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

TL;DR

This paper investigates Erdős's last equation, proving that the maximum solution value grows unbounded as the number of variables increases and classifying solutions with the largest variable up to 10.

Contribution

It establishes the unbounded growth of the maximum solution value and explicitly characterizes solutions with the largest variable up to 10.

Findings

Maximum solution value tends to infinity as n increases

All solutions with the largest variable up to 10 are classified

Proves the growth behavior of solutions for Erdős'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 $x_n\le 10$.