Number Theory
Description
Acquire knowledge of classical number theory, and form the skills of them application. Students will study the theory of numbers divisibility (numerical systems, greatest common divisor, least common multiple, prime numbers, arithmetical functions, continued fractions) and the congruence theory (Euler totient function, residue systems, Euler and Fermat theorems, congruence with unknowns, power residues systems).
Aim of the course
Main aim of the course is to provide the students with theoretical and practical knowledge and skills of classical number theory.
Prerequisites
High school mathematics knowledge.
Course content
1. Divisibility of integer numbers.
1.1. Main terms and theorems. Properties of divisibility.
1.2. Numerical systems.
1.3. Greatest common divisor.
1.4. Least common multiple.
1.5. Prime and composite numbers. Coprime numbers.
1.6. Factorization into primes.
1.7. Continued fractions.
2. Main functions in number theory.
2.1. Integer and fractional part of number.
2.2. Arithmetical and multiplicative functions.
2.3. Number and sum of divisors.
2.4. Möbius function.
2.5. Euler totient function.
3. Congruences.
3.1. Definition and properties of congruence.
3.2. Modular arithmetic.
3.3. Residue. Ring of residues.
3.4. Systems of residues.
3.5. Euler and Fermat theorems.
4. One variable congruences.
4.1. Main terms.
4.2. Linear congruences.
4.3. Solution of algebraic congruences.
4.4. Systems of linear congruences.
4.5. Congruences with prime power modulus.
4.6. Congruences with composite modulus.
4.7. Diophantine equations.
5. Higher order residues.
5.1. Index.
5.2. Primitive roots.
5.3. System fractions.
5.4. Length of period.
6. Application of number theory in other sciences.
Assesment Criteria
1. Student demonstrates the ability to illustrate main concepts with examples.
2. Student recognizes the type of congruences and solves them choosing the optimal method (GCD, continued fractions, etc.).
3. Understanding main point of task, student recognizes certain arithmetical function (-s) and applies them to solve practical exercise.
4. Operating on basic terms and propositions, student proves statements on divisibility of numbers, cases for solution of congruences and etc.
