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Let $k\geq 3$ and $f_{k,3}(x)$ denote the number of integers $\leq x$ which are the sum of three nonnegative $k$th powers. Is it true that \[f_{k,3}(x) \gg x^{3/k}\] or even $\gg_\epsilon x^{3/k-\epsilon}$?
#325
:
[ErGr80]
number theory
,
powers
Mahler and Erdős
[ErMa38]
proved that $f_{k,2}(x) \gg x^{2/k}$. For $k=3$ the best known is due to Wooley
[Wo15]
, \[f_{3,3}(x) \gg x^{0.917\cdots}.\]
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