Let K be an algebraic number field, Γ1, Γ2 two finitely generated multiplicative subgroups of K*, and a1, a2 two non-zero elements of K. It follows from the results of the preceding chapter that the equation

a1x1 + a2x2 = 1 in(x1, x2) ∈ Γ1 × Γ2

has only finitely many solutions, and effective upper bounds can be given for the heights of the solutions. These bounds are, however, too large for practical use, for finding all solutions of concrete equations of the above form. In this chapter a practical method will be provided to locate all the solutions to such equations, subject to the conditions that a1, a2 and the generators of Γ1 and Γ2 are effectively given and that the ranks of Γ1 and Γ2 are not too large, presently the bound is about 12. In particular, we present an efficient algorithm for solving completely S-unit equations in two unknowns.

The unknowns x1 and x2 can be represented as a power product of the generators of Γ1 and Γ2, respectively. Assuming that the generators of infinite order are multiplicatively independent, these representations are unique up to powers of roots of unity. Thus, we arrive at an exponential Diophantine equation of the form (5.1.3) below which has to be solved. As in Chapter 4, we first derive an explicit upper bound for the absolute values of the unknown exponents, using the best known Baker's type inequalities concerning linear forms in logarithms. In this way the existence of “large” solutions will be excluded. This part is an adaptation of Győory's method (Győory (1979)) who was the first to give explicit bounds for the solutions in case of S-unit equations over number fields. Then, in concrete cases, we can considerably reduce the obtained bound by means of deWeger's reduction techniques (deWeger (1987, 1989)) based on the LLL lattice basis reduction algorithm. This means that even “medium” sized solutions do not exist. Finally, some enumeration procedures due to Wildanger (1997, 2000) and Smart (1999) can be utilized to determine the “small” solutions under the reduced bound. We shall briefly illustrate the resolution process on two concrete equations. Of course, during the process some standard algebraic number-theoretical concepts and algorithms will also be needed, references to which, for convenience, are collected in Section 1.10.