# Fermat's Last Theorem for Fractional Exponents

Fermat’s Last Theorem says that for positive integers n > 2, x, y, z, there are no solutions to

$x^n+y^n=z^n$.

On the first day of my senior seminar on “The Big Questions,” the class asked me whether it remains true for rational exponents. Andrew Granville told me that it does and referred me to a very short, recent proof by Lenstra.

Next the students asked whether it remains true for real exponents. It is easy to see that it does not. Since

$4^2+5^2$ > $6^2$   but   $4^3+5^3$ < $6^3,$

by the Intermediate Value Theorem, for some $2

$4^n+5^n=6^n.$

(My students tell me that n is about 2.487939173.)

Now the question is, is there any nice specific such counterexample?