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Updated Oct 8

Problem 940

claims/: The 2 claim pages of Problem 940, one per claimant's result; the problem's standing derives from them.

Statement. Let $r\geq 3$ . A number $n$ is $r$ -powerful if for every prime $p$ which divides $n$ we have $p^r\mid n$ .

Are there infinitely many integers which are not the sum of at most $r$ many $r$ -powerful numbers? Does the set of integers which are the sum of at most $r$ $r$ -powerful numbers have density $0$ ?

Status. Open; the site's label is OPEN (page last edited 2025-11-03). The site's remarks record that the density-zero statement at $r=2$ was first proved by Baker and Brüdern [BaBr94], that at $r=3$ it is unknown even for sums of three cubes, and that Heath-Brown [He88] proves every large integer a sum of at most three $2$ -powerful numbers (Problem 941). The site's proof-claims tab carried one partial claim, filed 2026-09-06,.

Source. erdosproblems.com/940, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #940, https://www.erdosproblems.com/940.

References.

[BaBr94] Baker, R. C. and Brüdern, J., On sums of two squarefull numbers. Math. Proc. Cambridge Philos. Soc. 116 (1994), no. 1, 1-5.
[Er76d] Erdős, P., Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44.
[He88] Heath-Brown, D. R., Ternary quadratic forms and sums of three square-full numbers. Séminaire de Théorie des Nombres, Paris 1986–87, Progr. Math. 75, Birkhäuser Boston (1988), 137-163.

Formalization. Statement in formal-conjectures.

Current assessment

An unpublished manuscript of Beyer de Ryke (revised 26 July 2026) claims to prove that the positive integers that are not a sum of one, two or three $3$ -powerful numbers form a set of positive lower natural density; more precisely, for each $\varepsilon>0$ it gives a modulus $M\geq1$ and a residue $a$ coprime to $M$ such that the exactly-three-summand set has upper density at most $\varepsilon$ relative to the progression $a\pmod M$ ; see Beyer de Ryke (2026), Theorem 1.1 and Corollary 1.2. No review of the proof and no acceptance evidence (a refereed version or an independent review) is recorded. No source-supported result for $r\geq4$ is recorded. On the density question there is only a conditional result: Wang's Theorem 1.3 (arXiv:2108.03398, first posted 7 August 2021) shows, assuming three unproved conjectures on Hasse--Weil $L$ -functions (automorphy and a zero-free half-plane, a Ratios-type second-moment bound, and a square-free sieve), that the sums of three nonnegative cubes have positive lower density, so at $r=3$ the sums of at most three $3$ -powerful numbers would not have density $0$ ; it is recorded as a conditional claim on its claim page and settles nothing unconditionally. Beyond the site record, the site's proof-claims tab and the sources cited, no status search is recorded on this page.

The site's proof-claims tab (as of 2026-10-06) carries one partial proof claim, submitted 2026-09-06 by Basile Beyer de Ryke with a copy of the manuscript above, which the claimant describes as settling the infinitude question at $r=3$ and nothing else; it is the same result as the arXiv posting and is recorded, with both postings, as a pending (claimed) partial claim on its claim page. The site's label is OPEN (page last edited 2025-11-03) and the claim had no comments as of 2026-10-06. It does not settle the problem as stated: the density question is open for every $r$ and the infinitude question for every $r\ge4$ .

Known Results

For $r=3$ , the complement of the integers representable as sums of at most three $3$ -powerful numbers is claimed to have positive lower natural density, so infinitely many integers would not be such sums: Beyer de Ryke (2026, unpublished manuscript; proof not reviewed), a pending partial claim on its claim page.
For $r=3$ , conditionally on three unproved $L$ -function conjectures, the sums of at most three $3$ -powerful numbers have positive lower density, so the density question has answer no there: Wang (arXiv:2108.03398, Theorem 1.3), a conditional claim on its claim page.

