The primitive solutions to x^3+y^9=z^2

Details The primitive solutions to x^3+y^9=z^2 The primitive solutions to x^3+y^9=z^2

2003-10-31

arxiv.org/abs/math/0311002v1

J. Number Theory 111 (2005), no. 1, 179--189.

Mathematics

Number Theory

8 pages

Scientific paper

10.1016/j.jnt.2004.11.008

We determine the rational integers x,y,z such that x^3+y^9=z^2 and gcd(x,y,z)=1. First we determine a finite set of curves of genus 10 such that any primitive solution to x^3+y^9=z^2 corresponds to a rational point on one of those curves. We observe that each of these genus 10 curves covers an elliptic curve over some extension of Q. We use this cover to apply a Chabauty-like method to an embedding of the curve in the Weil restriction of the elliptic curve. This enables us to find all rational points and therefore deduce the primitive solutions to the original equation.

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