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What was the historical back ground that probably Fermat’s could had known about a much simpler proof than the first (Euler’s elementary proof of Fermat’s last theorem for $n = 3$), at least for the case that $3$ doesn’t divide $xyz$ ?

The proof is so elementary and based on using the general binomial theorem to reduce the equation to this simple form $$(x + y – z)^3 = 3(x + y)(z – x)(z – y)$$ where, and without loss of generality $z > y > x $ are three positive coprime integers pair wise,

The idea then is so simple, by using the little Fermat's theorem to show that the left hand side of the reduced equation is divisible by $27$, where as the right hand side is divisible only by $3^1$, since the prime factors of $$(x +y)(z -x)(z -y)$$ are some prime factors of $xyz$, which isn't divisible by $3$, therefore an immediate so rigorous and elegant proof for this case that is much simpler than Euler's elementary proof!

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closed as off-topic by Andrés E. Caicedo, J. W. Perry, VicAche, Gerald Edgar, Danu May 10 '16 at 8:07

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  • "This question does not appear to be about history of science and math, within the scope defined in the help center." – Andrés E. Caicedo, J. W. Perry, VicAche, Gerald Edgar, Danu
If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ I really didn't know that so elementary proof is so annoying, if this isn't clear enough to you, please feel free to inquire about any step you can't understand, I know it may be a little difficult for some school student from the first look, but I guarantee you full understanding if you would kindly ask, instead of down voting such a wonderful so elegant proof for this very simple case, you should realize that many around the world are quite educated of this proof from the internet only, so nothing to hide if you tend to eliminate it, at any case I do appreciate your complete silence and $\endgroup$ – bassam karzeddin Apr 24 '16 at 6:34
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    $\begingroup$ I did not downvote, but I think the the reason might be that the question appears to have little to do with history. Whether the proposed proof works should be discussed on Math SE, there is no historical source offered for the proof, or why anyone should think that Fermat had anything to do with it. Since he apparently thought that he had a proof for general $n$ it seems unlikely that he would be interested in special tricks for $n=3$. $\endgroup$ – Conifold Apr 25 '16 at 21:45
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    $\begingroup$ Thank you to break the silence around this elementary & sensitive issue, first we should remember that Fermat started with his famous words “a cube can’t be split into two cubes …”which implies that he started by the simplest case $n=3$, before he proved it rigorously to exponent of the form $4n$, then generalize it to all higher powers, otherwise how could he generalize it without understanding the simplest case then!, also note that peer reviewed so elementary and rigorous proof can simply be generalized to many higher powers with little more details than required for this case!, was ... $\endgroup$ – bassam karzeddin Apr 26 '16 at 7:16
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    $\begingroup$ was this so difficult to be recognized by Fermat even if not documented or even written some where?, about discussing this issue at MSE, isn't possible for me being not allowed presently, for my annoying questions or answers that don't please many (with no mentioned reasons), wonder why, as if mathematics is becoming how to please others, despite my real intention is to high light the fact and never to degrade others. $\endgroup$ – bassam karzeddin Apr 26 '16 at 7:28
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    $\begingroup$ The generalization I mean not mainly from cubic case, but a general similar formula I may present it later, also note that this mentioned above proof is rigorously valid even when $3^n$ divides $xyz$, where $n \neq 3k - 1$, then a strong contradiction is observed when equating the exponent of $3$ on both sides of equation, see here: math.stackexchange.com/questions/662313/… $\endgroup$ – bassam karzeddin Apr 27 '16 at 7:38
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I I think Fermat had the proof for n=3 based on the method of infinite descent.Pl.read on the internet'method of infinite descent and proof of Fermat's last theorem for n=3" This proof is in Netherlands text book on FLT in general.

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  • $\begingroup$ If what you claim is true, then why historian mathematicians consider Euler's proof for exponent $n = 3$, was the first prove (even with interesting mistake) nearly a century later of Fermat's conjecture? $\endgroup$ – bassam karzeddin May 9 '16 at 13:53
  • $\begingroup$ Please consider this elementary proof by Fermat: quora.com/… $\endgroup$ – bassam karzeddin May 9 '16 at 13:54
  • $\begingroup$ Can you please provide a link to how Fermat could miraculously could prove his last theorem for exponent $3$, where $3^k$ divides $xyz$, of course the first case follow from his only little theorem in half line proof only, $(x + y - z) = 0 \mod 3$, and note that this subject is so annoying to professionals or historians, that is why they are going soon to delete it, remember also, that there are so many sites that respect the member content and don't allow simply others even to touch it, but here rules are quite different! $\endgroup$ – bassam karzeddin May 11 '16 at 6:59

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