We study the Diophantine equation

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in integers x, y > 1, n > 2 and q [ges ] 2. Without loss of generality, we suppose that q is prime.

Saradha and Shorey [SS] considered equation (1·1) with x = z2 and proved that it has no solutions in the cases z > 31 or z ∈ {2, 3, 4, 8, 9, 16, 27}.Notice that the proof for z ∈ {2, 4, 8, 16} is elementary, whereas for z ∈ {3, 9, 27} the proof uses a result of Darmon–Merel [DM] which is a generalization of Wiles' result on Fermat's theorem. The purpose of the present paper is to treat completely the remaining cases and to prove the following result.

A longstanding conjecture claims that the Diophantine equation

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has finitely many solutions and, maybe, only those given by

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Among the known results, let us mention that Ljunggren [9] solved (1) completely when q = 2 and that very recently Bugeaud et al. [3] showed that (1) has no solution when x is a square. For more information and in particular for finiteness type results under some extra hypotheses, we refer the reader to Nagell [11], Shorey & Tijdeman [16, 17] and to the recent survey of Shorey [15]. One of the main tools used in most of the proofs is Baker's theory of linear form in archimedean logarithms of algebraic numbers and especially a dramatic sharpening obtained when the algebraic numbers involved are closed to 1. This was first noticed by Shorey [14], and has been applied in numerous works relating to (1) or to the Diophantine equation

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see for instance [1, 5, 10]. The purpose of the present work is to prove a similar sharpening for linear forms in p-adic logarithms and to show how it can be applied in the context of (1). Further, following previous investigations by Sander [12] and Saradha & Shorey [13], we derive from our results an irrationality statement for Mahler's numbers.

Let the elliptic curve E be defined by the equation

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with a1, …, a6 ∈ ℤ. Define a finite set of places S = {q1, …, qs−1, qs = ∞} of ℚ and put Q = max {q1, …, qs−1}. Let E(ℚ) denote the set of (x, y) ∈ ℚ2 satisfying (1) and the infinite point [Oscr ].

The multiplicative height of a rational point P = (x, y) ∈ E(ℚ) is defined as the following product over all places q of ℚ (including q = ∞):

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where the [mid ]x[mid ]qs are the normalized multiplicative absolute values of ℚ corresponding to the places q.

The main result of this paper establishes invariance under change of generators for automatic structures for monoids (a property that is well known to hold for automatic groups but fails for semigroups). This result is then applied to show that if a free product of two monoids is automatic, then so are both the free factors. Finally, the difference between automatic structures, in terms of monoid generating sets and semigroup generating sets, is discussed.

For each positive integer n, let Fn be a free group of rank n with basis (in other words, free generating set) {x1, …, xn} . If θ is an automorphism of Fn then {x1θ, …, xnθ} is also a basis of Fn and every basis of Fn has this form.

For any variety of groups [Vfr ], let [Vfr ](Fn) denote the verbal subgroup of Fn corresponding to [Vfr ]. (See [10] for information on varieties and related concepts.) Let Gn = Fn/[Vfr ](Fn). Then Gn is a relatively free group of rank n in the variety [Vfr ]. By a basis of Gn we mean a subset S such that every map of S into Gn extends, uniquely, to an endomorphism of Gn. Write x¯i = xi[Vfr ](Fn) for i = 1, …, n. Then {x¯1, …, x¯n} is a basis of Gn. If λ is an automorphism of Gn then {x¯1λ, …, x¯nλ} is also a basis of Gn and every basis of Gn has this form.

Any automorphism θ of Fn induces an automorphism θ of Gn in which x¯iθ = (xiθ)[Vfr ](Fn) for i = 1, …, n. Thus every basis of Fn induces a basis of Gn. The converse however is not always true; in general, there are automorphisms of Gn which are not induced by automorphisms of Fn.

If G is a finite group, X a generating set for G and R some set of relations holding among these generators (but not in general a full set of relations presenting G), we establish sufficient conditions for the group G* = 〈X/R〉 to be infinite and give a lower bound for the rank of the abelianization of the kernel of the natural map G*→G. Related results in the literature and possible directions for generalization are discussed.

In continuation of earlier work [7], we study in this paper the structure of cocommutative Hopf algebras of finite representation type. Since every such algebra is the group algebra H([Gscr ]) of a finite algebraic group [Gscr ], this problem has two interrelated aspects. On the one hand, one has to understand the structure of the group scheme [Gscr ], while on the other the Morita equivalence class of H([Gscr ]) has to be determined. Although we shall here be mainly concerned with the latter, structural information on the underlying group schemes enters crucially at several points.

We show that if G is a non-elementary torsion-free word hyperbolic group then there exists another word hyperbolic group G*, such that G is a subgroup of G* but G is not quasiconvex in G*. We also prove that any non-elementary subgroup of a torsion-free word hyperbolic group G contains a free group of rank 2 which is malnormal and quasiconvex in G.

A recent result of Bestvina and Brady [1, theorem 8·7], shows that one of two outstanding questions has a negative answer; either there exists a group of cohomological dimension 2 and geometric dimension 3 (a counterexample to the Eilenberg–Ganea Conjecture [4]), or there exists a nonaspherical subcomplex of an aspherical 2-complex (a counterexample to the Whitehead Conjecture [11]). More precisely, Bestvina and Brady construct a family of groups which are potential counterexamples to the Eilenberg–Ganea Conjecture, each of which has cohomological dimension 2. These are also examples of groups of type FP2 which are not finitely presented (see [1]). Dicks and Leary [3] give an explicit way of obtaining presentations (on finite generating sets) for these groups. For some of these examples, it is shown in [1] that any 2-dimensional classifying space would give rise to a counterexample to the Whitehead conjecture.

We will refer to the examples cited above as Bestvina–Brady groups. These come equipped with natural, nonpositively curved cubical 3-dimensional classifying complexes, which we will call Bestvina–Brady complexes. In this short note, we show that these Bestvina–Brady complexes are (up to homotopy equivalence) formed by applying the Quillen plus construction to certain finite 2-complexes. From this, together with known facts about 2-complexes with aspherical plus constructions, we recover the result of Bestvina and Brady [1] that the Bestvina–Brady groups act freely on acyclic 2-complexes, and hence have cohomological dimension at most 2. It also follows that these groups have free relation modules of finite rank and so are of type FF. Finally, we use our construction to give an alternative proof of the cited theorem of Bestvina and Brady; at least one of the Eilenberg–Ganea and Whitehead conjectures is false.

We do not use the full force of the Morse-theoretical techniques developed in [1], but will assume two results form that paper; the asphericity of the Bestvina–Brady complexes and the non-finite presentability of the Bestvina–Brady groups.

The automorphism group of a p-group acts naturally on the mod-p cohomology of the group. This action need not be faithful, but we show that it contains every simple module as a composition factor.

We define a sequence of purely algebraic invariants – namely, classes in the Quillen cohomology of the Π-algebra π∗X – for distinguishing between different homotopy types of spaces. Another sequence of such cohomology classes allows one to decide whether a given abstract Π-algebra can be realized as the homotopy Π-algebra of a space.

We construct univalent functions in the unit disc, whose coefficient sequences (an) have arbitrarily long intervals of zeros and at the same time arbitrarily long intervals where [mid ]an[mid ] > nεn holds, (εn) being an arbitrary prescribed sequence of positive numbers tending to zero. Furthermore we show that the initial interval of coefficients of such a function can be prescribed to be any interior point of the coefficient region.