Catalan without logarithmic forms (after Bugeaud, Hanrot and Mihăilescu)

  title={Catalan without logarithmic forms (after Bugeaud, Hanrot and Mihăilescu)},
  author={Yuri F. Bilu},
  journal={Journal de Theorie des Nombres de Bordeaux},
  • Y. Bilu
  • Published 2005
  • Mathematics
  • Journal de Theorie des Nombres de Bordeaux
C'est un rapport sur le travail recent de Bugeaud, Hanrot et Mihailescu, montrant qu'on peut demontrer l'hypothese de Catalan sans utiliser les formes logarithmiques, ni le calcul avec un ordinateur. 

Etudes sur les équations de Ramanujan-Nagell et de Nagell-Ljunggren ou semblables

Dans cette these, on etudie deux types d'equations diophantiennes. Une premiere partie de notre etude porte sur la resolution des equations dites de Ramanujan-Nagell $Cx^2+b^{2m}D=y^n$. Une deuxieme

Perfect Powers: Pillai's works and their developments

A perfect power is a positive integer of the form $a^x$ where $a\ge 1$ and $x\ge 2$ are rational integers. Subbayya Sivasankaranarayana Pillai wrote several papers on these numbers. In 1936 and again

Mihăilescu’s Ideal

Let (x, y, p, q) be a solution of Catalan’s equation (with x, y nonzero integers and p, q odd primes) and G the Galois group of the cyclotomic field \(K = \mathbb{Q}(\zeta _{p})\). From the previous

A Repulsion Motif in Diophantine Equations

It turns out that a similar phenomenon seems, conjecturally, to be at work for solutions which are close to being integral in another sense, and suggests a bound on the size of integral solutions in terms of the coefficients of the defining equation.

Progress in number theory in the years 1998-2009

We summarize the major results in number theory of the last decade.

Postęp w teorii liczb w latach 1998–2009

Osiem lat temu A. Schinzel opublikowal w Wiadomościach Matematycznych przegląd osiągniec teorii liczb w XX wieku. Niniejszy artykul stanowi rozszerzenie tego przeglądu o zaslugujące na uwage jedno

Introduction to Number Theory

One of the oldest branches of mathematics, number theory is a vast field devoted to studying the properties of whole numbers. Offering a flexible format for a one- or two-semester course,

Generalization of Ruderman's Problem to Imaginary Quadratic Fields

In 1974, H. Ruderman posed the following question: If (2 − 2)|(3 − 3), then does it follow that (2 − 2)|(x − x) for every integer x? This problem is still open. However, in 2011, M. R. Murty and V.



Catalan's conjecture (after Mihailescu), Seminaire Bourbaki, Expose 909, 55eme annee (20022003) ;

Je vous prie, Monsieur, de vouloir bien énoncer, dans votre recueil, le théorème suivant, que je crois vrai, bien que je n’aie pas encore réussi à le démontrer complètement: d’autres seront peut-être

Minorations pour l'équation de Catalan

Un nouveau critère pour l'équation de Catalan

Nous presentons dans ce travail une methode, issue de travaux de Bilu et de Bilu et Hanrot, qui permet, sous certaines conditions, de borner de facon bien plus precise que par la methode de Baker les

Formes linéaires en deux logarithmes et déterminants d′interpolation

Abstract We give a new lower bound for a linear form in two logarithms. This work combines the technique of interpolation determinants introduced by M. Laurent and a zero-lemma due to Y. Nesterenko.

Catalan's Equation Has No New Solution with Either Exponent Less Than 10651

Applying old and new theoretical results to a systematic computation, it is shown that Catalan's equation has only the obvious solutions when min{p, q} < 10651.

On the Equation ax - by = 1. II

The following conjecture was apparently first enunciated by Catalan (3) in 1844 but has never been proved.

Collected Papers

THIS volume is the first to be produced of the projected nine volumes of the collected papers of the late Prof. H. A. Lorentz. It contains a number of papersnineteen in all, mainly printed

2897. On the equation ax - by = 1

On the Diophantine Equation x2+y2+c = xyz