MUC33 Number theory

Faculty of Science
Autumn 2023
Extent and Intensity
2/2/0. 5 credit(s). Type of Completion: zk (examination).
Teacher(s)
Mgr. Michal Bulant, Ph.D. (lecturer)
prof. RNDr. Radan Kučera, DSc. (lecturer)
Mgr. Jan Vondruška (seminar tutor)
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
Wed 14:00–15:50 M1,01017
  • Timetable of Seminar Groups:
MUC33/01: Fri 12:00–13:50 M6,01011, M. Bulant
MUC33/02: Fri 10:00–11:50 M6,01011, 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
At the end of this course, students should be able to:
understand the basics of elementary number theory, especially basic facts about primes
use congruences
solve linear congruences and their systems and selected types of congruences of higher order
apply various methods for solving diophantine equations
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
explain time complexity of numerical operations on large numbers
describe basic principles and procedures of digital encryption and signing using number-theoretical 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.
Language of instruction
Czech
Follow-Up Courses
Study support
https://is.muni.cz/auth/el/sci/podzim2023/MUC33/index.qwarp
Further Comments
Study Materials
The course is taught annually.
Teacher's information
http://www.math.muni.cz/~bulik/vyuka/Algebra-2/
The course is also listed under the following terms Autumn 2019, autumn 2021, Autumn 2022, Autumn 2024.
  • Enrolment Statistics (Autumn 2023, recent)
  • Permalink: https://is.muni.cz/course/sci/autumn2023/MUC33