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Claim. Let $\mathcal F$ be the set of positive $3$-powerful (cube-full) integers, $1$ included. For every $\varepsilon>0$ there is a reduced residue class $a\pmod M$ on which the three-fold sumset $\mathcal F+\mathcal F+\mathcal F$ has upper relative density at most $\varepsilon$; consequently the positive integers that are not a sum of at most three $3$-powerful numbers have positive lower natural density, and in particular there are infinitely many of them. The result is Basile Beyer de Ryke, A density deficit for sums of three cube-full numbers, Theorem 1.1 and Corollary 1.2, an unpublished manuscript first posted as arXiv:2609.35772 on 2026-07-26 (its only version) and filed on the site's proof-claims tab on 2026-09-06 as a partial proof claim with a copy of the manuscript; both postings carry the same result and are recorded on this one page. The source card is beyer_de_ryke_2026_density_deficit_cube_full_sums.

Submission note. Posted to erdosproblems.com as a proof claim by Basile Beyer de Ryke (account Basile_Beyer_de_Ryke) on 6 September 2026:

I claim the following partial solution to Erdős Problem #940. For $r=3$, I prove that the positive integers which are not representable as sums of at most three cube-full integers have positive lower natural density. In particular, infinitely many integers are not sums of at most three cube-full integers. Thus this settles the infinitude question in Problem #940 for $r=3$, but does not settle the problem for $r\geq4$ or the density-zero question.

Covers. The first question of Problem 940 at $r=3$: there are infinitely many integers that are not the sum of at most three $3$-powerful numbers. The claim says nothing about any $r\ge4$ and does not settle the second question, whether the sums of at most three $3$-powerful numbers have density zero: a vanishing relative density on one progression whose modulus depends on $\varepsilon$ is compatible with the sums having positive density, which the manuscript records as unknown.

Method. Every cube-full number is uniquely $dx^3$ with $d=b^4c^5$, $bc$ squarefree, and the weights $\sum d^{-1/3}$ and $\sum3^{\omega(d)}d^{-1/3}$ over such $d$ converge. For a prime $q\equiv1\pmod3$ and a non-cube $u$ modulo $q$, the proportion of solutions of $x^3+y^3+z^3\equiv u$ is at most $1-2/q$. Primes splitting completely in the field generated by the cube roots of a finite set of canonical parts make every such part a nonzero cube modulo $q$; finitely many of them, chosen by Chebotarev's theorem with prescribed reciprocal sum, are combined into one reduced class on which the three-sum count is small, with the remaining parts controlled by the two tail weights.

Acceptance. None is recorded. The manuscript is not refereed, the site's label is OPEN (page last edited 2025-11-03) and the tab's claim had no comments as of 2026-10-06, no independent review is on record, and there is no Lean formalization of the manuscript's own argument. The statement was checked at claims level on the source card; no review of the proof is recorded.

Depends on. No other wiki page; the claim rests on the manuscript above.