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The Erdős–Moser equation $$1^k+2^k+\cdots +(m-1)^k=m^k$$ revisited using continued fractions. (English) Zbl 1231.11038
In the paper the authors investigate the non-trivial solutions of the equation $1^k+2^k+\cdots +(m-2)^k+(m-1)^k=m^k$ with $$k\geq 2$$. Conjecturally such solutions do not exist. L. Moser [Scripta Math. 19, 84–88 (1953; Zbl 0050.26604)] showed that if $$(m,k)$$ is a solution of this equation then $$m>10^{10^6}$$ and $$k$$ is even. Using the method of Moser this result cannot be improved on substantially. Now the authors prove that $$m>10^{10^9}$$ using an extensive continued fraction digits calculation of $$\frac{\log 2}{N}$$ where $$N$$ is an appropriate integer. The new method uses a large scale computation of a numerical constant.

##### MSC:
 11D61 Exponential Diophantine equations 11Y65 Continued fraction calculations (number-theoretic aspects) 11A55 Continued fractions
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