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Literatur
H. J. S. Smith,Collected Papers, vol. 1, pp. 455–509;Philosophical Transactions, vol. 157, pp. 255–298.
B. W. Jones,Trans. Amer. Math. Soc., vol. 33 (1931), pp. 92–110; alsoArnold Ross Proc. Nat. Acad. Sc., vol. 18 (1932), pp. 600–608.
Two forms are of the sameclass if one may be taken into the other by a linear trans formation withintegral coefficients and of determinant I; i. e. by a unimodular transformation.
B. W. Jones,Trans. Amer. Math. Soc., vol. 33 (1931), pp. 111–124.
L. E. Dickson,Annals of Math., (2), vol. 28 (1927), pp. 333–341.
A. Meyer gave a partial proof inJournal für Mathematik, vol. 108 (1891), pp. 125–139. For a complete proof with further references seeL. E. Dickson,Studies in the Theory of Numbers, chap. 4.
For references seeDickson,History of the Theory of Numbers, vol. 2.
S. Ramanujan,Proc. Cambridge Phil. Soc., vol. 19 (1916), pp. 11–21; alsoCollected Papers, pp. 169–178.
B. W. Jones, «The Representation of Integers by Positive Ternary Quadratic Forms», a University of Chicago thesis (1928), unpublished.
In the thesis the form (1, 5, 200) was erroneously reported to be regular. It fails to represent 44 and hence is irregular. The rest of the table has been checked and found to be correct.
that isx 2+y 2+16z 2. Similarlyax 2+by 2+cz 2+2ryz+28xz+2txy is denoted by (a, b, c, r, s, t).
Nazimoff,Applications of the Theory of Elliptic Functions to the Theory of Numbers (Russian) translated by Arnold Chaimovitch. The proof for this form was indicated by Nazimoff and carried out by Chaimovitch.
W. A. Tartakowsky,Comptes Rendus de l'Académie des Sciences, vol. 186 (1928), pp. 1337–1340, 1401–1403, 1684–1687. Errata in the second paper are corrected in vol. 187, p. 155. Complete paper in Bull. Ak. Sc. U. R. S. S. (7) (1929), pp. 111–22, 165–96.
For references seeDickson,History of the Theory of Numbers, vol. 2, pp. 261–3 and p. 268 respectively. For example Glaisher states the following inMessenger of Mathematics, new series vol. 6, (1877), p. 104: The excess of the number of representations of 8n+1 in the formx 2+4y 2+4z 2 withy andz even over the number of representations withy andz odd is zero if 8n+I is not a square and 2(−I)(s−1)/2 s if 8n+I=s 2.
Gordon Pall,Amer. Journ. of Math. (1937), vol. 59, pp. 895–913.
Eisenstein,Journal für Mathematik, vol. 41 (1851), pp. 141–190 gives a table for determinants from 1 to 100.Arnold Ross, inStudies in the Theory of Numbers byL. E. Dickson, pp. 181–185 has a table for determinants from 1 to 50.E. Borissow,Reduction of Positive Ternary Quadratic Forms by Selling's Method, with a Table of Reduced Forms for all Determinants from 1 to 200. St. Petersburg (1890), 1–108; tables 1–116 (Russian).B. W. Jones,A Table of Eisenstein reduced Positive Ternary Quadratic Forms of Determinant≦200 (1935), Bulletin No. 97 of the National Research Council.
H. J. S. Smith,Collected Papers, vol. 2, p. 635; alsoMémoires présentés par divers Savants à l'Académie des Sciences de l'Institut de France (2), vol. 29 (1887), No. 1, 72 pp. 22–38333.Acta mathematica. 70. Imprimé le 2 décembre 1938.
B. W. Jones,A New Definition of Genus... see earlier reference.
Forms so marked are in genera of more than one class.
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Jones, B.W., Pall, G. Regular and semi-regular positive ternary quadratic forms. Acta Math. 70, 165–191 (1939). https://doi.org/10.1007/BF02547347
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DOI: https://doi.org/10.1007/BF02547347