Auxiliary Results, Tools

In this Chapter we summarize the basic tools we use all through the book to solve our diophantine equations. The equations are usually reduced to so-called unit equations in two variables of type αu + βυ = 1 (cf. equation (2.5)) with given algebraic α, β, where u, υ 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 (Section 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 (Section 2.2) to reduce these bounds. The reduced bound is usually between 100 and 1000. These reduced bounds are quite modest, however if there are more than 4–5 of them, it is already impossible to test directly all possible exponents with absolute values under the reduced bound. Hence we have to apply certain enumeration methods (Section 2.3) to overcome this difficulty.