Sums of 19 biquadrates : on the representation of large integers

Jean-Marc Deshouillers; François Dress

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (1992)

  • Volume: 19, Issue: 1, page 113-153
  • ISSN: 0391-173X

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Deshouillers, Jean-Marc, and Dress, François. "Sums of 19 biquadrates : on the representation of large integers." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 19.1 (1992): 113-153. <http://eudml.org/doc/84116>.

@article{Deshouillers1992,
author = {Deshouillers, Jean-Marc, Dress, François},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
keywords = {representation of large integers; Waring's problem; sums of 19 biquadrates},
language = {eng},
number = {1},
pages = {113-153},
publisher = {Scuola normale superiore},
title = {Sums of 19 biquadrates : on the representation of large integers},
url = {http://eudml.org/doc/84116},
volume = {19},
year = {1992},
}

TY - JOUR
AU - Deshouillers, Jean-Marc
AU - Dress, François
TI - Sums of 19 biquadrates : on the representation of large integers
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 1992
PB - Scuola normale superiore
VL - 19
IS - 1
SP - 113
EP - 153
LA - eng
KW - representation of large integers; Waring's problem; sums of 19 biquadrates
UR - http://eudml.org/doc/84116
ER -

References

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  1. [1] F.C. Auluck, On Waring's problem for biquadrates. Proc. Nat. Acad. Sci. India Sect. A11 (1940), 437-450. Zbl0061.07405MR2342
  2. [2] R. Balasubramanian, On Waring's problem: g(4) ≤ 21. Hardy-Ramanujan Journ.2 (1979), 1-31. Zbl0422.10045
  3. [3] R. Balasubramanian, On Waring's problem: g(4) ≤ 20. Hardy-Ramanujan Journ.8 (1985), 1-40. Zbl0604.10027
  4. [4] R. Balasubramanian - J-M. Deshouillers - F. Dress, Problème de Waring pour les bicarrés, 1: Schéma de la solution. C.R. Acad. Sci. Paris303 (1986), 85-88. Zbl0594.10039MR853592
  5. [5] R. Balasubramanian - J-M. Deshouillers - F. Dress, Problème de Waring pour les bicarrés, 2: résultats auxiliaires pour le théorème asymptotique. C.R. Acad. Sci. Paris303 (1986), 161-163. Zbl0594.10040MR854724
  6. [6] Z.I. Borevich - I.R. Shafarevich, Number Theory. Acad. Press, New York, 1966. Zbl0145.04902MR195803
  7. [7] Chen Jing- RUN, Waring's problem for g(5) = 37. Sci. Sinica13 (1964), 1547-1568. Zbl0146.27302MR200236
  8. [8] H. Davenport, Sur les sommes de puissances entières. C.R. Acad. Sci. Paris207 (1938), 1366-1368. Zbl0019.39501JFM64.0123.02
  9. [9] H. Davenport, On Waring's problem for fourth powers. Ann. of Math.40 (1939), 731-747. Zbl0024.01402MR253JFM65.1149.02
  10. [10] F. Delmer - J-M. Deshouillers, On the computation of g(k) in Waring's problem. Math. Comp.54 (1990), 815-820. Zbl0701.11043MR1011440
  11. [11] J-M. Deshouillers, Problème de Waring pour les bicarrés. Sém. Th. Nb. Bordeaux, année 1984- 1985, exp. 14, 47 pp. Zbl0586.10027MR848372
  12. [12] J-M. Deshouillers, Le problème de Waring pour les bicarrés: le point en 1984. Sém. Th. anal. Nb. Paris, année 1984-1985. Zbl0586.10026MR848372
  13. [13] J-M. Deshouillers - F. Dress, Sommes de diviseurs et structures multiplicatives des entiers. Acta Arith.49 (1988), 341-375. Zbl0645.10038MR937932
  14. [14] J-M. Deshouillers, Sur la majoration de sommes de Weyl biquadratiques (to appear Annali Scuola Normale SuperiorePisa). Zbl0781.11032MR1197215
  15. [15] J-M. Deshouillers - F. Dress, Numerical results for sums of 5 and 7 biquadrates, and consequences for sums of 19 biquadrates (submitted to Math. Comp.). Zbl0879.11052
  16. [16] L.E. Dickson, Proof of the ideal Waring theorem for exponents 7-180, followed by Solution of Waring's problem. Amer. Journ. Math.58 (1936), 521-535. Zbl0014.25102MR1507175JFM62.0144.01
  17. [17] G.H. Hardy - J.E. Littlewood, Some problems in "Partitio Numerorum" (VI): Further researches omn Waring's problem. Math. Z.23 (1925), 1-37. Zbl51.0148.01MR1544728JFM51.0148.01
  18. [18] Hua Loo- KENG, On Waring's problem. Quart. Journ. Math.9 (1938), 199-202. Zbl0020.10504
  19. [19] B. Landreau, Moyennes de fonctions arithmétiques sur des suites de faible densité. Thèse 3ème cycle, Université Bordeaux I, 1987. 
  20. [20] V.I. Ne, V.L. Topunov, Estimation of the modulus of complete rational trigonometric sums of degree three and four (Russian). Trudy Mat. Inst. Steklov.158 (1981), 125-129, 229. Zbl0525.10019MR662840
  21. [21] S.S. Pillai, On Waring's problem. Journ. Indian Math. Soc.2 (1936), 16-44. Zbl0014.29404JFM63.0127.01
  22. [22] J.B. Rosser - L. Schoenfeld, Approximate formulas for some functions of prime numbers. Ill. Journ. Math.6 (1962), 64-94. Zbl0122.05001MR137689
  23. [23] H.E. Thomas, A numerical approach to Waring's problem for fourth powers. Ph.D. thesis, University of Michigan, 1973. 
  24. [24] H.E. Thomas, Waring's problem for 22 biquadrates. Trans. Amer. Math. Soc.193 (1974), 427-430. Zbl0294.10033MR342478
  25. [25] E.C. Titchmarsh, The Theory of the Riemann Zeta Function. Clarendon Press, Oxford, 1951. Zbl0042.07901MR46485
  26. [26] I.M. Vinogradov, A new solution of Waring's problem. Dokl. Akad. Nank SSSR(1934), vol. 2, 337-341. Zbl0010.00902JFM60.0133.01
  27. [27] I.M. Vinogradov, On the upper bound of G(n) in Waring's problem. Izv. Akad. Nauk SSSR10 (1934), 1455-1469. Zbl61.1061.02JFM61.1061.02

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