Abstract :
We consider the problem of finding (effectively) all the solutions to infinite families of Thue equations. We define a broad class of cubic Thue equations for which this is always possible, provided that the family satisfies a certain mild condition. The following theorem is representative of the sort of result one can obtain: Suppose that b is an integer ≥ 143. Then for all integers n ≥ 2, the Diophantine equation x(x − ny )(x − nby) + y3 = 1 has only the four solutions (1, 0), (0, 1), (n, 1), (nb, 1).
