Coloring the $n$-Smooth Numbers with $n$ Colors

Abstract

For which values of $n$ is it possible to color the positive integers using precisely $n$ colors in such a way that for any $a$, the numbers $a,2a,\dots,na$ all receive different colors? The third-named author posed the question around 2008-2009. Particular cases appeared in the Hungarian high school journal KöMaL in April 2010, and the general version appeared in May 2010 on MathOverflow, posted by D. Pálvölgyi. The question remains open. We discuss the known partial results and investigate a series of related matters attempting to understand the structure of these $n$-satisfactory colorings.
 Specifically, we show that there is an $n$-satisfactory coloring whenever there is an abelian group operation $\oplus$ on the set $\{1,2,\dots,n\}$ that is compatible with multiplication in the sense that whenever $i$, $j$ and $ij$ are in $\{1,\dots,n\}$, then $ij=i\oplus j$. This includes in particular the cases where $n+1$ is prime, or $2n+1$ is prime, or $n=p^2-p$ for some prime $p$, or there is a $k$ such that $q=nk+1$ is prime and $1^k,\dots,n^k$ are all distinct modulo $q$ (in which case we call $q$ a strong representative of order $n$). The colorings obtained by this process we call multiplicative. We also show that nonmultiplicative colorings exist for some values of $n$.
 There is an $n$-satisfactory coloring of $\mathbb Z^+$ if and only if there is such a coloring of the set $K_n$ of $n$-smooth numbers. We identify all $n$-satisfactory colorings for $n\leqslant 5$ and all multiplicative colorings for $n\leqslant 8$, and show that there are as many nonmultiplicative colorings of $K_n$ as there are real numbers for $n=6$ and 8. We show that if $n$ admits a strong representative $q$ then it admits infinitely many and in fact the set of such $q$ has positive natural density in the set of all primes.
 We also show that the question of whether there is an $n$-satisfactory coloring is equivalent to a problem about tilings, and use this to give a geometric characterization of multiplicative colorings.