On equations inS-units and the Thue-Mahler equation

Abstract

w 1. IntroductionSeveral classes of diophantine equations, such as the Thue-Mahler equation and certain generalisations of the Ramanujan-Nagell equation, can be reduced to certain linear equations in two S-units. Here S is a finite set of equivalence classes of valuations on a given algebraic number field K and an S-unit is an element eeK with the property that the only valuations on K which assume a value 4= 1 for e belong to equivalence classes from S. In this section we shall state a general result on the number of solutions of linear equations in two Sunits. Before we can state it, we have to introduce some notions on valuations. In w we shall discuss the consequences of our general result for generalisations of the Ramanujan-Nagell equation and in w167 3-4 we shall deal with the Thue-Mahler equation.Let K be an algebraic number field. By a prime on K we shall mean an equivalence class of non-trivial valuations on K. Usually we shall denote primes on Q by the letter p, on a given algebraic number field K by v and on an extension of K by V. The completion of K at the prime v is denoted by K v.We distinguish between finite primes, containing non-archimedean valuations, and infinite primes, containing archimedean valuations.

Representation of finite graphs as difference graphs of S-units. II

“…Consider first the cyclic graph 4 , v 1 }). As in the above proof, we use S and A as 'variables', to be changed during the algorithm.…”

“…We output A = {0, 1, 5, 22} and S := {2, 3, 11, 17}. One can easily check that with these choices, G S (A) is isomorphic to C 4 .…”

Integral points on elliptic curves over function fields

“…Hindry and Silverman [3] proved an analogue of Lang's conjecture for non-constant elliptic curves over zero-characteristic one-dimensional function fields. Influenced by the original work of Mason [5], we use a formula on 2-divison points given by Tan [7] and the method of Evertse [1,2] to prove another analogue of Lang's conjecture for these curves.…”

“…Let M K denote the set of all places of K. For a finite subset S of M K , denote by ^s the ring of 5-integers of K. Consider a non-constant elliptic curve E defined by W.-C. Chi, K. F. Lai and K.-S. Tan [2] The set of S-integral points of this curve is E(0 S ) = {P e E(K) :x(P),y(P) e 0 S ). Let A = -(4A 3 + TIB 1 ) be the discriminant of the equation (1) and @E/K be the divisor of the minimal discriminant of E/K. Then we have (2) (A) Denote by s, s\, s 2 the cardinality of 5, 5i and S 2 .…”

S-unit equations and their applications

“…By means of (a generalisation of) Theorem 2, Evertse (19] and later Evertse and Gyory [22] derived explicit upper bounds for the numbers of solutions of (11.1) and (11.1') which are independent of the coefficients of F. In [22], the bound 4n X 729(d+s+w(p)) has been obtained for the number of solutions of (11.l') where n = deg(F), g = [G: K] (hence 1 ~ g ~ n!) and w(/3) denotes the number of distinct prime ideal divisors of (/3).…”