Number Theory (MAY123)

General

School School of Science Department of Mathematics Undergraduate MAY123 1 Number Theory Lectures (Weekly Teaching Hours: 4, Credits: 7.5) General Background - Greek, English Yes (in English) See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes The main purpose of the course is the study of the structure and basic properties of natural numbers, and more generally of integers. This study is based on the fundamental concept of divisibility of integers, and the (unique) factorization of a natural number into prime factors. The most important ideas, concepts and results that allow us to understand the structure and fundamental properties of all positive integers with respect to divisibility, are as follows (Keywords of course): Divisibility, prime numbers, Euclidean algorithm, greatest common divisor and least common multiple. Congruences and systems of congruences, Chinese remainder theorem. Arithmetical functions and Moebius inversion formula. Euler’s φ-function. Theorems of Fermat, Euler and Wilson. Primitive mod p roots. Theory of indices and quadratic residues. Law of quadratic reciprocity. Applications to cryptosystems. We will formulate and prove several theorems concerning the structure of all integers through the concept of divisibility. During the course will analyse applications of Number Theory to other sciences, and particularly to Cryptography. This course is an introduction to the basic results, the basic methods, and the basic problems of elementary number theory, and requires no special knowledge of other subjects of the curriculum. At the end of the course we expect the student to (a) have understood the definitions and basic theorems concerning the divisibility structure of the integers which are discussed in the course, (b) to have understood how they are applied in discrete examples, (c) to be able to apply the material in order to extract new elementary conclusions, and finally (d) to perform some (no so obvious) calculations. The course aims to enable the undergraduate student to acquire the ability to analyse and synthesize basic knowledge of the theory of numbers, to apply basic examples in other areas, and in particular to solve concrete problems concerning properties of numbers occurring in everyday life. The contact of the undergraduate student with the ideas and concepts of number theory, (a) promotes the creative, analytical and deductive thinking and the ability to work independently, (b) improves his critical thinking and his ability to apply abstract knowledge in various field.

Syllabus

• Complex numbers.
• Divisibility.
• Congruences mod m.
• Chinese remainder theorem.
• Arithmetical functions and Moebius inversion formula.
• The theorems of Fermat, Euler and Wilson.
• Primitive roots mod p.
• The theory of indices and the Law of quadratic reciprocity.
• Applications to cryptography.

Teaching and Learning Methods - Evaluation

Delivery

Classroom (face-to-face)

Use of Information and Communications Technology
• Teaching Material: Teaching material in electronic form available at the home page of the course.
• Communication with the students:
1. Office hours for the students (questions and problem solving).
2. Email correspondence
3. Weekly updates of the homepage of the course.
Teaching Methods