I have very little experience with number theory and I am not in a place to read through Wiles' proof of Fermat's Last Theorem right now. However, I do want to get just a little bit of understanding of why 2 is different than other numbers in the context of the proof. To be more specific, presumably, at some point in Wiles' proof he must state a lemma or proposition or something that holds for all $n\in\mathbb{N}$ other than 2 (or something like that) so that his proof does not rule out the existence of Pythagorean triples. What is that lemma/proposition/step where 2 is excluded in such a way that Wiles' proof does not imply that Pythagorean triples are impossible?

To further clarify, I can give an example of the type of answer I am looking for. If I were asking about what makes 5 special when it comes to the unsolvability of the quintic (i.e why all of a sudden at degree 5 do general polynomial equations become unsolvable?), I would be looking for an answer roughly on the level of "$A_5$ is the smallest alternating group which is simple. The simplicity of $A_n$ for $n\geq 5$ is what makes quintics (and higher degree polynomials) fundamentally different from quartics and lower degree polynomials, and allows us to use Galois Theory to show that these higher degree polynomials are unsolvable."

So, what is it that separates 2 from every other number when it comes to Fermat's Last Theorem?

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    $\begingroup$ What separates $2$ in $x^n+y^n=z^n$ from other $n$? That we have non-trivial integer solutions by Pythagorean triples for $n=2$. Also $n=1$ has a lot of integer solutions. So not only $2$ is special, but also $1$. $\endgroup$ – Dietrich Burde Sep 1 at 14:33
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    $\begingroup$ This might be a question for Math Overflow instead. It sounds like you want an understanding of why Wiles' proof dismisses the $n = 2$ case. While you may find someone to answer that here, I think you'd have better luck finding someone with such specialist knowledge on Overflow. $\endgroup$ – Theo Bendit Sep 1 at 15:07
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    $\begingroup$ @TheoBendit I am also on MO and my experience is that such questions are not considered research level, because they are too broad and not specific in the technical details. The comparison "on the level of $A_5$" is not sufficient here. $\endgroup$ – Dietrich Burde Sep 1 at 15:29
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    $\begingroup$ I think the question is pretty clear, and is necessarily a bit vague. I'd also like to hear an answer. $\endgroup$ – Jair Taylor Sep 2 at 0:46
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    $\begingroup$ This really boils down to asking why the curve $$y^2=x(x-a^2)(x-b^2)$$ (for $a^2+b^2$ a perfect square) is not particularly interesting - in contrast, that is, with the (nonexistent) Frey curves. Note that I say this without knowing anything whatsoever about the topic; the point is just that since the shape of the proof of FLT is "the curve associated to a counterexample yileds a contradiction to Taniyama-Shimura," we know that the analogous curves associated to Pythagorean triples are fundamentally different. To say more I'd need to know things. $\endgroup$ – Noah Schweber Sep 2 at 0:59

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