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PERFECT POWERS THAT ARE SUMS OF CONSECUTIVE CUBES

Part of: Diophantine equations Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  05 October 2016

Michael A. Bennett ,
Vandita Patel  and
Samir Siksek
Michael A. Bennett
Affiliation:
Department of Mathematics, University of British Columbia, VancouverB.C., V6T 1Z2, Canada email [email protected]
Vandita Patel
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, U.K. email [email protected]
Samir Siksek
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, U.K. email [email protected]
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Abstract

Euler noted the relation $6^{3}\,=\,3^{3}+4^{3}+5^{3}$ and asked for other instances of cubes that are sums of consecutive cubes. Similar problems have been studied by Cunningham, Catalan, Gennochi, Lucas, Pagliani, Cassels, Uchiyama, Stroeker and Zhongfeng Zhang. In particular, Stroeker determined all squares that can be written as a sum of at most 50 consecutive cubes. We generalize Stroeker’s work by determining all perfect powers that are sums of at most 50 consecutive cubes. Our methods include descent, linear forms in two logarithms and Frey–Hellegouarch curves.

MSC classification

Primary: 11D61: Exponential equations
Secondary: 11D41: Higher degree equations; Fermat's equation 11F80: Galois representations 11F11: Holomorphic modular forms of integral weight
Type
Research Article
Information
Mathematika , Volume 63 , Issue 1 , 2017 , pp. 230 - 249
Copyright
Copyright © University College London 2016 

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