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

István Pink, Csaba Sándor

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 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 $x_n\le 10$.

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

Abstract

Let be a positive integer. The Diophantine equation , is called Erdős's last equation. We prove that as and determine all tuples with .

Paper Structure

This paper contains 6 sections, 8 theorems, 63 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Auxiliary results
  3. Proof of Theorem \ref{['thm1']}
  4. Proof of Theorem \ref{['thm2']}
  5. Proof of Theorem \ref{['thm3']}
  6. Proof of Theorem \ref{['thm4']}

Key Result

Theorem 1

Consider equation ELE. Then, we have $\liminf_{n\to \infty }\frac{g(n)}{\sqrt[3]{n}}\le 1$.

Theorems & Definitions (17)

  • Theorem 1
  • Theorem 2
  • Remark 1
  • Theorem 3
  • Remark 2
  • Theorem 4
  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • ...and 7 more