Hostname: page-component-84b7d79bbc-lrf7s Total loading time: 0 Render date: 2024-07-30T07:08:58.463Z Has data issue: false hasContentIssue false
  • English
  • Français

On a quartic diophantine equation

Published online by Cambridge University Press:  20 January 2009

R. J. Stroeker  and
B. M. M. De Weger
R. J. Stroeker
Affiliation:
Econometric InstituteErasmus UniversityP.O. Box 17383000 Dr RotterdamThe Netherlands E-mail address: stroeker@wis.few.eur.nl, dweger@wis.few.eur.nl
B. M. M. De Weger
Affiliation:
Econometric InstituteErasmus UniversityP.O. Box 17383000 Dr RotterdamThe Netherlands E-mail address: stroeker@wis.few.eur.nl, dweger@wis.few.eur.nl
Rights & Permissions [Opens in a new window]

Abstract

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.

In this paper we consider the quartic diophantine equation 3(y2 – 1) = 2x2(x2 – 1) in integers x and y. We show that this equation does not have any other solutions (x, y) with x≧0 than those given by x = 0,1,2,3,6,91. Two approaches are emphasized, one based on diophantine approximation techniques, the other depends on the structure of certain quartic number fields.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 39 , Issue 1 , February 1996 , pp. 97 - 114
Copyright
Copyright © Edinburgh Mathematical Society 1996

References

REFERENCES

1. Baker, A. and Wüstholz, G., Logarithmic forms and group varieties, J. Reine Angew. Math. 442 (1993), 19–62.Google Scholar
2. Diaconis, Persi and Graham, R. L., The Radon transform on , Pacific J. Math. 118 (2) (1985), 323–345.CrossRefGoogle Scholar
3. Stroeker, R. J., On quartic Thue equations with trivial solutions, Math. Comp. 52 (185) (1989), 175–187.CrossRefGoogle Scholar
4. Stroeker, R. J. and Tzanakis, N., On the application of Skolem's p-adic method to the solution of Thue equations, J. Number Theory 29 (2) (1988), 166–195.CrossRefGoogle Scholar
5. Stroeker, R. J. and Tzanakis, N., Solving elliptic diophantine equations by estimating linear forms in elliptic logarithms, Acta. Arith. 67 (2) (1994), 177–196.CrossRefGoogle Scholar
6. Tzanakis, N. and De Weger, B. M. M., On the practical solution of the Thue equation, J. Number Theory 31 (2) (1989), 99–132.CrossRefGoogle Scholar
7. De Weger, B. M. M., Algorithms for Diophantine Equations (PhD thesis, University of Leiden, 1987). Also appeared as CWI Tract 65, Centre for Mathematics and Computer Science, Amsterdam 1989.Google Scholar