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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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The Erdős–Moser equation $1^k+2^k+\dots +(m-1)^k=m^k$ revisited using continued fractions
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by Yves Gallot, Pieter Moree and Wadim Zudilin PDF
Math. Comp. 80 (2011), 1221-1237 Request permission

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
  • MR Author ID: 290905
  • 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
  • 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
  • © Copyright 2010 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Math. Comp. 80 (2011), 1221-1237
  • MSC (2010): Primary 11D61, 11Y65; Secondary 11A55, 11B83, 11K50, 11Y60, 41A60
  • DOI: https://doi.org/10.1090/S0025-5718-2010-02439-1
  • MathSciNet review: 2772120