The Erdős-Moser equation 1k+2k+...+(m-1)k=mk revisited using continued fractions

  title={The Erdős-Moser equation 1k+2k+...+(m-1)k=mk revisited using continued fractions},
  author={Yves Gallot and Pieter Moree and Wadim Zudilin},
  journal={Math. Comput.},
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… 

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