On the Representation of 1 by Binary Cubic Forms of Positive Discriminant

We looked for types of irreducible cubic forms with integer coefficients and of positive discriminant for which the Thue equation f(x, y) = 1 has many or large solutions. We give a way to design even non-reversible forms which have at least the four different solutions (1, 0), (x1, y0), (x2, y0), (x3, y0). Performance and results of a computer-search by which the solutions (x, y) up to a given bound on y of some 85.000 of these equations had been determined will be described.