Diophantine m-tuples and Elliptic Curves
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Beschreibung
This book provides an overview of the main results and problems concerning Diophantine m-tuples, i.e., sets of integers or rationals with the property that the product of any two of them is one less than a square, and their connections with elliptic curves. It presents the contributions of famous mathematicians of the past, like Diophantus, Fermat and Euler, as well as some recent results of the author and his collaborators.
The book presents fragments of the history of Diophantine m-tuples, emphasising the connections between Diophantine m-tuples and elliptic curves. It is shown how elliptic curves are used in solving some longstanding problems on Diophantine m-tuples, such as the existence of infinite families of rational Diophantine sextuples. On the other hand, rational Diophantine m-tuples are used to construct elliptic curves with interesting Mordell–Weil groups, including curves of record rank with a given torsion group. The book contains concrete algorithms and advice on how to use the software package PARI/GP for solving computational problems.
This book is primarily intended for researchers and graduate students in Diophantine equations and elliptic curves. However, it can be of interest to other mathematicians interested in number theory and arithmetic geometry. The prerequisites are on the level of a standard first course in elementary number theory. Background in elliptic curves, Diophantine equations and Diophantine approximations is provided.
This book provides an overview of the main results and problems concerning Diophantine m-tuples, i.e., sets of integers or rationals with the property that the product of any two of them is one less than a square, and their connections with elliptic curves. It presents the contributions of famous mathematicians of the past, like Diophantus, Fermat and Euler, as well as some recent results of the author and his collaborators.
The book presents fragments of the history of Diophantine m-tuples, emphasising the connections between Diophantine m-tuples and elliptic curves. It is shown how elliptic curves are used in solving some longstanding problems on Diophantine m-tuples, such as the existence of infinite families of rational Diophantine sextuples. On the other hand, rational Diophantine m-tuples are used to construct elliptic curves with interesting Mordell–Weil groups, including curves of record rank with agiven torsion group. The book contains concrete algorithms and advice on how to use the software package PARI/GP for solving computational problems.
This book is primarily intended for researchers and graduate students in Diophantine equations and elliptic curves. However, it can be of interest to other mathematicians interested in number theory and arithmetic geometry. The prerequisites are on the level of a standard first course in elementary number theory. Background in elliptic curves, Diophantine equations and Diophantine approximations is provided.
A unique mixture of classical mathematics and recent research Present solutions of problems that were open for centuries Provides an attractive introduction to elliptic curves for students
Autor*in
Andrej Dujella
Themen in »Diophantine m-tuples and Elliptic Curves«
Diophantine Equations Diophantine Quadruples Elliptic Curves Torsion Group Rank of Elliptic Curves Applications of Linear Forms in Logarithms Integer Points on Elliptic Curves History of Number Theory Open Problems in Number Theory
Stimmen zu »Diophantine m-tuples and Elliptic Curves«
“This book seamlessly integrates ideas that reveal both the theoretical and computational beauty and depth of the subject. Written in a clear and accessible manner, it is an invaluable resource for anyone interested in studying this topic. I thoroughly enjoyed reading this book.” (Gökhan Soydan, zbMATH 1561.11002, 2025)
“This book provides the essential motivational material for those new to the field of Diophantine m-tuples, effectively guiding readers to the forefront of current research. Graduates and researchers in number theory will find it an invaluable resource. Moreover, the book is engaging, well written, and includes numerous insightful exercises, making it an excellent tool for deepening understanding and fostering further exploration of this fascinating subject.” (Dimitrios Poulakis, Mathematical Reviews, July, 2025)
“The present book is a monograph on a very specialized topic in number theory, namely Diophantine m-tuples. … this book is very complex, containing results from a very interesting and current research field. I think that this book deserved to be published by the Springer publishing house and I think it will be a benchmark for current research in number theory.” (Diana Savin, Bulletin of the Transilvania University of Brasov, Vol. 4 (1), 2024)
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