Problem 769

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 $c(n) \gg n^n$ ?

Source: erdosproblems.com/769

Number theory
Geometry

Wiki page

Status

Open

No claim settles this problem.

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.

Problem 769

Status

Proof claims

Discussion