Andrej Dujella:

Number Theory

Textbook of the University of Zagreb
Publisher: Školska knjiga, Zagreb, 2021.
Translated by Petra Švob
ISBN: 978-953-0-30897-8
621 pages, 17 × 24 cm

cover of the book Number Theory

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Front matter (Prefaces to the Croatian and English editions, Contents) and back matter (References, Notation Index, Subject Index).

Look inside the book at e-sfera.

Andrej Dujella's monograph Number Theory translated from Croatian, an article by Darko Žubrinić on Croatian World Network

Preliminary book review in Zentralblatt MATH

In libraries: MPIM Bonn, TU Graz, Klagenfurt, Zagreb-HAZU, Zagreb-PMF-MO.

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.

In the English edition, there are only minor changes compared with the Croatian version. Several misprints noticed by the author and the readers were corrected. Some information and references were updated, in particular, those related to elliptic curves rank records and new constructions of families of rational Diophantine sextuples. At just a few places in the Croatian version of the book only the references to literature in Croatian were given; these references were expanded in the English edition with the appropriate recommendations of literature in English. The list of references has been expanded to include some recent books and papers, as well as some references which were mentioned in the text of the Croatian edition but were not included in the list of references.


    Preface to the Croatian edition

    Preface to the English edition

    1. Introduction
          1.1. Peano's axioms
          1.2. Principle of mathematical induction
          1.3. Fibonacci numbers
          1.4. Exercises

    2. Divisibility
          2.1. Greatest common divisor
          2.2. Euclid's algorithm
          2.3. Primes
          2.4. Exercises

    3. Congruences
          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 π(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

    13. Approximation of algebraic numbers
          13.1. Liouville's theorem
          13.2. Roth's theorem
          13.3. The hypergeometric method
          13.4. Approximation by quadratic irrationals
          13.5. Polynomial root separation
          13.6. Exercises

    14. Diophantine equations II
          14.1. Thue equations
          14.2. Tzanakis' method
          14.3. Linear forms in logarithms
          14.4. Baker-Davenport reduction
          14.5. LLL reduction
          14.6. Diophantine m-tuples
          14.7. Exercises

    15. Elliptic curves
          15.1. Introduction to elliptic curves
          15.2. Equations of elliptic curves
          15.3. Torsion group
          15.4. Canonical height and Mordell-Weil theorem
          15.5. Rank of elliptic curves
          15.6. Finite fields
          15.7. Elliptic curves over finite fields
          15.8. Applications of elliptic curves in cryptography
          15.9. Primality proving using elliptic curves
          15.10. Elliptic curve factorization method
          15.11. Exercises

    16. Diophantine problems and elliptic curves
          16.1. Congruent numbers
          16.2. Mordell's equation
          16.3. Applications of factorization in quadratic field
          16.4. Transformation of elliptic curves to Thue equations
          16.5. Algorithm for solving Thue equations
          16.6. abc conjecture
          16.7. Diophantine m-tuples and elliptic curves
          16.8. Exercises


    Notation Index

    Subject Index

You may send your comments, remarks and suggestions on the book by e-mail to I will be grateful to anyone who points out inaccuracies or errors in the book.

Errata et addenda

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