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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: nbruin@wi.leidenuniv.nl
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Abstract

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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