OPEN
This is open, and cannot be resolved with a finite computation.
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 $\gg_\epsilon x^{3/k-\epsilon}$?
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],\[f_{3,3}(x) \gg x^{0.917\cdots}.\]This problem has been
formalised in Lean as part of the
Google DeepMind Formal Conjectures project.
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When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #325, https://www.erdosproblems.com/325, accessed 2026-01-16
