The Erdős--Moser equation $1^k+2^k+...+(m-1)^k=m^k$ revisited using continued fractions
This work advances a long-standing number theory problem by providing a much larger lower bound, though the result is incremental for the specific equation.
The authors improved the lower bound for m in the Erdős–Moser equation from 10^(9.3×10^6) to 10^(10^9) by using a continued fraction approach involving log2, achieving a significant computational milestone.
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.