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

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.Google Scholar

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

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

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

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.Google Scholar

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

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

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

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

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

See however the following note of Herr Ostrowski.Google Scholar

The accompanying asymptotic formula is of course new.Google Scholar