TITLE: Fermat's equation with real exponents
QUESTION [7 upvotes]: 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 $r\in\Bbb R\setminus\Bbb N$ ?

REPLY [6 votes]: 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.