Advertisement

Journal of Mathematical Sciences

, Volume 137, Issue 2, pp 4699–4715 | Cite as

On Waring’s problem (elementary methods)

  • Yu. V. Nesterenko
Article
  • 72 Downloads

Abstract

The paper deals with two elementary methods for solving Waring’s problem on the representation of numbers as a sum of equal exponents in powers of natural numbers. The first method is an elementary version of the original Hilbert’s proof, and the second one simplifies and makes more precise the elementary Linnik’s proof based on the estimate of the number of solutions of a certain system of Diophantine equations. Bibliography: 21 titles.

Keywords

Positive Integer Nonnegative Integer Recurrent Equation Diophantine Equation Continue Fraction Expansion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    B. M. Bredikhin and T. I. Grishina, “An elementary estimate of G(n) in Waring’s problem,” Mat. Zametki, 24, No. 1, 7–18.Google Scholar
  2. 2.
    B. M. Bredikhin and Yu. V. Linnik, “New method in the analytical number theory,” in: Actual Problems in Analytic Number Theory, Minsk (1974), pp. 5–22.Google Scholar
  3. 3.
    A. O. Gel’fond and Yu. V. Linnik, Elementary Methods in Analytic Number Theory [in Russian], Fizmatlit, Moscow (1962).Google Scholar
  4. 4.
    W. B. Jones and W. J. Thron, Continued Fractions, Addison-Wesley (1980).Google Scholar
  5. 5.
    D. S. Kuznetsov, Special Functions [in Russian], Vysshaya Shkola, Moscow (1965).Google Scholar
  6. 6.
    Yu. V. Linnik, “An elementary proof of Waring’s problem by using Shnirelman’s method,” Mat. Sb., 12, No. 2, 225–230 (1943).zbMATHMathSciNetGoogle Scholar
  7. 7.
    Yu. V. Linnik, “On the decomposition of large numbers into seven cubes,” Mat. Sb., 12, No. 2, 218–224 (1943).zbMATHMathSciNetGoogle Scholar
  8. 8.
    Hua Loo Keng, The Method of Trigonometric Sums and Its Applications in Number Theory [Russian translation], Mir, Moscow (1964).Google Scholar
  9. 9.
    L. G. Shnirelman, “Uber additive Eigenschaften von Zahlen,” Math. Ann., 107, 649–690 (1933).MathSciNetCrossRefGoogle Scholar
  10. 10.
    L. E. Dickson, History of the Theory of Numbers, Vol. 2, Chelsea, New York (1971).Google Scholar
  11. 11.
    G. Frobenius, “Uber den Stridsbergschen Beweis des Waringschen Satzes,” Sitzungsber. Akad. Wiss. Berlin, 666–670 (1912).Google Scholar
  12. 12.
    F. Hausdorff, “Zur Hilbertschen Lözung des Waringschen Problems,” Math. Ann., 67, 301–305 (1909).zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    D. Hilbert, “Beweis für Darstellbarkeit deer ganzen Zahlen durch eine feste Anzahl n-ter Potenzen (Waringsches Problem),” Math. Ann., 67, 281–300 (1909).zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Hua Loo Keng, Introduction to Number Theory, Springer, Berlin (1982).zbMATHGoogle Scholar
  15. 15.
    A. Hurwitz, “Über die Darstellung der ganzen Zahlen als Summen von n ten Potenzen ganzer Zahlen,” Math. Ann., 65, No. 3, 424–427 (1908).zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    A. Hurwitz, “Über definite Polynome,” Math. Ann., 73, 173–176 (1912).zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    R. Remak, “Bemerkung zu Herrn Stridsbergs Beweis des Waringschen Theorems,” Math. Ann., 72, 153–156 (1912).zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    E. Stridsberg, “Sur la démonstration de M. Hilbert du théorème de Waring,” Math. Ann., 72, No. 2, 145–152 (1912).zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    G. L. Watson, “A proof of the seven cube theorem,” J. London Math. Soc., 26, 153–156 (1951).zbMATHMathSciNetGoogle Scholar
  20. 20.
    T. D. Wooley, “Large improvements in Waring’s problem,” Ann. Math., 135(2), No. 1, 131–164 (1992).zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    H. Davenport, Analytic Methods for Diophantine Equations and Diophantine Inequalities, Second edition, Cambridge University Press (2005).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • Yu. V. Nesterenko
    • 1
  1. 1.Moscow Lomonosov State UniversityMoscowRussia

Personalised recommendations