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Solvability of Diophantine Equations |
Solution methods for Diophantine Equations 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 x^{l }+ B y^{m}^{
}= C z^{n}^{ }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 x^{n} – B y^{n} = 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 x^{2
}+ 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 ( 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. Dr. J.H. Evertse, Univ. Prof. dr. R. Tijdeman,
Univ. [Back] |