Greedy sums of distinct squares

Montgomery, Hugh; Vorhauer, Ulrike

doi:10.1090/S0025-5718-03-01513-8

Greedy sums of distinct squares
HTML articles powered by AMS MathViewer

by Hugh L. Montgomery and Ulrike M. A. Vorhauer PDF

Math. Comp. 73 (2004), 493-513 Request permission

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

George E. Andrews, Richard Askey, and Ranjan Roy, Special functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR 1688958, DOI 10.1017/CBO9781107325937
Paul T. Bateman, Adolf J. Hildebrand, and George B. Purdy, Sums of distinct squares, Acta Arith. 67 (1994), no. 4, 349–380. MR 1301824, DOI 10.4064/aa-67-4-349-380
Richard Bellman and Kenneth L. Cooke, Differential-difference equations, Academic Press, New York-London, 1963. MR 0147745
Hubert Delange, Sur la fonction sommatoire de la fonction“somme des chiffres”, Enseign. Math. (2) 21 (1975), no. 1, 31–47 (French). MR 379414
Philippe Flajolet, Peter Grabner, Peter Kirschenhofer, Helmut Prodinger, and Robert F. Tichy, Mellin transforms and asymptotics: digital sums, Theoret. Comput. Sci. 123 (1994), no. 2, 291–314. MR 1256203, DOI 10.1016/0304-3975(92)00065-Y
Franz Halter-Koch, Darstellung natürlicher Zahlen als Summe von Quadraten, Acta Arith. 42 (1982/83), no. 1, 11–20 (German). MR 678994, DOI 10.4064/aa-42-1-11-20
G. Pall, On sums of squares, Amer. Math. Monthly 40 (1933), 10–18.
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.