Discrete Mathematics 1

Course guarantor:

Zuzana Masáková

Lecturer:

Lubomíra Dvořáková, Zuzana Masáková, Edita Pelantová

Tutor:

Lubomíra Dvořáková, Zuzana Masáková, Edita Pelantová

Supervisor:

Department of Mathematics

Synopsis:

The seminar is devoted to elementary number theory and applications. In addition to the material discussed, it includes individual problem solving.

Requirements:

Participation in seminars is mandatory, two absences are allowed. Absences beyond the allowed ones increase the point threshold that must be achieved to obtain a credit.

Several short tests are written during the semester. Credit can be obtained based on the results of these tests. A student who does not get enough points from these tests writes a final exam, for which one term will be announced. A student who cannot participate in the final exam for serious reasons will be given the opportunity in one substitute term.

Credit points can be obtained for activity during lessons. The instructor decides on the allocation of such points.

Syllabus of lectures:

1. Divisibility, congruences, Femat's little theorem.

2. Linear diofantic equations, linear congruences, Chinese remainder theorem.

3. Euler's function, Euler´s theorem, Moebius function, inclusion exclusion principle.

4. Perfect numbers, Mersenne's primes, Fermat's numbers.

5. Primality testing. Public key cryptographic systems: RSA, knapsack problem.

Syllabus of tutorials:

Study Objective:

Acquired knowledge: Students learn to solve some types of elementary number theoretical problems.

Acquired skills: The emphasis is put on correct formulation of mathematical ideas and logic process.

Study materials:

Ronald L. Graham, Donald E. Knuth, Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, Reading, Massachusetts: Addison-Wesley, 1994

J. Herman, R. Kučera, J. Šimša,

Equations and Inequalities: Elementary Problems and Theorems in Algebra and Number Theory. 1. vyd. New York : Springer-Verlag,

2000. 355 s. Canadian Mathematical Society Books in Math.

P. Erdös, J. Surányi, Topics in the Theory of Numbers,

Springer-Verlag, 2001.

M. Křížek, F. Luca, L. Somer,

17 Lectures on Fermat Numbers: From Number Theory to Geometry, CMS Books in Mathematics, vol. 9, Springer-Verlag, New

York, 2001.

Note:

Time-table for winter semester 2025/2026:

Time-table is not available yet

Time-table for summer semester 2025/2026:

Time-table is not available yet

The course is a part of the following study plans:

• Aplikovaná algebra a analýza (compulsory course in the program)
• Aplikace informatiky v přírodních vědách (compulsory course in the program)
• Aplikované matematicko-stochastické metody (elective course)
• Aplikovaná informatika (compulsory course in the program)
• Fyzikální inženýrství - Fyzika plazmatu a termojaderné fúze (elective course)
• Matematické inženýrství - Matematická fyzika (elective course)
• Matematické inženýrství - Matematická informatika (PS)
• Matematické inženýrství - Matematické modelování (elective course)