1. Introduction. Many diophantine problems can be reduced to (ordinary) unit equations and S-unit equations in two unknowns (for references, see e.g. [15] (with explicit constants). These led to a lot of applications.The purpose of the present paper is to considerably improve (in completely explicit form) the above-mentioned estimates in terms of the cardinality of S and of the parameters involved (degree, unit rank, regulator, class number) of the ground field. To obtain these improvements we use, among other things, some recent improvements of Waldschmidt [26] and Kunrui Yu [27] concerning linear forms in logarithms, some recent estimates of Brindza [5] and Hajdu [18] for fundamental systems of S-units, some upper and lower bounds for S-regulators (cf. Lemma 3 of this paper) and an idea of Schmidt [23]. Further, in our arguments we pay a particular attention to the dependence on the parameters in question. As a consequence of our result, we derive explicit bounds for the solutions of homogeneous linear equations of three terms in S-integers of bounded S-norm. These improve some earlier estimates of Győry [13], [14].An application of our improvements is given in [17] to decomposable form equations (including Thue equations, norm form equations and discriminant form equations) in S-integers of a number field. Some other applications will be published in two further works.

We show that if $k$ is a positive integer, then there are, under certain technical hypotheses, only finitely many coprime positive $k$-term arithmetic progressions whose product is a perfect power. If $4 \leq k \leq 11$, we obtain the more precise conclusion that there are, in fact, no such progressions. Our proofs exploit the modularity of Galois representations corresponding to certain Frey curves, together with a variety of results, classical and modern, on solvability of ternary Diophantine equations. As a straightforward corollary of our work, we sharpen and generalize a theorem of Sander on rational points on superelliptic curves.

The main purpose of this paper is to considerably improve (in completely explicit form) the best known effective upper bounds for the solutions of S-unit equations and decomposable form equations. , their proofs rely on Baker's method and its p-adic analogue as well as certain quantitative results concerning fundamental/independent systems of units. In our Theorems 1 and 2 we improve upon the best known estimates for S-unit equations in terms of the parameters of S and the ground field K. As a consequence of Theorem 2 we deduce a completely explicit result (cf. Corollary 2) in the direction of the abc conjecture over number fields.To prove our results we use, among other things, some recent improvements due to Matveev [25] and Yu [33] concerning linear forms in logarithms of algebraic numbers, a recent theorem of Loher and Masser [22] on multiplicatively independent algebraic numbers, and our improved estimates for fundamental/independent systems of S-units. In proving our Theorem 1 we follow the arguments of [5] with some refinements and utilize the improvements mentioned above.In the bound in Theorem 1 there is a factor of the form s 2s , where s denotes the cardinality of S. This factor arises from the use of estimates concerning fundamental S-units. To avoid such a factor in Theorem 2, we

In this paper, we resolve a conjecture of Schäffer on the solvability of Diophantine equations of the shape 1 k + 2 k + · · · + x k = y n , for 1 k 11. Our method, which may, with a modicum of effort, be extended to higher values of k, combines a wide variety of techniques, classical and modern, in Diophantine analysis.