Hostname: page-component-848d4c4894-nr4z6 Total loading time: 0 Render date: 2024-05-18T13:31:40.355Z Has data issue: false hasContentIssue false
  • English
  • Français

Classes of Equations of the Type y2 = x3 + k Having No Rational Solutions

Published online by Cambridge University Press:  22 January 2016

Hugh M. Edgar
Hugh M. Edgar*
Affiliation:
San Jose State College, San Jose 14, Calif., U.S.A.
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The equation y2 = x3 + k, k an integer, has been discussed by many authors. Mordell [1] has found many classes of k values for which the equation has no integral solutions. Fueter [2], Mordell [3] and Chang [4] have found classes of k values for which the equation has no rational solutions. The following two theorems exhibit two more sets of conditions which give rise to classes of k values for which the corresponding equations have no rational solutions.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 28 , October 1966 , pp. 49 - 58
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1966

References

[1] Mordell, L. J., Proc. London Math. Soc., Hodgson, London, 1914, Volume 13, The Diophantine equation y 2−k = x 3 , Pages 60–80.Google Scholar
[2] Fueter, R., Commentarii Mathematici Helvetici, Societate Mathematica Helvetica, Zurich, 1930, Volume 2, Ueber kubische diophantische Gleichungen, Pages 69–89.Google Scholar
[3] Mordell, L. J., Archiv fur Mathematik og Naturvidenskab B.I.L., NR 6, Oslo, 1947, On some diophantine equations y 2 = x 3 + k with no rational solutions, Pages 143–150.Google Scholar
[4] Chang, K. L., The Quarterly Journal of Mathematics, Oxford University Press, Oxford, 1948, Volume 19. On some diophantine equations y2 = x3+k with no rational solutions, Pages 181–188.Google Scholar
[5] Landau, E. G. H., Vorlesungen über Zahlentheorie, Chelsea Publishing Company, New York, 1947, Page 172, Theorem 872.Google Scholar
[6] Landau, E. G. H., Vorlesungen über Zahlentheorie, Chelsea Publishing Company, New York, 1947, Page 178, Theorem 879.Google Scholar
[7] Landau, E. G. H., Vorlesungen über Zahlentheorie, Chelsea Publishing Company, New York, 1947, Page 180, Theorem 880.Google Scholar