Number Theory I introduces the fundamental concepts and results regarding the integers. The course begins with the study of divisibility, covering the Principle of Mathematical Induction, the Division Algorithm, prime numbers, and the Fundamental Theorem of Arithmetic, along with essential arithmetic functions like Euler's phi function and the Möbius inversion formula.
A major focus is the theory of congruences, including modular arithmetic, linear Diophantine equations, and the Chinese Remainder Theorem. We also study classical results such as the theorems of Fermat, Euler, and Wilson, which provide the theoretical basis for modern applications in cryptography.
The final part of the course explores quadratic residues and the Law of Quadratic Reciprocity, using Legendre and Jacobi symbols. Additional topics include the method of infinite descent, sums of squares, Pythagorean triples, and a brief introduction to elliptic curves.
This document compiles selected notes and examples developed during the course. While not intended as a comprehensive textbook, these materials provide detailed explanations of key topics and are designed to serve as a robust supplementary resource for students.
As with the notes, this section contains a compilation of various homework assignments and problem sets. Although they do not follow a unified structure, they may serve as valuable practice material for anyone who wishes to reinforce their understanding of the topics covered in the course.
