Number Theory and Ordinal Arithmetic


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

General characterization





Responsible teacher

Joaquim Eurico Anes Duarte Nogueira


Weekly - 4

Total - 252

Teaching language



No prerequisites


- 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