On the largest value of the solutions of Erdős's last equation

Abstract

Let $n$ be a positive integer. The Diophantine equation $n(x_1+x_2+\cdots +x_n)=x_1{x}_2\dots x_n$, $1\le x_1\le x_2\le \cdots \le x_n$ is called Erdős'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 $x_n\le 10$.

14 pages