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OPEN
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 $f_{k,k}(x)=o(x)$.