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Erdős #769 · number theory / geometry

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

MSC 11 · Number theory

Inferred from the community field; finer classification has not been reviewed.

Source reports OPENAs of 2026-09-08Source & evidence ↓
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Problem statement

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

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Background

A problem first investigated by Hadwiger, who proved the lower bound

c(n)≥2n+2n−1.c(n) \geq 2^n+2^{n-1}.

It is easy to see that c(2)=6c(2)=6. Meier conjectured c(3)=48c(3)=48. Burgess and Erdős [Er74b] proved

c(n)≪nn+1.c(n) \ll n^{n+1}.

Erdős wrote 'I am certain that if n+1n+1 is a prime then c(n)>nnc(n)>n^n.'

Hudelson [Hu98] proved that if (2n−1,3n−1)=1(2^n-1,3^n-1)=1 then c(n)<6nc(n) < 6^n, and in general c(n)≪(2n)n−1c(n) \ll (2n)^{n-1}. Connor and Marmorino [CoMa18] proved that

c(n)≥2n+1−1c(n) \geq 2^{n+1}-1

for all n≥3n\geq 3,

c(n)≤1.8nn+1c(n) \leq 1.8n^{n+1}

if n+1n+1 is prime, and

c(n)≤e2nnc(n) \leq e^2n^n

otherwise.

Progress & source assessment

The source labels this entry OPEN in the 2026-09-08 snapshot. Editorial progress notes are reproduced as source commentary, not independently verified results. Prize amounts and Lean badges at the source are not local awards or verification.

References
  1. [1]Thomas Bloom (ed.), Erdős Problems #769 link
  2. [2]Upstream Lean 4 statement for Erdős #769 linkExternal statement reference. Not compiled or verified by this platform; upstream files may contain open proof obligations.
  3. [3]Formal Conjectures — Erdős Problem 769 link

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