Andrej Dujella: Diophantine m-tuples and Elliptic Curves, published by Springer, Cham, 2024.

http://www.croatia.org/crown/articles/11414/1/Andrej-Dujella-Diophantine-m-tuples-and-Elliptic-Curves-published-by-Springer-Cham-2024.html

By Nenad N. Bach and Darko ®ubriniæ

Published on 07/3/2024

Andrej Dujella is a leading Croatian expert in Number Theory. 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. |

Most of my 13 PhD students had theses related to Diophantine m-tuples (and several of their PhD students, too)

Front matter of the monograph (highly recommended!): [PDF]

Back matter (References and Index): [PDF]

Professor Andrej Dujella, distinguished Croatian mathematician,

is a Fellow of Croatian Academy of Sciences and Arts

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. More information |

Professor Andrej Dujella is since 2017 a recipient of the honorary doctorate (Doctor Honoris Causa)

from the University of Debrecen, Hungary, due to his very fruitful collaboration with Hungarian

mathematicians. Hungary has a long-standing tradition in Number Theory.

Professor Andrej Dujella is working at the University of Zagreb. He is a member of the Croatian Academy of Sciences. He started to work on diophantine sets as a PhD student. These sets are named after the Greek mathematician Diophantus of Alexandria. Multiplying any two of their elements and adding one gives a square. The most important result of professor Dujella is that there are only finitely many diophantine quintuples and they are effectively enumerable. Its ingenious proof combines elementary number-theoretic considerations with A. Baker's method, which is one of the most modern tools of diophantine number theory. Professor Dujella generalized the notion of diophantine sets in several directions. After his efforts this has become very fashionable; many Hungarian, American, Austrian, French and Japanese researchers are working on the topic. He is not only a leading expert of number theory, but has important results on weak parameters of the RSA cryptosystem, which is fundamental for the security of the internet. Professor Dujella has led a mutually fruitful collaboration with the Faculty of Informatics and with the number theory group of the University of Debrecen (in Hungary) for more than 20 years. He has published joint papers with seven mathematicians from Debrecen. He led several joint Hungarian-Croatian research projects and was the co-organizer of the Hungarian-Croatian Workshop on Mathematics and Informatics. Source - University of Debrecen, Hungary |

Autobiography of Professor Andrej Dujella, in Croatian