Computing all integer solutions of a genus 1 equation

Abstract. 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,wheref ∈ Z[u, v] is irreducible over Q, defines a curve of genus 1, 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. 1