On Large Rational Solutions of Cubic Thue Equations: What Thue Did to Pell

The paper is largely expository. The first part is devoted to studying integer solutions to Pell’s Equation: u2 − dv2 = 1. The authors present the classic construction of a fundamental solution via continued fractions, from which all solutions can be derived. The primary focus of the second part is on rational solutions to the Thue’s equation, u3−dv3 = 1. The authors explain why these rational solutions correspond to rational points on the elliptic curve, y2 = x3 − 432d2 (as a special case of a more general result). They then cite the famous result of Mordell, which says that the set of rational points of an elliptic curve over Q is a finitely generated abelian group, listing the rank for the above family of curves when 1 ≤ d ≤ 100. At this point the paper comes a little closer to the frontier (perhaps partially original?), asking a specific question about the topological distribution of solutions to the above Thue’s equation. Specifically, if one has a point (x, y) of infinite order on the associated elliptic curve, can it be used to produce a sequence (un, vn) of solutions for which |un|, |vn| → ∞? The authors answer this question in the affirmative, by giving an algorithm based on the elliptic logarithm and (again) continued fractions. The paper concludes by explicitly calculating such a sequence of solutions to the specific Thue equation, u3 − 7v3 = 1.