MUC33 Number theory

Faculty of Science
Autumn 2024
Extent and Intensity
2/2/0. 5 credit(s). Type of Completion: zk (examination).
In-person direct teaching
Teacher(s)
Mgr. Michal Bulant, Ph.D. (lecturer)
Guaranteed by
Mgr. Michal Bulant, Ph.D.
Department of Mathematics and Statistics – Departments – Faculty of Science
Supplier department: Department of Mathematics and Statistics – Departments – Faculty of Science
Timetable
Mon 10:00–11:50 M1,01017
  • Timetable of Seminar Groups:
MUC33/01: Fri 12:00–13:50 M1,01017, M. Bulant
MUC33/02: Fri 10:00–11:50 M1,01017, M. Bulant
Prerequisites
Basics of divisibility.
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
Course objectives
The aim of the course is to develop knowledge and skills in number theory, especially those that can be used while teaching in secondary school. In doing so, it builds on the knowledge of algebra, which is deepened and applied specifically to the field of integers. In order to solve basic Diophantine equations, the theory of solving congruences is built up while deepening knowledge of divisibility and prime numbers. Teaching methods are used that emphasise developing didactic competences of the students-future teachers, among other things through continuous assignments and feedback, the use of software tools and the description of practical applications of number theory.
Learning outcomes
At the end of this course, students should be able to:
understand the basics of elementary number theory
use properly congruences
solve linear congruences and their systems and selected types of congruences of higher order, in particular quadratic and binomial
evaluate and explain the time complexity of numerical operations on large numbers
describe basic principles and procedures of digital encryption and signature schemes using number theory methods
apply various methods for solving diophantine equations
Syllabus
  • Elementary number theory (prime numbers, congruences, Fermat theorem, Euler theorem).
  • Congruences in one variable (linear congruences, algebraic congruences, primitive root). Quadratic congruences, Legendre symbol, quadratic reciprocity law.
  • Applications of number theory
  • Diophantine equations (linear diophantine equations, elementary methods for solving of some special-type diophantine equations).
Literature
    recommended literature
  • HERMAN, Jiří, Radan KUČERA and Jaromír ŠIMŠA. Metody řešení matematických úloh. Vydání druhé přepracovan. V Brně: Masarykova univerzita, 1996, 278 stran. ISBN 8021012021. info
  • SLOVÁK, Jan, Martin PANÁK and Michal BULANT. Matematika drsně a svižně (Brisk Guide to Mathematics). 1st ed. Brno: Masarykova univerzita, 2013, 773 pp. ISBN 978-80-210-6307-5. Available from: https://dx.doi.org/10.5817/CZ.MUNI.O210-6308-2013. Základní učebnice matematiky pro vysokoškolské studium info
    not specified
  • IRELAND, Kenneth F. and Michael I. ROSEN. A classical introduction to modern number theory. 2nd ed. New York: Springer, 1990, xiv, 389. ISBN 038797329X. info
Teaching methods
Lectures: theoretical explanation with practical examples Exercises: solving problems for understanding of basic concepts and theorems, contains also some basic applications (e.g. public-key cryptography) Homeworks and their reflection
Assessment methods
Mid-term exam (1/3 points), final written and oral exam. Small portion of points will be assigned by means of homeworks.
Náhradní absolvování (in Czech)
V případě zahraničního výjezdu je možné předmět absolvovat v~náhradní podobě. Po návratu z~výjezdu je možné napsat vnitrosemestrální písemku a zpracovat náhradní úkol. Následně proběhne dle domluvy náhradní termín zkoušky
Language of instruction
Czech
Follow-Up Courses
Study support
https://is.muni.cz/auth/el/sci/podzim2024/MUC33/index.qwarp
Further Comments
Study Materials
The course is taught annually.
The course is also listed under the following terms Autumn 2019, autumn 2021, Autumn 2022, Autumn 2023.
  • Enrolment Statistics (recent)
  • Permalink: https://is.muni.cz/course/sci/autumn2024/MUC33