Ateneo - UPO

Elementary theory of numbers. Some basic ideas of linear algebra.

Recommended texts:
+ Bramanti, Pagani, Salsa, "Analisi matematica 1 con elementi di
geometria e algebra lineare", Ed. Zanichelli.
+ Teacher's notes available on the DIR platform.

The course aims to provide students with the knowledge of the main elements of elementary number theory and some notions of linear algebra. The aim of the teaching is that students acquire and know how to use an appropriate mathematical language in relation to topics covered in the course. The educational objective of the teaching is to develop the ability to model simple situations with the help of basic ideas and methods of algebra (elementary number theory) and linear algebra.

Reading and production of short texts in italian. Basic properties of integers. Basic properties of the cartesian plane.

Face-to-face lectures. Teaching will take place through lectures on the blackboard with theoretical part and exercises. Each topic of the course is introduced by means of a general discussion which aims to make it as understandable as possible to students. In a second moment the basic notions of each topic are introduced; they are subsequently followed by examples with the aim of clarifying their meaning. The last part is dedicated to the exercises that will be carried out by the teacher with the active involvement of students to stimulate their critical sense and independent judgment.

The control of ongoing learning will be carried out by carrying out exercises in the classroom with the active participation of students. In addition to the books suggested for the theory and exercises, any further material for the preparation of the exam will be provided during the course.
The algebra and geometry module grade is a weighted average with the logic module grade to obtain the final discrete mathematics grade.
Students with physical disabilities, Learning Disabilities or Special Education Needs can request specific services and tools via the __Staff Sviluppo e Coordinamento Carriere e Servizi alle Studentesse e agli Studenti__, consulting the University webpage: https://www.uniupo.it/en/services/servicesstudents-physical-or-learning- disabilities.
Students with disabilities, learning disabilities or special education needs, once they have contacted the University Staff, can refer to the
tutor in charge of the course to define the examination modalities.

The exam consists of a written test containing four exercises, two of which on Modular Arithmetic topics and two on Linear Algebra topics. In each written test, most of the topics contained in the course are covered. The presence of questions on the theoretical part is not excluded. The written test lasts one hour and is considered passed if at least one exercise of Modular Arithmetic and one of Linear Algebra are correctly carried out. During the written test, the use of a calculator is allowed.

Integers. Divisibility and primes. Greatest common divisor. Euclid's algorithm. Least common multiple. Diophantine equations. Congruences and residue classes. Divisibility criteria. Representation of number in basis other than ten. Chinese remainder theorem. Sharing secrets by means of systems of congruences. Euler's theorem. The problem of the discrete logarithm. The Diffie-Hellman protocol. The RSA protocol. Linear systems. Gauss' elimination method. Matrices, determinants, rank, inverse matrix. Vector spaces, bases, linear dependence. Linear functions, scalar product. Applications to linear systems.

Knowledge and understanding of some basic concepts of elementary number theory and linear algebra, and ability to apply them to simple problems. These notions must necessarily become part of the cultural background of the students to allow them to be able to face more easily some applicative exams in which the aforementioned notions find a direct application as in the case of Physics. Ability to evaluate the
appropriateness of an algorithm that applies these concepts. Ability to communicate the procedures adopted in a simple and clear way.