# Fermat's equation with real exponents

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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@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