Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-25T07:42:02.284Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Diophantine equation z2 = x4 + Dx2y2 + y4

Published online by Cambridge University Press:  18 May 2009

J. H. E. Cohn
J. H. E. Cohn
Affiliation:
Department of Mathematics, Royal Holloway University of London, Egham, Surrey, TW20 0EX, England
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 of the title in positive integers x, y, z where D is a given integer has been considered for some 300 years [4, pp 634–639]. As observed by V. A. Lebesgue, and probably known to Euler, if x, y, z is one non-trivial solution i.e., one with xy(x2 – y2) ≠0, another is given by . It then follows that there are infinitely many such with (x, y) = 1. The question that remains is to determine for which values of D such solutions exist.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 36 , Issue 3 , September 1994 , pp. 283 - 285
Copyright
Copyright © Glasgow Mathematical Journal Trust 1994

References

REFERENCES

1.Brown, E., x 4 + d x 2y 2 + y 4 = z 2: some cases with only trivial solutions—and a solution Euler missed, Glasgow Math. J. 31 (1989), 297–307.CrossRefGoogle Scholar
2.Cohn, J. H. E., Squares in arithmetical progressions I, Math. Scand. 52 (1983) 5–19.CrossRefGoogle Scholar
3.Cohn, J. H. E., Squares in arithmetical progressions II, Math. Scand. 52 (1983) 20–23.CrossRefGoogle Scholar
4.Dickson, L. E., History of the theory of numbers, II, (Chelsea Publishing Co., New York, 1952).Google Scholar
5.Pocklington, H. C., Some diophantine impossibilities, Proc. Cambridge Phil. Soc. 17 (1914) 108–121.Google Scholar