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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.

References [Enhancements On Off] (What's this?)

  • 1. G. Andrews, R. Askey and R. Roy, Special functions, Encyclopedia of Mathematics and its Applications 71, Cambridge University Press, Cambridge, 1999. MR 2000g:33001
  • 2. P.T. Bateman, A.J. Hildebrand and G.B. Purdy, Sums of distinct squares, Acta Arith. 67 (1994), 349-380. MR 95j:11092
  • 3. R. Bellman and K.L. Cooke, Differential-difference equations, Mathematics in Science and Engineering, vol. 6, Academic Press, New York, 1963. MR 26:5259
  • 4. H. Delange, Sur la fonction sommatoire de la fonction ``somme de chiffres'', Enseignement Math. (2) 21 (1975), 31-47. MR 52:319
  • 5. P. Flajolet, P. Grabner, P. Kirschenhofer, H. Prodinger and R. Tichy, Mellin transforms and asymptotics: digital sums, Theoret. Comput. Sci. 123 (1994), 291-314. MR 94m:11090
  • 6. F. Halter-Koch, Darstellung natürlicher Zahlen als Summe von Quadraten, Acta Arith. 42 (1982), 11-20. MR 84b:10025
  • 7. G. Pall, On sums of squares, Amer. Math. Monthly 40 (1933), 10-18.
  • 8. E.M.Wright, The representation of a number as a sum of five or more squares, Quart. J. Math. Oxford Ser. 4 (1933), 37-51.

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Additional Information

Hugh L. Montgomery
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1109

Ulrike M. A. Vorhauer
Affiliation: Department of Mathematics, Kent State University, Kent, Ohio 44242

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

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