1. komponenta

Lecture type	Total
Lectures	30
Exercises	30

COURSE AIMS AND OBJECTIVES: The course will cover the basic notions from elementary number theory and selected topics from other areas of number theory.

COURSE DESCRIPTION AND SYLLABUS:
1. Divisibility. Greatest common divisor. Euclidean algorithm. Primes. (2 weeks)
2. Congruences. Chinese remainder theorem. Euler's theorem. Wilson's theorem. Hensel's lemma. Primitive roots and indices. (2 weeks)
3. Quadratic residues. Legendre symbol. Quadratic reciprocity law. Jacobi symbol. Divisibility properties of Fibonacci numbers. (1 week)
4. Quadratic forms. Reduction of binary quadratic forms. Sums of two and four squares. (1 week)
5. Arithmetic functions. Multiplicative functions. Asymptotic estimates for arithmetic functions. Distribution of primes. Riemann zeta function. (2 weeks)
6. Diophantine approximations. Dirichlet's theorem. Continued fractions. Law of best approximation. Liouville's theorem. (2 weeks)
7. Diophantine equations. Linear diophantine equations. Pythagorean triples. Pell equation. Elliptic curves. (2 weeks)
8. Quadratic fields. Units and primes in quadratic fields. Applications to Diophantine equations. (1 week)

1. An Introduction to the Theory Numbers, I. Niven, H. S. Zuckerman, H. L. Montgomery, John Wiley & Sons, 1991.
2. Elementary Number Theory and Its Applications, K. H. Rosen, Addison - Wesley, 1993.
3. A Concise Introduction to the Theory of Numbers, A. Baker, Cambridge University Press, 1994.
4. Teorija brojeva, A. Dujella, Školska knjiga, 2019.
5. Number Theory, A. Dujella, Školska knjiga, 2021.
6. The Higher Arithmetic, H. Davenport, Cambridge University Press, 1999.
7. Introduction to Number Theory, H. L. Keng, Springer Verlag, 1982.
8. A Classical Introduction to Modern Number Theory, K. Ireland, M. Rosen, Springer Verlag, 1998.
9. Introduction to Number Theory, T. Nagell, Chelsea, 1981.
10. Elementarna matematika 2, B. Pavković, D. Veljan, Školska knjiga, Zagreb, 1995.
11. Elementary Theory of Numbers, W. Sierpinski, PNW, Varšava; North Holland, Amsterdam, 1987.
12. Elements of Number Theory, I. M. Vinogradov, Dover, 1954.

Enrollment :
Passed : Mathematical analysis 2

Mandatory course - Regular study - Mathematics