Topics in Discrete Mathematics
Code
8503
Academic unit
Faculdade de Ciências e Tecnologia
Department
Departamento de Matemática
Credits
3.0
Teacher in charge
Ana Cristina Malheiro Casimiro
Weekly hours
3
Teaching language
Português
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. Congruences
Definition and properties. Representation in basis n. Ring of integers modulo n. Wilson''s Theorem. Wolstenholme''s Theorem.
3. Revisions in Algebraic structures
Groups, rings and fields.
4. Euler function and the Theorem of Euler -Fermat
Systems of (invertible) residues modulo n, Euler-Fermat''s Theorem and Fermat''s little Theorem, Wilson''s Theorem,
5.Linear congruences and the Chinese remainder Theorem.
6. Criptography RSA
7. Arithmetic functions
(Totally) Multiplicative functions. (Dirichlet) Convolution product.
8.Diofantine Equations
Pitagoric triples. 4 squares sum and Waring''s problem.
Bibliography
- Teoria dos Números um passeio com primos e outros números familiares pelo mundo inteiro, Fabio Martinez, Carlos Gustavo Moreira, Nicolau Saldanha e Eduardo Tengan, IMPA, 2015, ISBN: 978-85-244-0416-0, 4ª edição
-Introdução à Teoria de Números, Filipe Oliveira, FCT - Universidade Nova de Lisboa
- A Classical Introduction to Modern Number Theory, Kenneth Ireland, Michael Rosen, Springer New York, 1990, ISBN: 978-1-4419-3094-1
– An introduction to the theory of numbers, HARDY, G. H., WRIGHT, E. M. 3ª ed., Oxford, at the Clarendon Press, 1954.
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
The evaluation is carried out through Continuous evaluation or Exam evaluation.
Continuous evaluation
(A) The student has to study a topic of the subjest syllabus and must present it orally (1h) and written.
(B) At the end of the semester it will be a 2 hours test, with consultation, about the program explained during the classes.
The classification of continuous evaluation is obtained by the average of the grades obtained in (A) and in (B), these are graded between 0 and 20 values. The student is approved in the course if the average is greater than or equal to 9.5 values.
Exam
All the students enrolled in the course that have suspended in the continuous evaluation may submit to the Exam (3h).
Grade improvement
Students have the right to improve grade by enrollment within the established deadlines, at the time of the Exam. In this case, they may take the 3-hour Exam.
Final considerations
In all that this Regulation is missing, the FCT-UNL General Regulations are valid.