Auxiliary Results and Tools

In this chapter we summarize the basic tools we use all over the book to solve our Diophantine equations. The equations are usually reduced to the so-called unit equations Unit equation of type α u + β v = 1 $$\displaystyle \alpha u+\beta v =1 $$ (cf. Eq. (2.5)) with given algebraic α, β, where u, v are unknown units in a number field. These units are written as a power product of the generators of the unit group and the unknown exponents are to be determined. Baker’s method (Sect. 2.1) is used to give an initial upper bound for the unknowns, which is of magnitude 1018 for the simplest Thue equations but 10100 is also not unusual for more complicated equations. We apply numerical Diophantine approximation techniques based on the LLL basis reduction algorithm (Sect. 2.2) to reduce these bounds. The reduced bounds are usually between 100 and 1000. These reduced bounds are quite modest; however, if there are more than four to five unknown exponents, it is already impossible to test directly all possible exponents with absolute values under the reduced bound. Hence it is crucial to apply some efficient enumeration methods (Sect. 2.3) to overcome this difficulty. In fact we observe that the majority of computations are spent by proving that there are no “large” solutions. According to our experience the solutions of the equations we consider are usually “small” and can be found sometimes quite easily and fast.