# Number Theory and Ordinal Arithmetic

## Objectives

The student is supposed to learn the basic methods of number theory and transfinite arithmetic and the corresponding mathematical proofs.

## General characterization

12919

9.0

##### Responsible teacher

Joaquim Eurico Anes Duarte Nogueira

Weekly - 4

Total - 252

Português

No prerequisites

### Bibliography

- Gareth Jones, Josefine Jones, Elementary Number Theory, Springer 1998
- William Stein, Elementary Number Theory: Primes, Congruences, and Secrets, Springer 2008
- H. Davenport, The Higher arithmetic, Cambridge 2008
- H. B. Enderton, Elements of Set Theory, Academic Press, 1977
- A. J. Franco de Oliveira, Teoria de Conjuntos, Livraria Escolar Editora, 1982
- P. R. Halmos, Naive Set Theory, Springer, 1998

### Teaching method

Classes consist on two different aspects: an oral explanation which is illustrated by examples and the resolution, by the students, of proposed exercises. Students can ask for any questions either in class or during office hours.

### Evaluation method

2 Tests during the semester or Final Exam

## Subject matter

- Prime numbers, divisibility and the fundamental theorem of arithmetic.

- Maximum common divisor and Euclid''s algorithm. Diophantine linear equation.

- Congruences. Fermat''s theorem. Euler''s theorem.

- Chinese remainder theorem. Primitive Roots. Quadratic reciprocity theorem.

Nonlinear Diophantine Equations. Pell''s equation. Pythagorean triples. Fermat''s infinite descent method. Elliptic Curves.

- Intuitive theory of cardinal numbers.

- Ordering cardinal numbers.

- Axiom of Choice, Zermelo’s Well-Ordering Theorem and Zorn’s Lemma.

- Cardinal arithmetic and Continuum Hypothesis.

- Goodstein’s Theorem

## Programs

Programs where the course is taught: