Skip to main content
Log in

The rational solutions of the diophantine equation

  • Published:
Acta Mathematica

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

References

  1. Billing, G., “Ueber kubische Diophantische Gleichungen mit endlich vielen Lösungen”. Comm. Math. Helv. 9 (1936–37), pp. 161–165.

    Article  Google Scholar 

  2. —, “Beiträge zur arithmetischen Theorie ebener kubischer Kurven”. Nova Acta Reg. Soc. Scient. Upsaliensis. Ser. IV vol. XI (1938) No. 1, pp. 1–165.

    Google Scholar 

  3. Brunner, O. “Lösungseigenschaften d. kubischen Diophantischen GleichungZ 3Y 2=D”. Inaugural dissertation. Zürich 1933.

  4. —, “Weitere Untersuchungen über die Kubische Diophantische GleichungZ 3Y 2=D”. Comm. Math. Helv. 7 (1934). p. 67.

    Article  MathSciNet  Google Scholar 

  5. Chang Kuo-Lung, “On some Diophantine equationsy 2=x 3+k with no rational solutions”. Quarterly J. (Oxford) 19 (1948), pp. 181–188.

    Article  Google Scholar 

  6. Châtelet, F. “Points exceptionnels d'un cubique de Weierstrass”. Comptes Rendus (Paris). 210 (1940), p. 90.

    Google Scholar 

  7. — “Groupe exceptionel d'une classe de cubiques”. Comptes Rendus (Paris). 210 (1940), p. 200.

    Google Scholar 

  8. Dedekind, R., “Ueber reine kubische Körper”. J. f. Math. 121 (1900). 40–123 and Ges. Math. Werke (Braunschweig, 1930). Vol. II, pp. 148–234.

    Google Scholar 

  9. Delaunay, B. N. andFaddeev, D. K., “Theory of irrationals of the third degree” (in Russian). Travaux de l'Institut Stekloff XI (1940).

  10. Euler, L. “Theoremata quorundarum arithmeticorum demonstratio” Theorema 10. Opera Omnia Ser. 1 vol. 2 (=Commentationes Arithmeticae vol. 1) (Lipsiae et Berolini, MCMXV) pp. 38–58 especially pp. 56–58.

  11. Faddeev, D. K., “The equationx 3+y 3=Az 3”. (in Russian). Travaux de l'Institut Stekloff V (1934). pp. 25–40.

    Google Scholar 

  12. Fueter, R. “Ueder kubische Diophantische Gleichungen”. Comm. Math. Helv. 2 (1930). pp. 69–89.

    Article  MathSciNet  MATH  Google Scholar 

  13. Hasse, H. “Bericht über neuere Untersuchungen und Probleme aus d. Theorie d. algebraischen Zahlkörpern”, Teil 1: Klassenkörpertheorie”. (Berlin, 1930).

  14. Hasse, H. “Klassenkörpertheorie”. (Marburg, 1933. Cyclostyled).

  15. Hilbert, D. “Ges. Abhandlungen Erster Band, Zahlentheorie”. (Berlin, 1932).

  16. Holzer, L. “Ueber die Gleichungx 3+y 3=Cz 3”. J. f. Math. 159 (1928). pp. 93–100

    MATH  Google Scholar 

  17. Hurwitz, A. “Ueber ternäre Diophantische Gleichungen dritten Grades”. Vierteljahrschrift d. Naturf. Ges. in Zürich. 62 (1917). pp. 207–229.

    Google Scholar 

  18. Lind, C. E. “Ein Analogon zu einem Nagell'schen Satze über kubische Diophantische Gleichungen”. Comm. Math. Helv. 9 (1936–7). pp. 156–160.

    Article  Google Scholar 

  19. Lucas, E. “Sur l'analyse indéterminé du troisième degré”. Nouv. Annales de Math. 2, sér. 17 (1878). pp. 507–14.

    Google Scholar 

  20. Lutz, E. “Sur l'équationy 2=x 3Ax−B dans les corpsp-adiques”. J. f. Math. 177 (1937). pp. 238–247.

    Google Scholar 

  21. — “(Same title). Comptes Rendus (Paris) 203 (1936). pp. 20–22. A short resumé of the above but followed by an interesting comment by A. Weil.

    MATH  Google Scholar 

  22. Markoff, A. “Sur les nombres entiers dépendents d'une racine cubique d'un nombre entier”. Mém. de l'Acad. Imp. des Sciences de St. Pétersboug. VII Série, Tome XXXVIII No. 9 (1882).

  23. Mordell, L. J., “The Diophantine equationy 2k=x 3”. Proc. London Math. Soc. 13 (1914) pp. 60–80. (The closing paragraphs of this paper are misleading since the author was unaware that there is never more than a finite number of integer solutions of the title equation. cf. next paper).

