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## Proof that 2^(1/n) is irrational

After an interesting conversation with somebody recently I was looking around and aggregating simple mathematical facts that have somewhat crazy proofs. Like for all such questions, I found a great MathOverflow thread and decided to share this gem from there:

Fact: $\sqrt[n] {2}$ is irrational for any integer $n \geq 3$.

Proof: Suppose it is not. Then $\displaystyle \sqrt[n] {2} = \frac{p}{q}$, then $2 q^n = p^n$, or $p^n = q^n + q^n$ contradicting Fermat’s Last Theorem.

Although a commenter there mentions that the argument is essentially circular (which I find fascinating), but other than that it made me laugh, what I find interesting about this answer and an accompanying comment by Greg Kuperberg is that it made me realize that Fermat’s Last Theorem is not strong enough to imply the irrationality of $\sqrt{2}$.