Problem 769

Erdős Problem #769: declared status 'open'. Formalized: no. 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 $c(n) \gg n^n$? Current best: A problem first investigated by Hadwiger, who proved the lower bound$$c(n) \geq 2^n+2^{n-1}.$$It is easy to see that $c(2)=6$. Connor and Marmorino [CoMa18] proved that$$c(n) \geq 2^{n+1}-1$$for all $n\geq 3$,$$c(n) \leq 1.8n^{n+1}$$if $n+1$ is prime, and$$c(n) \leq e^2n^n$$otherwise. Prize: no. OEIS: A014544. Tags: geometry, number theory.