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