For integer $k > 1$, not every number can be written as a sum of $k$ $k$th powers of non-negative integers, but if we consider $\sum_{j=1}^k n_j^k$ where all $n_j \leq N$, then the number of unique combinations of $n_j$ we can get should be asymptotically like $N^k / k!$, and if almost all of them give unique sums then we would have that the fraction of non-negative that are $\leq kN^k$ that can be written as a sum of $k$ $k$th powers is at least something like $1/(k\cdot k!)$. Let $f_k(M)$ be the fraction of integers between $0$ and $M$ that can be written as a sum of $k$ $k$th powers of non-negative integers. What is $\alpha_k = \lim_{M \to \infty} f_k(M)$? Even an answer for $k = 2$ or $k = 3$ would be great. For $k=2$, we have the extension of Fermat's theorem, that positive odd $n$ is expressible as a sum of two squares if and only if all primes congruent to $3$ mod $4$ in the prime factorization of $n$ appear with even exponents.
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$\begingroup$ This is a hard problem. The answer is known only for $k=2$ (it is zero); for $k\ge 3$, it is conjectured (but not proven) that the limit is positive. The natural density for $k=3$ (i.e. for sums of three cubes) is thought to be about $0.09994$. $\endgroup$– mjqxxxxSep 27, 2013 at 15:54
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$\begingroup$ Thanks, I'm a bit surprised the answer is $0$ for $k=2$. If you list as an answer and for $k=2$ either post a reference or reproduce the proof, I'll upvote and accept since I didn't realize this was such a hard problem. $\endgroup$– user2566092Sep 27, 2013 at 15:57
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$\begingroup$ Here you can find the original reference for k=2: Landau, E.: Über die Einteilung der positiven ganzen Zahlen in vier Klassen nach der Mindestzahl der zu ihrer additiven Zusammensetzung erforderlichen Quadrate. Archiv der Math. und Phys. 13(3), 305–312 (1908); Collected Works, Mirsky, L., Schoenberg, I.J., Schwarz, W., Wefelscheid, H. (eds.) vol. 4, pp. 59–66. Thales Verlag (1985) $\endgroup$– Paolo LeonettiJul 30, 2016 at 13:24
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