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Question -1 What level of math do I need to understand the proof Andrew Wiles wrote? Am I supposed to be a mathematics professor? For example, I don't understand anything from these pages. That's a really bad feeling.

Question-2 This question may not look nice. It's ridiculous to ask this question. One of our mathematics teachers said in the course: "Andrew Wiles's proof was not actually approved. Only the mathematicians accepted this as true." Of course, I don't believe it. But there was a doubt in me.

Anyway, my main question is the first question I ask.

For example, I don't know anything about these mathematical notations:

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Thank you very much.

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    $\begingroup$ You need a PhD in elliptic number theory even to start reading his proof. Moreover, he brings together just about every single branch of modern mathematics to pull of this proof. Fewer than about 0.5% of all mathematicians world-wide are qualified even to read the thing, much less verify it. $\endgroup$ Commented Sep 19, 2018 at 20:17
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    $\begingroup$ The proof is very complicated indeed. You need not have a bad feeling if you don't understand it. If the proof were easier, it would probably have been found much earlier. $\endgroup$
    – Peter
    Commented Sep 19, 2018 at 20:20
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    $\begingroup$ I am not able to approve Wiles' proof but I would not go so far that I don't believe that it is valid. This is oversceptical to my opinion. In fact, I even did not understand Dirichlet's proof that a+nb contains infinite many primes, whenever a and b are coprime, but I have no problem to accept this theorem. $\endgroup$
    – Peter
    Commented Sep 19, 2018 at 20:23
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    $\begingroup$ It's certainly possible that Wiles proof is wrong. At this level, with very few people able to review it, there's a chance that some error has been missed. We always prefer simpler, shorter proofs for this reason. Simpler and shorter proofs also feel more concrete, in some cases, as the proofs that use general abstract theorems seem almost mystical to those of us not able to understand them. $\endgroup$ Commented Sep 19, 2018 at 20:30
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    $\begingroup$ @Peter I really have no idea. It depends on how many other mathematicians have read the proof and understood it, and I don't have a way to guess that number. The difference between, say, 10 readers and 200 readers is pretty big. $\endgroup$ Commented Sep 19, 2018 at 20:34

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According to a report by J-P. Bourguignon, the director of the IHES (the French institute for math. and theoretical physics which, I think, all researchers know of), the total number of mathematicians world-wide at the end of the 20th century was as high as 80,000. And the term "mathematician" was reserved to those having a level of formation equivalent to a PHD, and whose professional activity genuinely makes room to research in math. or assimilation of math. results. So, even with such a restrictive definition, the 0,5% evoked by @Adrien Keister amounts to roughly 400 persons who could have a good enough global (= not necessarily detailed) comprehension of what Wiles' proof is about. Of course the number of experts able to follow and/or use (hence check) Wiles' results should be actually smaller, but not so small as to allow doubts. By the way, u your professor's satement: "Andrew Wiles's proof was not actually approved. Only the mathematicians accepted this as true" looks like non sense to me.

More precisely : 1) Outside the usual "popularization" comments, there have been many serious published accounts at post-PHD level, meant to explain the backbone of Wiles' demonstration, by known workers in the field such as Boston, Frey, Rubin, Stein, Stevens, etc. 2) At the research level, many academic sessions or seminars (in particular the reference "Séminaire Bourbaki", with a talk given by J-P. Serre) world-wide have been devoted to Wiles' work 3) Much more important, Wiles' results are not limited to FLT, which in itself is only an isolated riddle (deeply buried, but nonetheless a riddle). Wiles' real achievement was the proof of the Shimura-Taniyama-Weil conjecture (= the "modularity" of elliptic curves defined over $\mathbf Q$). His (many) new results have been "tested" in the sense that they paved the way to the proof of Serre's conjecture on "modular representations", from which FLT can now be derived by a half-page long argument ! This, together with STW, is considered as the first step toward the realization of the Langlands program.

At this level, as @Thomas Andrew said, it's certainly possible that some error has been missed. But in my opinion it would be merely an error, not a gap, because of the "coherence" of Wiles' proof.

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@AdrienKeister has answered your first question: you need some very specialized advanced number theory.

Your second question is a little more interesting. Whether or not a mathematical theorem has been "proved" depends on what the mathematicians capable of understanding the proof think of the proof. There is no more objective standard than that. But this criterion isn't nearly as subjective as it may seem. As pointed out in one of the comments, Wiles' first proposed proof wasn't complete. The referees (mathematicians chosen by the journal to which it was submitted, and others who looked) found a gap. Wiles (and Taylor) were able to fill that gap with more work. If they hadn't, the question might still be open.

There are rare instances of theorems whose proofs were accepted but found faulty years later. Almost all the time what mathematicians "accept as true" stays accepted.

(Note: there is a current area of research that works with programming computers to prove theorems. That's a philosophical direction beyond an answer to your question.)

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    $\begingroup$ More important, for this question, is that the software can verify proofs. But the proofs have to be entered to verify them. $\endgroup$ Commented Sep 20, 2018 at 4:42

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