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### Number theory

 Code: 92892 ECTS: 6.0 Lecturers in charge: prof. dr. sc. Andrej Dujella - Lectures prof. dr. sc. Filip Najman - Lectures Lecturers: Adrian Beker - Exercises Petar Orlić, mag. math. - Exercises English level: 1,0,0 All teaching activities will be held in Croatian. However, foreign students in mixed groups will have the opportunity to attend additional office hours with the lecturer and teaching assistants in English to help master the course materials. Additionally, the lecturer will refer foreign students to the corresponding literature in English, as well as give them the possibility of taking the associated exams in English.

### 1. komponenta

Lecture typeTotal
Lectures 30
Exercises 30
Description:
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)
Literature:
1. I. Niven, H. S. Zuckerman, H. L. Montgomery: An Introduction to the Theory Numbers
2. K. H. Rosen: Elementary Number Theory and Its Applications
3. H. Davenport: The Higher Arithmetic
4. A. Baker: A Concise Introduction to the Theory of Numbers
5. H. L. Keng: Introduction to Number Theory
6. K. Ireland, M. Rosen: A Classical Introduction to Modern Number Theory
7. T. Nagell: Introduction to Number Theory
8. B. Pavković, D. Veljan: Elementarna matematika 2
9. W. Sierpinski: Elementary Theory of Numbers
10. I. M. Vinogradov: Elements of Number Theory
Prerequisit for:
Enrollment :
Passed : Mathematical analysis 2
 4. semester Mandatory course - Regular study - Mathematics
Consultations schedule:

### Content

Link to the course web page: https://web.math.pmf.unizg.hr/~duje/utb.html