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Remembering that a STEP mathematics question which guided us through showing $n^3 + (n-3)^3 = (n+3)^3$ has no solutions in the integers could be sledgehammered with Fermat's Last Theorem, I wondered if there are other fun ways to apply Fermat's Last Theorem.

Another example:

Claim $ \sqrt[n]{2}$ is irrational for all integers n > 3

Proof Suppose to the contrary, that $ \sqrt[n]{2} = \frac{p}{q}$ for integers p and q. Then $q^n + q^n = p^n $. A contradiction (Wiles, 95).

Any other examples? I'd be happy to see applications of other big theorems (such as the Catalan conjecture/Mihăilescu's theorem). Fermat fits naturally because of the gulf in the elementariness of the statement and the difficulty of the proof.

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    More generally $\sqrt[n]{a^n+b^n}$ is irrational if $a,b>0$, otherwise $p^n=(qa)^n + (qb)^n$.2012-04-12
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    I'm sure there are examples that use the classification of finite simple groups.2012-04-12
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    See a comment on circularity of this proof: http://mathoverflow.net/questions/42512/awfully-sophisticated-proof-for-simple-facts/42519#425192012-04-12

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