On the largest value of the solutions of Erd\H{o}s's last equation

Abstract

Let $n$ be a positive integer. The Diophantine equation $n(x_1+x_2+\dots +x_n)=x_1x_2\dots x_n$, $1 \le x_1\le x_2\le \dots \le x_n$ is called Erd\H{o}s's last equation. We prove that $x_n\to \infty $ as $n\to \infty$ and determine all tuples $(n,x_1,\dots ,x_n)$ with $x_n\le 10$.