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An Analytical Exploration of the Erdös-Moser Equation $ \sum_{i=1}^{m-1} i^k = m^k $ Using Approximation Methods

Guillaume Lambard

Abstract

The Erdös-Moser equation $ \sum_{i=1}^{m - 1} i^k = m^k $ is a longstanding challenge in number theory, with the only known integer solution being $ (k,m) = (1,3) $. Here, we investigate whether other solutions might exist by using the Euler-MacLaurin formula to approximate the discrete sum $ S(m-1,k) $ with a continuous function $ S_{\mathbb{R}}(m-1,k) $. We then analyze the resulting approximate polynomial $ P_{\mathbb{R}}(m) = S_{\mathbb{R}}(m-1,k) - m^k $ under the rational root theorem to look for integer roots. Our approximation confirms that for $ k=1 $, the only solution is $ m=3 $, and for $ k \geq 2 $ it suggests there are no further positive integer solutions. However, because Diophantine problems demand exactness, any omission of correction terms in the Euler-MacLaurin formula could mask genuine solutions. Thus, while our method offers valuable insights into the behavior of the Erdös-Moser equation and illustrates the analytical challenges involved, it does not constitute a definitive proof. We discuss the implications of these findings and emphasize that fully rigorous approaches, potentially incorporating prime-power constraints, are needed to conclusively resolve the conjecture.

An Analytical Exploration of the Erdös-Moser Equation $ \sum_{i=1}^{m-1} i^k = m^k $ Using Approximation Methods

Abstract

The Erdös-Moser equation is a longstanding challenge in number theory, with the only known integer solution being . Here, we investigate whether other solutions might exist by using the Euler-MacLaurin formula to approximate the discrete sum with a continuous function . We then analyze the resulting approximate polynomial under the rational root theorem to look for integer roots. Our approximation confirms that for , the only solution is , and for it suggests there are no further positive integer solutions. However, because Diophantine problems demand exactness, any omission of correction terms in the Euler-MacLaurin formula could mask genuine solutions. Thus, while our method offers valuable insights into the behavior of the Erdös-Moser equation and illustrates the analytical challenges involved, it does not constitute a definitive proof. We discuss the implications of these findings and emphasize that fully rigorous approaches, potentially incorporating prime-power constraints, are needed to conclusively resolve the conjecture.

Paper Structure

This paper contains 77 sections, 14 theorems, 156 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Extending the Sum $S(m - 1, k)$ to the $\mathbb{R}^{*}_{+}$ domain
  5. Approximating $S(m - 1, k)$ Using the Euler-MacLaurin Formula
  6. Integral Term
  7. Boundary Terms
  8. Approximation of $S(m - 1, k)$
  9. Ensuring Continuity and Differentiability of $S_{\mathbb{R}}(m - 1, k)$
  10. Analysis of the Error Term and Justification for Its Omission
  11. The Euler-MacLaurin Formula with Non-Zero Correction Terms
  12. Omission of the First Correction Term
  13. Computing the First Correction Term $C$
  14. Magnitude of the Correction Term $C$
  15. Comparison with Leading Terms
  16. ...and 62 more sections

Key Result

Lemma 1

Let $m \geq 2$ and $k \geq 1$ be positive integers. The discrete sum $S(m - 1, k) = \sum_{i=1}^{m - 1} i^{k}$ can be approximated using the Euler-MacLaurin formula gould1963maclaurinknopp1990theoryapostol2013introduction as follows: where $f(x) = x^{k}$. Evaluating the integral and boundary terms yields:

Figures (2)

  • Figure 1: Top: Plots of $S_{\mathbb{R}}(m-1,k)$ (continuous line), $m^k$ (dashed line), and $\sum_{i=1}^{m-1} i^k$ (dotted line) as a function of $k \in [2,102]$ (color-coded from purple to yellow) and $m \in [3,200]$ (as the variable on the x-axis). Bottom: Plots of $S_{\mathbb{R}}(m-1,k) - m^k$ (continuous line), $S_{\mathbb{R}}(m-1,k) - m^k + C$ (dashed line), and $\sum_{i=1}^{m-1} i^k - m^k$ (dotted line) as a function of $k \in [2,102]$ (color-coded from purple to yellow) and $m \in [3,200]$ (as the variable on the x-axis).
  • Figure 2: Plot of $P_{\mathbb{R}}(m)$ as a function of $k$ for the various rational roots $m_{0}$ described in \ref{['sec:sum_rat_roots']}

Theorems & Definitions (29)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Conjecture 1
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • ...and 19 more