|Andrej Dujella, of the Croatian Academy of Sciences and Arts, is distinguished Croatian mathematician, expert in Number Theory. His monograph Teorija brojeva, written in Croatian, saw its translation into English in 2021: Number Theory. It is primarily intended for students of mathematics and related faculties who attend courses in number theory and its applications. However, it can also be useful to advanced high school students who are preparing for mathematics competitions in which at all levels, from the school level to international competitions, number theory has a significant role, and for doctoral students and scientists in the fields of number theory, algebra and cryptography.|
Description of the monograph by Andrej Dujella
Number theory is a branch of mathematics that is primarily focused on the study of positive integers, or natural numbers, and their properties such as divisibility, prime factorization, or solvability of equations in integers. Number theory has a very long and diverse history, and some of the greatest mathematicians of all time, such as Euclid, Euler and Gauss, have made significant contributions to it. Throughout its long history, number theory has often been considered as the "purest" branch of mathematics in the sense that it was the furthest from any concrete application. However, a significant change took place in the mid-1970s, and nowadays, number theory is one of the most important branches of mathematics for applications in cryptography and secure information exchange.
This book is based on teaching materials from the courses Number Theory and Elementary Number Theory, which are taught at the undergraduate level studies at the Department of Mathematics, Faculty of Science, University of Zagreb, and the courses Diophantine Equations and Diophantine Approximations and Applications, which were taught at the doctoral program of mathematics at that faculty. The book thoroughly covers the content of these courses, but it also contains other related topics such as elliptic curves, which are the subject of the last two chapters in the book. The book also provides an insight into subjects that were and are at the centre of research interest of the author of the book and other members of the Croatian group in number theory, gathered around the Seminar on Number Theory and Algebra.
This book is primarily intended for students of mathematics and related faculties who attend courses in number theory and its applications. However, it can also be useful to advanced high school students who are preparing for mathematics competitions in which at all levels, from the school level to international competitions, number theory has a significant role, and for doctoral students and scientists in the fields of number theory, algebra and cryptography.
Preface to the Croatian edition
Preface to the English edition
1. Introduction1.1. Peano's axioms
Finnish mathematician Pentti Haukkanen wrote a review about the book for Zentralblatt MATH, available at
Here, we cite his opinion about it:
This book is a beautiful invitation to number theory. It provides interesting connections between various fields of number theory. Proofs are presented in a concise form. I think that this is a useful opus for a wide branch of readership interested in number theory.
Presentation of Dujella's book Number Theory published in EMS Magazine (edition of European Mathematical Society), written by professor Jean-Paul Allouche (IMJ-PRG, Sorbonne, Paris):
An excerpt from the presentation:
A very recent book, entitled Number Theory and based on teaching materials, has been written by A. Dujella. Devoted to several subfields of this domain, this book is both extremely nice to read and to work from.
this book addresses many jewels of number theory. This is done in a particularly appealing way, mostly elementary when possible, with many well-chosen examples and attractive exercises. I arbitrarily choose two delightful examples, the kind of Ă˘elementaryĂ˘ statements that a beginner could attack, but whose proofs require some ingenuity, namely the unexpected statements 4.6 and 4.7.
The book also comprises short historical indications and 426 references. It really made me think of my first reading of Hardy and Wright, and I almost felt regret that I cannot start studying Number Theory again from scratch, but using this book! I highly recommend it not only to neophytes, but also to more "established" scientists who would like to start learning Number Theory, or to refresh and increase their knowledge of the field in an entertaining and subtle way.
Mathematical Association of America
Review of Andrej Dujella's monograph "Number Theory" written by Caleb McWorther
... The book presents an appropriately challenging, focused read that covers a large swath of topics. This makes the book very useful for a more advanced undergraduate number theory course, a department having several undergraduate number theory offerings, or a two-sequence course in number theory. The level of simultaneous breadth and depth offered in this book simply cannot be found in other standard texts like Burton's Elementary Number Theory, Barnett's Elements of Number Theory, or Adler and Coury's The Theory of Numbers. For a similar treatment of the expansiveness of number theory, one would have to turn to Ireland and Rosen's A Classical Introduction to Modern Number Theory, which would not be appropriate for many undergraduate audiences. This book could also be perfectly used in a single semester graduate course in number theory wishing to give students a "birds eye view" of number theory to direct their interests. Wonderfully for both audiences, the book contains plenty of recent results to capture the reader's attention and give a flavor for modern research. Although not overly abundant, the exercises are not guilty of superfluous repetition and are instead focused on pushing the essentials for each topic. Many chapters address open problems that an instructor could assign students to examine. Above all, Number Theory does a masterful job at capturing the subtle and graceful intertwining of the analytic, the algebraic, and the combinatorial with more traditional elementary number theory.
Academicians Marko Tadić and Andrej Dujella (Mathematics Department of the School of Science), Professor Dražen Adamović (editor in chief of Glasnik matematički),
and Professor Tomislav Došlić (Faculty of Civil Enginnering), all of them from the University of Zagreb. Photo taken after a lecture of
Tomislav Šikić about Srinivasa Ramanujan, the most famous Indian mathematician in history.