Number Theory

Recommended Prerequisites

Good knowledge of High School Mathematics.

Teaching Methods

Classes are expository and include examples to clarify and motivate abstract concepts, and exercises for applying the acquired knowledge. Substancial part of the time is devoted to problem-solving. Optional challenging problems are also proposed. During the semester, students may use tutorial time to clarify their difficulties and train their skills in problem solving.

Learning Outcomes

The knowledge and the ability to prove results in elementary number theory and the capacity to apply them on problem solving. To develop a rigorous mathematical thinking.

The main competencies to be developed are: calculus ability; knowledge of mathematical results; generalization and abstraction; ability to formulate and solve problems; logical reasoning; rigorous and clear mathematical writing as well as oral communication; critical and independent thinking; research and independent learning; imagination and creativity.

Work Placement(s)

No

Syllabus

The integer numbers. Well ordering and mathematical induction principles. Divisibility. Euclidean  algorithm. Primes. Fundamental Theorem of Arithmetic. Congruences. Fermat’s Little theorem, Euler’s theorem, and Wilson’s theorem. Chinese remainder theorem. Lagrange’s theorem.  Primitive roots. Arithmetic functions. Diophantine equations. Applications of number theory (for example, the basic version of the RSA encryption method).

Head Lecturer(s)

Cristina Helena de Matos Caldeira

Assessment Methods

Avaliação
Exam(100%) or Midterm exam (60%) + Test (40%): 100.0%

Bibliography

G. A. Jones, J. M. Jones, Elementary Number Theory, Springer-Verlag, 1998

 

I. Niven, H. Zuckerman, H. Montgomery, An Introduction to the Theory of Numbers, 5th ed, John Wiley & Sons, 1991

 

J. F. Queiró, Teoria dos Números, 2008