OPEN
This is open, and cannot be resolved with a finite computation.
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 $x$?
This would have significant applications to Waring's problem. Erdős and Graham describe this as 'unattackable by the methods at our disposal'. The case $k=2$ was resolved by Landau, who showed\[f_{2,2}(x) \sim \frac{cx}{\sqrt{\log x}}\]for some constant $c>0$.
For $k>2$ it is not known if $f_{k,k}(x)=o(x)$.
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T. F. Bloom, Erdős Problem #323, https://www.erdosproblems.com/323, accessed 2026-01-16
