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Greedy sums of distinct squares
Author(s):
Hugh
L.
Montgomery;
Ulrike
M. A.
Vorhauer.
Journal:
Math. Comp.
73
(2004),
493-513.
MSC (2000):
Primary 11B83, 11A63, 11Y70, 34J10
Posted:
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  's, or several  '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:
-
- 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
Email:
hlm@umich.edu
Ulrike
M. A.
Vorhauer
Affiliation:
Department of Mathematics, Kent State University, Kent, Ohio 44242
Email:
vorhauer@math.kent.edu
DOI:
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
Posted:
June 19, 2003
Additional Notes:
The first author was supported in part by NSF Grant DMS 0070720
Copyright of article:
Copyright
2003,
American Mathematical Society
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