Günter Lettl

Simple families of Thue inequalities

We use the hypergeometric method to solve three families of Thue inequalities of degree 3, 4 and 6, respectively, each of which is parametrized by an integral parameter. We obtain bounds for the solutions, which are astonishingly small compared to similar results which use estimates of linear forms in logarithms.

A lower bound for the class number of certain cubic number fields

Let K be a cyclic number field with generating polynomial X3 a 3 32 - a + 3 X - 1 2 2 and conductor m. We will derive a lower bound for the class number of these fields and list all such fields with prime conductor m = (a2 + 27)/4 or m = (1 + 27b2)/4 and small class number. 1. Introduction. Let h.,, denote the class number of the cyclotomic field Q(,m), and h +, the class number of its maximal real subfield Q(cos(2T/m)). It is a well-known conjecture of Vandiver that p + h+ holds for all primes p E P. This is a customary assumption for proving the second case of Fermats Last Theorem (for more details see Washington (16)). Since h+ grows slowly (h+ = 1 for p 1 the exact value of h + is known without using GRH. Masley suggested that P P perhaps h + 1. Using the quadratic subfield, Ankeny, Chowla and Hasse (1) showed that h + > 1 if p belongs to certain quadratic sequences in N, and S.-D. Lang (9) and Takeuchi (15) found more such sequences. Similar results were

A parametric family of quintic Thue equations

For an integral parameter t ∈ Z we investigate the family of Thue equations F(x,y) = x 5 + (t - 1) 2 x 4 y - (2t 3 + 4t + 4)x 3 y 2 + (t 4 + t 3 + 2t 2 + 4t - 3)x 2 y 3 + (t 3 + t 2 + 5t + 3)xy 4 + y 5 = ±1, originating from Emma Lehmers family of quintic fields, and show that for |t| > 3.28.10 15 the only solutions are the trivial ones with x= 0 or y = 0. Our arguments contain some new ideas in comparison with the standard methods for Thue families, which gives this family a special interest.

THUE EQUATIONS ASSOCIATED WITH ANKENY–BRAUER–CHOWLA NUMBER FIELDS

For a wide class of one-parameter families of Thue equations of arbitrary degree n [ges ]3 all solutions are determined if the parameter is sufficiently large. The result is based on the Lang–Waldschmidt conjecture, on the primitivity of the associated number fields and on an index bound, which does not depend on the coefficients. By applying the theory of Hilbertian fields and results on thin sets, primitivity is proved for almost all choices (in the sense of density) of the parameters.

Subsemigroups of finitely generated groups with divisor-theory

In this paper we will characterize all subsemigroups of finitely generated abelian groups, for which there exists a divisor-theory. Besides an explicit geometrical construction of the divisor-theory is given, and it is shown that any finitely generated abelian group occurs as the divisor-class-group of some semigroup.

Finding fundamental units in algebraic number fields

Recently Cusick [4] presented a very elegant and short proof of the fact that a pair of fundamental units of a totally real cubic or quartic number field can be found by taking ‘successive minima’ of the function tr(e 2 ), where e runs through the group of units and tr denotes the absolute trace. Hidden in Cusicks proof there is a general theorem, which shows how strictly convex functions can be used to find lattice-vectors extensible to a basis of a given geometrical lattice, and which we state and prove in § 3. A result analogous to Cusicks for some families of functions related to tr (e 2 ) is given in Theorem 1, thereby improving results of Brunotte and Halter-Koch [2], [5]. For a survey of unit groups of rank 2 and more literature the reader is also referred to [2].