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Bounds on sums of cubes

Published online by Cambridge University Press:  17 February 2025

A. F. Beardon
A. F. Beardon*
Affiliation:
DPMMS, University of Cambridge, CB3 0WB e-mail: afb@dpmms.cam.ac.uk

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Here we are interested in the familiar identity but only in as far as it provides the solution (1, 2, …, n) of the Diophantine equation (1) where n and the aj are positive integers. In fact, not only is (1, 2, …, n) a solution of (1), it has the remarkable property that, up to a permutation of its entries, it is the only solution of (1) in which the aj are distinct (see Section 4). Thus any other solution will necessarily contain repetitions among the aj and, for this reason, a solution must be defined to be a sequence (which allows repetitions) rather than a set. However, we shall not distinguish between two solutions whose entries are permutations of each other.

Information

Type
Articles
Information
The Mathematical Gazette , Volume 109 , Issue 574 , March 2025, pp. 39 - 46
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Copyright
© The Authors, 2025 Published by Cambridge University Press on behalf of The Mathematical Association

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