Acta Mathematica

, Volume 70, Issue 1, pp 165–191 | Cite as

Regular and semi-regular positive ternary quadratic forms

  • Burton W. Jones
  • Gordon Pall


Quadratic Residue Proper Solution Positive Form Regular Form Pure Quaternion 
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  1. 1.
    H. J. S. Smith,Collected Papers, vol. 1, pp. 455–509;Philosophical Transactions, vol. 157, pp. 255–298.Google Scholar
  2. 2.
    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.CrossRefMathSciNetGoogle Scholar
  3. 3.
    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.Google Scholar
  4. 1h.
    B. W. Jones,Trans. Amer. Math. Soc., vol. 33 (1931), pp. 111–124.CrossRefMathSciNetGoogle Scholar
  5. 2.
    L. E. Dickson,Annals of Math., (2), vol. 28 (1927), pp. 333–341.MathSciNetGoogle Scholar
  6. 3.
    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.Google Scholar
  7. 4.
    For references seeDickson,History of the Theory of Numbers, vol. 2.Google Scholar
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  9. 6.
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  10. 7.
    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.Google Scholar
  11. 1.
    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).Google Scholar
  12. 2.
    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.Google Scholar
  13. 1i.
    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.Google Scholar
  14. 2.
    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.Google Scholar
  15. 3.
    Gordon Pall,Amer. Journ. of Math. (1937), vol. 59, pp. 895–913.CrossRefMathSciNetGoogle Scholar
  16. 1j.
    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.Google Scholar
  17. 2.
    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.Google Scholar
  18. 1.
    B. W. Jones,A New Definition of Genus... see earlier reference.Google Scholar
  19. 1.
    Forms so marked are in genera of more than one class.Google Scholar

Copyright information

© Almqvist & Wiksells Boktryckeri-A.-B. 1939

Authors and Affiliations

  • Burton W. Jones
    • 1
  • Gordon Pall
    • 2
  1. 1.PrincetonU.S.A.
  2. 2.MontrealCanada

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