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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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Pedro
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    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. :) – Thomas Andrews Feb 18 '16 at 03:37
  • @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. – Pedro Feb 18 '16 at 03:44
  • @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 http://math.stackexchange.com/questions/1191176/irrationality-of-sqrtn2/1191180#1191180 – reuns Feb 18 '16 at 03:48
  • @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. – Pedro Feb 18 '16 at 03:58
  • I'm quite sure you didn't think to what means "logically valid". – reuns Feb 18 '16 at 04:01
  • The proof of FLT probably doesn't depend on the irrationality of $\sqrt[n]{2}$ directly. However, it definitely use some fundamental property of integers, in particular, $\mathbb{Z}$ is a UFD. Since one can show $\sqrt[n]{2}$ easily by this sort of properties. One can say that the proof is logical valid but highly non efficient. One can simplify the complete proof (i.e. the one start from the peano axioms of integers) by getting rid of all uses of FLT in the middle. – achille hui Feb 18 '16 at 04:03
  • @user1952009 What does "logically valid" mean? – Pedro Feb 18 '16 at 04:06
  • @frank000 This link is in my post. – Pedro Feb 18 '16 at 04:07
  • @Pedro : I'd say that one problem/difficulty in logic is that the difference between the theorems and their proofs is not really emphasized, so once $T$ is proved, using $T$ to prove $T$ is perfectly valid. – reuns Feb 18 '16 at 04:15
  • @user1952009 I should not have used the phrase "logically valid". In fact, I want to know if the proof is circular (as suggested in the linked comments). – Pedro Feb 18 '16 at 12:00
  • @Pedro and what does mean a proof is circular ? – reuns Feb 18 '16 at 12:07
  • @user1952009 https://proofwiki.org/wiki/Definition:Circular_Proof – Pedro Feb 18 '16 at 12:09
  • @Pedro : so it is not a valid proof. obviously, Andrew Wiles gave a valid proof of the FLT.... – reuns Feb 18 '16 at 12:10

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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.

Paul Sinclair
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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.

Dan Brumleve
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