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

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 $2k\mathrm{/}(2m-3)$ is a convergent of $\log 2$ and making an extensive continued fraction digits calculation of $(\log 2)\mathrm{/}N$, 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.