A Comprehensive Guide to Number Theory with Beginning Number Theory by Neville Robbins
Beginning Number Theory Neville Robbins Pdf Files
If you are interested in learning about number theory, one of the oldest and most fascinating branches of mathematics, you might want to check out Beginning Number Theory, a book written by Neville Robbins. This book is a comprehensive introduction to the major topics and applications of number theory, written in an accessible and engaging style. In this article, we will give you an overview of what number theory is, what are the main topics covered in the book, who is Neville Robbins and why did he write this book, how to use this book for learning number theory, and what are the benefits of reading this book. By the end of this article, you will have a better idea of whether this book is suitable for you and how it can help you explore the fascinating world of numbers.
Beginning Number Theory Neville Robbins Pdf Files
What is Number Theory?
Number theory is the study of the properties and relationships of integers, or whole numbers. It is one of the oldest branches of mathematics, dating back to ancient times when people started to explore patterns and puzzles involving numbers. For example, some of the questions that number theorists ask are:
How can we tell if a number is divisible by another number?
How can we find prime numbers, or numbers that have only two factors: 1 and themselves?
How can we solve equations involving unknown integers?
How can we represent numbers using different bases or systems?
How can we use numbers to encrypt or decrypt messages?
Number theory is often considered as a pure or abstract branch of mathematics, because it deals with concepts that are not directly related to physical reality. However, number theory also has many applications and connections to other fields of science, engineering, cryptography, computer science, music, art, and more. For example, some of the applications of number theory are:
Using prime numbers to create secure encryption systems that protect online transactions and communications.
Using modular arithmetic to create checksums or error-detection codes that ensure data integrity.
Using continued fractions to approximate irrational numbers or transcendental numbers with rational numbers.
Using Fibonacci numbers to model natural phenomena such as plant growth or animal populations.
Using quadratic residues to create visual patterns or artistic designs.
What are the main topics covered in Beginning Number Theory?
Beginning Number Theory is a textbook that covers all of the major topics in a classic number theory course. It is divided into 12 chapters, each focusing on a specific topic or theme. The chapters are:
Divisibility: This chapter introduces the concept of divisibility and its properties. It also covers topics such as greatest common divisors (GCD), least common multiples (LCM), Euclidean algorithm, linear Diophantine equations, and Chinese remainder theorem.
Primes: This chapter explores the properties and distribution of prime numbers. It also covers topics such as prime factorization, Fermat numbers, Mersenne numbers, perfect numbers, and Goldbach's conjecture.
Congruences: This chapter introduces the concept of congruences and modular arithmetic. It also covers topics such as Euler's theorem, Fermat's little theorem, Wilson's theorem, and pseudoprimes.
Arithmetic Functions: This chapter studies some important functions that are defined on the integers, such as the divisor function, the totient function, the Möbius function, and the Liouville function. It also covers topics such as multiplicative functions, Dirichlet convolution, and Riemann zeta function.
Primitive Roots and Indices: This chapter investigates the concept of primitive roots and indices, which are useful for solving congruences of higher degrees. It also covers topics such as primitive root theorem, index arithmetic, discrete logarithm problem, and quadratic reciprocity law.
Quadratic Congruences: This chapter focuses on solving congruences involving quadratic equations. It also covers topics such as Legendre symbol, Jacobi symbol, quadratic residues, and Pell's equation.
Sums of Squares: This chapter explores the problem of representing integers as sums of squares. It also covers topics such as Fermat's two-square theorem, Lagrange's four-square theorem, Waring's problem, and partition function.
Continued Fractions: This chapter introduces the concept of continued fractions and their properties. It also covers topics such as simple continued fractions, convergents, best approximations, periodic continued fractions, and quadratic irrationals.
Nonlinear Diophantine Equations: This chapter deals with some famous nonlinear Diophantine equations that have interesting solutions or unsolved problems. It also covers topics such as Pythagorean triples, Fermat's last theorem, Catalan's conjecture, and elliptic curves.
Computational Number Theory: This chapter discusses some algorithms and methods that are used to perform computations in number theory. It also covers topics such as primality testing, factorization methods, RSA encryption system, and digital signatures.
Cryptology: This chapter applies some concepts and techniques from number theory to the field of cryptology, or the science of secret communication. It also covers topics such as substitution ciphers, transposition ciphers, Vigenère cipher, Hill cipher, affine cipher, and ElGamal encryption system.
Appendices: This section contains some additional material that supplements the main text. It includes some open questions in elementary number theory, some tables of useful data, a bibliography of references, and an index of terms.
Who is Neville Robbins and why did he write this book?
