the fact that conics are rational curves, though) and go very wrong right away. This one, at least, has some nice pictures.2012-02-08
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    Let's be clear, I do understand that the writing style of the author is very bad (I have a PhD and understand how to write good mathematics). But I have put in some effort to struggle through his reasoning and have not so far found any major problem in his approach.2012-02-08
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    Voting to close. Please see this: http://meta.math.stackexchange.com/questions/2290/is-it-ok-to-ask-about-the-correctness-of-preprints-of-crank-friendly-topics. If you have any specific mathematical points you want to discuss, please post that. A blanket "is it correct?" type of question is liable to be closed.2012-02-08

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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.

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    But the author _does_ state that eq (4) follows from the assumption that (x,y,z) is a pythagorean triple. Since the proof is a bit contorted I am hesitant to say so but I think that focusing your criticism on eq (4) is wrong.2012-02-08
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    Please read:http://meta.math.stackexchange.com/questions/2290/is-it-ok-to-ask-about-the-correctness-of-preprints-of-crank-friendly-topics before you consider answering such questions. Thanks. (Note: just requesting you to read it, You are free to post as you wish).2012-02-08
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    Fair enough. Happy for this to be closed. I'll work at the proof a bit more and if necessary post a specific question.2012-02-08
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    @Robert The flaw to me looks like the author is presenting a specific parametrization of the Pythagorean triples (it looks like just a variant on the usual $(m^2-n^2, 2mn, m^2+n^2)$ to me, with the 'root' $r$ looking like $m^2-2mn$ at first glance though I haven't really gone through the arithmetic closely) and then saying that this parametrization doesn't extend to the higher-powers case - but there's no reason to believe that any parametrization would have to look similar to that, and ample reason not to.2012-02-08
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    @Aryabhata: Thanks for pointing that out; I wasn't aware of that thread. To my mind, the current form of the question, which it had when you voted to close it (though not when I answered it), is rather close to this example you gave there: "**GOOD**: This new paper claiming a big result is beyond my ken to read. Before I invest the time to learn all this stuff and try to read it, I am curious: have there been any discussions of it? *Answer*: We can answer your question here." Where do you see the decisive difference?2012-02-08
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    @joriki: You are right. This question as it stands now, it just asking for prior discussion and should be answerable without discussion/opinions etc.2012-02-08
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    @joriki: I am beginning to think we should remove that portion, as having that almost always seems to lead to someone giving an answer with their own opinion disproof (rather than a link to prior discussion with some credibility), forcing people to go read the preprint, if they wish to vote/downvote.2012-02-08
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    I just realized, we need to edit the title of the question before we can reopen it.2012-02-08
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    Did my edit of the question (I was responding to the comment from Rasmus) cause this confusion?2012-02-08
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    @Steven Agree that he gives a parameterization. But the extension to higher powers is not based on a parameterization. Rather, he chops up a wedge of depth $z^{n-2}$ in a certain way and then proposes we look at slices of the wedge. The chopping has been done so that when we look at the slices we recognize shapes from his earlier analysis for Pythagorean triples. All in all, he is projecting the case for a general $n$ onto the case for $n=2$.2012-02-08
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    @Steven Hmmm. Just had a re-read of your comment and am wondering whether my previous comment was based on a misunderstanding of your comment. _Is that confusing or what?_ Oh dear I am losing credibility with this question ;-)2012-02-08
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    @Robert: He isn't projecting the case for general $n$ onto the case for $n=2$. He's projecting it into two dimensions and drawing figures that look like the ones he drew for $n=2$, but the relationship $x^2+y^2=z^2$ has no basis in this case, and the corresponding relationship involving $r$ is just introduced without any justification.2012-02-08