Computing all integer solutions of a genus 1 equation
The Elliptic Logarithm Method has been applied with great success to the problem of computing all integer solutions of equations of degree 3 and 4 defining elliptic curves. We extend this method to include any equation f(u,v)=0 that defines a curve of genus 1. Here f is a polynomial with integer coefficients and irreducible over the algebraic closure of the rationals, but is otherwise of arbitrary shape and degree. We give a detailed description of the general features of our approach, and conclude with two rather unusual examples corresponding to equations of degree 5 and degree 9.
|Keywords||Dophantine equation, Elliptic curve, Elliptic logarithm|
Stroeker, R.J., & Tzanakis, N.. (2001). Computing all integer solutions of a genus 1 equation (No. EI 2001-44). Report / Econometric Institute, Erasmus University Rotterdam. Retrieved from http://hdl.handle.net/1765/592