On Some Diophantine Equations y2 = x3 + k with no Rational Solutions (II)

  • L. J. Mordell

Abstract

The simplest Diophantine equation of degree greater than 2 is the equation
$$ y^2 = x^3 + k $$
(1)
where k is an integer. Two problems arise according as rational solutions or integral solutions are required. It is the problem of rational solutions which will be discussed here. No finite algorithm is known for finding solutions if they exist, except for special values of k.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    G. Billing, Beiträge zur arithmetischen Theorie ebener kubischer Kurven, Nova Acta Reg. Soc. Scient. Upsaliensis, Ser. IV, XI, Nr. 1 (1938), 1–165.Google Scholar
  2. [2]
    J. W. S. Cassels, The rational solutions of the Diophantine equation y 2 = x 3d, Acta Mathematica 82 (1950), 243–273.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    J. B. Birch and H. P. F. Swinnerton-Dyer, Notes on elliptic curves, J. reine angew. Math. 212 (1963), 7–25.MathSciNetMATHGoogle Scholar
  4. [4]
    R. Fueter, Über kubische diophantische Gleichungen, Comm. Math. Helv. 2 (1930), 69–89.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    O. Brunner, Lösungseigenschaften der kubischen diophantischen Gleichung z 3y 2 = D, Inauguraldissertation, Zürich 1933.Google Scholar
  6. [6]
    L. J. Mordell, One some Diophantine equations y 2 = x 3 + k with no rational solutions, Arch. for Math. og Naturvidenskap 6 (1947).Google Scholar
  7. [7]
    Kuo-Lung Chang, One some Diophantine equations y 2 = x 3 + k with no rational solutions, Quarterly J. of Math. 19 (1948), 181–188.MATHCrossRefGoogle Scholar
  8. [8]
    A. Scholz, Über die Beziehung der Klassenzahlen quadratischer Körper zueinander, J. reine angew. Math.166 (1932), 201–203.Google Scholar

Copyright information

© Springer Science+Business Media New York 1969

Authors and Affiliations

  • L. J. Mordell
    • 1
  1. 1.CambridgeUK

Personalised recommendations