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### Elementary number theory

 Code: 36962 ECTS: 5.0 Lecturers in charge: izv. prof. dr. sc. Zrinka Franušić - Lectures doc. dr. sc. Tomislav Pejković - Lectures Lecturers: izv. prof. dr. sc. Zrinka Franušić - Exercises doc. dr. sc. Tomislav Pejković - 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. Special attention will be given to topics relevant for mathematical competitions.

COURSE DESCRIPTION AND SYLLABUS:
1. Divisibility. Greatest common divisor. Euclidean algorithm. Primes. Unique factorization.
2. Congruences. Linear congruences. Linear diophantine equations. Chinese remainder theorem. Euler's theorem. Wilson's theorem. Primitive roots.
3. Problems from mathematical competitions I.
4. Arithmetic functions. Floor function. Number and sum of divisors. Euler's function. Mobius function. Distribution of primes.
5. Quadratic residues and quadratic forms. Legendre symbol. Quadratic reciprocity law. Sums of two and four squares.
6. Diophantine equations. Pythagorean triples. Pell equation. Continued fractions. Diophantine approximations.
7. Problems from mathematical competitions II.
8. Applications of number theory. Introduction to cryptography. Primality tests. Factoring methods.
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 : Introduction to mathematics
 2. semester Izborni matematički predmet 1 - Regular study - Mathematics and Physics Education 4. semester Izborni matematički predmet 2 - Regular study - Mathematics and Physics Education
Consultations schedule:

### Content

Link to the course Web page: https://www.pmf.unizg.hr/math/predmet/etb