Pillai's conjecture revisited

We prove a generalization of an old conjecture of Pillai (now a theorem of Stroeker and Tijdeman) to the effect that the Diophantine equation 3^x−2^y=c has, for |c|>13, at most one solution in positive integers x and y. In fact, we show that if N and c are positive integers with N?2, then the equation |(N+1)^x−N^y|=c has at most one solution in positive integers x and y, unless (N,c)∈{(2,1),(2,5),(2,7),(2,13),(2,23),(3,13)}. Our proof uses the hypergeometric method of Thue and Siegel and avoids application of lower bounds for linear forms in logarithms of algebraic numbers.     

On the second Goldbach conjecture

The results in this paper suggest that Goldbach's conjecture that every odd number is the sum of three primes is true even under the requirement that two of the primes be the same and the third be arbitrarily small.     

A conjecture of Beauville and Catanese revisited

A theorem of Green, Lazarsfeld and Simpson (formerly a conjecture of Beauville and Catanese) states that certain naturally defined subvarieties of the Picard variety of a smooth projective complex variety are unions of translates of abelian subvarieties by torsion points. Their proof uses analytic methods. We refine and give a completely new proof of their result. Our proof combines galois-theoretic methods and algebraic geometry in positive characteristic. When the variety has a model over a function field and its Picard variety has no isotrivial factors, we show how to replace the galois-theoretic results we need by results from model theory (mathematical logic). Furthermore, we prove partial analogs of the conjecture of Beauville and Catanese in positive characteristic.     

The second Brauer-Thrall conjecture for isolated singularities of excellent hypersurfaces

We give an extension of Dieterich's Theorems on the structure of the Auslander-Reiten quiver of an isolated singularity. The first author was supported by a fellowship of the Alexander von Humboldt-Stiftung during the last period of preparation of this paper.     

Artin's primitive root conjecture for function fields revisited

Artin's primitive root conjecture for function fields was proved by Bilharz in his thesis in 1937, conditionally on the proof of the Riemann hypothesis for function fields over finite fields, which was proved later by Weil in 1948. In this paper, we provide a simple proof of Artin's primitive root conjecture for function fields which does not use the Riemann hypothesis for function fields but rather modifies the classical argument of Hadamard and de la Vallée Poussin in their 1896 proof of the prime number theorem.     

The scope of Feferman’s semi-intuitionistic set theories and his second conjecture

The paper is concerned with the scope of semi-intuitionistic set theories that relate to various foundational stances. It also provides a proof for a second conjecture of Feferman’s that relates the concepts for which the law of excluded middle obtains to those that are absolute with regard to the relevant test structures, or more precisely of ${\Delta }_1$ complexity. The latter is then used to show that a plethora of statements is indeterminate with respect to various semi-intuitionistic set theories.     

A direct attempt to goldbach problem

The so-called principal term of Goldbach problem, i.e., the principal order of the number of representations of an even number as a sum of two primes, is evaluated here in a way quite different from the classical circle method. It is amazing that the author did this with the knowledge just at the PNT (Prime Number Theorem) level, even without using high-level theorems in analytic number theory as usual, e.g., Siegel-Walfisz Theorem.     

Essential dimension and the second serre conjecture for exceptional groups

We study the essential dimension of exceptional connected simply connected algebraic groups over algebraically closed fields. In the present paper, we find upper estimates for the essential dimensions of the groupsF ₄ ,E ₆ , andE ₇ . For the groupF ₄ , the upper estimate, thus obtained coincides with the known lower estimate. We also prove the second Serre conjecture for the groupE ₆ and for a function field. Translated fromMatematicheskie Zametki, Vol. 68, No. 4, pp. 539–547, October, 2000.     

Existential monadic second order logic of undirected graphs: The Le Bars conjecture is false

In 2001, J.-M. Le Bars disproved the zero-one law (that says that every sentence from a certain logic is either true asymptotically almost surely (a.a.s.), or false a.a.s.) for existential monadic second order sentences (EMSO) on undirected graphs. He proved that there exists an EMSO sentence ? such that $\mathsf{P}(G_n??)$ does not converge as $n\to \infty$ (here, the probability distribution is uniform over the set of all graphs on the labeled set of vertices $\{1,\dots ,n\}$). In the same paper, he conjectured that, for EMSO sentences with 2 first order variables, the zero-one law holds. In this paper, we disprove this conjecture.     

On the Bieberbach conjecture for functions with a small second coefficient

In the following we prove that for a given univalent function such that |a ₂|<1.05, |a _n|<n for eachn. This is an improvement of the result in [1].     

The borel conjecture

We show the Borel Conjecture is consistent with the continuum large.     

The rigidity conjecture

A central question in dynamics is whether the topology of a system determines its geometry. This is known as rigidity. Under mild topological conditions rigidity holds for many classical cases, including: Kleinian groups, circle diffeomorphisms, unimodal interval maps, critical circle maps, and circle maps with a break point. More recent developments show that under similar topological conditions, rigidity does not hold for slightly more general systems. In this paper we state a conjecture which describes how topological classes are organized into rigidity classes.     

The desmic conjecture

The twelve point Desmic configuration in Euclidean three space is composed of three finite sets with the property that any line intersecting points of two of the sets also intersects the remaining set. The Desmic conjecture asserts that this is the only such configuration. In this paper the Desmic conjecture is proven.     

The Wills conjecture

Two strengthenings of the Wills conjecture, an extension of Bonnesen's inradius inequality to -dimensional space, are obtained. One is the sharpest of the known strengthenings of the conjecture in three dimensions; the other employs techniques which are fundamentally different from those utilized in the other proofs.