# On Hilbert’s solution of Waring’s problem

Open Mathematics (2011)

- Volume: 9, Issue: 2, page 294-301
- ISSN: 2391-5455

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topPaul Pollack. "On Hilbert’s solution of Waring’s problem." Open Mathematics 9.2 (2011): 294-301. <http://eudml.org/doc/269093>.

@article{PaulPollack2011,

abstract = {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.},

author = {Paul Pollack},

journal = {Open Mathematics},

keywords = {Waring’s problem; Hilbert-Waring theorem; Additive number theory; Elementary methods; sums of -th powers of positive integers; variant of Waring's problem; polynomial identities; elementary methods},

language = {eng},

number = {2},

pages = {294-301},

title = {On Hilbert’s solution of Waring’s problem},

url = {http://eudml.org/doc/269093},

volume = {9},

year = {2011},

}

TY - JOUR

AU - Paul Pollack

TI - On Hilbert’s solution of Waring’s problem

JO - Open Mathematics

PY - 2011

VL - 9

IS - 2

SP - 294

EP - 301

AB - 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.

LA - eng

KW - Waring’s problem; Hilbert-Waring theorem; Additive number theory; Elementary methods; sums of -th powers of positive integers; variant of Waring's problem; polynomial identities; elementary methods

UR - http://eudml.org/doc/269093

ER -

## References

top- [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) Zbl0384.10010
- [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 Zbl0247.10029
- [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 Zbl47.0114.02
- [6] Hardy G.H., Wright E.M., An Introduction to the Theory of Numbers, 6th ed., Oxford University Press, Oxford, 2008 Zbl1159.11001
- [7] Hausdorff F., Zur Hilbertschen Lösung des Waringschen Problems, Math. Ann., 1909, 67(3), 301–305 http://dx.doi.org/10.1007/BF01450406 Zbl40.0237.01
- [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) Zbl0063.03580
- [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) Zbl1072.11073
- [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 Zbl0094.02702
- [12] Rieger G.J., Zur Hilbertschen Lösung des Waringschen Problems: Abschätzung von g(n), Mitt. Math. Sem. Giessen, 1953, #44 Zbl0053.35902
- [13] Rieger G.J., Zur Hilbertschen Lösung des Waringschen Problems: Abschätzung von g(n), Arch. Math. (Basel), 1953, 4, 275–281 Zbl0053.35902
- [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 Zbl0868.11046
- [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 Zbl0010.10306

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