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Representation by Quadratic Forms

Published online by Cambridge University Press:  20 November 2018

Gordon Pall
Gordon Pall*
Affiliation:
Illinois Institute of Technology, Chicago
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1. Introduction. The elementary portions of the theory of integral representation of numbers or forms by quadratic forms will be somewhat simplified and generalized in this article. This indicates certain directions in which new applications can be made. The applications made here will be largely to the representation of numbers or binary quadratic forms by ternary quadratic forms. Particularly, we shall obtain the correct estimate (Theorem 10) needed to fill a lacuna in certain work of U. V. Linnik [1] on the representation of large numbers by ternary quadratic forms. Since Linnik applied his theorem on ternaries to prove [9] that every large number is a sum of at most seven positive cubes, a lacuna in this proof can now be regarded as filled.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 1 , Issue 4 , 01 August 1949 , pp. 344 - 364
Copyright
Copyright © Canadian Mathematical Society 1949

References

[1] Linnik, U. V., “On the Representation of Large Numbers by Positive Ternary Quadratic Forms,” Bull. Acad. Sci. USSR, math, ser., vol. 4 (1940), 363-402.Google Scholar
[2] Kloosterman, H. D., Acta Math., vol. 49 (1926), 407-464.Google Scholar
[3] Tartakowsky, W., Bull. Acad. Sci. Leningrad (7) 2 (1929), 111-122 and 165-196.Google Scholar
[4] Pall, G., Amer. J. Math., vol. 68 (1946), 47–58; Ross, A. E. and Pall, G., Amer. J. Math., vol. 68 (1946), 59-65.Google Scholar
[5] Jones, B. W. and G. Pall, , Acta Math., vol. 70 (1939), 165-191.Google Scholar
[6] Glaisher, J. W. L., Quart, J. Math., vol. 20 (1885), 94.Google Scholar
[7] Venkov, B. A., Elementary Theory of Numbers (Russian), 1937.Google Scholar
[8] Pall, G., Amer. J. Math., vol. 64 (1942), 503-513.Google Scholar
[9] Linnik, U. V., Rec. Math. [Mat. Sbornik] N.S., vol. 12 (54), (1943), 218-224.Google Scholar
[10] Siegel, C. L., Ann. of Math., vol. 36 (1935), 527-606.Google Scholar
[11] Hermite, C., J. für Mathematik, vol. 47 (1850), 192.Google Scholar
[12] Smith, H. J. S., Collected Mathematical Papers (Oxford, 1894).Google Scholar
[13] C. F. Gauss, , Disquisitiones Arithmeticae, 1801, Arts. 278-283.Google Scholar
[14] Hardy, G. H. and Wright, E. M., The Theory of Numbers (Clarendon Press, 1938), p. 259.Google Scholar
[15] Schild, A., Can. J. Math., vol. 1 (1949), 47.Google Scholar