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A Repulsion Motif in Diophantine Equations

by Graham Everest (East Anglia University) and Tom Ward (East Anglia University)

Award: Lester R. Ford

Year of Award: 2012

Publication Information: The American Mathematical Monthly, vol. 118, no. 7, August-September 2011, pp. 584-598

Summary (From the Prizes and Awards booklet, MathFest 2012)

Beginning with \(y^2 + 2 = x^3\), the authors entice the reader with the distinguished history of this equation along with the surprising sizes of solutions. The authors then lead the reader forward in time, effectively offering a "speed dating" tour of highlights in Diophantine equations, such as the abc conjecture, the Baker-Stark methods, the recent proof of the Catalan conjecture, and the geometry of elliptic curves. They introduce key definitions and themes of Diophantine equations in simple concrete contexts, hinting at the complexity that a fully general description would involve. The authors weave several themes throughout the article, such as the interplay of computation/conjecture/theory, or the "familiar refrain" that an effective (bounded) search may still be an impracticable one.

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About the Authors: (From the Prizes and Awards booklet, MathFest 2012)

Tom Ward has worked at the University of East Anglia since 1992, and is currently Pro-Vice-Chancellor (Academic) with responsibility for teaching and learning and the student experience. He attended Waterford-Kamhlaba School in Swaziland, where he encountered several inspirational mathematics and physics teachers who nurtured an interest started by his physicist parents. After studying at the University of Warwick, he worked at College Park and Ohio State University before returning to England. He works in ergodic theory, and enjoyed a long collaboration with Graham Everest, studying dynamical systems from a number-theoretic point of view and number theory from a dynamical point of view. He has written several books, including Recurrence sequences with Everest, Alf van der Poorten, and Igor Shparlinski, Heights of polynomials and entropy in algebraic dynamics with Everest, and Ergodic theory with a view towards number theory with Manfred Einsiedler.

Graham Everest, who was elected a member of the London Mathematical Society in 1983, died on 30 July 2010, aged 52.

Thomas Ward writes: Graham's talent for mathematics took him to Bedford College and doctoral study under the supervision of Colin Bushnell at King's College London. He joined the University of East Anglia as a lecturer in 1983, and spent his whole career there.

His research appeared in the form of some 70 research papers and three monographs, and spanned diverse areas of number theory. Three themes informed his research. First, the impact of twentieth-century developments in Diophantine analysis and transcendence theory on counting problems and questions in algebraic number theory. Second, the fascinating arithmetic properties of recurrence sequences, including classical questions in the spirit of Mersenne, Lehmer, Zsigmondy, and so on, as well as more modern developments on bilinear sequences and elliptic divisibility sequences. Third, Graham had an abiding interest in all aspects of the interaction between number theory and dynamical systems.

As a researcher Graham brought great joy and creativity to his work, and the generosity of his approach to mathematics will be familiar to his thirty co-authors. Graham was a dedicated teacher and supervisor, and many generations of students will remember the energy and enthusiasm of his lectures. His belief in the transforming power of higher education was recognized in the form of a UEA Excellence in Teaching award in 2005.

Subject classification(s): Algebra and Number Theory | Number Theory | Diophantine Equations
Publication Date: 
Wednesday, August 8, 2012