First of all, I am no number theorist, so this question may be a little dummy.
The two squares theorem imply that $m = x^2 + y^2$ for some (possible zero) integer numbers $x,y$ iff $m$ factors as $m = a b^2$, where $a$ has no prime factor $\equiv 3 \mod 4$ (see, eg., these notes)
The following question seems natural to me: Is there such a simple characterization for numbers expressed as $m = x^n + y^n$ for $n > 2$? Since this seems related to Fermat's Last Theorem, maybe there is only partial (necessary or sufficient) conditions?
I am also interested in possible generalizations (characterizations of numbers that may be expressed as $m = x_1^n + \ldots + x_k^n$ ). What is proven/conjectured about this and is there any good reference with these statements?