# Topics in Discrete Mathematics

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## General characterization

8503

3.0

##### Responsible teacher

Jorge Manuel Leocádio André

##### Hours

Weekly - 3

Total - Available soon

Português

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.

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## 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.

## Programs

Programs where the course is taught: