The Erdős--Moser equation $1^k+2^k+...+(m-1{)}^k=m^k$ revisited using continued fractions

Abstract

If the equation of the title has an integer solution with $k\ge 2$, then $m>10^{9.3\cdot 10^6}$. 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 $\log 2$ and making an extensive continued fraction digits calculation of $(\log 2)/N$, 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.

17 pages

Cited by in corpus (2)

• On stronger conjectures that imply the Erdös-Moser conjecture
• Forbidden integer ratios of consecutive power sums