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Claim. P. Erdős, Remarks on some problems in number theory, Math. Balkanica 4 (1974), 197--202, papers presented at the Fifth Balkan Mathematical Congress, gives on p. 199 an improvement, due to Burgess and himself, of Meier's upper bound for the least $c(n)$ such that the unit $n$-cube splits into $k$ homothetic cubes for every $k\ge c(n)$, the quantity of Problem 769. Its display (2) is

$$c(n)\le(2^n-2)\bigl((n+1)^n-2\bigr)-1.$$

The proof replaces one cube of a decomposition by $k^n$ smaller cubes, so every $\sum_{k=2}^{n+1}c_k(k^n-1)$ with $c_k\ge1$ is a tile count; the numbers $k^n-1$, $2\le k\le n+1$, have no common factor (the paper's lemma), and a theorem of Brauer on representations by a set of coprime integers bounds the largest gap. The paper adds that a sharper theorem of Brauer gives $c(n)<\alpha n^{n+1}$ for an absolute constant $\alpha$, which it calls not hard to prove and does not prove. The source card is Erdős (1974).

Covers. The displayed upper bound for every $n$, as part of the request for good bounds; the bound $c(n)\ll n^{n+1}$ is stated without proof. Not covered: the order of $c(n)$ and the question whether $c(n)\gg n^n$.

Depends on. No page of this wiki; the paper's lemma is restated on the source card's lemma page.

Acceptance. None on record. The volume collects the papers of a congress, with no evidence that it was refereed, and the site, which credits the bound $c(n)\ll n^{n+1}$ to Burgess and Erdős, labels the problem OPEN.