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It's common to see the Fermat Last Theorem being used to prove the irrationality of $\sqrt[n]{2}$. In fact, according this post, the said proof appeared in American Mathematical Monthly.

On the other hand, I have seen two objections on this approach: DanielLittlewood's comment here and BCnrd's comment here.

Since I'm not able to judge by myself, I'd like a definitive answer: Is this proof logically valid?

Edit

It seems the phrase "logically valid" is not appropriate for the context. In fact, I want to know if the proof is circular (as suggested in the linked comments).

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    $\begingroup$ Depends on whether the proof of Fermat uses the irrationality of roots like this in it. It's certainly not the best way to prove it. :) $\endgroup$ – Thomas Andrews Feb 18 '16 at 3:37
  • $\begingroup$ @ThomasAndrews "whether the proof of Fermat uses the irrationality of roots" is the big question. If so, worst than a bad way, the "proof" isn't a proof. $\endgroup$ – Pedro Feb 18 '16 at 3:44
  • $\begingroup$ @Pedro : you don't even understand your own question : where is your problem ? write it down you'll see there is no problem at all, and everything is explained in the discussion you linked math.stackexchange.com/questions/1191176/… $\endgroup$ – reuns Feb 18 '16 at 3:48
  • $\begingroup$ @user1952009 My problem is described in bold letters in the post. In my opinion, the information in the linked post are not enough to answer the question. $\endgroup$ – Pedro Feb 18 '16 at 3:58
  • $\begingroup$ I'm quite sure you didn't think to what means "logically valid". $\endgroup$ – reuns Feb 18 '16 at 4:01
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The proof is logically valid. Even if Wiles' proof depends on $\sqrt[n] 2$ being irrational, this does not make the proof circular - just redundant. The reason is, the irrationality of $\sqrt[n] 2$ can be easily proven without resorting to Fermat's last theorem. Thus Wiles' proof would still hold without this proof of the irrationality of $\sqrt[n] 2$.

Concerning BCnrd's comment, it only points out that the irrationality of $\sqrt[n] 2$ is easily proven on the way to proving FLT.

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It is a logically valid proof. BCnrd's comment describes it as "circular", but I guess what he really means is something like "redundant". A proof is a sequence of formulas each of which is a direct consequence of formulas appearing earlier in the sequence, and the theorem it proves (its conclusion) is the last formula in the sequence. Essentially we are talking about a proof that first includes a formula $A$ meaning $\sqrt[n] 2$ is irrational for $n \gt 1$, then later has a formula $B$ expressing FLT (perhaps depending on $A$), then proves $B \rightarrow A$, and finally concludes with $A$ by modus ponens. It is still a valid proof of $A$ even though $A$ appears multiple times and the proof could have concluded at any of those earlier times.

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