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Diophantine m-tuples and Elliptic Curves

  • Book
  • May 2024

Overview

Authors:
  1. Andrej Dujella

    1. Department of Mathematics, Faculty of Science, University of Zagreb, Zagreb, Croatia

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  • 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

Part of the book series: Developments in Mathematics (DEVM, volume 79)

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Keywords

  • 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

About this book

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.

Authors and Affiliations

  • Department of Mathematics, Faculty of Science, University of Zagreb, Zagreb, Croatia

    Andrej Dujella

About the author

Andrej Dujella is a professor of mathematics at the University of Zagreb, Fellow of the Croatian Academy of Sciences and Arts and Doctor Honoris Causa of University of Debrecen. His research interests include Diophantine equations, elliptic curves, polynomial root separation, and applications of Diophantine approximation to cryptography.

Bibliographic Information

  • Book Title: Diophantine m-tuples and Elliptic Curves

  • Authors: Andrej Dujella

  • Series Title: Developments in Mathematics

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024

  • Hardcover ISBN: 978-3-031-56723-0Due: 19 May 2024

  • Softcover ISBN: 978-3-031-56726-1Due: 19 May 2024

  • eBook ISBN: 978-3-031-56724-7Due: 19 May 2024

  • Series ISSN: 1389-2177

  • Series E-ISSN: 2197-795X

  • Edition Number: 1

  • Number of Pages: XI, 335

  • Number of Illustrations: 7 b/w illustrations

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