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

Abstract

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

1 page, accepted for publication in the IJNT