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PERFECT POWERS THAT ARE SUMS OF CONSECUTIVE CUBES

Published online by Cambridge University Press:  05 October 2016

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.

Type
Research Article
Copyright
Copyright © University College London 2016 

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