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

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

2009-07-08

arxiv.org/abs/0907.1356v1

Math. Comp. 80 (2011), no. 274, 1221--1237

Mathematics

Number Theory

17 pages

Scientific paper

10.1090/S0025-5718-2010-02439-1

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.

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