Decomposing cubes into smaller cubes
Given a positive integer \(n\), let us define \(D(n)\) as the set of integers \(k\) such that an \(n\)-dimensional cube can be decomposed into \(k\) smaller \(n\)-dimensional cubes. In addition, let \(c(n)\) be the smallest integer (it is known to exist, for example \(c(2)=6\)) such that \(k\geq c(n) \Rightarrow k\in D(n)\). This paper presents some results dealing with \(c(n)\). In particular, it is proven that \(c(n)\geq 2^{n+1}+1\) for every \(n\geq 3\) and also that \(c(n)\leq e^2n^n\) if \(n+1\) is not a prime and \(c(n)\leq 1.8n^{n+1}\) if \(n+1\) is a prime.
scientific article; zbMATH DE number 6876764