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Content archived on 2024-05-29

Modular and classical approaches to Diophantine Equations


Since ancient Greeks, mathematicians have been interested in what we now call Diophantine problems. A Diophantine problem is an equation where one is interested in finding solutions that are whole numbers.
The most famous Diophantine problem of all time is Fermat's Last Theorem. This problem attracted the attention of huge numbers of both professional and amateur mathematicians for over 350 years, and was finally solved by Andrew Wiles in 1994. We call the approach taken in the proof of Fermat's Last Theorem, the 'Modular Approach'. We call the earlier (pre-Wiles) approaches, the classical approaches.
The EU is the world leader in the classical approaches, but not in the 'modular approach'.
The aims of this project include
(i) To improve our understanding of the information given by the modular approach.
(ii) To set out a coherent strategy for supplementing the local information given by the modular approach with the global information obtained from classical approaches, such as Mordell-Weil groups, hypergeom etric methods, Runge's method, etc.
(iii) To identify and solve interesting and outstanding Diophantine problems that can be tackled by the innovations introduced in (i), (ii) The project will involve extensive collaborations. It will be a link between lea ding British research groups and those centres of excellence in Diophantine equations in France, Germany, Spain, The Netherlands, Greece, Croatia, Hungary and elsewhere in Europe.
These collaborations and interactions will reduce the fragmentation of the subject across Europe. Dr. Siksek is a leading European expert on Diophantine equations (and the modular approach in particular), who has been away for 7 years. He has, in collaboration with other European researchers, used the modular approach to solve several famous open Diophantine problems. His return to the UK and reintegration into European research will help take the EU a step closer to becoming the leader in the 'modular approach'.

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Mathematics Institute Kirby Corner Road, University House
United Kingdom

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