# 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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## Abstract

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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.

## How to cite

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Paul 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

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1. [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
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13. [13] Rieger G.J., Zur Hilbertschen Lösung des Waringschen Problems: Abschätzung von g(n), Arch. Math. (Basel), 1953, 4, 275–281 Zbl0053.35902
14. [14] Rieger G.J., Zum Waringschen Problem für algebraische Zahlen and Polynome, J. Reine Angew. Math., 1955, 195, 108–120
15. [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. [16] Vaughan R.C., The Hardy-Littlewood Method, 2nd ed., Cambridge Tracts in Math., 125, Cambridge University Press, Cambridge, 1997 Zbl0868.11046
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