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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?

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IIRC there's some specific small integer (around $21$?) for which it's not known whether or not it's a sum of three cubes, or something like that. So it seems very little is known. Matiyasevich's theorem is also relevant. – Qiaochu Yuan Feb 26 '14 at 1:50
Some discussion (gives $33$ as a number which is not known to be a sum of three cubes): – Qiaochu Yuan Feb 26 '14 at 1:57
You may be thinking of $33$. Not $22$, which is excluded mod $9$. – Noam D. Elkies Feb 26 '14 at 1:57
Sorry, you answered as I was typing, and anyway you wrote $21$, not $22$. For the record, trying for $|a^3+b^3-c^3| = 21$ with $0 \leq a,b \leq 100$ finds $$ 21 = 16^3 - 14^3 - 11^3 = 28^3 + 85^3 - 86^3 = 49^3 + 97^3 - 101^3. $$ – Noam D. Elkies Feb 26 '14 at 4:30

This is not an answer but a series of comments.

Here's an obstacle stopping you from straightforwardly generalizing the classical based on the Gaussian integers to, say, sums of two cubes. The problem is that $x^3 + y^3$ is not the norm of an element of a number field. If it were, the number field would be $\mathbb{Q}(\zeta_6)$, where $\zeta_6$ is a primitive sixth root of unity, but this is a quadratic number field and the norm of, say, $x + y \zeta_6$ is actually the quadratic factor $x^2 - xy + y^2$ of $x^3 + y^3$ rather than the whole thing.

The ring where norms give $x^3 + y^3$ is instead the ring $\mathbb{Q}[\zeta]/(\zeta^3 + 1)$, which unfortunately is not a field, so it's unclear in what sense the "algebraic integers" $\mathbb{Z}[\zeta]/(\zeta^3 + 1)$ inside it can have a reasonable theory of prime factorization. Moreover, the general element of this ring has the form $x + y \zeta + z \zeta^2$, with three parameters, and the corresponding norm is

$$x^3 + y^3 + z^3 - 3xyz$$

so even if one has understood this norm one still has to take into account the additional constraint that $z = 0$.

It's also worth pointing out that $m = x^3 + y^3$ describes a curve of genus $1$ for almost all values of $m$ and IIRC it's already unknown whether there exists an algorithm taking as input such a curve and determining whether it has an integer point or not. Bjorn Poonen has written about related topics; see, for example, this expository article.

Some other relevant stuff:

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So, even the case $x^3 + y^3$ seems an open problem... I am also interested in what would be the state-of-the-art of such characterizations (maybe for some specific $n$ or the "sharpest" necessary/sufficient conditions?) – Campello Feb 26 '14 at 19:34

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