Problem 769

../

claims/: The 6 claim pages of Problem 769, one per claimant's result; the problem's standing derives from them.

---

Statement. Let $c(n)$ be minimal such that if $k\geq c(n)$ then the $n$-dimensional unit cube can be decomposed into $k$ homothetic $n$-dimensional cubes. Give good bounds for $c(n)$ - in particular, is it true that $c(n) \gg n^n$?

Status. Open on the site: the erdosproblems.com page labels the problem OPEN (page last edited 1 October 2025), and its proof-claims tab carries two partial proof claims, Jeffrey Zeng's listing of 24 July 2026 and Samuel Korsky's manuscript of 5 August 2026; a third partial claim, the Lean disproof of the $n^n$ bound that Star Fleet Math lists with a report dated 14 July 2026 and the formal-conjectures catalog links, is not on the tab. Each is recorded on its own page under claims without being adopted; the standing in the frontmatter is derived from the claim pages.

Source. erdosproblems.com/769, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #769, https://www.erdosproblems.com/769.

References.

• [CoMa18] Connor, Peter and Marmorino, Phillip, Decomposing cubes into smaller cubes. J. Geom. (2018), Paper No. 19, 11.
• [Er74b] Erdős, P., Remarks on some problems in number theory. Math. Balkanica (1974), 197-202.
• [Hu98] Hudelson, Matthew, Dissecting $d$-cubes into smaller $d$-cubes. J. Combin. Theory Ser. A (1998), 190-200.

Formalization. Statement in formal-conjectures, pinned at the catalog's revision of 2026-09-18: the theorem erdos_769, stating that the answer to $c(n)\gg n^n$ is no, is tagged research solved with a sorry proof and a formal_proof attribute naming Star Fleet Math's Lean disproof, linked since 2026-08-07, while the variant erdos_769.variants.growth_rate, asking whether $\log c(n)/(n\log n)$ has a limit, is tagged research open; the claim page Star Fleet Math links that proof.

Current assessment

• Question and standing. The site formulation above asks for the order of $c(n)$, the least $k_0$ such that the unit $n$-cube splits into $k$ homothetic cubes for every $k\ge k_0$, and in particular whether $c(n)\gg n^n$. No claim settles the problem, so it is open. Three pending partial claims bear on the particular question, each giving a negative answer. The earliest is Star Fleet Math's Lean development, dated 14 July 2026, whose theorem states that no absolute constant bounds $c(n)/n^n$ below, through the threshold $n\,2^n\lceil49n/100\rceil^n+2$ for odd $n\ge201$; Zeng's listing claims $c(n)\le(S(n)-1)(2^n-2)(S(n)^n-1)+1$ for odd $n$, with $S(n)=\lfloor\sqrt{4n}\rfloor+1$, and Korsky's manuscript claims $c(n)\le n^{(1/(4\sqrt e)+\varepsilon)n}$ for large odd $n$, with a polylogarithmic base under the generalized Riemann hypothesis, and $c(n)\ge((1-1/\log_23)n-\log_2n-O(1))2^n$ for even $n$. Each claim gives $c(n)=o(n^n)$ along the odd integers, so if any one holds the uniform bound $c(n)\gg n^n$ fails; none is reviewed; Zeng and Korsky call theirs partial, and the lean-proofs index calls Star Fleet Math's partial, though Star Fleet Math's own report calls it a negative resolution. None improves the upper bound in the case $n+1$ prime, where Erdős expected $c(n)>n^n$ and where the known upper bound is of order $n^{n+1}$; Korsky's lower bound alone reaches that case, since it holds for every large even $n$.
• Known results. The site records the value $c(2)=6$, Meier's conjecture $c(3)=48$ and Hadwiger's lower bound $c(n)\ge2^n+2^{n-1}$; [Er74b] records that bound without a reference, and the site cites none, so it has no claim page. Three published bounds have partial claim pages: Burgess and Erdős [Er74b], $c(n)\le(2^n-2)((n+1)^n-2)-1$, with $c(n)\ll n^{n+1}$ stated without proof, in a congress volume and so claimed; Hudelson [Hu98], $c(n)\ll(2n)^{n-1}$ in general and $c(n)<6^n$ when $\gcd(2^n-1,3^n-1)=1$; and Connor and Marmorino [CoMa18], $c(n)\ge2^{n+1}-1$ for $n\ge3$, $c(n)\le1.8n^{n+1}$ when $n+1$ is prime and $c(n)\le e^2n^n$ otherwise; the last two are refereed. The arithmetic behind the upper bounds is the threshold $h(n)$ of Problem 770: a prime $p$ with $p-1\mid n$ divides every increment $m^n-1$ with $m<p$, so the grid refinements up to size $M$ can only reach every large tile count once $M\ge h(n)$. Erdős's 1974 paper has the source card erdos_1974_remarks_problems_number_theory.
• A release item that claims nothing here. The OpenAI Math Release's preprints The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane Re(s) > 7/8 (30 September 2026; card openai_2026_quasi_riemann_hypothesis_zero_free_half_plane_7_8), the companion manuscript The Quasi-Riemann Hypothesis (5 October 2026; card openai_2026_quasi_riemann_hypothesis_zero_free_half_plane_11_12), which claims by a different proof the weaker half-plane $\operatorname{Re}s>11/12$, and Uniform exclusion of Landau–Siegel zeros (1 October 2026; card openai_2026_uniform_exclusion_landau_siegel_zeros), with Lean in the release's formalization, state a zero-free half-plane $\operatorname{Re}s>7/8$ (or $11/12$) for every Dirichlet $L$-function and a uniform gap for real zeros. They name no Erdős problem and say nothing about $c(n)$; such a half-plane would be an input to the character-nonresidue step of Korsky's argument, whose exponent rests on Pollack's Burgess-scale estimate, but no one has written that deduction, so the item is background here and gives no claim page.
• Status search. The site's page and proof-claims tab the formal-conjectures catalog and the community database (which lists the problem as formalized) are the sources of the claims above. No broader literature search is recorded.
• Proof coverage and review. The $h(n)$ step of Zeng's argument, the bounds on the collective gcd threshold, has an author-recorded own-words reconstruction on the card partial_threshold_theorem, unreviewed; the numerical-semigroup step from $h(n)$ to the bound on $c(n)$, Korsky's arguments and Star Fleet Math's Lean development are unchecked in this corpus.

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.

• erdos_1974_remarks_problems_number_theory
• erdos_1974_remarks_problems_number_theory / lemma_p199
• openai_2026_quasi_riemann_hypothesis_zero_free_half_plane_11_12
• openai_2026_quasi_riemann_hypothesis_zero_free_half_plane_11_12 / theorem_1_1
• openai_2026_quasi_riemann_hypothesis_zero_free_half_plane_7_8
• openai_2026_quasi_riemann_hypothesis_zero_free_half_plane_7_8 / corollary_1_2
• openai_2026_uniform_exclusion_landau_siegel_zeros