Topics in Discrete Mathematics
Objectives
Available soon
General characterization
Code
8503
Credits
3.0
Responsible teacher
Jorge Manuel Leocádio André
Hours
Weekly - 3
Total - Available soon
Teaching language
Português
Prerequisites
Available soon
Bibliography
- Introdução à Teoria de Números, Filipe Oliveira, FCT - Universidade Nova de Lisboa
- Elementary Number Theory, Gareth Jones and Josephine Jones, Springer Undergraduate Mathematics Series
- An introduction to the Theory of Numbers, G.H. Hardy, Oxford Science Publication
- Introduction à la théorie des nombres, JM De Koninck & A Mercier, Collection Universitaire de Mathématiques, 1994.
- The theory of numbers: a text and source book of problems, Andrew Adler, John E. Coury, Barlett Publishers, 1995
Teaching method
During the Theoretical-Practical classes, the different contents of this course will be exposed. Students will be asked to solve exercises and elaborate proofs of some of the different resuts presented.
Any questions or doubts will be adressed during the classes, during the weeekly sessions specially programmed to it or even at special sessions previously arranged between professors and students.
Evaluation method
Available soon
Subject matter
1. Divisibility in the ring of integers
Euclidian division, Euclides'''' algorythm, Bezout''''s Theorem and Euclides'''' Lemma, prime numbers and the Fundamental Theorem of Arythmetic.
2. Revisions in Algebraic structures
Groups, rings, normal subgroups and ideals, quotient groups and rings.
3. Congruences
Systems of residues, Euler''''s Theorem and Fermat''''s little Theorem, Wilson''''s Theorem, linear congruences and the Chinese remainder Theorem.
4. Arithmetic functions
Convolution product, Euler''''s function and other classical results.
5. Criptography
Caesar cipher and RSA system .
6. Waring''''s problem
Study of the equations a^2+b^2=n, x^2+y^2=z^2, x^4+y^4=z^2. Waring´s Theorem.