Does Fermat's theorem hold for rational numbers less than one?
That is: Can you find integers x, y and r such that $x^n+y^n=r^n$ for a given rational number less than 1?
Everything I find online says no, but I claim that I can find and infinite number of integers that satisfy the equation for n=1/3 and n=1/7. For example:
$2^{1/3}+54^{1/3}=128^{1/3}$ and $2^{1/7}+256^{1/7}=4374^{1/7}$

wtf is going on?