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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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    My question was marked as a duplicate of a question that was posed 5 months *after* mine was. math.stackexchange watchdogs like time paradoxes.2016-08-22

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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.2012-12-14
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    @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 such that $4^r+5^r = 6^r.$2012-12-14
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    ah, ok, sure. Your "the theorem is false" somehow made me think that you were taking a fixed value of $r$.2012-12-14