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Some problems of “partitio numerorum”: II. Proof that every large number is the sum of at most 21 biquadrates

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References

  1. G. H. Hardy and J. E. Littlewood, Some problems of ‘Partitio numerorum’; I: A new solution of Waring's Problem, Göttinger Nachrichten 1920, S. 33–54. We shall refer to this memoir as W. P.

  2. A. J. Kempner, Über das Waringsche Problem und einige Verallgemeinerungen, Inaugural-Dissertation, Göttingen 1912.

  3. W. S. Baer, Beiträge zum Waringschen Problem, Inaugural-Dissertation, Göttingen 1913.

  4. The formula (2. 11) would lead only toG(4)≦33, in itself a new result.

  5. For a formal proof of this result see D. Cauer, Neue Anwendungen der Pfeifferschen Methode zur Abschätzung zahlentheoretischer Funktionen, Inaugural-Dissertation, Göttingen 1914, S. 38. Fork=2 (when the result includesa fortiori the corresponding results for 4, 6, ...) see E. Landau, Über die Anzahl der Gitterpunkte in gewissen Bereichen, Göttinger Nachrichten, 1912, S. 750.

  6. The symbol π is used in this sense down to the end of 5. 2, after which it is used in the ordinary sense.

  7. It should be observed that, owing to the vanishing ofA π ak andA π ak+μ whenn does not satisfy certain congruence conditions, χπ is in all cases afinite series; but this is irrelevant for our argument.

  8. See H. Weber, Lehrbuch der Algebra, Bd.1, S. 584. In Weber's notation,S p,π is one of the numbers\(\zeta = 4\eta + 1 = \sqrt n + (i,\eta ) + ( - i,\eta )\)

  9. From this point onwards π is used in the ordinary sense.

  10. Weber, l. c. Lehrbuch der Algebra, Bd.1, p. 584.

  11. See however the following note of Herr Ostrowski.

  12. The accompanying asymptotic formula is of course new.

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  1. Oxford

    G. H. Hardy

  2. Cambridge

    J. E. Littlewood

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  1. G. H. Hardy
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Hardy, G.H., Littlewood, J.E. Some problems of “partitio numerorum”: II. Proof that every large number is the sum of at most 21 biquadrates. Math Z 9, 14–27 (1921). https://doi.org/10.1007/BF01378332

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