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 $c(2)=6$. Meier conjectured $c(3)=48$. Burgess and Erdős [Er74b] proved\[c(n) \ll n^{n+1}.\]Erdős wrote 'I am certain that if $n+1$ is a prime then $c(n)>n^n$.'

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

View the LaTeX source

This page was last edited 01 October 2025.