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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 amethod due to L. Moser (1953). This approach cannot be improved to reach thebenchmark $m>10^{10^7}$. Here we achieve $m>10^{10^9}$ by showing that$2k/(2m-3)$ is a convergent of $\log2$ and making an extensive continuedfraction digits calculation of $(\log2)/N$, with $N$ an appropriate integer.This method is very different from that of Moser. Indeed, our result seems togive one of very few instances where a large scale computation of a numericalconstant has an application.

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