Cubic Thue inequalities with negative discriminant Original Research Article

We give an upper bound for the solutions of the family of cubic Thue inequalities x3+axy2+by3less-than-or-equals, slantk when a is positive and larger than a certain value depending on b. For the case k=a+b+1 and agreater-or-equal, slanted360b4 we show that these inequalities have only trivial solutions. For the case k=a+b+1 and b=1,2, we solve these inequalities for all agreater-or-equal, slanted1. Our method is based on Padé approximations using Rickertʹs integrals. We also use a generalization of Legendreʹs theorem on continued fractions.