Problem 323

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claims/: The 1 claim page of Problem 323, one per claimant's result; the problem's standing derives from them.

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Statement. Let $1\leq m\leq k$ and $f_{k,m}(x)$ denote the number of integers $\leq x$ which are the sum of $m$ many nonnegative $k$th powers. Is it true that

$$f_{k,k}(x) \gg_\epsilon x^{1-\epsilon}$$

for all $\epsilon>0$? Is it true that if $m<k$ then

$$f_{k,m}(x) \gg x^{m/k}$$

for sufficiently large $x$?

Status. Open. The site labels the problem OPEN. Its commentary credits Landau with resolving the case $k=2$, through his asymptotic $f_{2,2}(x)\sim cx/\sqrt{\log x}$, recorded as an accepted partial claim on Landau's claim page; for $k>2$ the site records that it is not even known whether $f_{k,k}(x)=o(x)$.

Source. erdosproblems.com/323, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #323, https://www.erdosproblems.com/323.

References.

• [ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). Library card: erdos_1980_old_new_problems_results_combinatorial_number_theory.

Formalization. Statement in formal-conjectures.

Current assessment

The site's formulation asks two questions about $f_{k,m}(x)$, the number of integers up to $x$ that are sums of $m$ nonnegative $k$th powers: whether $f_{k,k}(x)\gg_\epsilon x^{1-\epsilon}$ for every $\epsilon>0$, and whether $f_{k,m}(x)\gg x^{m/k}$ for $m<k$ and large $x$. The problem comes from Erdős and Graham's 1980 monograph [ErGr80], where the authors call it unattackable by the methods available to them; the site's commentary notes its bearing on Waring's problem.

The one settled case is $k=2$: Landau's theorem of 1908, that the integers up to $x$ which are sums of two squares number asymptotically $cx/\sqrt{\log x}$, gives the first question a yes at $k=2$, and at $k=2$ the second question is the trivial count of squares. This is the accepted partial claim on Landau's claim page, accepted on the refereed paper alone, with its proof not checked in this corpus. For $k\ge3$ neither question is settled, and the site records that it is not known whether $f_{k,k}(x)=o(x)$. No claim about any $k\ge3$ is recorded, so the open standing of the remaining cases rests on the site's label.