Solution methods for Diophantine Equations

Lorentz Center, Leiden, The Netherlands

An instructional Diamant-Stieltjes conference from 7 to 11 May, 2007

A workshop from 14 to 16 May, 2007

**Scientific context**

Two of the highlights of mathematics research in the past 12 years are the proof of Fermat's Last Theorem by Wiles and Taylor in 1995 that for n > 3 the sum of two positive n-th powers cannot be an n-th power and Mihailescu's settling of the Catalan conjecture in 2002 that 8 and 9 are the only positive consecutive integers which are both perfect powers (made accessible by Bilu).

Less known is that this is part of a big development in resolving Diophantine equations.

For example, several other equations of the form A xl + B ym = C zn have been completely solved or parametrized (Beukers, Zagier, Edwards, Serre, Darmon, Merel, Ribet, Bennett, Skinner, and several others). It has been proved that the equation

A xn – B yn = 1 (n > 2) has at most one solution in integers x, y. (This is an extension of a result of Siegel was proved by Bennett.). All perfect powers of the form x2 + 7 have been determined (this problem goes back to Ramanujan), it has been proved that there are no perfect powers of the form 11...1, and that there are no other perfect powers among the Fibonacci numbers other than 1, 8 and 144. (The last three statements were proved by Bugeaud, Mignotte and Siksek).

Many of these questions have only been answered after being open for decades. It is remarkable that several of these problems have been solved by a combination of methods from diophantine approximation and arithmetic algebraic geometry. In this context the most relevant methods from diophantine approximation are the hypergeometric method and the method on linear forms in logarithms of algebraic numbers and the most relevant methods from algebraic geometry the modular method and the Chabauty method.

**Instructional Conference**

The Diamant-Stieltjes week is primarily intended for Master and Ph.D.-students and post-docs (at most 35 years of age) who are interested in the subject. Each morning from Monday till Thursday there will be two or three lectures given by experts in the field. these lectures will be given by Mike Bennett (UBC, Vancouver), Nils Bruin (SF, Vancouver), Yann Bugeaud (Strasbourg) and Samir Siksek (Warwick). Mike Bennett is the Kloosterman Professor 2007 and will be in Leiden in April and May, 2007.

In the afternoon small groups will work on problems related to the morning lectures and the solutions will be discussed in a problem session at the end of each afternoon. On Friday there will be four one-hour lectures dealing with recent results obtained by the treated methods.

This instructional conference will enable young researchers to learn (more about) the various techniques which are nowadays used to solve Diophantine equations and to meet active researchers in the field. They meet others in their age group working on related problems. It should enable the participants to attend the subsequent workshop with better understanding of the lectures and the discussions.

**The workshop**

At the workshop the participants can present their results and discuss them informally. It will be a meeting place for researchers from different fields. Although there are often conferences for researchers in Diophantine approximation and for researchers in arithmetic algebraic geometry, it is rare to have a conference where researchers from both areas meet, although they work on the same type of problems. The workshop will help to bridge the gap and to stimulate new research.

The workshop is targeted to the lecturers and participants of the first week as well as for senior researchers who only attend the second week. For the workshop we shall invite about 10 persons who have made significant contributions to the subject of the conference. Other researchers are welcome at their own cost.

Prof. dr. F. Beukers, Univ. Utrecht

Dr. J.H. Evertse, Univ. Leiden

Prof. dr. R. Tijdeman, Univ. Leiden.

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