Introduction to Number Theory (Number and Proofs)


If you’ve ever wondered if 46,548 is divisible by 7, then Introduction to Number Theory is for you.

Introduction to Number Theory is a friendly guide for the study of integers, proof writing, mathematical induction, divisibility, congruence equations, and more. This book is full of examples and exercises, perfect for undergraduate students studying math and computer science, as well as advanced high school students studying mathematics.

Introduction to Number Theory teaches you core principles, each with a dedicated chapter:

  • Natural Numbers
  • Integers
  • Congruence

An upgrade from the first edition, Edition 2 includes more exercises and helpful examples laid out in an easy-to-follow fashion, as well as entire sections dedicated to the infamous Fibonacci numbers, Euler’s function and theorem, and the Tonelli Shanks Algorithm.

Sold By: D. A. Smith
ISBN: 9781951290009 Category:


If you’ve ever wondered if 46,548 is divisible by 7, then Introduction to Number Theory is for you. Based on 15 years experience teaching mathematics at the University of Texas at Arlington, author David Andrew Smith uses Introduction to Number Theory to guide you through proof writing, mathematical induction, divisibility, congruence equations, and more.

In short, Number Theory is the study of integers, or whole numbers like 1, 2, 3, and 4. While these are simple numbers to the naked eye, their properties are quite complex.

Therefore, most universities and high schools developed Number Theory courses. Introduction to Number Theory is the perfect supplemental read for a Number Theory course and a great complement to the course textbook. Designed specifically for undergraduate students (studying math or computer science) or advanced high school students, this book offers a comprehensive and strong Number Theory base for self-study and is both example and exercise-heavy.

Number Theory Content

Smith guides you through core principles over three chapters:

  • Chapter 1: The Natural Numbers
    • How to write proofs
    • Fibonacci numbers
    • Mathematical induction
  • Chapter 2: Introducing the Integers
    • Divisibility
    • Prime numbers
    • Greatest common divisors
    • Fundamental theorem of arithmetic
    • How to apply induction
  • Chapter 3: Congruence
    • Linear congruence equations and how to solve
    • Systems of congruence equations and how to solve
    • Polynomials and how to solve them
    • Divisibility tests
    • Euler’s function and theorem
    • Law of quadratic reciprocity
    • Tonelli Shanks Algorithm

Each chapter includes theorems, the actual arithmetic, and many examples of solutions and real-life applications.

Setting this book apart from other Number Theory books, is its logical, step-by-step guide to the mathematics. Hence, the beginning of the book will help you grasp the basics and then continue to strengthen your knowledge with modern day applications. Additionally, the Tonelli Shanks Algorithm is a bonus featured section because of its uniqueness in solving more complicated quadratic equations.

For example, do you know what day January 1, 1970 was? Was it a Wednesday or Sunday? Congruence formulas of Number Theory will help you solve that question in seconds!

Above all, Introduction to Number Theory will help you make sense of the numbers and avoid spending extra hours in the library, deciphering your textbook. Number Theory can be magical with the right understanding and application. As a result, it’s provided us with many common luxuries like online shopping, credit cards, and even encryption and internet security. This book will provide a core foundation to go on and study more advanced Number Theory courses, cryptography, and more.

140 pages.

About the Author

David Andrew Smith is a mathematics professor at the University of Texas at Arlington. He holds his Bachelors of Science and Masters of Science in Mathematics from the University of Texas at Arlington. An award-winning teacher and lifelong student of math, he is also the author of Calculus 1, Calculus 2, Introduction to Linear Algebra, Multivariate Calculus, Proofs, and more. Named “Student of the Year” in 1993 and 1994, and “Professor of the Year” in 2012, Smith is passionate about writing and teaching mathematics to help students better grasp the fundamentals and learn to trust the process.

The Natural Numbers

In the first chapter The Natural Numbers, readers will find two sections which are Mathematical Induction and Fibonacci Numbers. In the first section, we see a collection of other topics, such as the strong form of mathematical induction, the well-ordering axiom (principle), and the relationship between the two. The rest of the chapter is about Fibonacci numbers. The author introduces them, some identities, and gives several examples. Then, the author moves on to the Euler-Binet formula, the prime conjecture, and closes with how the Fibonacci sequence can grow. 

Introducing the Integers

The second chapter covers six main topics. They include divisibility, prime numbers, greatest common divisors, Euclidean Algorithm, Fundamental Theorem of Arithmetic, and Linear Diophantine Equations. In the first subsection, you will find that the author defines divisibility, then delves into the divisibility lemmas. After that, the Division algorithm is explained while using it in several examples. The section on prime numbers deals with the sieve of Eratosthenes, prime divisors, and infinitude of primes. 

The third section on greatest common divisors has two subdivisions, i.e., Bezout’s identity and relatively prime integers. After the author introduces the Euclidean Algorithm in the next subsection, several lemmas are studied. The penultimate part of this chapter is about the Fundamental Theorem of Arithmetic. Within it, the author characterizes primes, provides proof of the Fundamental Theorem, and then visits the topic of least common multiples. The final section of the second chapter deals with Linear Diophantine Equations of different types, i.e., two and multivariable, and how to solve them. 

Any Introduction to Number Theory Should Concentrate on Congruence

The third chapter covers nine main topics and is the heart of the text. Of those, the first topic, the introduction to congruence, deals with other subtopics, which are the lemmas, least positive residues, and modular arithmetic. Then in the next section on linear congruence equations, the author describes how they can be solved and the inverse of an integer. After the exercises relevant to those topics, a detailed section on the Chinese Remainder Theorem is presented. 

The author presents polynomial congruence equations, many examples, and how to solve them in the fourth part of the last chapter. Under it, you will also find congruence reduction and Hensel’s Lifting Theorem. Within the fifth part of the chapter are applications of congruence, such as divisibility tests and days of the week. It ends on practice questions, as well. The sixth part is based on special congruence equation. It delves into two theorems: Wilson’s and Fermat’s, before culminating on exercises.

The subsequent part of the third chapter covers Euler’s function, its properties, and how to solve equations related to it. It also details Euler’s theorem before ending on practice questions. The penultimate portion of this topic is about quadratic congruence equations. It is quite detailed since it includes general quadratic congruence, residues, characters, and the law of reciprocity. Besides those, you will also find portions dedicated to the Legendre Symbol and its properties and Euler’s and Gauss’s Criterion. 

The final section is dedicated to Shank’s algorithm, several examples, and how it may be used to solve quadratic congruences.


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