Density of sums of three $k$-th powers: is $f_{k,3}(x)\gg x^{3/k}$? (Erdős #325)
Statement
Fix $k\geq 3$ and let $f_{k,3}(x)$ denote the number of integers $\leq x$ that are a sum of three nonnegative $k$-th powers. Is it true that $$f_{k,3}(x)\gg x^{3/k},$$ or even $f_{k,3}(x)\gg_\epsilon x^{3/k-\epsilon}$ for every $\epsilon>0$?
Acceptance. FULLY RESOLVES: a proof (machine-checkable Lean/Coq preferred — the statement is formalized in the DeepMind formal-conjectures repository — else a complete written proof) that $f_{k,3}(x)\gg x^{3/k}$ (or $\gg_\epsilon x^{3/k-\epsilon}$) for the stated $k$, or a disproof. ADVANCES: (a) for $k=3$, improve the density exponent strictly above Wooley's $f_{3,3}(x)\gg x^{0.917\cdots}$ stated in the background (target $x^{1}$), with proof; (b) prove $f_{k,3}(x)\gg x^{3/k-\epsilon}$ for some $k\geq 4$ where it is currently unknown, with proof; (c) prove the exact exponent $3/k$ for a new $k$, with proof; (d) rigorously compute $f_{k,3}(x)$ over a large range and report a certified empirical exponent with the sieving code (evidence, not a proof). Deliver the proof, the improved-exponent proof, or the sieving code plus certified counts.
Background
Posed by Erdős and Graham [ErGr80]. The exponent $3/k$ is the expected order, since there are $\asymp x^{3/k}$ triples of $k$-th powers with sum $\leq x$. Mahler and Erdős [ErMa38] proved the two-summand analogue $f_{k,2}(x)\gg x^{2/k}$. For three cubes ($k=3$) the conjecture is $f_{3,3}(x)\gg x$; the best known lower bound is Wooley's [Wo15] $f_{3,3}(x)\gg x^{0.917\cdots}$, short of the conjectured $x^{1}$. This is exactly the $m=3$ case of the broader venue problem Erdős #323, which asks $f_{k,m}(x)\gg x^{m/k}$ for all $m<k$ (and $\gg x^{1-\epsilon}$ for $m=k$); #325 is the sharper, more tractable slice with a concrete numerical frontier. The statement is formalised in Lean in the Google DeepMind Formal Conjectures project. Related OEIS sequences A004825, A004832, A004843 (integers that are sums of three nonnegative $k$-th powers, for small $k$). Listed as open on erdosproblems.com/325 (fetched 2026-07-21, status 'open', tagged 'number theory | powers'). No prize is recorded. Attacker's tool: sieve the threefold sumset of $k$-th powers up to $x$ to compute $f_{k,3}(x)$ exactly and estimate its exponent against the $3/k$ prediction; the $k=3$ frontier is analytic — improve Wooley's $0.917$ via the circle method / efficient congruencing.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #325 (T. F. Bloom) | website |
| REF-02 | Lean formalization of Erdős #325 (DeepMind formal-conjectures) | website |
| REF-03 | OEIS A004825 — integers that are the sum of three nonnegative k-th powers (per erdosproblems.com/325) | website |
| REF-04 | OEIS A004832 — integers that are the sum of three nonnegative fourth powers (per erdosproblems.com/325) | website |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.