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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Greedy sums of distinct squares


Authors: Hugh L. Montgomery and Ulrike M. A. Vorhauer
Journal: Math. Comp. 73 (2004), 493-513
MSC (2000): Primary 11B83, 11A63, 11Y70, 34J10
Published electronically: June 19, 2003
MathSciNet review: 2034134
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Abstract: When a positive integer is expressed as a sum of squares, with each successive summand as large as possible, the summands decrease rapidly in size until the very end, where one may find two $4$'s, or several $1$'s. We find that the set of integers for which the summands are distinct does not have a natural density but that the counting function oscillates in a predictable way.


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Additional Information

Hugh L. Montgomery
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1109
Email: hlm@umich.edu

Ulrike M. A. Vorhauer
Affiliation: Department of Mathematics, Kent State University, Kent, Ohio 44242
Email: vorhauer@math.kent.edu

DOI: http://dx.doi.org/10.1090/S0025-5718-03-01513-8
PII: S 0025-5718(03)01513-8
Keywords: Greedy algorithm, differential-difference equations
Received by editor(s): May 18, 2001
Published electronically: June 19, 2003
Additional Notes: The first author was supported in part by NSF Grant DMS 0070720
Article copyright: © Copyright 2003 American Mathematical Society