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


Authors: Yves Gallot, Pieter Moree and Wadim Zudilin
Journal: Math. Comp. 80 (2011), 1221-1237
MSC (2010): Primary 11D61, 11Y65; Secondary 11A55, 11B83, 11K50, 11Y60, 41A60
Published electronically: November 22, 2010
MathSciNet review: 2772120
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Abstract: If the equation of the title has an integer solution with $ k\ge 2$, then $ m>10^{9.3\cdot 10^6}$. 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 $ \log 2$ and making an extensive continued fraction digits calculation of $ (\log 2)/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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Additional Information

Yves Gallot
Affiliation: 12 bis rue Perrey, 31400 Toulouse, France
Email: galloty@orange.fr

Pieter Moree
Affiliation: Max-Planck-Institut für Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany
Email: moree@mpim-bonn.mpg.de

Wadim Zudilin
Affiliation: School of Mathematical and Physical Sciences, University of Newcastle, Callaghan NSW 2308, Australia
Email: wadim.zudilin@newcastle.edu.au

DOI: http://dx.doi.org/10.1090/S0025-5718-2010-02439-1
Received by editor(s): July 7, 2009
Received by editor(s) in revised form: April 9, 2010
Published electronically: November 22, 2010
Additional Notes: This research was carried out while the third author was visiting the Max Planck Institute for Mathematics (MPIM) and the Hausdorff Center for Mathematics (HCM) and was financially supported by these institutions. He and the second author thank the MPIM and HCM for providing such a nice research environment
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.