Product representations of perfect powers

Abstract

Let $\ddot{I}{}_k(N)$ denote the maximum size of a set $A\subseteq \{1,2,\dots ,N\}$ such that no product of $k$ distinct elements of $A$ is a perfect $d$-th power. In this short note, we prove that $\ddot{I}{}_d(N)=\sum _{k=1}^{d-1}\ddot{I}€\left(\frac{N}{k}\right)+O_d(\ddot{I}€(N^{1/2}))$, furthermore, for prime power $d$ and sufficiently large $N$ we have $\ddot{I}{}_d(N)=\sum _{k=1}^{d-1}\ddot{I}€\left(\frac{N}{k}\right)$. This answers a question of Verstraëte.