Product representation of perfect cubes

Abstract

Let $F_{k,d}\left(n\right)$ be the maximal size of a set $A\subseteq \left[n\right]$ such that the equation $a_1{a}_2\cdots a_k=x^d,a_1<a_2<\cdots <a_k$has no solution with $a_1,a_2,\dots ,a_k\in A$ and integer $x$. Erdős, Sárközy and T. Sós studied $F_{k,2}$, and gave bounds when $k=2,3,4,6$ and also in the general case. We study the problem for $d=3$, and provide bounds for $k=2,3,4,6$ and 9, as well as in the general case. In particular, we refute an 18-year-old conjecture of Verstraëte.

We also introduce another function $f_{k,d}$ closely related to $F_{k,d}$: While the original problem requires $a_1,\dots ,a_k$ to all be distinct, we can relax this and only require that the multiset of the $a_i$’s cannot be partitioned into $d$-tuples where each $d$-tuple consists of $d$ copies of the same number.