Problem 323

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 $x$ ?

Source: erdosproblems.com/323

Number theory
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Open

No claim settles this problem.

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 (1908); for $k>2$ the site records that it is not even known whether $f_{k,k}(x)=o(x)$ .

Problem 323

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