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Andrej Dujella's monograph Number Theory translated from Croatian
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.
3.1. Definition and properties of congruences 3.2. Tests of divisibility 3.3. Linear congruences 3.4. Chinese remainder theorem 3.5. Reduced residue system 3.6. Congruences with a prime modulus 3.7. Primitive roots and indices 3.8. Representations of rational numbers by decimals 3.9. Pseudoprimes 3.10. Exercises
4. Quadratic residues
4.1. Legendre's symbol 4.2. Law of quadratic reciprocity 4.3. Computing square roots modulo p 4.4. Jacobi's symbol 4.5. Divisibility of Fibonacci numbers 4.6. Exercises
5. Quadratic forms
5.1. Sums of two squares 5.2. Positive definite binary quadratic forms 5.3. Sums of four squares 5.4. Sums of three squares 5.5. Exercises
6. Arithmetical functions
6.1. Greatest integer function 6.2. Multiplicative functions 6.3. Asymptotic estimates for arithmetic functions 6.4. Dirichlet product 6.5. Exercises
7. Distribution of primes
7.1. Elementary estimates for the function Ď(x) 7.2. Chebyshev functions 7.3. The Riemann zeta-function 7.4. Dirichlet characters 7.5. Primes in arithmetic progressions 7.6. Exercises
8. Diophantine approximation
8.1. Dirichlet's theorem 8.2. Farey sequences 8.3. Continued fractions 8.4. Continued fraction and approximations to irrational numbers 8.5. Equivalent numbers 8.6. Periodic continued fractions 8.7. Newton's approximants 8.8. Simultaneous approximations 8.9. LLL algorithm 8.10. Exercises
9. Applications of diophantine approximation to cryptography
9.1. A very short introduction to cryptography 9.2. RSA cryptosystem 9.3. Wiener's attack on RSA 9.4. Attacks on RSA using the LLL algorithm 9.5. Coppersmith's theorem 9.6. Exercises
10. Diophantine equations I
10.1. Linear Diophantine equations 10.2. Pythagorean triangles 10.3. Pell's equation 10.4. Continued fractions and Pell's equation 10.5. Pellian equation 10.6. Squares in the Fibonacci sequence 10.7. Ternary quadratic forms 10.8. Local-global principle 10.9. Exercises
11. Polynomials
11.1. Divisibility of polynomials 11.2. Polynomial roots 11.3. Irreducibility of polynomials 11.4. Polynomial decomposition 11.5. Symmetric polynomials 11.6. Exercises
12. Algebraic numbers
12.1. Quadratic fields 12.2. Algebraic number fields 12.3. Algebraic integers 12.4. Ideals 12.5. Units and ideal classes 12.6. Exercises
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):
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 "elementary" 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.
... 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.
The Mathematical Intelligencer
This extremely important book first appeared in a Croatian edition in 2019 and has now been translated into English (along with a few minor corrections and updates) for this edition.
It is particularly well suited for students doing independent research or thesis work, and it is also ideal as a text for specialized graduate courses in number theory. Moreover, anyone seriously interested in number theory will discover in this book a veritable gold mine of information about the current state of affairs in several areas of modern number theory.
While reading this book I also had great fun discovering many things about numbers I had never thought of before.
A more usual and informal appearance of Professor Andrej Dujella.
This video contains his lively lecture about Number Theoretic problems that he has studied,
originating from Diophantus (Ancient Greek mathematician), Euler and Fermat.
On the left Professor Andrej Dujella, since 2012 a member of Croatian Academy of Sciences and Arts, distinguished expert in Number Theory.
And excellent singer. He is talking to Dr. Vera Tonić, who is working in the field of topology. On his left Professors Goran Lešaja, Georgia Southern University, USA, and Sonja Štimac, University of Zagreb, one of invited lecturers,
expert in Dynamical Systems. Far on the left, Professor Andrej Ščedrov, expert in Mathematical Logic, Penn State University, USA,
who completed his studies of Mathematics at the University of Zagreb.
AcademiciansMarko Tadić and Andrej Dujella(Mathematics Department of the School of Science), Professor Dražen Adamović (editor in chief ofGlasnik 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.
