Problem Statement
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 $\gg_\epsilon x^{3/k-\epsilon}$?
Categories:
Number Theory Powers
Progress
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.Source: erdosproblems.com/325 | Last verified: January 14, 2026