The Erd\H{o}s--Moser equation $1^k+2^k+...+(m-1)^k=m^k$ revisited using continued fractions

TL;DR

This paper improves the lower bound for solutions to the Erdős–Moser equation by using continued fractions of log 2, surpassing previous computational limits and demonstrating a novel application of numerical constant calculations.

Contribution

It introduces a new method based on continued fractions to establish a higher lower bound for solutions, surpassing prior approaches by Moser.

Findings

Established that if solutions exist, m > 10^{10^9}

Connected the problem to convergents of log 2 in continued fractions

Performed extensive computations of continued fraction digits of log 2

Abstract

If the equation of the title has an integer solution with $k\ge2$, then $m>10^{9.3\cdot10^6}$. This was the current best result and proved using a method due to L. Moser (1953). This approach cannot be improved to reach the benchmark $m>10^{10^7}$. Here we achieve $m>10^{10^9}$ by showing that $2k/(2m-3)$ is a convergent of $\log2$ and making an extensive continued fraction digits calculation of $(\log2)/N$, with $N$ an appropriate integer. This method is very different from that of Moser. Indeed, our result seems to give one of very few instances where a large scale computation of a numerical constant has an application.