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Just out of curiosity : has the equation $$ x^r+y^r=z^r,\qquad(x,y,z)\in\Bbb Z^3,\quad r\in\Bbb R, $$ been studied? Any non trivial result for $r\in\Bbb R\setminus\Bbb N$ ?

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up vote 6 down vote accepted

For information on rational exponents, see here. In particular, see this paper. Clearly there are real values for which the theorem is false, by a continuity argument.

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I'm not sure what is the continuity argument you are thinking of. – Andrea Mori Dec 14 '12 at 7:47
@AndreaMori Define for example $f(r) = 4^r+5^r-6^r$, which is continuous in $r.$ Then $f(2) >0 $ and $f(3)< 0$ so by the intermediate value theorem there is some $2<r<3$ such that $4^r+5^r = 6^r.$ – Ragib Zaman Dec 14 '12 at 13:42
ah, ok, sure. Your "the theorem is false" somehow made me think that you were taking a fixed value of $r$. – Andrea Mori Dec 14 '12 at 13:55

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