Neville Robbins is a professor emeritus of mathematics at California State University in Northridge. He has a PhD in mathematics from UCLA and has taught various courses in mathematics for over 40 years. He is also the author of several other books on mathematics and computer science.
Robbins wrote Beginning Number Theory to provide a comprehensive yet accessible introduction to number theory for undergraduate students. He wanted to present the subject in a way that is both rigorous and entertaining, blending in numerous applications and special treatments of number theory. He also wanted to include many exercises, examples, case studies, historical notes, and modern developments to enhance the learning experience of the readers.
How to use this book for learning number theory?
If you want to use Beginning Number Theory for learning number theory, you should have some basic knowledge of algebra and calculus. You should also have access to a computer or a calculator that can perform arithmetic operations on large numbers.
The book is designed to be used in a one-semester course on number theory. However, you can also use it for self-study or reference. The book is structured in a logical and progressive way, starting from the basic concepts and moving on to more advanced topics. Each chapter contains a clear exposition of the main results and proofs, followed by several examples and applications that illustrate the concepts. Each chapter also contains a set of exercises that test your understanding and skills. Some exercises require you to use a computer or a calculator to perform computations or simulations. Some exercises are marked with an asterisk (*) to indicate that they are more challenging or require more background knowledge.
What are the benefits of reading this book?
Reading Beginning Number Theory can offer you many benefits, whether you are a student, a teacher, a researcher, or a hobbyist. Some of the benefits are:
You will learn about the beauty and elegance of number theory, one of the most fascinating and ancient branches of mathematics.
You will develop your logical thinking and problem-solving skills, as you encounter various proofs, puzzles, and challenges involving numbers.
You will discover the applications and relevance of number theory to various fields and problems, such as cryptography, computer science, music, art, and more.
You will appreciate the history and development of number theory, as you learn about the contributions and discoveries of many famous mathematicians and scientists.
You will expand your mathematical knowledge and curiosity, as you explore some open questions and unsolved problems in number theory.
In conclusion, Beginning Number Theory is a comprehensive and accessible introduction to number theory that covers all of the major topics and applications of the subject. It is written by Neville Robbins, a professor emeritus of mathematics who has over 40 years of teaching experience. It is suitable for use in a one-semester course on number theory or for self-study or reference. It is rich with interesting facts, historical notes, and modern developments, yet is clear and manageable for introductory students. It is supported with numerous exercises, examples, case studies, and resources that enhance the learning experience of the readers. If you are interested in learning about number theory, this book is a great choice for you.
Where can I find the pdf files of Beginning Number Theory?
You can find the pdf files of Beginning Number Theory on some online platforms or websites that offer free or paid access to academic books or articles. However, you should be careful about the legality and quality of these sources. Alternatively, you can buy or borrow a hard copy of the book from a bookstore or a library.
How can I check my answers to the exercises in Beginning Number Theory?
Some of the exercises in Beginning Number Theory have hints or solutions provided at the end of the book. For the rest of the exercises, you can check your answers by consulting other sources such as textbooks, websites, or online forums that discuss number theory. You can also ask your instructor or peers for help or feedback.
What are some other books that I can read to learn more about number theory?
There are many other books that you can read to learn more about number theory. Some of them are:
An Introduction to the Theory of Numbers by G.H. Hardy and E.M. Wright
A Friendly Introduction to Number Theory by Joseph H. Silverman
A Classical Introduction to Modern Number Theory by Kenneth Ireland and Michael Rosen
The Higher Arithmetic: An Introduction to the Theory of Numbers by H. Davenport
The Art of Problem Solving: Introduction to Number Theory by Mathew Crawford
How can I practice my skills in number theory?
You can practice your skills in number theory by solving more problems and challenges involving numbers. You can find these problems in books, websites, online platforms, contests, or journals that focus on number theory or mathematics in general. Some examples are:
Project Euler: A website that offers hundreds of problems that require mathematical and programming skills to solve.
Art of Problem Solving: A website that offers resources, forums, courses, and contests for students who love math.
Mathcounts: A national program that provides fun and challenging math activities for middle school students.
MAA Math Competitions: A series of math competitions for high school students organized by the Mathematical Association of America.
Mathematics of Computation: A journal that publishes research papers on computational mathematics and number theory.
What are some careers that involve number theory?
There are many careers that involve number theory or require some knowledge of number theory. Some of them are:
Cryptographer: A person who designs or analyzes encryption systems that protect data and communication.
Computer Scientist: A person who studies or develops algorithms, software, hardware, or systems that involve computation or data processing.
Mathematician: A person who researches or teaches mathematics or applies mathematics to various fields and problems.
Statistician: A person who collects, analyzes, or interprets data using mathematical methods and models.
Teacher: A person who educates or instructs students in mathematics or other subjects.