    Google Scholar 

  24. — “Not on the integer solutions of the equationEy 2=Ax 3+Bx 2+Cx+D”. Mess. of Math. 51 (1922). pp. 169–171.

    Google Scholar 

  25. — “On the rational solutions of the indeterminate equation of the third or fourth degree”. Proc. Cambridge Phil. Soc. 21 (1922). pp. 179–182.

    Google Scholar 

  26. Mordell, L. J. “On some Diophantine equationsy 2=x 3+k with no rational solutions”. Archiv for Math. og Natur. B. I. L. Nr. 6, 1947.

  27. Mordell, L. J. “A chapter in the theory of numbers”. (Cambridge, 1947).

  28. Nagell, T., “Ueber die Einheiten in reinen kubischen Zahlkörpern”. Skrifter Videnskapselskapet. Christiania, 1922.

  29. — “Ueber die rationalen Punkte auf einigen kubischen Kurven”. Tôhoku Math. J. 24 (1924). 48–53.

    MATH  Google Scholar 

  30. — “Sur les propriétés arithmétiques des cubiques planes du premier genre”. Acta math. 52 (1928–9). pp. 93–126.

    MathSciNet  Google Scholar 

  31. Nagell, T. “L'ahalyse indéterminé du degré supérieur”. Mem. des Sci. Math. 39 (1929).

  32. Nagell, T. “Solutions de quelques problèmes dans la théorie arithmétique des cubiques planes du premier genre”. Skrifter utg. av d. Norske Videnskabs-Akademi. I. Math.-Naturv. Klasse 1935 No. 1.

  33. — “Bermerkungen über d. Diophantische Gleichungx 3+y 3=Az”. Arkiv f. Math. Astronomik och Fysik 25B No. 5 (1935). pp. 1–6.

    Google Scholar 

  34. — “Ueber die Lösbarkeit gewisser Diophantischer Gleichungen dritten Grades”. Comm. Math. Helv. 9 (1936–7). pp. 31–39.

    Article  MathSciNet  Google Scholar 

  35. Nagell, T. “Sur la résolubilité des équations Diophantiennes cubiques à deux inconnus dans un domaine relativement algébrique”. Nova Act. Reg. Soc. Sci. Upsaliensis, Series IV, Vol. 13 (1940). No. 3.

  36. Pépin, Father. “Sur la décomposition d'un nombre entier en une somme de deux cubes rationnels”. Journal de Math. (Liouville) IIe Sér. t. 15 (1870). pp. 217–236.

    Google Scholar 

  37. — “Sur certains nombres complexes compris dans la formule\(a + b\sqrt { - c^{''} } \). Journal de Math. (Liouville) IIIe Sér. t. 1 (1875). pp. 317–372 especially pp. 360–372.

    Google Scholar 

  38. Poincaré, H. “Sur les propriétés arithmétiques des courbes algébriques”. Journal de Math. (Liouville) Ve Sér. t. 7 (1901). pp. 161–234.

    Google Scholar 

  39. Reid, L. W. “Tafel der Klassenzahlen für kubischen Zahlkörpern”. Dissertation, Göttingen 1899. Abstract in Amer. J. Math. 23 (1901). pp. 68–84.

  40. Sylvester, J. J. “On certain ternary cubic-form equations”. Collected Math. Papers (Cambridge 1909) vol. HI. pp. 312–391 especially pp. 312–313 and pp. 347–350. Originally appeared Amer. J. Math. vol. II (1878). pp. 280–285, pp. 257–393 and vol. III (1880) pp. 58–88, pp. 179–189.

    Google Scholar 

  41. Weil, A. “L'arithmétique sur les courbes algébriques”. Acta Math. 52 (1928–29). pp. 281–315.

    MathSciNet  Google Scholar 

  42. — “Sur un théorème de Mordell”. Bull. des Sci. Math. 2e Sér. 54 (1930). pp. 182–191.

    Google Scholar 

  43. Weyl, H. “Algebraic theory of numbers”. (Princeton, 1940).

  44. Whittaker, E. T. andWatson, G. N. “A course of Modern Analysis”. (Cambridge. 4th Edition 1927).

  45. Wolfe, C. “On the indeterminate cubic equationx 3+Dy 2+D 2 z 3-3Dxyz=1”. Univ. of California Pub. in Math. 1 No. 16 (1923). pp. 359–369.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cassels, J.W.S. The rational solutions of the diophantine equation. Acta Math. 82, 243–273 (1950). https://doi.org/10.1007/BF02398279

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02398279

Keywords

Navigation