TLÜ ÕIS

Number Theory

Course code

MLM6206.DT

old course code

MLM6206

Course title in Estonian

Arvuteooria

Course title in English

Number Theory

ECTS credits

4.0

Assessment form

Examination

lecturer of 2025/2026 Autumn semester

Not opened for teaching. Click the study programme link below to see the nominal division schedule.

lecturer of 2025/2026 Spring semester

Not opened for teaching. Click the study programme link below to see the nominal division schedule.

Course aims

To provide a structured overview of elementary number theory, with a focus on selected areas particularly relevant to practical applications.

Brief description of the course

Introduction to number theory. Basic properties of divisibility. Number theoretic functions. Continued fractions. Linear diophantine equations. Continued fractions and approximation of real numbers. Basic properties of congruences. Modular arithmetic. Linear congruences. The Chinese remainder theorem. High-order congruences. Quadratic residues. Orders and primitive roots. Index Calculus. Some applications of Number Theory in cryptography.

Learning outcomes in the course

Upon completing the course the student:
- defines and explains the fundamental concepts of divisibility (e.g., divisibility, greatest common divisor, least common multiple, prime numbers) and proves their key properties;
- applies the euclidean algorithm and the sieve of eratosthenes to solve number-theoretic problems;
- expands rational and irrational numbers into continued fractions, determines convergents and best approximations, and solves related diophantine equations;
- identifies and evaluates the main number-theoretic functions, proves their properties and computes function values;
- solves congruences and systems of congruences, including applications of the Chinese remainder theorem, and justifies the use of appropriate solution techniques;
- states the definitions of quadratic residues, the Legendre and Jacobi symbols; proves key results and solves standard problems in quadratic residue theory;
- defines the concepts of primitive root and index, proves fundamental theorems and applies them in relevant contexts.

Teacher

Tatjana Tamberg

Study programmes containing that course