Related papers: Perfect Powers: Pillai's works and their developme…
A qualitative (and selective) discussion of current activities and problems in the field is given.
Throughout this book, we discuss some open problems in various branches of science, including mathematics, theoretical physics, astro-physics, geophysics etc. It is of our hope that some of the problems discussed in this book will find…
We develop a variety of new techniques to treat Diophantine equations of the shape $x^2+D =y^n$, based upon bounds for linear forms in $p$-adic and complex logarithms, the modularity of Galois representations attached to Frey-Hellegouarch…
In this article, we consider the Diophantine equation $\sigma_{2}(n)-n^2=An+B$ with $A=P^2\pm2$. For some $B$, we show that except for finitely many computable solutions in the range $n\leq(|A|+|B|)^{3}$, all the solutions are expressible…
We survey old and recent results on the problem of finding a complete set of rules describing the behavior of the power function, i.e. the function which takes a cardinal $\kappa$ to the cardinality of its power $2^\kappa$.
This article shines new light on the classical problem of tiling rectangles with squares efficiently with a novel method. With a twist on the traditional approach of resistor networks, we provide new and improved results on the matter using…
We develop machinery to explicitly determine, in many instances, when the difference $x^2-y^n$ is divisible only by powers of a given fixed prime. This combines a wide variety of techniques from Diophantine approximation (bounds for linear…
We discuss some easy statements dealing with linear inhomogeneous Diophantine approximation. Surprisingly, we did not find some of them in the literature.
The symposium has shown the dynamism of this rapidly evolving discipline. I shall concentrate here on some highlights and some complementary informations. I conclude on open questions with some perspectives on solar & stellar activity and…
This note pushes further the discussion about relations between Dirichlet improvable, badly approximable and singular points held in recent joint work with Beresnevich, Guan, Velani and Ramirez, by considering Diophantine sets extending the…
We discuss the problem of finding optimal exponents in Diophantine estimates involving one real number and, in some cases where such an exponent is known, present some properties of the corresponding extremal numbers.
We survey some results on the structure of the groups which are definable in theories of fields involved in the applications of model theory to Diophantine geometry. We focus more particularly on separably closed fields of finite degree of…
This paper, written in relation to the Current Developments in Mathematics 2012 Conference, discusses the recent papers on perfectoid spaces. Apart from giving an introduction to their content, it includes some open questions, as well as…
We give a simple rigourous treatment of the classical results of the abelian sandpile model. Although we treat results which are well-known in the physics literature, in many cases we did not find complete proofs in the literature. The…
A motivated q-extension of the values of the Riemann zeta function at positive integers is presented. Several irrationality and transcendence results as well as new general problems for these q-zeta values are stated.
We present a brief introduction to analytic capacity, with an emphasis on its numerical computation. We also discuss several related open problems.
We study formal power series which can be interpreted as interpolations of Fibonacci and Lucas polynomials with even (or odd) indices.
This note discusses elliptic functions in Ramanujan's work.
We simplify the proof of some widely used theoretical theorems, extending their applicability, while correcting some erroneous results. We also generalize key results and present new results that contribute to the development of the theory.…
We show that the intuitionistic first-order theory of equality has continuum many complete extensions. We also study the Vitali equivalence relation and show there are many intuitionistically precise versions of it.