Highest perfect power of a product of integers less than $x$

TL;DR

This paper determines the asymptotic behavior of the highest perfect power obtainable from the product of a subset of integers less than x, revealing a precise exponential decay rate involving logarithmic factors.

Contribution

It establishes an exact asymptotic formula for the maximum perfect power from products of integers below x, a problem previously not precisely quantified.

Findings

Derived the asymptotic formula for w(x) involving exponential decay

Showed the dominant growth rate is governed by a specific logarithmic expression

Provided insights into the structure of products forming perfect powers

Abstract

For $x\geq 3$, we define $w(x)$ as the highest integer $w$ for which there exist integers $m, y\geq 1$ and $1\leq n_1<\dots<n_m\leq x$ such that $n_1\cdots n_m=y^w$. We show that \[w(x)=x\exp\big(-(\sqrt{2}+o(1))\sqrt{\log x\log\log x}\big).\]