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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

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

References

Notation Index

Subject Index

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.