Principles of Number Theory

The goal of this book is to provide the basics of number theory, where we cover the fundamental principles of this branch of mathematical sciences, which is used in most other fields of mathematics.

We have employed various proof methods to give students the opportunity to practice these techniques, and each section is followed by a set of exercises, with each chapter concluding with exercises related to its topics.

This book includes finding the greatest common divisor and least common multiple and their applications. It also covers algorithms (division, Euclid's algorithm), linear combinations, natural numbers, and the method of mathematical induction in proofs. Additionally, it includes prime numbers and the factorization of any integer into prime factors. It discusses number-theoretic functions, perfect numbers, and some other number-theoretic functions. A chapter is dedicated to congruences, especially linear congruences, and another chapter deals with real numbers as both rational and irrational numbers, attempting to represent them through simple continued fractions. We conclude with a chapter on Diophantine equations.

Finally, we hope that we have succeeded in enriching the Arabic library and that this work will be beneficial to mathematics students.