On Hilbert’s solution of Waring’s problem

• Autorzy: Paul Pollack
• Języki publikacji: EN

Abstrakty

EN

In 1909, Hilbert proved that for each fixed k, there is a number g with the following property: Every integer N ≥ 0 has a representation in the form N = x 1k + x 2k + … + x gk, where the x i are nonnegative integers. This resolved a conjecture of Edward Waring from 1770. Hilbert’s proof is somewhat unsatisfying, in that no method is given for finding a value of g corresponding to a given k. In his doctoral thesis, Rieger showed that by a suitable modification of Hilbert’s proof, one can give explicit bounds on the least permissible value of g. We show how to modify Rieger’s argument, using ideas of F. Dress, to obtain a better explicit bound. While far stronger bounds are available from the powerful Hardy-Littlewood circle method, it seems of some methodological interest to examine how far elementary techniques of this nature can be pushed.

Słowa kluczowe

EN

Waring’s problem; Hilbert-Waring theorem; Additive number theory; Elementary methods

Kategorie tematyczne

• 11P05: Waring's problem and variants

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2011
• Tom: 9
• Numer: 2
• Strony: 294-301

Daty

wydano

2011-04-01

online

2011-02-18

Twórcy

autor

Paul Pollack

• University of Illinois

Bibliografia

• [1] Bredikhin B.M., Grishina T.I., An elementary estimate of G(n) in Waring’s problem, Mat. Zametki, 1978, 24(1), 7–18 (in Russian)
• [2] Dress F., Méthodes élémentaires dans le problème de Waring pour les entiers, Journées Arithmétiques Françaises, Mai 1971, Université de Provence, Marseille, 1971
• [3] Dress F., Théorie additive des nombres, problème de Waring et théorème de Hilbert, Enseignement Math., 1972, 18, 175–190, 301–302
• [4] Hardy G.H., Some Famous Problems of the Theory of Numbers and in Particular Waring’s Problem, Clarendon Press, Oxford, 1920
• [5] Hardy G.H., Littlewood J.E., Some problems of “Partitio Numerorum” I: a new solution of Waring’s problem, Göttingen Nachr., 1920, 33–54
• [6] Hardy G.H., Wright E.M., An Introduction to the Theory of Numbers, 6th ed., Oxford University Press, Oxford, 2008
• [7] Hausdorff F., Zur Hilbertschen Lösung des Waringschen Problems, Math. Ann., 1909, 67(3), 301–305 http://dx.doi.org/10.1007/BF01450406
• [8] Hua L.K., Introduction to Number Theory, Springer, Berlin-New York, 1982
• [9] Linnik Yu.V., An elementary solution of the problem of Waring by Schnirelman’s method, Mat. Sb., 1943, 12(54)(2), 225–230 (in Russian)
• [10] Nesterenko Yu.V., On Waring’s problem (elementary methods), Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 2005, 322, Trudy po Teorii Chisel, 149–175 (in Russian)
• [11] Newman D.J., A simplified proof of Waring’s conjecture, Michigan Math. J., 1960, 7(3), 291–295 http://dx.doi.org/10.1307/mmj/1028998439
• [12] Rieger G.J., Zur Hilbertschen Lösung des Waringschen Problems: Abschätzung von g(n), Mitt. Math. Sem. Giessen, 1953, #44
• [13] Rieger G.J., Zur Hilbertschen Lösung des Waringschen Problems: Abschätzung von g(n), Arch. Math. (Basel), 1953, 4, 275–281
• [14] Rieger G.J., Zum Waringschen Problem für algebraische Zahlen and Polynome, J. Reine Angew. Math., 1955, 195, 108–120
• [15] Stridsberg E., Sur la démonstration de M. Hilbert du théorème de Waring, Math. Ann., 1912, 72(2), 145–152 http://dx.doi.org/10.1007/BF01667319
• [16] Vaughan R.C., The Hardy-Littlewood Method, 2nd ed., Cambridge Tracts in Math., 125, Cambridge University Press, Cambridge, 1997
• [17] Waring E., Meditationes Algebraicæ, American Mathematical Society, Providence, 1991
• [18] Wright E.M., An easier Waring’s problem, J. London Math. Soc., 1934, 9(4), 267–272 http://dx.doi.org/10.1112/jlms/s1-9.4.267

Identyfikatory

DOI

10.2478/s11533-011-0009-z