Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-23T15:48:24.551Z Has data issue: false hasContentIssue false
  • English
  • Français

The Diophantine Equations x2± y4=±z6 and x2+y8= z3

Published online by Cambridge University Press:  04 December 2007

NILS BRUIN
NILS BRUIN
Affiliation:
Department of Mathematics and Computer Science, University of Leiden, Leiden, The Netherlands e-mail: [email protected]
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 article we determine all solutions to the equation xp+yq=zr, (p,q,r)∈{(2,4,6), (2,6,4), (4,6,2), (2,8,3)} in coprime integers x,y,z. First we determine a set of curves of genus 2, such that every solution corresponds to a rational point on one of these curves. Then we determine the rational points on these curves using either covers of rank 0 elliptic curves or a method known as effective Chabauty which works if the Mordell–Weil rank of the jacobian is smaller than the dimension.

Keywords

rational points on algebraic curveseffective Chabautygeneralised Fermat-equationsgenus 2 curves.
Type
Research Article
Information
Compositio Mathematica , Volume 118 , Issue 3 , September 1999 , pp. 305 - 321
Copyright
© 1999 Kluwer Academic Publishers