Product representations of perfect powers

TL;DR

This paper determines the maximum size of subsets of {1,...,N} avoiding products of k elements being perfect d-th powers, providing exact formulas for prime power d and answering a question by Verstra"ete.

Contribution

It establishes precise asymptotic formulas for the size of such sets, extending previous understanding of product-avoiding subsets and resolving an open question.

Findings

Exact asymptotic formula for (N) when d is a prime power.

Asymptotic expression involving the prime counting function (N/k).

Resolution of Verstrate's question on the structure of these sets.

Abstract

Let $\rho_k(N)$ 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 $\rho _d(N)=\sum\limits_{k=1}^{d-1}\pi\left( \frac{N}{k} \right) +O_d(\pi (N^{1/2}))$, furthermore, for prime power $d$ and sufficiently large $N$ we have $\rho _d(N)=\sum\limits_{k=1}^{d-1}\pi\left( \frac{N}{k} \right)$. This answers a question of Verstra\"ete.