Croatian

Number Theory

Lectures: Andrej Dujella, Filip Najman

Contents

Divisibility. Greatest common divisor. Euclidean algorithm. Primes.

Congruences. Euler's theorem. Chinese remainder theorem. Primitive roots and indices.

Quadratic residues. Legendre symbol. Quadratic reciprocity law. Divisibility properties of Fibonacci numbers.

Quadratic forms. Reduction of binary quadratic forms. Sums of two and four squares.

Arithmetic functions. Euler and Möbius functions. Distribution of primes. Asymptotic estimates for arithmetic functions.

Diophantine approximation. Dirichlet's theorem. Continued fractions. Law of best approximation. Liouville's theorem.

Diophantine equations. Linear diophantine equations. Pythagorean triples. Pell equation. Elliptic curves.

Quadratic fields. Units and primes in quadratic fields. Applications to Diophantine equations.

Basic references

1. A. Dujella: Teorija brojeva, Skolska knjiga, Zagreb, 2019.

2. A. Dujella: Number Theory, Skolska knjiga, Zagreb, 2021.

3. A. Baker: A Concise Introduction to the Theory of Numbers, Cambridge University Press, Cambridge, 1994.

4. I. Niven, H. S. Zuckerman, H. L. Montgomery: An Introduction to the Theory of Numbers, Wiley, New York, 1991.

5. K. H. Rosen: Elementary Number Theory and Its Applications, Addison-Wesley, Reading, 1993.

Additional references

1. A. Baker: A Comprehensive Course in Number Theory, Cambridge University Press, Cambridge, 2012.

2. K. Chandrasekharan: Introduction to Analytic Number Theory, Springer-Verlag, Berlin, 1968.

3. H. Davenport: The Higher Arithmetic, Cambridge University Press, Cambridge, 1999.

4. A. Dujella, M. Maretic: Kriptografija, Element, Zagreb, 2007.

5. G. H. Hardy, E. M. Wright: An Introduction to the Theory of Numbers, Oxford University Press, Oxford, 1980.

6. Hua Loo Keng: Introduction to Number Theory, Springer-Verlag, Berlin, 1982.

7. B. Hutz, An Experimental Introduction to Number Theory, American Mathematical Society, Providence, 2018.

8. B. Ibrahimpasic: Uvod u teoriju brojeva, Pedagoski fakultet Bihac, 2014.

9. K. Ireland, M. Rosen: A Classical Introduction to Modern Number Theory, Springer-Verlag, New York, 1998.

10. M. Jukic Bokun, I. Soldo: Zbirka zadataka iz teorije brojeva, Sveuciliste Josipa Jurja Strossmayera u Osijeku - Fakultet primijenjene matematike i informatike, Osijek, 2023.

11. J.-M. de Koninck, A. Mercier: 1001 Problems in Classical Number Theory, American Mathematical Society, 2007.

12. W. J. LeVeque: Elementary Theory of Numbers, Dover, New York, 1990.

13. T. Nagell: Introduction to Number Theory, Chelsea, New York, 1981.

14. H. E. Rose: A Course in Number Theory, Oxford University Press, Oxford, 1995.

15. W. M. Schmidt: Diophantine Approximation, Springer-Verlag, Berlin, 1996.

16. W. Sierpinski: Elementary Theory of Numbers, PNW, Warszawa; North Holland, Amsterdam, 1987.

17. I. M. Vinogradov: Elements of Number Theory, Dover, New York, 1954.

Lecture notes
(in pdf format; in Croatian)

Student seminar - Elliptic curves and their applications in cryptography (2002/2003)

Exams

Some (useful) links

Number Theory Web (maintained by Keith Matthews)
Number theory groups and seminars
Number Theory Listserver Archives
Cryptography - Undergraduate course (Andrej Dujella)
Number Theory in Cryptography - Graduate course (Andrej Dujella)
Diophantine equations - Graduate course (Andrej Dujella)
Algorithms for Elliptic Curves - Graduate course (Andrej Dujella)
Diophantine approximations and applications - Graduate course (Andrej Dujella)
Seminar on Number Theory and Algebra
Recommended readings for graduate students in number theory
Diophantine m-tuples page (Andrej Dujella)
High rank elliptic curves with prescribed torsion (Andrej Dujella)
The Prime Pages (Chris Caldwell)
The ABC Conjecture Home Page (Abderrahmane Nitaj)
Fibonacci Numbers and the Golden Section (Ron Knott)
Kevin Brown's number theory page
Number Theory and PARI/GP
Online mathematical journal math.e

Web pages of some number theory courses:

W. Chen: Elementary Number Theory, Imperial College, University of London
M. Filaseta: Elementary Number Theory, University of South Carolina
W. Stein: Elementary Number Theory, Harvard University
D. Burde: Algebraic Number Theory, University of Vienna
P. L. Clark: Number Theory, University of Georgia
F. Lemmermeyer: Algebraic Number Theory, Bilkent University
J. Milne: Algebraic Number Theory, University of Michigan
W. Stein: Introduction to Algebraic Number Theory, Harvard University
W. Chen: Distribution of Prime Numbers, Imperial College, University of London
N. Elkies: Introduction to Analytic Number Theory, Harvard University
K. Kedlaya: Analytic Number Theory, MIT