Research

The research folder for Problem 940 holds reading notes on the sources cited above.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (55)

Wang's conditional positive density of sums of three cubes Beyer de Ryke's density deficit for sums of three cube-full numbers problems/diophantine_problems/E0940/claims Every large integer is a sum of at most three powerful numbers Diophantine Problems and Powers Distinct grade of the Problem 939 reconstruction reviews Sums of powerful numbers Beyer de Ryke: A density deficit for sums of three cube-full numbers Browning–Heath-Brown: Counting rational points on quadric surfaces Browning–Munshi–Wang: Beyond the square-root barrier: cubic forms of Perazzo type Browning–Verzobio: Sums of three powerful numbers Colliot-Thélène–Skorobogatov: Brauer groups of schemes Colliot-Thélène–Skorobogatov: Comparing the two Brauer groups, II Elman–Karpenko–Merkurjev: Chapters 17–18 of The Algebraic and Geometric Theory of Quadratic Forms Ghidelli: Arbitrarily long gaps between the values of positive-definite cubic and biquadratic diagonal forms Heath-Brown: The density of rational points on cubic surfaces Voight: Quaternion algebras over global fields Wang: Sums of cubes and the Ratios Conjectures Diophantine Problems and Powers Beyer de Ryke: A density deficit for sums of three cube-full numbers Browning–Heath-Brown: Counting rational points on quadric surfaces Browning–Munshi–Wang: Beyond the square-root barrier: cubic forms of Perazzo type Browning–Verzobio: Sums of three powerful numbers Ghidelli: Arbitrarily long gaps between the values of positive-definite cubic and biquadratic diagonal forms Lemma 5.4 (p. 12): prime number theorem for nontrivial unitary Hecke characters, after Kubilyus Proposition 5.2 (p. 10): power residue symbols and normalized Jacobi sums as Hecke characters Proposition 8.5 (p. 21): few congruence solutions force half of a truncated progression to start gaps Theorem 1.1 (p. 2): long gaps between the values of a positive ternary cubic diagonal form Theorem 1.2 (p. 2): long gaps between the values of a non-exceptional positive quaternary biquadratic diagonal form Theorem 6.2 (p. 13): a biquadratic diagonal form is exceptional exactly when r_F(0,q) >= q^3 at every good prime Theorem 8.8 (p. 22): a positive proportion of gap starts for cubic and non-exceptional biquadratic diagonal forms Heath-Brown: The density of rational points on cubic surfaces Corollary (p. 3): integers with two representations as a sum of two cubes Theorem 1: non-trivial points on cubic surfaces with three rational coplanar lines Salberger: Counting rational points on projective varieties Corollary 0.7 (pp. 1094-1095): O_d(B^{3/sqrt d}(log B)^4+1) points on a diagonal surface off its two-term zero loci Corollary 6.5 (p. 1122): primitive solutions of a diagonal quaternary equation with a_0x_0^d+a_jx_j^d nonzero number O_d(B^{3/sqrt d}(log B)^4+1)Theorem 0.4 (p. 1093): zeros of f in [-B,B]^n number O(B^{n-2+eps}) for degree at least 4 and O(B^{n-3+2/sqrt 3+eps}) for cubics Theorem 1.2 (p. 1098): a box version of the global determinant method, one hypersurface of degree O((V/T^{1/d})^{1/rd^{1/r}} log V+1)Wang: Sums of cubes and the Ratios Conjectures Corollary 1.7 (p. 5): conditionally, E_{F,w}(X) = o(X^3) for x_1^3 + ... + x_6^3 and every smooth weight Theorem 1.3 (p. 3): conditionally, N_F(X) << X^3 for x_1^3 + ... + x_6^3 and F_0(S^3) has positive lower density when S does Theorem 1.6 (pp. 4--5): conditionally, Hooley's asymptotic for diagonal cubics in six variables, and 100% of a not = ±4 mod 9 are sums of three cubes Theorem 1.9 (p. 6): conditionally, a power saving E_{F,w}(X) << X^{3 - delta} for diagonal cubics in six variables Colliot-Thélène–Skorobogatov: Brauer groups of schemes Theorem 3.5.4 (p. 88): Br(X) injects into Br(F) for a geometrically locally factorial integral scheme Theorems 3.7.1 and 3.7.2 (pp. 93–94): purity for the l-primary Brauer group of a regular integral scheme Colliot-Thélène–Skorobogatov: Comparing the two Brauer groups, II Theorem 4.3.10 (p. 114): finite cokernel of Br(X) to the Galois-invariant geometric Brauer group in characteristic zero Elman–Karpenko–Merkurjev: Sections 17–18 of The Algebraic and Geometric Theory of Quadratic Forms

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